Gaussian Gabor frames, Seshadri constants and generalized Buser–Sarnak invariants

Abstract

We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander’s ${\overline{\partial }}$-$L^2$ estimate with singular weight, Demailly’s Calabi–Yau method for Kähler currents and a Kähler-variant generalization of the symplectic embedding theorem of McDuff–Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson–Lempert method and the Ohsawa–Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings’ theta metric formula for the Arakelov Green functions.

1 Introduction

Let $\Lambda$ be a lattice in ${\mathbb {R}}^{2n}$ and

$$\begin{aligned} {\mathfrak {H}}:=\{\Omega \in \mathfrak {gl}(n,{\mathbb {C}}): \Omega =\Omega ^T, \ \text {Im}\, \Omega \ \text {is positive definite} \} \end{aligned}$$

(1.1)

be the Siegel upper half-space. Fix $\Omega \in {\mathfrak {H}}$, a Gaussian Gabor system, denoted by $(g_\Omega ,\Lambda )$, is the family of functions $\{\pi _\lambda g_\Omega \}_{\lambda \in \Lambda }$, where $\Lambda$ is a lattice in ${\mathbb {R}}^{2n}$ and

$$\begin{aligned} (\pi _\lambda g_\Omega )(t):=e^{2\pi i \xi ^T t}g_\Omega (t-x), \ \ \lambda :=(\xi ,x) \in \Lambda , \ \ \xi ^T t:=\sum _{j=1}^n \xi _j t_j, \end{aligned}$$

denotes a time-frequency shift of the Gaussian $g_\Omega (t):=\overline{e^{\pi i t^T\Omega t}}$. Letting I denote the identity matrix, we note that $g_{iI}(t)$ is the standard Gaussian $e^{-\pi |t|^2}$. The frame set of $g_\Omega$ is the set of all lattices $\Lambda$ in ${\mathbb {R}}^{2n}$ such that $\{\pi _\lambda g_\Omega \}_{\lambda \in \Lambda }$ is a frame for $L^2({\mathbb {R}}^n)$. Here, by a frame, we mean there exist positive constants A, B (called frame bounds) such that

$$ A\Vert f\Vert ^2\le \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_\Omega )|^2\le B\Vert f\Vert ^2\quad \text {for}~~f\in L^2({\mathbb {R}}^n),$$

where $(\cdot , \cdot )$ denotes the $L^2$-inner product. Using the Bargmann transform, one may reformulate the frame property for Gaussian Gabor frames in terms of sampling property of the Bargmann–Fock space in complex analysis. In the one-dimensional case there is a density criterion for the interpolation problem in the Bargmann-Fock space due to Lyubarskii and Seip-Wallstén, which implies a seminal result in the theory of Gaussian Gabor frames. Namely, that $\{\pi _\lambda g_{iI}\}_{\lambda \in \Lambda }$ is a Gabor frame if and only if the covolume $|\Lambda |<1$, [Lyu92, Sei92, SW92]. Recent progress on the description of the frame set of a Gabor atom has been made for totally positive functions [GS13, GRS18] and for rational functions [BKL21]. Note that all aforementioned results on frame sets for Gabor systems are for uniformly discrete point sets in the plane. The generalization to the higher-dimensional case has been one of the most intriguing problems in the study of Gaussian Gabor frames since the methods in [Lyu92, Sei92, SW92] do not have natural counterparts in the theory of several complex variables. The reason being that the theory of sampling and interpolation in several complex variables [MT00, Lin01, Gro11, GL20, LM09] is far more intricate than in the one-dimensional case (see [PR13, section 3]) and despite considerable effort not well understood. In particular, the following central problem in multivariate Gaussian Gabor frame theory is still open.

Problem A

Is there an equivalent Gabor frame criterion for $(g_{iI}, \Lambda )$ only in terms of the covolume $|\Lambda |$ for almost all lattices $\Lambda$ in ${\mathbb {R}}^{2n}$ ($n>1$)?

We obtain the following partial result, which is a direct consequence of Proposition 1.4 and our Hörmander criterion in Sect. 1.2.

Theorem 1.1

(First main theorem). Fix $\Omega \in {\mathfrak {H}}$, if $|\Lambda |<\frac{n!}{n^n}$ then $(g_\Omega ,\Lambda )$ is a Gabor frame for almost all $\Lambda$ in ${\mathbb {R}}^{2n}$. More precisely, $(g_\Omega ,\Lambda )$ is a Gabor frame if $|\Lambda |<\frac{n!}{n^n}$ and $(\Omega , \Lambda )$ is a transcendental pair (see the definition and the remark below for explicit examples of the transcendental pairs).

Definition 1.1

Let $\Lambda ^\circ :=\{(\eta , y)\in {\mathbb {R}}^n \times {\mathbb {R}}^n: \xi ^T y -x^T\eta \in {\mathbb {Z}}, \ \forall \ (\xi ,x) \in \Lambda \}$ denote the symplectic dual of $\Lambda$ (also known as the adjoint lattice of $\Lambda$). Put

$$\begin{aligned} \Gamma _{\Omega , \Lambda ^\circ }:=\{(\textrm{Im}\, \Omega )^{-1/2} z \in {\mathbb {C}}^n: z=\eta +\Omega y, \ \ (\eta , y)\in \Lambda ^\circ \}, \end{aligned}$$

(1.2)

where $(\textrm{Im}\, \Omega )^{-1/2}$ denotes the unique positive definite matrix whose square equals $(\textrm{Im}\, \Omega )^{-1}$. We call $(\Omega , \Lambda )$ a transcendental pair if the complex torus ${\mathbb {C}}^n/ \Gamma _{\Omega , \Lambda ^\circ }$ has no analytic subvariety of dimension $1\le d <n$ (see Definition 1.4 for a related notion).

Remark

From the definition we know that all $(\Omega , \Lambda )$ are transcendental in case $n=1$. In case $n=2$, by [Sha13, p. 161] we know that $(\Omega , \Lambda )$ is a transcendental pair if ${\mathbb {C}}^2/ \Gamma _{\Omega , \Lambda ^\circ }$ is biholomorphic to ${\mathbb {C}}^2/\Gamma$ for some lattice

$$\begin{aligned} \Gamma := {\mathbb {Z}}(1,0)+{\mathbb {Z}}(0,1) + {\mathbb {Z}}(a,b)+\mathbb Z(c,d), \ \ a,b,c,d\in {\mathbb {C}}, \end{aligned}$$

with the set $\{1, a, b, c, d, ad-bc\}$ being linearly independent over ${\mathbb {Z}}$ (for example, if

$$\begin{aligned} (a, b, c, d)=(\pi , i \pi ^2, i\pi ^3, \pi ^4), \ (\sqrt{2}, i\sqrt{3}, i\sqrt{5}, \sqrt{7}), etc \end{aligned}$$

then $\Gamma$ is transcendental). In fact, the argument in [Sha13, page 160–163] can also be used to prove that $(\Omega , \Lambda )$ is transcendental for almost all lattices $\Lambda$ in ${\mathbb {C}}^n$. The proof of our first main theorem also suggests the following conjecture, which would answer our Problem A (just take $\Omega =iI$).

Conjecture A

Let $(\Omega , \Lambda )$ be a transcendental pair (see Definition 1.1). Then $(g_\Omega ,\Lambda )$ is a Gabor frame for $L^2({\mathbb {R}}^n)$ if and only if $|\Lambda |<\frac{n!}{n^n}$.

Remark

Since all $(\Omega , \Lambda )$ are transcendental in case $n=1$, the above conjecture is precisely the Lyubarskii-Seip-Wallstén theorem [Lyu92, Sei92, SW92] in the one dimensional case. In case $n>1$, the only known necessary condition is $|\Lambda |<1$, which is a consequence of the Balian-Low type theorems (see [AFK14, GHO19] or a complex analysis proof of $|\Lambda |\le 1$ by Lindholm [Lin01]).

Our second main result is a multivariate Gaussian Gabor frame criterion for general lattices (see the remark after Theorem A in Sect. 1.2 for the proof and Corollary 1.8 for applications).

Theorem 1.2

(Second main theorem). Fix $\Omega \in {\mathfrak {H}}$ and a lattice $\Lambda$ in $\mathbb R^{2n}$. Assume that there exist $r>1$, $\beta =(\beta _1,\cdots , \beta _n)\in {\mathbb {R}}^n$ with

$$\begin{aligned} \beta _1+\cdots +\beta _n=1, \ \ \beta _j >0, \ 1\le j\le n, \end{aligned}$$

and a holomorphic injection f from the ellipsoid

$$\begin{aligned} B_r^\beta :=\left\{ z\in {\mathbb {C}}^n: \pi \,\sum _{j=1}^n \beta _j |z_j|^2 <r^2 \right\} \end{aligned}$$

to the torus $X:={\mathbb {C}}^n/ \Gamma _{\Omega , \Lambda ^\circ }$ (see (1.2)) such that

$$\begin{aligned} f^*(\omega +i\partial {\overline{\partial }}\phi )= \frac{i}{2} \sum _{j=1}^n dz_j \wedge d{\bar{z}}_j \ \text {on }B_r^\beta \end{aligned}$$

for some smooth function $\phi$ on X, where $\omega :=\frac{i}{2} \sum _{j=1}^n dw_j \wedge d{\bar{w}}_j$ is the Euclidean Kähler form on X (note that the holomorphic cotangent bundle of X is trivial with global frame $\{dw_j\}$). Then $(g_\Omega ,\Lambda )$ is a Gabor frame.

Remark

In case $(\Omega , \Lambda )$ is transcendental the above theorem is equivalent to our first main theorem. In order to prove this equivalence, we generalize (see Theorem A in Sect. 1.2) McDuff–Polterovich’s result [MP94] (see Theorem 3.4) to all Kähler ellipsoid embeddings (see Sect. 3.1).

In the one-dimensional case, we also obtain the following frame bound estimates (see Theorem B in Sect. 1.3 for more results), which can be seen as an effective version of [BGL10, Theorem 1.1].

Theorem 1.3

(Third main theorem). Let $\Lambda$ be a lattice in ${\mathbb {R}}\times {\mathbb {R}}$. Suppose that the lattice

$$\begin{aligned} \Gamma :=\{z \in {\mathbb {C}}: z=\eta +i y, \ \ (\eta , y)\in \Lambda ^\circ \}, \end{aligned}$$

in ${\mathbb {C}}$ is generated by $\{1, \tau \}$ with $\textrm{Im}\, \tau >1$. Then for all $f\in L^2({\mathbb {R}})$ with $||f||=1$, we have

$$\begin{aligned} \frac{4\pi (\textrm{Im}\, \tau -1)|\eta (\tau )|^6 }{\left( \sum _{n\in {\mathbb {Z}}} e^{-\pi n^2 \textrm{Im}\, \tau }\right) ^2} \le \sqrt{2}\cdot \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_{iI})|^2 \le \frac{\textrm{Im}\, \tau }{1-e^{-C}}, \ \ C:=\frac{\pi }{4} \cdot \inf _{0\ne \lambda \in \Gamma } |\lambda |^2, \end{aligned}$$

(1.3)

where $\eta (\tau ):=e^{\pi i\tau /12} \,\Pi _{n=1}^\infty (1-e^{2\pi i n \tau })$ is the Dedekind eta function.

The above three main theorems are special cases of the Hörmander criterion (Theorem 1.7), Theorems A and B in Sect. 1.2. The whole paper is organized as follows.

1.1 Background

Our starting point is the following observation, which is an extension of a well-known duality result for $\Omega =iI$ [Jan82, Lyu92, Sei92].

Proposition 1.4

$(g_\Omega , \Lambda )$ is a Gabor frame for $L^2({\mathbb {R}}^n)$ if and only if $\Gamma _{\Omega , \Lambda ^\circ }$ (see (1.2)) is a set of interpolation for the Bargmann–Fock space ${\mathcal {F}}^2$ (see (1.4) and Definition 1.2 in Sect. 1.2).

Our first main theorem is an extension of the approach by Berndtsson–Ortega Cerdà [BO95] for one-dimensional Gaussian Gabor frames to the multivariate case by utilizing the theory of Hörmander’s $L^2$-estimates for ${\overline{\partial }}$ in the higher-dimensional case, which has been developed during the past two decades and has received quite some attention [OT87, Ohs94, Dem92, Ber10, Ber06, Ber09, Blo13, GZ15, BL16]. The new idea is to apply Demailly’s mass concentration technique [Dem93] (see [Tos16] for a nice survey).

A crucial notion in the proof of our second main theorem is a generalized extremal type Seshadri constant (see Definition 1.5 and Theorem 3.3 for the extremal property), which replaces the covolume of a lattice in the one-dimensional case. Note also that the Seshadri constant has a quite different flavor than the (Beurling) densities used in the discussion of Gabor frames, since it actually takes into account the volumina of all subvarieties of the complex torus (see Theorem 3.9) and not just of the whole complex torus.

Our third main theorem is based on a recent result of Berndtsson–Lempert [BL16], which also yields explicit estimates for the multivariate Gaussian Gabor frame bounds (see Theorems 5.1 and 5.2).

A short account of our other related results is given in Sects. 1.2 and 1.3 below (the Hörmander criterion, Theorems A, B are among the most crucial ones).

1.2 Part I: Interpolation in Bargmann–Fock space

Consider the Bargmann–Fock space

$$\begin{aligned} {\mathcal {F}}^2:=\{F\in {\mathcal {O}}({\mathbb {C}}^n): ||F||^2:=\int _{{\mathbb {C}}^n} |F(z)|^2 e^{-\pi |z|^2} <\infty \}, \end{aligned}$$

(1.4)

where ${\mathcal {O}}({\mathbb {C}}^n)$ denotes the space of holomorphic functions on ${\mathbb {C}}^n$ and we omit the Lebesgue measure in the integral. Let $\Gamma$ be a lattice in ${\mathbb {C}}^n$.

Definition 1.2

We call $\Gamma$ a set of interpolation for ${\mathcal {F}}^2$ if there exists a constant $C>0$ such that for every sequence of complex numbers $a=\{a_\lambda \}_{\lambda \in \Gamma }$ with $\sum _{\lambda \in \Gamma } |a_\lambda |^2 e^{-\pi |\lambda |^2}=1,$ there exists $F\in {\mathcal {F}}^2$ such that $F(\lambda )=a_\lambda$ for all $\lambda \in \Gamma$ and $||F||^2 \le C$.

Denote by $|\Gamma |$ (the covolume of $\Gamma$) the volume of the torus $X:={\mathbb {C}}^n/\Gamma$ with respect to the Lebesgue measure. In the one-dimensional case, Lyubarskii, Seip and Wallstén [Lyu92, Sei92, SW92] (see [OS98] for the most general one-dimensional generalization) proved that

Theorem 1.5

A lattice $\Gamma$ in ${\mathbb {C}}$ is a set of interpolation for ${\mathcal {F}}^2$ if and only if $|\Gamma |>1$.

Another proof of the “sufficient" part of the above theorem was given by Berndtsson and Ortega Cerdà in [BO95] using the Hörmander ${\overline{\partial }}$ theory. In order to make best use of the Hörmander theory, we shall introduce the following notion of the Hörmander constant, which is an analogue of the Seshadri constant introduced by Demailly in [Dem92].

Definition 1.3

Let $\Gamma$ be a lattice in ${\mathbb {C}}^n$. The Hörmander constant of $\Gamma$ is defined by

$$\begin{aligned} \iota _\Gamma :=&\sup \{\gamma \ge 0: \text {there exists a }\Gamma \text {-invariant function }\psi \text { on }{\mathbb {C}}^n\text { such that } \psi +\pi |z|^2\text { is psh and} \\&\psi =\gamma \log (|z_1|^{2/\beta _1}+\cdots + |z_n|^{2/\beta _n})\text { near }z=0\text { for some }\beta _j>0, \beta _1+\cdots +\beta _n=1\}, \end{aligned}$$

where “$\Gamma$-invariant" means that $\psi (z+\lambda ) =\psi (z)$ for all $z\in {\mathbb {C}}^n$ and $\lambda \in \Gamma$; “psh" means subharmonic on each embedded disc.

By using of a result of Tosatti (see [Tos18, Theorem 4.6]), we can prove the following:

Proposition 1.6

Assume that the only positive dimensional analytic subvariety of $X:={\mathbb {C}}^n/\Gamma$ is X itself, then

$$\begin{aligned} \iota _\Gamma = \frac{(n!)^{1/n}}{n} |\Gamma |^{1/n}. \end{aligned}$$

(1.5)

The above result suggests the following definition.

Definition 1.4

A lattice $\Gamma$ in ${\mathbb {C}}^n$ is said to be transcendental if the only positive dimensional analytic subvariety of $X:={\mathbb {C}}^n/\Gamma$ is X itself.

By Proposition 1.4 and Proposition 1.6, we know that the following criterion implies our first main result — Theorem 1.1.

Theorem 1.7

(The Hörmader criterion). Let $\Gamma$ be a lattice in ${\mathbb {C}}^n$. If $\iota _\Gamma >1$ then $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$. In particular, if $\Gamma$ is transcendental and $|\Gamma |>\frac{n^n}{n!}$, then $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$.

In order to use the above criterion for non-transcendental lattices we have to investigate the Hörmander constant in more depth. Our main idea is to introduce the following definition.

Definition 1.5

Let $(X, \omega )$ be an n-dimensional compact Kähler manifold. Fix $x\in X$ and take a holomorphic coordinate chart $z=\{z_j\}$ near x such that $z(x)=0$. Put

$$\begin{aligned} T_\beta :=\log (|z_1|^{2/\beta _1}+\cdots + |z_n|^{2/\beta _n}), \ \ \ \beta _j >0, \ \beta _1+\cdots +\beta _n=1. \end{aligned}$$

(1.6)

The $\beta$-Seshadri constant of $(X, \omega )$ at $x\in X$ is defined by

$$\begin{aligned} \epsilon _x(\omega ;\beta ):=&\sup \{\gamma \ge 0: \text {there exists an }\omega \text {-psh function }\psi \\&\text {on }X\text { such that }\psi =\gamma T_\beta \text { near }x\}, \end{aligned}$$

where “$\omega$-psh" means that $\psi$ is upper semi continuous on X with

$$\begin{aligned} \omega +dd^c \psi \ge 0, \ d^c:=(\partial -{\overline{\partial }})/(4\pi i), \end{aligned}$$

in the sense of currents on X.

Remark

In case $X={\mathbb {C}}^n/\Gamma$ and $\omega = dd^c(\pi |z|^2)$, we know that $\epsilon _x(\omega ;\beta )$ does not depend on $x\in X$. Comparing the above definition with Definition 1.3, we further get

$$\begin{aligned} \iota _\Gamma = \sup _{\beta _1+\cdots +\beta _n=1,\, \beta _j >0, \, 1\le j\le n} \epsilon _x(\omega ;\beta ). \end{aligned}$$

(1.7)

In case $\beta _1=\cdots =\beta _n=1/n$, we know that $n \epsilon _x(\omega ;\beta )$ equals the classical Seshadri constant of Demailly (see [Tos18, section 4.4]). A famous result of McDuff–Polterovich is a symplectic embedding formula [MP94] for the classical Seshadri constant. Theorem 1.2 follows from the following generalization of McDuff–Polterovich’s result to all $\beta$-Seshadri constants.

Theorem A

Let $(X, \omega )$ be a compact Kähler manifold. Denote by ${\mathcal {K}}_\omega$ the space of Kähler metrics for the cohomology class $[\omega ]$. Fix $x\in X$ and $\beta _j >0$, $1\le j\le n$ with $\beta _1+\cdots +\beta _n=1$. Then the $\beta$-Seshadri constant $\epsilon _x(\omega ;\beta )$ is equal to the following $\beta$-Kähler width

where “" means that there exists a holomorphic injection $f: B^\beta _r \rightarrow X$ such that $f(0)=x$ and $f^*({\tilde{\omega }})=\frac{i}{2} \sum _{j=1}^n dz_j \wedge d{\bar{z}}_j$.

Remark

The above theorem implies Theorem 1.2. In fact, by the above theorem and (1.7), the assumption in Theorem 1.2 implies that $\iota _\Gamma >1$. Hence our second main theorem follows from the Hörmander criterion, Theorem 1.7 and Proposition 1.4.

In case $X={\mathbb {C}}^n/\Gamma$ and $\omega =dd^c(\pi |z|^2)$ is the Euclidean Kähler form, we know that $B_r^\beta$ is included in X if and only if $(\gamma + B_r^\beta ) \cap B_r^\beta =\emptyset , \ \ \forall \ 0\ne \gamma \in \Gamma ,$ which is equivalent to that

$$\begin{aligned} r \le \frac{1}{2} \sqrt{\inf _{0\ne z\in \Gamma } \sum _{j=1}^n\beta _j |z_j|^2}. \end{aligned}$$

Hence Theorem A gives

$$\begin{aligned} \epsilon _x(\omega ;\beta )= c_x(\omega ;\beta ) \ge \pi r^2, \ \ \forall \ r \le \frac{1}{2} \sqrt{\inf _{0\ne z\in \Gamma } \sum _{j=1}^n\beta _j |z_j|^2}. \end{aligned}$$

Put

$$\begin{aligned} {\mathcal {B}}:=\{\beta \in {\mathbb {R}}^n: \beta _j\ge 0, \ 1\le j\le n, \ \beta _1+\cdots +\beta _n=1\}, \end{aligned}$$

(1.8)

then (1.7) gives

$$\begin{aligned} \iota _\Gamma \ge \frac{\pi }{4}\sup _{\beta \in {\mathcal {B}}} \inf _{0\ne z\in \Gamma } \sum _{j=1}^n\beta _j |z_j|^2. \end{aligned}$$

Apply the Hörmander criterion above we get:

Corollary 1.8

Let $\Gamma$ be a lattice in ${\mathbb {C}}^n$. If $\sup _{\beta \in {\mathcal {B}}} \inf _{0\ne z\in \Gamma } \sum _{j=1}^n\beta _j |z_j|^2 > \frac{4}{\pi },$ then the Hörmander constant $\iota _\Gamma >1$ (see Definition 1.3) and $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$.

Remark

In case all $\beta _j$ are equal to 1/n,

$$\begin{aligned} n\cdot \inf _{0\ne z\in \Gamma } \sum _{j=1}^n\beta _j |z_j|^2= \inf _{0\ne z\in \Gamma } |z|^2 \end{aligned}$$

(1.9)

is known as the Buser–Sarnak invariant $m(\Gamma )$ (see [BS94, Laz96, Laz04, Theorem 5.3.6]) of $\Gamma$. For general $\beta \in {\mathcal {B}}$ we call

$$\begin{aligned} m_\beta (\Gamma ):= \inf _{0\ne z\in \Gamma } \sum _{j=1}^n\beta _j |z_j|^2 \end{aligned}$$

(1.10)

the $\beta$-Buser–Sarnak invariant of $\Gamma$. In general, we have that $\sup _{\beta \in {\mathcal {B}}}m_\beta (\Gamma )>m(\Gamma )/n$. For example, if

$$\begin{aligned} \Gamma := {\mathbb {Z}}(A,0)+{\mathbb {Z}}( Ai, 0)+ {\mathbb {Z}}(0, B)+{\mathbb {Z}}(0, Bi), \ \ A>B>0, \end{aligned}$$

then a direct computation gives

$$\begin{aligned} \sup _{\beta \in {\mathcal {B}}}m_\beta (\Gamma )=\frac{A^2B^2}{A^2+B^2} > \frac{B^2}{2}= \frac{m(\Gamma )}{2}. \end{aligned}$$

1.3 Part II: Gaussian Gabor frames

In this section we shall show how to apply the preceding results on sets of interpolation in ${\mathcal {F}}^2$ in Gabor analysis. We use the same symbol $\Gamma _{\Omega , \Lambda ^\circ }$ in (1.2) to denote the underlying lattice

$$\begin{aligned} \{(\eta , y)\in {\mathbb {R}}^n \times {\mathbb {R}}^n: \eta +i y\in \Gamma _{\Omega , \Lambda ^\circ }\} \end{aligned}$$

in ${\mathbb {R}}^n\times {\mathbb {R}}^n$. By a direct computation, we know that the symplectic dual of $\Gamma _{\Omega , \Lambda ^\circ }$ is equal to

$$\begin{aligned} \Gamma _{\Omega , \Lambda }:=\{(\textrm{Im}\, \Omega )^{-1/2} (\xi +\textrm{Re}\,\Omega x, \textrm{Im}\, \Omega x) \in {\mathbb {R}}^n\times {\mathbb {R}}^n: (\xi , x)\in \Lambda \}. \end{aligned}$$

Hence Proposition 1.4 gives the following:

Corollary 1.9

$(g_\Omega ,\Lambda )$ defines a frame in $L^2({\mathbb {R}}^n)$ if and only if $(g_{iI}, \Gamma _{\Omega , \Lambda })$ does.

Remark

Notice that $\Gamma _{\Omega , \Lambda }$ is equal to $f_\Omega (\Lambda )$, where

$$\begin{aligned} f_\Omega (\xi , x):=(\textrm{Im}\, \Omega )^{-1/2} (\xi +\textrm{Re}\,\Omega x, \textrm{Im}\, \Omega x) \end{aligned}$$

is a linear mapping preserving the standard symplectic form $\omega :=d\xi ^T\wedge dx$ on ${\mathbb {R}}^n\times {\mathbb {R}}^n$. In dimension one, we know that $(g_\Omega , \Lambda )$ defines a frame in $L^2({\mathbb {R}})$ if and only if $(g_{iI}, \Lambda )$ defines a frame in $L^2({\mathbb {R}})$ by the Theorem of Lyubarskii-Seip-Wallstén. However, the following result implies that this is not the case for multivariate Gabor systems.

Theorem 1.10

The Gabor system $(g_{iI}, ({\mathbb {Z}}\oplus \frac{1}{2}{\mathbb {Z}})^2)$ does not give a frame in $L^2({\mathbb {R}}^2)$. Note that we have

• (1) There exists an ${\mathbb {R}}$-linear isomorphism f of ${\mathbb {R}}^4$ preserving the standard symplectic form $d\xi ^T\wedge dx$ on ${\mathbb {R}}^2\times {\mathbb {R}}^2$ such that $( g_{iI}, f({\mathbb {Z}}\oplus \frac{1}{2}{\mathbb {Z}})^2)$ does give a frame in $L^2({\mathbb {R}}^2)$;

• (2) There exists $\Omega \in {\mathfrak {H}}$ such that $( g_\Omega , ({\mathbb {Z}}\oplus \frac{1}{2}{\mathbb {Z}})^2)$ does give a frame in $L^2({\mathbb {R}}^2)$. Moreover, the set $\{\Omega \in {\mathfrak {H}}: ( g_\Omega , ({\mathbb {Z}}\oplus \frac{1}{2}{\mathbb {Z}})^2)\text { is not a frame}\}$ is included in a closed analytic subset of the Siegel upper half-space ${\mathfrak {H}}$.

In the one-dimensional case, we obtain the following estimates thanks to Faltings’ Green function formula [Fal84] (see also Sect. 5.3).

Theorem B

Let $\Lambda$ be a lattice in ${\mathbb {R}}\times {\mathbb {R}}$. Put

$$\begin{aligned} \Gamma :=\{z \in {\mathbb {C}}: z=\eta +i y, \ \ (\eta , y)\in \Lambda ^\circ \}, \ \ C:=\frac{\pi }{4} \cdot \inf _{0\ne \lambda \in \Gamma } |\lambda |^2. \end{aligned}$$

Then $|\Lambda |^{-1}=|\Gamma | \ge C$ and we have the following estimates:

• (1) If $C \ge 2$ then for all $f\in L^2({\mathbb {R}})$ with $||f||=1$, we have

  $$\begin{aligned} \frac{e}{4|\Lambda |} \le \sqrt{2}\cdot \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_{iI})|^2 \le \frac{1}{(1-e^{-C})|\Lambda |}. \end{aligned}$$

  (1.11)

• (2) If $1<C< 2$ then for all $f\in L^2({\mathbb {R}})$ with $||f||=1$, we have

  $$\begin{aligned} \frac{(C-1) e}{C^2|\Lambda |} \le \sqrt{2}\cdot \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_{iI})|^2 \le \frac{1}{(1-e^{-C})|\Lambda |}. \end{aligned}$$

  (1.12)

• (3) The most general case. Suppose that $\Lambda$’s symplectic dual lattice $\Gamma$ in ${\mathbb {C}}$ is generated by $\{a, \tau \}$ with $a>0$ and $a\,\textrm{Im}\, \tau >1$. Then for all $f\in L^2({\mathbb {R}})$ with $||f||=1$, we have

  $$\begin{aligned} \frac{4\pi (a\, \textrm{Im}\, \tau -1)|\eta (\tau /a)|^6 }{\left( \sum _{n\in {\mathbb {Z}}} e^{-\pi n^2 \textrm{Im}\, \tau /a}\right) ^2} \le \sqrt{2}\cdot \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_{iI})|^2 \le \frac{a\,\textrm{Im}\, \tau }{1-e^{-C}}, \end{aligned}$$

  (1.13)

  where $\eta (\tau ):=e^{\pi i\tau /12} \,\Pi _{n=1}^\infty (1-e^{2\pi i n \tau })$ is the Dedekind eta function.

Remark

With the notation in (3), we have

$$\begin{aligned} |\Lambda |^{-1}=|\Gamma |=a\,\textrm{Im}\,\tau , \end{aligned}$$

thus one may look at (1.13) as an effective version of [BGL10, Theorem 1.1] (or the corresponding Theorem 1.5). By our definition (1.9) of the Buser–Sarnak constant, we have

$$\begin{aligned} C=\frac{\pi }{4} m(\Gamma ). \end{aligned}$$

The lower bound estimate in (1.13) is based on a precise Robin constant estimate (see Theorem 5.5). For the upper bound, note that by [Sie89, Theorem 37], we have

$$\begin{aligned} C \le \frac{\pi }{2\sqrt{3}}\, |\Gamma |, \end{aligned}$$

where equality holds if and only if $\Gamma$ is the hexagonal lattice. Hence the best upper bound in (1.13) (for fixed $a\,\textrm{Im}\,\tau$) is attached if and only if $\Gamma$ is the hexagonal lattice. This fact is compatible with the Strohmer–Beaver conjecture (see [FS17, section 2.4]). However, the upper bound in (1.13) is not optimal in general.

2 Preliminaries

2.1 Gabor transform and Bargmann transform

For $\Omega \in {\mathfrak {H}}$ (see 1.1) we identify ${\mathbb {R}}^n\times {\mathbb {R}}^n$ with ${\mathbb {C}}^n$ via $(\xi ,x)\mapsto z:=\xi +\Omega x$. We call the short-time Fourier transform of $f\in L^2({\mathbb {R}}^n)$ with respect to

$$\begin{aligned} g_{\Omega }(t)=\overline{e^{\pi i t^T\Omega t}} \end{aligned}$$

the Gabor transform of f:

$$\begin{aligned} {\mathcal {V}}_{g_\Omega }f (\xi ,x)=\int _{{\mathbb {R}}^n} f(t) e^{-2\pi i \xi ^T t} \overline{g_{\Omega }(t-x)} dt_1\cdots dt_n. \end{aligned}$$

The latter can be related to the $\Omega$-Bargmann transform

$$\begin{aligned} {\mathcal {B}}_\Omega f(z):=\int _{{\mathbb {R}}^n} f(t) e^{\pi i t^T\Omega t}e^{-2\pi i z^T t} dt_1\cdots dt_n. \end{aligned}$$

as follows:

$$\begin{aligned} {\mathcal {V}}_{g_\Omega }f (\xi ,x)= {\mathcal {B}}_\Omega f(z) e^{\pi i x^T\Omega x}. \end{aligned}$$

By Moyal’s identity we have (note than $||g_\Omega ||^2=(2^n \det (\textrm{Im}\,\Omega ))^{-1/2}$)

$$\begin{aligned} \int _{{\mathbb {C}}^n} |{\mathcal {B}}_\Omega f(z)|^2 e^{-2\pi x^T \mathrm{Im\,}\Omega x} d\xi _1 \wedge dx_1 \wedge \cdots \wedge d\xi _n \wedge dx_n =(2^n \det (\textrm{Im}\,\Omega ))^{-1/2}\cdot ||f||^2. \end{aligned}$$

(2.1)

Since $z=\xi +\Omega x$ implies that

$$\begin{aligned} d\xi _1 \wedge dx_1 \wedge \cdots \wedge d\xi _n \wedge dx_n = (\det (\textrm{Im}\,\Omega ))^{-1} \left( \frac{i}{2} \partial {\overline{\partial }}|z|^2\right) ^n, \end{aligned}$$

we know that (2.1) gives

$$\begin{aligned} (\det (\textrm{Im}\,\Omega ))^{-1} ||{\mathcal {B}}_\Omega f||_\Omega ^2 =(2^n \det (\textrm{Im}\,\Omega ))^{-1/2}\cdot ||f||^2, \end{aligned}$$

(2.2)

where

$$\begin{aligned} ||{\mathcal {B}}_\Omega f||_\Omega ^2:=\int _{{\mathbb {C}}^n} |{\mathcal {B}}_\Omega f(z)|^2 e^{-2\pi x^T \mathrm{Im\,}\Omega x} \left( \frac{i}{2} \partial {\overline{\partial }}|z|^2\right) ^n. \end{aligned}$$

Sometimes we shall omit the Lebesgue volume form $\left( \frac{i}{2} \partial {\overline{\partial }}|z|^2\right) ^n$ in the above. Now we are ready to introduce the following definition.

Definition 2.1

Fix $\Omega \in {\mathfrak {H}}$ and put (notice that $(\textrm{Im}\, \Omega ) x= \textrm{Im}\, z$)

$$\begin{aligned} \phi _\Omega (z):=x^T \mathrm{Im\,}\Omega \, x= (\textrm{Im}\,z)^T(\textrm{Im}\, \Omega )^{-1} \textrm{Im}\, z. \end{aligned}$$

We call the space of holomorphic functions F on ${\mathbb {C}}^n$ with

$$\begin{aligned} ||F||^2_\Omega :=\int _{{\mathbb {C}}^n} |F(z)|^2 e^{-2\pi \phi _\Omega (z)} <\infty , \end{aligned}$$

the $\Omega$-Bargmann–Fock space and denote it by ${\mathcal {F}}^2_\Omega$.

Remark

In case $\Omega = i I_n$, where $I_n$ denotes the identity matrix, ${\mathcal {F}}^2_\Omega$ is precisely the following classical Bargmann–Fock space (see [Mum79, page 7] for the related Von-Neumann–Stone theorem)

$$\begin{aligned} {\mathcal {B}}:=\left\{ F\in {\mathcal {O}}({\mathbb {C}}^n): \int _{{\mathbb {C}}^n} |F(z)|^2 e^{-2\pi |\textrm{Im}\, z|^2} \, d\xi _1\cdots d\xi _n dx_1\cdots dx_n <\infty \right\} . \end{aligned}$$

(2.3)

In time-frequency analysis, the Bargmann–Fock space ${\mathcal {F}}^2$ in (1.4) is more widely used. But these two spaces are naturally isomorphic to each other since

$$\begin{aligned} |F(z)|^2 e^{-2\pi \phi _{i I_n}(z)} = |F(z) e^{-\pi z^T z/2}|^2 e^{-\pi z^T z}. \end{aligned}$$

Let us briefly recall the basics of Gaussian Gabor frames: We associate to the Gabor system $(g_\Omega ,\Lambda )$ the following operators:

• analysis operator: $C^\Lambda _{g_\Omega }$ is a map from $L^2({\mathbb {R}}^n)$ to $l^2(\Lambda )$ defined by $f\mapsto \{(f, \pi _\lambda g_\Omega )\}_{\lambda \in \Lambda }$;

• synthesis operator: $D_{g_\Omega }^\Lambda$ is a map from $l^2(\Lambda )$ to $L^2({\mathbb {R}}^n)$ given by $D_{g_\Omega }^\Lambda c=\sum _{\lambda \in \Lambda } c_\lambda \pi _\lambda {g_\Omega }$;

• frame operator: $S_{g_\Omega }^\Lambda :=D_{g_\Omega }^\Lambda \circ C^\Lambda _{{g_\Omega }}$ is an operator on $L^2({\mathbb {R}}^n)$:

  $$\begin{aligned} S_g^\Lambda f=\sum _{\lambda \in \Lambda } (f, \pi _\lambda g_\Omega ) \pi _\lambda g_\Omega . \end{aligned}$$

An elementary computation shows that $(C^\Lambda _{g_\Omega })^*=D_{g_\Omega }^\Lambda$ and thus $S_{g_\Omega }^\Lambda :=(C_{g_\Omega }^\Lambda )^*\circ C^\Lambda _{g_\Omega }$ is a selfadjoint operator. The following result is well known, see [Gro01].

Lemma 2.1

For an arbitrary lattice $\Lambda$ in ${\mathbb {R}}^{n} \times {\mathbb {R}}^n$, the coefficient operator $C^\Lambda _{g_\Omega }$ from $L^2({\mathbb {R}}^n)$ to $l^2(\Lambda )$ is bounded.

Proof

We shall give a proof for readers’ convenience. Using the $\Omega$-Bargmann transform, it suffices to show that

$$\begin{aligned} \sum _{z\in \Gamma } |F(z)|^2e^{-2\pi (\textrm{Im}\,z)^T(\textrm{Im}\, \Omega )^{-1} \textrm{Im}\, z} \le C \, ||F||^2_\Omega , \ \ \forall \ F\in {\mathcal {F}}^2_\Omega , \end{aligned}$$

where $\Gamma :=\{z=\xi +\Omega x: (\xi ,x)\in \Lambda \}$. Since for $w:=(\textrm{Im}\, \Omega )^{-1/2}z$, we have

$$\begin{aligned} |F(z)|^2 e^{-2\pi (\textrm{Im}\,z)^T(\textrm{Im}\, \Omega )^{-1} \textrm{Im}\, z}=|F((\textrm{Im}\, \Omega )^{1/2}w) e^{\frac{\pi }{2} w^T w}|^2 e^{-\pi |w|^2}, \ \end{aligned}$$

we know the above inequality is equivalent to that

$$\begin{aligned} \sum _{w\in \Gamma '} |F(w)|^2e^{-\pi |w|^2} \le C \, ||F||^2, \ \ \forall \ F\in {\mathcal {F}}^2, \end{aligned}$$

where $\Gamma ':=\{w=(\textrm{Im}\, \Omega )^{-1/2}(\xi +\Omega x): (\xi ,x)\in \Lambda \}$. By the submean inequality, we know that the above inequality is true for

$$\begin{aligned} C=\int _{|w|<R} e^{-\pi |w|^2}, \ \ R:=\inf _{w\in \Gamma ', \, w\ne 0} |w|/2, \end{aligned}$$

hence the lemma follows. $\square$

There is a fundamental duality theory (see [DLL95, Jan95, RS97, CKL04] and [JL20, Theorem 4.22]) that links the Gabor system $(g_\Omega ,\Lambda )$ with another Gabor system associated to the symplectic dual lattice/adjoint lattice defined in Definition 1.1.

Theorem 2.2

(Duality Theorem). $(g_\Omega ,\Lambda )$ is a Gabor frame for $L^2({\mathbb {R}}^n)$ with bounds A and B if and only if $\{\pi _{\lambda ^\circ }g_\Omega \}_{\lambda ^\circ \in \Lambda ^\circ }$ is a Riesz sequence with bounds $A|\Lambda |$ and $B|\Lambda |$, i.e. we have

$$\begin{aligned} A|\Lambda |\,\Vert c\Vert ^2\le \Vert \sum _{\lambda ^\circ \in \Lambda ^\circ }c_{\lambda ^\circ }\pi _{\lambda ^\circ }g_\Omega \Vert ^2 \le B|\Lambda |\,\Vert c\Vert ^2 \end{aligned}$$

for all $c\in \ell ^2(\Lambda ^\circ )$.

There is an intricate link between Gabor analysis and Bargmann-Fock spaces: a Gaussian Gabor system $(g_\Omega ,\Lambda )$ is a frame if and only if $\Lambda$ is a set of sampling for ${\mathcal {F}}^2_\Omega$, and $(g_\Omega ,\Lambda ^\circ )$ is a Riesz basis for its closed linear span if and only if $\Lambda ^\circ$ is a set of interpolation for ${\mathcal {F}}^2_\Omega$, see [GL20] for the standard case $\Omega =iI$.

Motivated by this we introduce the following well known notions:

Definition 2.2

Let $T: H_1 \rightarrow H_2$ be a bounded ${\mathbb {C}}$-linear map between two complex Hilbert spaces. Then

• (1) T is called sampling if there exist constants $A, B>0$ such that

  $$\begin{aligned} A\,||f||^2 \le ||Tf||^2 \le B\,||f||^2, \ \ \forall \ f\in H_1; \end{aligned}$$

  (2.4)

• (2) T is referred to as interpolating if T is surjective and there exist constants $A, B>0$ such that

  $$\begin{aligned} A\,||f_c||^2 \le ||c||^2 \le B\, ||f_c||^2, \ \ \ \forall \ c\in H_2, \end{aligned}$$

  (2.5)

  where $f_c$ denotes the (unique) solution of $T(\cdot )=c$ with minimal norm.

The constants A, B above are called the sampling (interpolating) bounds.

Proposition 2.3

T is sampling with (2.4) if and only if $T^*$ is interpolating with (2.5).

Proof

Assume that T is sampling with (2.4). Then the eigenvalues of $T^*T$ lie in [A, B], thus $T^*T$ has an inverse, say $S:=(T^*T)^{-1}$, which implies that $T^* T S f=f, \ \ \forall \ c\in H_1$. Thus $T^*: H_2\rightarrow H_1$ is surjective and the minimal solution of $T^*(\cdot )=f$ is TSf (note that TSf is minimal since $TSf \bot \ker \, T^*$). We need to show that

$$\begin{aligned} A||TSf||^2 \le ||f||^2 \le B |||TSf||^2. \end{aligned}$$

In fact, $S^{-1}\ge A\,I$ implies that

$$\begin{aligned} ||TSf||^2=(TSf, TSf)=(Sf, T^*TSf)=(Sf, f)\le \frac{1}{A} ||f||^2, \end{aligned}$$

thus $A||TSf||^2 \le ||f||^2$. Moreover, $f=T^*TSf$ implies

$$\begin{aligned} ||f||=\sup _{||g||=1} (T^*TSf, g)&=\sup _{||g||=1} (TSf, Tg) \\&\le ||TSf|| \cdot \sup _{||g||=1} ||Tg|| \le \sqrt{B} ||TSf||, \end{aligned}$$

which gives $||f||^2 \le B ||TSf||^2$. This establishes one direction and the other direction may be deduced in a similar manner. $\square$

The following theorem follows directly from Theorem 2.2 and Definition 2.2.

Theorem 2.4

Let $\Lambda$ be a lattice in ${\mathbb {R}}^{n} \times {\mathbb {R}}^n$. Then $C_{g_\Omega }^\Lambda$ is sampling with

$$\begin{aligned} A\, I \le S_{g_\Omega }^\Lambda \le B\,I \end{aligned}$$

if and only if $D_{g_\Omega }^{\Lambda ^\circ }$ is sampling with

$$\begin{aligned} |\Lambda |\cdot A \le (S_{g_\Omega }^{\Lambda ^\circ })^* \le |\Lambda |\cdot B. \end{aligned}$$

Notice that $C_{g_\Omega }^\Lambda$ is sampling if and only $(g_\Omega ,\Lambda )$ defines a frame in $L^2({\mathbb {R}}^n)$. The density theorem for Gabor frames states that if $C_{g_\Omega }^\Lambda$ is sampling, then $|\Lambda | \le 1$. Furthermore, a Balian-Low type theorem (see [AFK14, Theorem 1.5] or [GHO19] for related results associated to general Fock spaces) further gives:

Theorem 2.5

Given a lattice $\Lambda$ in ${\mathbb {R}}^{n} \times {\mathbb {R}}^n$. If $C_{g_\Omega }^\Lambda$ is sampling then $|\Lambda |<1$.

The above two theorems and Proposition 2.3 imply

Corollary 2.6

Given a lattice $\Lambda$ in ${\mathbb {R}}^{n} \times {\mathbb {R}}^n$. Then $C_{g_\Omega }^\Lambda$ is sampling if and only if $C_{g_\Omega }^{\Lambda ^\circ }$ is interpolation. Moreover, the interpolation bounds are a scalar multiple of the sampling bounds. In particular, $C_{g_\Omega }^\Lambda$ can not be both sampling and interpolation.

Proof

The first part follows directly from Theorem 2.4 and Proposition 2.3. For the second part, notice that if $C_{g_\Omega }^\Lambda$ is both sampling and interpolation, we must have

$$\begin{aligned} |\Lambda |<1, \ \ |\Lambda ^\circ |<1, \end{aligned}$$

which is a contradiction since $|\Lambda | \cdot |\Lambda ^\circ |=1$. $\square$

Remark

In case $n=1$ and

$$\begin{aligned} g(t)=\overline{e^{\pi i a t^2}}, \end{aligned}$$

for some complex number a with $\textrm{Im} \,a>0$, we known that $C_{g}^\Lambda$ is sampling if and only if $|\Lambda | <1$ (see [Lyu92, Sei92, SW92]). For general n, $g_\Omega (t):= \overline{e^{\pi i t^T\Omega t}}$, Theorem 1.10 implies that there exists a lattice $\Lambda$ in ${\mathbb {R}}^{n} \times {\mathbb {R}}^n$ such that $C_{g_{\Omega }}^\Lambda$ is sampling (resp. interpolation) for some $\Omega \in {\mathfrak {H}}$ but not for all $\Omega \in {\mathfrak {H}}$. On the other hand, if $C_{g_{\Omega _0}}^\Lambda$ is sampling for some $\Omega _0\in {\mathfrak {H}}$ then by Theorem 1.3 in [AFK14], we know that $C_{g_\Omega }^\Lambda$ is sampling if $\Omega$ is very close to $\Omega _0$.

2.2 Proof of Proposition 1.4

Proof of Proposition 1.4

By our definition, $(g_\Omega ,\Lambda )$ defines a frame in $L^2({\mathbb {R}}^n)$ if and only if $C_{g_\Omega }^\Lambda$ is sampling, which is equivalent to that $C_{g_\Omega }^{\Lambda ^\circ }$ is interpolation (see Corollary 2.6). Using the $\Omega$ Bargmann transform, we know that $C_{g_\Omega }^{\Lambda ^\circ }$ is interpolation if and only if $C_{g_\Omega }^{\Lambda ^\circ }$ is bounded and

$$\begin{aligned} \Gamma := \{\xi +\Omega x\in {\mathbb {C}}^n: (\xi , x)\in \Lambda ^\circ \} \end{aligned}$$

is a set of interpolation for ${\mathcal {F}}^2_\Omega$. By Lemma 2.1, we know that $C_{g_\Omega }^{\Lambda ^\circ }$ is always bounded, hence $(g_\Omega ,\Lambda )$ defines a frame in $L^2({\mathbb {R}}^n)$ if and only if $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2_\Omega$. Notice that

$$\begin{aligned} |F(z)|^2 e^{-2\pi \phi _\Omega (z)}=|F((\textrm{Im}\, \Omega )^{1/2}w) e^{\frac{\pi }{2} w^T w}|^2 e^{-\pi |w|^2}, \ \ w:=(\textrm{Im}\, \Omega )^{-1/2}z \end{aligned}$$

implies that

$$\begin{aligned} F(z) \mapsto F((\textrm{Im}\, \Omega )^{1/2}w) e^{\frac{\pi }{2} w^T w} \end{aligned}$$

(2.6)

defines an isomorphism from ${\mathcal {F}}^2_\Omega$ to ${\mathcal {F}}^2$. Thus $(g_\Omega ,\Lambda )$ defines a frame in $L^2({\mathbb {R}}^n)$ if and only if

$$\begin{aligned} (\textrm{Im}\, \Omega )^{-1/2}\Gamma = \Gamma _{\Omega , \Lambda ^\circ } \end{aligned}$$

is a set of interpolation for ${\mathcal {F}}^2$. $\square$

The duality principle Theorem 2.4 further implies:

Theorem 2.7

With the notation in the above proof, the following statements are equivalent:

• (1) $\Gamma _{\Omega , \Lambda ^\circ }$ is a set of interpolation for ${\mathcal {F}}^2$ and for all $F\in {\mathcal {F}}^2$ with

  $$\begin{aligned} \sum _{\gamma \in \Gamma _{\Omega , \Lambda ^\circ }} |F(\gamma )|^2e^{-\pi |\gamma |^2} =1, \end{aligned}$$

  we have

  $$\begin{aligned} A\le \inf _{F' \in {\mathcal {F}}^2,\, F'=F \, \text {on} \,\Gamma _{\Omega , \Lambda ^\circ } } ||F'||^2 \le B; \end{aligned}$$

• (2) $\Gamma$ is a set of interpolation for ${\mathcal {F}}_\Omega ^2$ and for all $F\in {\mathcal {F}}_\Omega ^2$ with

  $$\begin{aligned} \sum _{\gamma \in \Gamma } |F(\gamma )|^2e^{-2\pi \phi _\Omega (\gamma )} =1, \end{aligned}$$

  we have

  $$\begin{aligned} A \cdot \det (\textrm{Im}\,\Omega )\le \inf _{F' \in {\mathcal {F}}_\Omega ^2,\, F'=F \, \text {on} \,\Gamma } ||F'||^2 \le B \cdot \det (\textrm{Im}\,\Omega ); \end{aligned}$$

• (3) $(\Lambda , g_\Omega )$ defines a frame in $L^2({\mathbb {R}}^n)$ and for all $f\in L^2({\mathbb {R}}^n)$, $||f||=1$,

  $$\begin{aligned} \frac{(B\cdot |\Lambda |)^{-1}}{\sqrt{2^n\det (\textrm{Im}\,\Omega )}} \le \sum _{\lambda \in \Lambda } |(f, \pi _\lambda g_\Omega )|^2 \le \frac{(A\cdot |\Lambda |)^{-1}}{\sqrt{2^n\det (\textrm{Im}\,\Omega )}}. \end{aligned}$$

Proof

(2.6) implies $(1)\Leftrightarrow (2)$. By (2.2), we know that (2) is equivalent to that $C_{g_\Omega }^{\Lambda ^\circ }$ is interpolation (see 2.5) with

$$\begin{aligned} \frac{B^{-1}\cdot ||f_c||^2}{\sqrt{2^n\det (\textrm{Im}\,\Omega )}} \le ||c||^2 \le \frac{A^{-1}\cdot ||f_c||^2}{\sqrt{2^n\det (\textrm{Im}\,\Omega )}}, \ \ \ \forall \ c\in l^2. \end{aligned}$$

By Proposition 2.3, the above inequality is equivalent to that

$$\begin{aligned} \frac{B^{-1}}{\sqrt{2^n\det (\textrm{Im}\,\Omega )}}\,I \le (S_{g_\Omega }^{\Lambda ^\circ })^* \le \frac{A^{-1}}{\sqrt{2^n\det (\textrm{Im}\,\Omega )}}\,I. \end{aligned}$$

Thus Theorem 2.4 gives $(2)\Leftrightarrow (3)$. $\square$

2.3 Proof of the Hörmander criterion (Theorem 1.7)

2.3.1 $L^2$-estimate for the ${\overline{\partial }}$-equation

We shall use the following special case of Hörmander’s theorem (see [Dem, page 378, Theorem 6.5], see also Chapter 4 in [Hor90]):

Theorem 2.8

Fix a smooth (0, 1)-form v with ${\overline{\partial }}v=0$ on ${\mathbb {C}}^n$. Let $\phi$ be a plurisubharmonic function such that $\phi -\delta |z|^2$ is also plurisubhamonic on ${\mathbb {C}}^n$ for some positive constant $\delta$. Then there is a smooth function a on ${\mathbb {C}}^n$ such that ${\overline{\partial }}u=v$ and

$$\begin{aligned} \int _{{\mathbb {C}}^n} |u|^2 e^{-\phi } \le \frac{1}{\delta }\int _{{\mathbb {C}}^n} |v|^2 \, e^{-\phi }, \end{aligned}$$

where $|v|^2:=\sum |v_{{\bar{j}}}|^2$ for $v=\sum v_{{\bar{j}}} d{\bar{z}}_j$.

Proof of the Hörmander criterion (Theorem 1.7)

Notice that the $\beta$-Seshadri constant does not depend on the choose of $x\in {\mathbb {C}}^n/\Gamma$. Thus, if the Hörmander constant is bigger than one then there exist $\gamma >1$ and an $\omega _\textrm{euc}$-psh function $\psi$ on ${\mathbb {C}}^n/\Gamma$ such that $\psi =\gamma T_\beta$ near $0\in {\mathbb {C}}^n/\Gamma$ for some $\beta \in {\mathcal {B}}$. Let

$$\begin{aligned} p: {\mathbb {C}}^n \rightarrow {\mathbb {C}}^n/\Gamma , \end{aligned}$$

be the natural quotient mapping. Fix $c=\{c_\lambda \}$ such that

$$\begin{aligned} \sum _{\lambda \in \Gamma } |c_\lambda |^2 e^{-\pi |\lambda |^2}=1. \end{aligned}$$

Let us apply Theorem 2.8 to

$$\begin{aligned} \phi (z):=\pi |z|^2+\frac{\psi (p(z))}{\gamma }, \ \ \ v(z):= \sum _{\lambda \in \Gamma } c_\lambda e^{\pi {\bar{\lambda }}(z-\lambda )} {\overline{\partial }}\chi (|z-\lambda |), \end{aligned}$$

where $\chi$ is a smooth function on ${\mathbb {R}}$ that is equal to 1 near the origin and equals to 0 outside a smooth ball of radius r. Let us take r such that

$$\begin{aligned} \{z\in {\mathbb {C}}^n: |z-\lambda |<r\} \cap \{z\in {\mathbb {C}}^n: |z-\lambda '|<r\}=\emptyset , \ \ \forall \ \lambda \ne \lambda ' \in \Gamma . \end{aligned}$$

Then we know that v is smooth, ${\overline{\partial }}v=0$ and

$$\begin{aligned} \int _{{\mathbb {C}}^n} |v|^2 e^{-\phi } < C, \end{aligned}$$

for some constant thay does not depend on the sequence $c=\{c_\lambda \}$. Moreover, since $\psi$ is $\omega _\textrm{euc}$-psh, we know that $\phi (z)-(1-\gamma ^{-1})\pi |z|^2$ is plurisubharmonic. Thus Theorem 2.8 implies that there exists a smooth function u such that ${\overline{\partial }}u=v$ and

$$\begin{aligned} \int _{{\mathbb {C}}^n} |u|^2 e^{-\phi }<\frac{C}{(1-\gamma ^{-1})\pi }. \end{aligned}$$

(2.7)

By a direct computation we know that $e^{-T_\beta }$ is not integrable near $0\in {\mathbb {C}}^n/\Gamma$, hence $e^{-\phi }$ is not integrable near $\Gamma$ and (2.7) implies that u vanishes at $\Gamma$. Take

$$\begin{aligned} F(z)= \sum _{\lambda \in \Gamma } c_\lambda e^{\pi {\bar{\lambda }}(z-\lambda )} \chi (|z-\lambda |) - u(z), \end{aligned}$$

we know that F is holomorphic in ${\mathbb {C}}^n$,

$$\begin{aligned} \int _{{\mathbb {C}}^n} |F(z)|^2 e^{-\pi |z|^2} <C_1, \end{aligned}$$

for some constant $C_1$ does not depend on c (notice that $\psi$ is bounded from above) and $F(\lambda )=c_\lambda$ for all $\lambda \in \Gamma$. Thus $\Gamma$ is a set of interpolation. The final statement is a direct consequence pf Proposition 1.6, which will be proved in Sect. 3.1. $\square$

In order to estimate the $L^2$ norm of the extension F in the above proof, we shall introduce the following Ohsawa–Takegoshi type theorem [OT87] proved by Berndtsson and Lempert (see [BL16, Theorem 3.8], the main theorem in [GZ15] and [Blo13] for related results).

Theorem 2.9

Let $\Gamma$ be a lattice in ${\mathbb {C}}^n$. Assume that there exists a non positive $\Gamma$ invariant function $\psi$ on ${\mathbb {C}}^n$ such that $\psi (z)+\pi |z|^2$ is plurisubharmonic on ${\mathbb {C}}^n$, $\psi$ is smooth outside $\Gamma$ and $\psi (z) - \gamma \log |z|^2$ is bounded near the origin for some constant $\gamma >n$. Then for every sequence of complex numbers $\{c_{\lambda }\}_{\lambda \in \Gamma }$ with $\sum _{\lambda \in \Gamma } |c_\lambda |^2 e^{-\pi |\lambda |^2}=1,$ there exists $F\in {\mathcal {F}}^2$ such that $F(\lambda )=c_\lambda$ for all $\lambda \in \Gamma$ and

$$\begin{aligned} ||F||^2 \le \left( 1-\frac{n}{\gamma }\right) ^{-1} \cdot \frac{\pi ^n}{n!} \cdot e^{-\frac{n}{\gamma }\rho }, \ \ \ \rho := \liminf _{z\rightarrow 0} \psi (z) - \gamma \log |z|^2. \end{aligned}$$

Proof

Since $\psi$ is $\Gamma$ invariant, we know that $\psi$ has isolated order $\gamma$ log poles at $\Gamma$, one may use Ohsawa–Takegoshi extension theorem to extend $L^2$ functions from $\Gamma$ to ${\mathbb {C}}^n$. Denote by F the extension with minimal $L^2$ norm. By our assumption

$$\begin{aligned} i\partial {\overline{\partial }}\left( \frac{n}{\gamma } \psi (z)+\pi |z|^2\right) \ge \left( 1-\frac{n}{\gamma }\right) i\partial {\overline{\partial }}(\pi |z|^2). \end{aligned}$$

Hence [BL16, Theorem 3.8] or the main theorem in [GZ15] implies

$$\begin{aligned} \left( 1-\frac{n}{\gamma }\right) \int _{{\mathbb {C}}^n}|F|^2 e^{-\pi |z|^2} \le \frac{\pi ^n}{n!} \limsup _{z\rightarrow 0} e^{-\frac{n}{\gamma }(\psi (z)-\gamma \log |z|^2)}, \end{aligned}$$

thus our theorem follows. $\square$

2.4 Transcendental lattices and jet interpolations

Let us first introduce the following definition for jet interpolations.

Definition 2.3

Let $k\ge 0$ be an integer. Let $\Gamma$ be a lattice in ${\mathbb {C}}^n$. Put

$$\begin{aligned} N_k:=\{\alpha =(\alpha _1, \ldots , \alpha _n)\in {\mathbb {Z}}_n: \alpha _j\ge 0, \ \sum \alpha _j \le k\}. \end{aligned}$$

We say that $\Gamma$ is a set of k-jet interpolation for ${\mathcal {F}}^2$ if there exists a constant $C>0$ such that for every sequence of complex numbers $\{c_{\lambda , \alpha }\}_{\lambda \in \Gamma , \alpha \in N_k }$ with $\sum _{\lambda \in \Gamma , \alpha \in N_k} |c_{\lambda , \alpha }|^2 e^{-\pi |\lambda |^2}=1,$ there exists $F\in {\mathcal {F}}^2$ with

$$\begin{aligned} \left( e^{\pi |z|^2}\partial ^{\alpha } (e^{-\pi |z|^2} F)\right) \big |_{z=\lambda } = c_{\lambda ,\alpha }, \ \ \forall \,\lambda \in \Gamma , \,\alpha \in N_k, \ \ \partial ^\alpha f:=\frac{\partial ^{\alpha _1+\cdots +\alpha _n} f}{\partial z_1^{\alpha _1}\cdots \partial z_n^{\alpha _n}}, \end{aligned}$$

and $||F||^2 \le C$.

The proof of the Hörmander criterion above also implies the following result.

Theorem 2.10

Let $k\ge 0$ be an integer. Let $\Gamma$ be a transcendental lattice in ${\mathbb {C}}^n$. Assume that

$$\begin{aligned} |\Gamma | > \frac{(n+k)^n}{n!}, \end{aligned}$$

(2.8)

then $\Gamma$ is a set of k-jet interpolation for ${\mathcal {F}}^2$.

Proof

Since $\Gamma$ is transcendental, by (3.2), we know that (2.8) implies that there exists a non positive $\Gamma$ invariant function $\psi$ on ${\mathbb {C}}^n$ such that $\psi (z)+\pi |z|^2$ is plurisubharmonic on ${\mathbb {C}}^n$, $\psi$ is smooth outside $\Gamma$ and $\psi (z) - \gamma \log |z|^2$ is bounded near the origin for some constant $\gamma >n+k$. Thus the Hörmander $L^2$ estimate with singular weight $\psi$ (similar to the proof of the Hörmander criterion above) gives the above theorem. $\square$

Remark

If $\Gamma$ is transcendental with $|\Gamma |>\frac{n^n}{n!}$ then

$$\begin{aligned} \big | \sqrt{\frac{n+k}{n}} \,\Gamma \big |> \frac{(n+k)^n}{n!}, \end{aligned}$$

thus we know that $\sqrt{\frac{n+k}{n}} \,\Gamma$ is a set of k-jet interpolation for ${\mathcal {F}}^2$. In one dimensional case, we have the following theorem [GL09].

Theorem 2.11

Let $\Gamma$ be a lattice in ${\mathbb {C}}$. Then the followings are equivalent:

• (1) $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$;

• (2) $\sqrt{k+1}\,\Gamma$ is a set of k-jet interpolation for ${\mathcal {F}}^2$ for some positive integer k;

• (3) $|\Gamma |>1$.

For the higher-dimensional cases, we can prove the following result.

Theorem 2.12

Let $\Gamma$ be a transcendental lattice in ${\mathbb {C}}^n$. If $\sqrt{\frac{n+k}{n}} \,\Gamma$ is a set of k-jet interpolation for ${\mathcal {F}}^2$ for some non-negative integer k then $|\Gamma | \ge \frac{(k+1)^n}{(n+k)^n}\frac{n^n}{n!}$.

Proof

Put $\nabla ^\alpha F:=e^{\pi |z|^2}\partial ^{\alpha } (e^{-\pi |z|^2} F)$ and $\Gamma _k:=\sqrt{\frac{n+k}{n}} \,\Gamma$. Assume that $\sqrt{\frac{n+k}{n}} \,\Gamma$ is a set of k-jet interpolation for ${\mathcal {F}}^2$. Let us define

$$\begin{aligned} G(z):=\sup \{|F(z)|^2e^{-\pi |z|^2}: F\in {\mathcal {F}}^2, \ \nabla ^\alpha F(\lambda )=0, \ \forall \ \lambda \in \Gamma _k\}. \end{aligned}$$

(2.9)

Then by the Balian-Low type theorem, we know that G is not identically zero on ${\mathbb {C}}^n$. We claim that G is $\Gamma _k$ invariant. In fact, if we put

$$\begin{aligned} T_\lambda F(z):=F(z+\lambda ) e^{-\pi |\lambda |^2/2}e^{-\pi z{\bar{\lambda }}}. \end{aligned}$$

Then $F \mapsto T_\lambda F$ is an isomorphism on ${\mathcal {F}}^2$ with $||F||=||T_\lambda F||$, $|T_\lambda F(0)|^2=|F(\lambda )|^2e^{-\pi |\lambda |^2}$ and

$$\begin{aligned} \nabla ^\alpha (T_\lambda F)(0)=0 \Leftrightarrow \nabla ^\alpha F(\lambda )=0. \end{aligned}$$

Hence G is $\Gamma _k$ invariant. Put $\psi =\log G$, we know that $\psi$ is $\Gamma$ invariant, $\psi (z)+\pi |z|^2$ is plurisubharmonic on ${\mathbb {C}}^n$ and $\psi (z)-(k+1)\log |z|^2$ is bounded above near $z=0$. Since $\Gamma$ is transcendental, we know that $\Gamma _k$ is also transcendental, thus the $\Gamma _k$ invariant analytic set $\{\psi =-\infty \}$ is discrete. Hence (3.2) gives that

$$\begin{aligned} (n! |\Gamma _k|)^{1/n} \ge k+1, \end{aligned}$$

from which our theorem follows. $\square$

Remark

The above theorem is our motivation for the conjecture A after Theorem 1.1, moreover, notice that $\lim _{k\rightarrow \infty } \frac{k+1}{n+k}=1$, the above theorem also suggests the following higher-dimensional analogue of Theorem 2.11.

Conjecture B

Let $\Gamma$ be a transcendental lattice in ${\mathbb {C}}^n$. Then the followings are equivalent:

• (1) $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$;

• (2) $\sqrt{\frac{n+k}{n}} \,\Gamma$ is a set of k-jet interpolation for ${\mathcal {F}}^2$ for some positive integer k;

• (3) $|\Gamma |>\frac{n^n}{n!}$.

Remark

From Proposition 1.4, we know that the above conjecture implies conjecture A.

3 Hörmander constants and Kähler embeddings

3.1 Hörmander constants and proof of Proposition 1.6

In Definitions 1.3 and 1.5 we have defined the Hörmander constants and the $\beta$-Seshadri constants for an n-dimensional compact Kähler manifold $(X, \omega )$. If all $\beta _j=1/n$ and $\omega \in c_1(L)$ for some ample line bundle L then we have

$$\begin{aligned} n \epsilon _x(\omega ;\beta )= \epsilon _x(\omega ), \end{aligned}$$

(3.1)

where $\epsilon _x(\omega )$ denotes the Seshadri constant introduced by Demailly in [Dem92] (in fact, from (6.2) in [Dem92], we have $n \epsilon _x(\omega ;\beta )=\gamma (L,x)$, but Theorem 6.4 in [Dem92] tells us that $\gamma (L,x)$ is precisely the Seshadri constant used in algebraic geometry when L is ample). In general, the condition $\sum \beta _j=1$ is used to make sure that

$$\begin{aligned} \sup \{c\ge 0: e^{-cT_\beta } \ \text {is integrable near }z=0\}=1. \end{aligned}$$

Remark

For transcendental $\omega$ on a general compact Kähler manifold, we know that (see Theorem 3.2 below for the proof and generalizations) $n \epsilon _x(\omega ;\beta )$ is equal to the generalized Seshadri constant (also denoted by $\epsilon _x(\omega )$) defined by Tosatti in [Tos18, section 4.4]. In this general case, we shall prove the following result.

Proposition 3.1

Let $(X, \omega )$ be an n-dimensional compact Kähler manifold. Assume that X has no non-trivial analytic subvarieties, then

$$\begin{aligned} \sup _{\beta \in {\mathcal {B}}} \epsilon _x(\omega ; \beta ) = \frac{\epsilon _x(\omega )}{n} =\frac{ \left( \int _X \omega ^n\right) ^{1/n}}{n}. \end{aligned}$$

(3.2)

Proof

Note that for every $\beta \in {\mathcal {B}}$, by [Dem, page 167, Corollary 7.4] (our definition of $dd^c$ in Definition 1.5 is half of the one there) we have

$$\begin{aligned} (dd^c T_\beta )^n_x =(\beta _1\cdots \beta _n)^{-1}, \ \ \ \ (dd^c T_\beta )^n_x:=\lim _{r\rightarrow 0} \int _{|z-x|<r} (dd^c T_\beta )^n, \end{aligned}$$

(3.3)

which gives

$$\begin{aligned} \int _X \omega ^n \ge \epsilon _x(\omega ; \beta )^n (\beta _1\cdots \beta _n)^{-1} \ge \epsilon _x(\omega ; \beta )^n n^n. \end{aligned}$$

Hence

$$\begin{aligned} \sup _{\beta \in {\mathcal {B}}} \epsilon _x(\omega ; \beta ) \le \frac{ \left( \int _X \omega ^n\right) ^{1/n}}{n}. \end{aligned}$$

(3.4)

On the other hand, we have the following identity proved by Tosatti in [Tos18, Theorem 4.6]

$$\begin{aligned} \epsilon _x(\omega )=\inf _{V\ni x} \left( \frac{\int _V \omega ^{\dim V}}{\textrm{mult}_x V}\right) ^{\frac{1}{\dim V}}, \end{aligned}$$

where the infimum runs over all positive-dimensional irreducible analytic subvarieties V containing x and $\textrm{mult}_x V$ denotes the multiplicity of V at x. Hence if X has no non-trivial subvarieties then (put $\beta _0= (1/n,\cdots , 1/n)$, use (3.1) and the remark above)

$$\begin{aligned} \sup _{\beta \in {\mathcal {B}}} \epsilon _x(\omega ; \beta ) \ge \epsilon _x(\omega ; \beta _0) =\frac{\epsilon _x(\omega )}{n} = \frac{ \left( \int _X \omega ^n\right) ^{1/n}}{n}. \end{aligned}$$

The above inequality and (3.4) together imply (3.2). $\square$

Proof of Proposition 1.6

Apply (3.2) to the case that $X={\mathbb {C}}^n /\Gamma$ and $\omega = dd^c(\pi |z|^2)$, we get immediately Proposition 1.6 (note that in this case

$$\begin{aligned} \sup _{\beta \in {\mathcal {B}}} \epsilon _x(\omega ; \beta ) = \iota _\Gamma \end{aligned}$$

and $\int _X \omega ^n = n! \,|\Gamma |$). $\square$

3.2 Relation with the s-invariant

Our $\beta$-Seshadri constant is closely related to the s-invariant introduced by Cutkosky, Ein and Lazarsfeld in [CEL01].

Theorem 3.2

Let $(X, \omega )$ be an n-dimensional compact Kähler manifold. Assume that $\omega$ lies in the first Chern class of a holomorphic line bundle L on X. Fix $\beta =(\beta _1,\cdots , \beta _n)\in {\mathbb {R}}^n$ such that all $\beta _j^{-1}$ are positive integers. Then

$$\begin{aligned} \epsilon _x(\omega ;\beta )=\frac{1}{s_{L}({\mathcal {I}}_\beta )} , \end{aligned}$$

(3.5)

where ${\mathcal {I}}_\beta$ is the ideal of ${\mathcal {O}}_X$ generated by $\{z_1^{1/\beta _1}, \cdots , z_n^{1/\beta _n}\}$ and

$$\begin{aligned} s_{L}({\mathcal {I}}_\beta ):=\min \{s\in {\mathbb {R}}: \mu ^*(sL)-E \ \text {is nef}\} \end{aligned}$$

is the s-invariant of ${\mathcal {I}}_\beta$ with respect to L (see Definition 5.4.1 in [Laz04]), where $\mu$ is the blowing-up of X along ${\mathcal {I}}_\beta$ with exceptional divisor E.

Proof

First let us prove $\epsilon _x(\omega ;\beta )\ge \frac{1}{s_{L}({\mathcal {I}}_\beta )}$. Note that L is ample since $\omega$ is positive, hence

$$\begin{aligned} \frac{1}{s_{L}({\mathcal {I}}_\beta )}= \sup \{\gamma \ge 0: \mu ^*L- \gamma E \ \text {is ample}\}. \end{aligned}$$

By Example 5.4.10 in [Laz04], one may replace $\mu$ by a desingularization $f: Y\rightarrow X$ of ${\mathcal {I}}_\beta$ with exceptional divisor F, more precisely, we have

$$\begin{aligned} \frac{1}{s_{L}({\mathcal {I}}_\beta )}= \sup \{\gamma \ge 0: f^*L- \gamma F \ \text {is ample}\}. \end{aligned}$$

which implies that for every $\gamma <\frac{1}{s_{L}({\mathcal {I}}_\beta )}$ there is a singular metric $e^{-\phi }$ on $f^*L$ with $\gamma$-log pole along F such that $i\partial {\overline{\partial }}\phi >0$ on Y. Then the weight $f_*\phi$ on L will have the $\gamma T_\beta$-singularity, from which we know that $\epsilon _x(\omega ;\beta )\ge \gamma$. Hence $\epsilon _x(\omega ;\beta )\ge \frac{1}{s_{L}({\mathcal {I}}_\beta )}$.

Now let us prove that $\epsilon _x(\omega ;\beta )\le \frac{1}{s_{L}({\mathcal {I}}_\beta )}$. For every $\gamma < \epsilon _x(\omega ;\beta )$, we can find a singular metric $e^{-\psi }$ on L with $\gamma T_\beta$-singularity such that $i\partial {\overline{\partial }}\psi >0$. Then $e^{-f^*\psi }$ defines a singular metric on $f^*L$ with $\gamma$-log pole along F such that $i\partial {\overline{\partial }}(f^*\psi )>0$ on Y, from which we know that $\epsilon _x(\omega ;\beta )\le \frac{1}{s_{L}({\mathcal {I}}_\beta )}$. $\square$

Remark

Since ${\mathcal {I}}_\beta$ are special monomial ideals, it is not hard to find the explicit desingularizations. Let us look at the simple example $\beta =(1,\frac{1}{2})$. Then, in this case, we have

$$\begin{aligned} {\mathcal {I}}_\beta = \textrm{Span}\{z_1, z_2^2\}. \end{aligned}$$

First, one may blow up the origin, so $z_1=uv, z_2=u$ gives

$$\begin{aligned} \textrm{Bl}_0 {\mathcal {I}}_\beta = \textrm{Span}\{u v, u^2\}, \end{aligned}$$

then we can blow up the point $(u, v)=(0,0):={\tilde{0}}$, so $v= ts, u=s$ gives

$$\begin{aligned} \textrm{Bl}_{{\tilde{0}}}\textrm{Bl}_0 {\mathcal {I}}_\beta = \textrm{Span}\{s^2\}, \end{aligned}$$

from which we know that $F= 2\cdot |\{s=0\}|$.

3.3 Extremal property of the $\beta$-Seshadri constant

For s-invariant of a general monomial ideal

$$\begin{aligned} {\mathcal {I}}_P:= \textrm{Span}\{z^{\alpha ^1}, \ldots , z^{\alpha ^k}\}, \ \ \alpha ^j \in {\mathbb {Z}}_{>0}^n, \ \ z^{\alpha ^j}:=z_1^{\alpha ^j_1} \cdots z_n^{\alpha ^j_n}, \end{aligned}$$

with isolated zero set $\{x\}$, where P is the Newton polytope defined by

$$\begin{aligned} P:= \text {convex hull of }\cup _{1\le j \le k} P_{\alpha ^j}, \ \ P_{\alpha ^j}:=\{x\in {\mathbb {R}}^n: x_l \ge \alpha ^j_l, \ 1\le l\le n \}, \end{aligned}$$

one may correspondingly define the P-Seshadri constant

$$\begin{aligned} \epsilon _x(\omega ;P):= \sup \{\gamma \ge 0:&\ \ \text {there exists an }\omega \text {-psh function }\psi \nonumber \\&\text {on }X\text { with }\psi =\gamma T_P\text { near }x\}, \end{aligned}$$

(3.6)

where

$$\begin{aligned} T_P:= \log \sum _{\alpha \in v(P)}|z^\alpha |^2, \ \ v(P) \ \text {denotes the set of vertices of }P. \end{aligned}$$

Then the proof of Theorem 3.2 also implies

$$\begin{aligned} \epsilon _x(\omega ;P)=\frac{1}{s_{L}({\mathcal {I}}_P)}. \end{aligned}$$

(3.7)

The reason why we only use $\beta$-Seshadri constants in this paper is that they have the following extremal property.

Theorem 3.3

$\sup \{ \epsilon _x(\omega ;P): (1,\ldots , 1)\text { lies in the boundary of }P\}$ equals $\sup _{\sum \beta _j=1}\epsilon _x(\omega ;\beta )$.

Proof

The proof follows from a very simple fact: for an arbitrary Newton polytope P such that $(1,\ldots , 1)$ lies in the boundary of P, one can always find $\beta \in {\mathbb {R}}_{>0}^n$ with $\sum \beta _j =1$ and

$$\begin{aligned} P\subset P_\beta :=\{x\in {\mathbb {R}}_{\ge 0}^n: \beta _1 x_1+\cdots +\beta _n x_n \ge 1\}, \end{aligned}$$

hence

$$\begin{aligned} \epsilon _x(\omega ;P) \le \epsilon _x(\omega ; P_\beta ) = \epsilon _x(\omega ; \beta ) \end{aligned}$$

gives the theorem. $\square$

Remark

The condition that $(1,\ldots , 1)$ lies in the boundary of P is equivalent to the following identity

$$\begin{aligned} \sup \{c\ge 0: e^{-cT_P} \ \text {is integrable near }z=0\}=1, \end{aligned}$$

see [How01, Gue12] for the proof and related results.

3.4 McDuff–Polterovich’s theorem

McDuff–Polterovich [MP94] proved that the Seshadri constant

$$\begin{aligned} \epsilon _x(\omega ):=&\sup \{\gamma \ge 0: \text {there exists an } \omega \text {-psh function }\psi \\&\text {on }X\text { with }\psi =\gamma \log |z|^2\text { near }x\} \end{aligned}$$

is always no bigger than the following Gromov width of $(X, \omega )$ (see [LMS13, Theorem 1.1] and [EV16])

where means there exist a smooth injection $f: B_r \hookrightarrow X$ such that $f^*(\omega )=\omega _\textrm{euc}:=\frac{i}{2} \sum _{j=1}^n dz_j \wedge d{\bar{z}}_j.$ In fact, the proof in [MP94] also gives the following stronger result:

Theorem 3.4

Let $(X, \omega )$ be a compact Kähler manifold. Denote by ${\mathcal {K}}_\omega$ the space of Kähler metrics in the cohomology class $[\omega ]$. Then the Seshadri constant $\epsilon _x(\omega )$ is equal to the following Kähler width

where

$$\begin{aligned} B_r:=\left\{ z\in {\mathbb {C}}^n: \sum _{j=1}^n |z_j|^2 <r^2 \right\} , \end{aligned}$$

“" means that there exists an holomorphic injection $f: B_r \hookrightarrow X$ such that $f(0)=x$ and $f^*({\tilde{\omega }})=\omega _\textrm{euc}$.

Proof

The proof here is different from McDuff–Polterovich’s approach in [MP94]. Our main idea is to use the following plurisubharmonic function on ${\mathbb {C}}^n$

$$\begin{aligned} \psi _r(z):= {\left\{ \begin{array}{ll} \pi r^2 \left( \log \frac{|z|^2}{r^2} +1\right) &{} |z|<r, \\ \pi |z|^2 &{} |z| \ge r, \end{array}\right. } \end{aligned}$$

(3.8)

which satisfies

• (L1) $\psi _r(z)-\pi r^2 \log |z|^2$ is bounded near 0.

• (L2) $\psi _r(z) \le \pi |z|^2$ on ${\mathbb {C}}^n$ and $\{z\in {\mathbb {C}}^n:\psi _r(z)<\pi |z|^2\}=B_r$.

• (L3) For every $w\in B_r$ and $0<\delta <1$ we have

  $$\begin{aligned} \psi _r(f_\delta (w))= \psi _r(w) + \pi r^2 \log (\delta ^2), \end{aligned}$$

  where $f_\delta (w):=(\delta w_1, \ldots , \delta w_n).$

We shall use (L1)-(L3) to prove (P1), (P2) below which imply the theorem:

• (P1) If $c_x(\omega ) > \pi r^2$ then $\epsilon _x(\omega ) \ge \pi r^2$.

• (P2) If $\epsilon _x(\omega ) > \pi r^2$ then $c_x(\omega ) \ge \pi r^2$.

Proof of (P1): If $c_x(\omega ) > \pi r^2$ then then one may think of $B_r$ as a Kähler subset of X. Let us define ${\tilde{\psi }}$ such that ${\tilde{\psi }}=\psi _r-\pi |z|^2$ on $B_r$ and ${\tilde{\psi }}=0$ outside $B_r$ in X. Then (L2) implies that ${\tilde{\psi }}$ is $\omega$-psh (note that $\omega _\textrm{euc}=dd^c(\pi |z|^2)$). Fix a small $\varepsilon >0$ (assume that $\varepsilon <\pi r^2$), then by (L1) one may take a sufficiently big $C>0$ such that

$$\begin{aligned} \psi (z):=\max \{{\tilde{\psi }}(z)+C, (\pi r^2-\varepsilon )\log |z|^2\} \end{aligned}$$

(3.9)

is equal to ${\tilde{\psi }}+C$ near the boundary of $B_r$. Hence $\psi$ extends to an $\omega$-psh function on X. Notice that $\psi = (\pi r^2-\varepsilon )\log |z|^2$ near $z(x)=0$, hence $\epsilon _x(\omega ) \ge \pi r^2-\varepsilon$ for all $\varepsilon >0$ and (P1) holds.

Proof of (P2): To simplify the notation we take $r=1$. If $\epsilon _x(\omega ) > \pi$ then one may use the construction in (3.9) to produce an $\omega$-psh function $\psi$ smooth on $X\setminus \{x\}$ such that

$$\begin{aligned} (*) \qquad \omega +dd^c\psi =dd^c (\psi _1+|z|^2)\text { near } z(x)=0, \end{aligned}$$

and $\omega +dd^c\psi >0$ on X, where $\psi _1$ in defined in (3.8). For $w\in B_1$ and $0<\delta <1$ let us define

$$\begin{aligned} \phi (f_\delta (w)):= \textrm{Max} \left\{ \psi _1(f_\delta (w)) +|f_\delta (w)|^2, \pi (|w|^2+2\log \delta )\right\} , \end{aligned}$$

where $\textrm{Max}$ denotes a regularized max function. By (L3) and (L2)

$$\begin{aligned} \psi _1(f_\delta (w) ) = \psi _1(w ) +\pi \log (\delta ^2) \le \pi ( |w|^2+2\log \delta ) \end{aligned}$$

for $w\in \overline{B_1}$, with identity holds if and only if $w\in \partial B_1$; together with $(*)$, we know that for every $0<\gamma <1$, one may take a small $\delta$ such that $dd^c \phi$ extends to a Kähler form ${\tilde{\omega }} \in {\mathcal {K}}_\omega$ with

$$\begin{aligned} f_\delta ^*({\tilde{\omega }})=f_\delta ^* dd^c \phi = dd^c(\phi (f_\delta (w)))=dd^c(\pi |w|^2) \ \ \text {on} \ B_{\gamma }. \end{aligned}$$

Letting $\gamma \rightarrow 1$ we finally get $c_x(\omega )\ge \pi$. The proof is complete. $\square$

In order to generalize the above proof to general $\beta$-Seshadri constants, we need to construct the associated $\beta$-version of $\psi _r$ in (3.8) (see Lemma 3.8 below). In the next subsection, we shall use the Legendre transform theory to decode the construction.

3.5 Iterated Legendre transform and proof of Theorem A

The main ingredient in our proof of Theorem A is the theory of iterated Legendre transform.

Definition 3.1

Let $\phi$ be a smooth convex function on ${\mathbb {R}}^n$. We call

$$\begin{aligned} \phi ^*(\alpha ):=\sup _{y\in {\mathbb {R}}^n} \alpha \cdot y-\phi (y) \end{aligned}$$

the Legendre transform of $\phi$. Let $A\subset {\mathbb {R}}^n$ be a closed set. We call

$$\begin{aligned} \phi _A(x):=\sup _{\alpha \in A} \alpha \cdot x- \phi ^*(\alpha ) \end{aligned}$$

the iterated Legendre transform of $\phi$ with respect to A.

Remark

Notice that

$$\begin{aligned} \phi (x)+ \phi ^*(\alpha ) \ge \alpha \cdot x, \end{aligned}$$

hence we always have

$$\begin{aligned} \phi _A \le \phi . \end{aligned}$$

Lemma 3.5

Let $\phi$ be a smooth convex function on ${\mathbb {R}}^n$. Then

$$\begin{aligned} \phi (x)+ \phi ^*(\alpha )= \alpha \cdot x \end{aligned}$$

if and only if $\alpha =\nabla \phi (x)$, where

$$\begin{aligned} \nabla \phi (x):=\left( \frac{\partial \phi }{\partial x_1}(x), \ldots , \frac{\partial \phi }{\partial x_n}(x) \right) . \end{aligned}$$

Proof

It follows from the fact that x is the maximum point of the following concave function

$$\begin{aligned} \psi ^\alpha : y \mapsto \alpha \cdot y-\phi (y) \end{aligned}$$

if and only if x is a critical point of $\psi ^\alpha$. $\square$

Proposition 3.6

If $\phi$ is smooth strictly convex and $A \subset {\mathbb {R}}^n$ is closed then

$$\begin{aligned} \{x\in {\mathbb {R}}^n: \phi (x)=\phi _A (x)\} = \{x\in {\mathbb {R}}^n: \nabla \phi (x) \in A\}. \end{aligned}$$

Proof

By the above lemma, we have

$$\begin{aligned} \alpha \cdot x -\phi ^*(\alpha ) \le \phi (x) \end{aligned}$$

with identity holds if and only if $\alpha =\nabla \phi (x)$. Hence $\nabla \phi (x)$ is the unique maximum point of the following function

$$\begin{aligned} \rho ^x: \alpha \mapsto \alpha \cdot x -\phi ^*(\alpha ). \end{aligned}$$

The proof of Proposition 2.2 in [Wan18] implies that $\rho ^x$ is smooth strictly concave on $\nabla \phi ({\mathbb {R}}^n)$. Hence the supremum of $\rho ^x$ on the complement of any small ball around $\nabla \phi (x)$ must be strictly smaller than $\phi (x)$. By our assumption, A is closed, thus $\phi (x)=\phi _A (x)$ if and only if $\nabla \phi (x) \in A$. $\square$

Definition 3.2

Let $\phi$ be a smooth strictly convex function on ${\mathbb {R}}^n$ and A be a closed set in ${\mathbb {R}}^n$. We call

$$\begin{aligned} \Omega _A(\phi ):=\{x\in {\mathbb {R}}^n: \phi _A (x)<\phi (x)\} \end{aligned}$$

the Hele–Shaw domain of $(\phi , A)$.

The above proposition implies that

$$\begin{aligned} \Omega _A(\phi )= (\nabla \phi )^{-1} ({\mathbb {R}}^n\setminus A). \end{aligned}$$

(3.10)

Our key observation is the following:

Theorem 3.7

Assume that $\phi$ is smooth strictly convex and $A \subset {\mathbb {R}}^n$ is closed. If $x\in \Omega _A(\phi )$ then

$$\begin{aligned} \phi _A(x)=\sup _{\alpha \in \partial A} \alpha \cdot x-\phi ^*(\alpha ), \end{aligned}$$

(3.11)

where $\partial A$ denotes the boundary of A.

Proof

Since $\nabla \phi (x)$ is the unique maximum point of the following concave function

$$\begin{aligned} \rho ^x: \alpha \mapsto \alpha \cdot x -\phi ^*(\alpha ) \end{aligned}$$

and $\nabla \phi (x) \notin A$, we know that for every point $a\in A$,

$$\begin{aligned} f(t):= \rho ^x (t\nabla \phi (x) +(1-t)a ) \end{aligned}$$

is increasing on $t\in [0,1]$. Take ${\hat{t}} \in [0,1]$ such that

$$\begin{aligned} {\hat{t}}\nabla \phi (x) +(1-{\hat{t}})a \in \partial A, \end{aligned}$$

we know that

$$\begin{aligned} \rho ^x(a) =f(0) \le f({\hat{t}}) \le \sup _{\alpha \in \partial A} \rho ^x(\alpha ). \end{aligned}$$

Hence the theorem follows. $\square$

Lemma 3.8

With the notation in Theorem A, for each $r>0$ there exists $\psi _r \in \textrm{psh}({\mathbb {C}}^n)$ such that

• (L1) $\psi _r-\pi r^2 T_\beta$ is bounded near 0.

• (L2) $\psi _r(z) \le \pi |z|^2$ on ${\mathbb {C}}^n$ and $\{z\in {\mathbb {C}}^n: \psi _r(z)<\pi |z|^2\}=B_r^\beta$.

• (L3) For every $w\in B_r^\beta$ and $0<\delta <1$ we have

  $$\begin{aligned} \psi _r(f_\delta (w))= \psi _r(w) + \pi r^2 \log (\delta ^2), \end{aligned}$$

  where $f_\delta (w):=(\delta ^{\beta _1}w_1, \ldots , \delta ^{\beta _n}w_n).$

Proof

Let us consider

$$\begin{aligned} \psi _r(z):=\phi _A(\log |z_1|^2, \ldots , \log |z_n|^2), \end{aligned}$$

where

$$\begin{aligned} \phi (x):=\pi (e^{x_1}+\cdots +e^{x_n}), \ \ A:=\{\alpha \in {\mathbb {R}}^n: \alpha \cdot \beta \ge \pi r^2\}. \end{aligned}$$

Then we have

$$\begin{aligned} \nabla \phi (x) =\pi (e^{x_1}, \ldots , e^{x_n}) \end{aligned}$$

and

$$\begin{aligned} \phi ^*(\alpha )= {\left\{ \begin{array}{ll} \sum _{j=1}^n \left( \alpha _j \log \frac{\alpha _j}{\pi }-\alpha _j \right) &{} \ \forall \ \alpha _j \ge 0, \\ \infty &{} \ \exists \ \alpha _j < 0, \end{array}\right. } \end{aligned}$$

where $0\log 0:=0$. Hence

$$\begin{aligned} \phi _A(x)=\sup _{\alpha \in A_+} \alpha \cdot x-\sum _{j=1}^n \left( \alpha _j \log \frac{\alpha _j}{\pi }-\alpha _j \right) , \end{aligned}$$

where

$$\begin{aligned} A_+:=\{\alpha \in [0, \infty )^n: \alpha \cdot \beta \ge \pi r^2\}. \end{aligned}$$

Proof of (L1): Notice that $\psi _r-\pi r^2 T_\beta$ is bounded near 0 if and only if $\phi _A(x)-\sup _{\alpha \in A_+} \alpha \cdot x$ is bounded on $(-\infty , 0]^n$. The above formula for $\phi _A$ implies that

$$\begin{aligned} \inf _{1\le j\le n} \left( \frac{\pi r^2}{\beta _j}-\frac{\pi r^2}{\beta _j} \log \frac{ r^2}{\beta _j}\right) \le \phi _A(x)-\sup _{\alpha \in A_+} \alpha \cdot x \le n\pi \end{aligned}$$

for all $x\in (-\infty , 0]^n$. Hence (L1) follows.

Proof of (L2): Follows directly from (3.10).

Proof of (L3): Notice that (3.11) implies

$$\begin{aligned} \phi _A(x+(\log (\delta ^2))\beta )=\phi _A (x)+ \pi r^2 \log (\delta ^2), \ \ \ \forall \ x\in \Omega _A(\phi ), \end{aligned}$$

from which (L3) follows. $\square$

Proof of Theorem A

Similar as the proof of Theorem 3.4 (replace $B_r$ by $B_r^\beta$), we know that the lemma above gives Theorem A. $\square$

3.6 A partial converse of the Hörmander criterion

For general higher dimensional cases, we do not know whether the Hörmander criterion is an equivalent criterion or not. Based on Demailly–Păun’s generalized Nakai–Moishezon ampleness criterion [DP04], a theorem of Nakamaye [Nak96, Oht19], Lindholm’s result [Lin01] and the Balian-Low type theorem (see [Hei07, Theorem 10] and [AFK14, Theorem 1.5]), we obtain the following partial converse of the Hörmander criterion.

Theorem 3.9

Let $\Gamma$ be a lattice in ${\mathbb {C}}^n$. Denote by $\iota _\Gamma$ the Hörmander constant of $({\mathbb {C}}^n/\Gamma , \omega _\textrm{euc})$.

• (1) If $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$ and all irreducible analytic subvarieties of X are translates of complex tori then $\iota _\Gamma >1/n$.

• (2) If $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$ and the only positive dimensional irreducible analytic subvariety of X is X itself then

  $$\begin{aligned} \iota _\Gamma =\frac{(n!\,|\Gamma |)^{1/n} }{n} > \frac{(n!)^{1/n} }{n}. \end{aligned}$$

• (3) Assume that $\omega _\textrm{euc}$ is rational on $\Gamma$ or the Picard number of X is $n^2$. If $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$ then $\iota _\Gamma >1/(n\,e)$.

Proof

Denote by 0 the unit of the torus and write $\omega :=\omega _\textrm{euc}$. The main idea is to use the following Demailly–Păun identity proved by Tosatti in [Tos18, Theorem 4.6]

$$\begin{aligned} \epsilon _0(\omega )=\inf _{V\ni 0} \left( \frac{\int _V \omega ^{\dim V}}{\textrm{mult}_0 V}\right) ^{\frac{1}{\dim V}}, \end{aligned}$$

where the infimum runs over all positive-dimensional irreducible analytic subvarieties V containing 0, and $\textrm{mult}_0 V$ denotes the multiplicity of V at 0. Now let us prove Theorem (1), by our assumption, it suffices to show that for all complex subtorus

$$\begin{aligned} \int _V \omega ^{d} >1, \ \ d:=\dim V. \end{aligned}$$

Choose a ${\mathbb {C}}$ linear subspace E of ${\mathbb {C}}^n$ such that $V=E/(E\cap \Gamma )$. Notice that if $\Gamma$ is a set of interpolation for ${\mathcal {F}}^2$, then $E\cap \Gamma$ is a set of interpolation for ${\mathcal {F}}^2|_{E}$. Thus the Balian–Low type theorem implies that

$$\begin{aligned} |V|=\int _V \omega ^{d} /d! >1, \end{aligned}$$

which gives $\epsilon _0(\omega )>1$, hence (1) follows. Now assume further that X has no non-trivial subvarieties, then (3.2) implies

$$\begin{aligned} (n\iota _\Gamma )^n/n!= |X|=|\Gamma |, \end{aligned}$$

thus (2) follows directly from the Balian–Low type theorem. To prove (3), we shall use the following inequality (see [Ito20, Lemma 3.2])

$$\begin{aligned} \epsilon _0(\omega )\ge \inf _{V\ni 0} \frac{\left( \int _V \omega ^{\dim V}\right) ^{\frac{1}{\dim V}}}{\dim V}, \ \ \text {if } \omega \text { is integral on }\Gamma , \end{aligned}$$

(3.12)

where the infimum runs over all positive-dimensional abelian subvarieties V containing 0. Notice that the right hand side of (3.12) is 1-homogeneous with respect to $\omega$, we know that (3.12) also holds for all $\omega =c\omega '$, where $\omega '$ is integral on $\Gamma$. In particular, it holds true if $\omega$ is rational on $\Gamma$. In case the Picard number of X is $n^2$, we know that $\omega$ can be approximated by rational $\omega '$, hence (3.12) is true for all $\omega$. By Balian–Low type theorem, we have

$$\begin{aligned} \int _V \omega ^{d} > d!. \end{aligned}$$

By Stirling’s approximation, we have

$$\begin{aligned} (d!)^{1/d}/d \ge e^{-1}, \end{aligned}$$

hence (3) follows. $\square$

3.7 Hörmander constants and densities of general discrete sets

Definition 3.3

Let S be a discrete set in ${\mathbb {C}}^n$. Let $\psi$ be a non positive function such that $\psi +\pi |z|^2$ is plurisubharmonic on ${\mathbb {C}}^n$. Let $\gamma$ be positive number. We call $(\psi , \gamma )$ an S-admissible pair if $\psi$ is smooth outside S, $e^{-\psi /\gamma }$ is not integrable near every point in S and there exists a small constant $\varepsilon _0>0$ such that

$$\begin{aligned} \inf _{\varepsilon<|z-\lambda |<2\varepsilon \ \text {for some} \ \lambda \in S} \psi (z) >-\infty , \qquad \ \forall \ 0<\epsilon \le \epsilon _0. \end{aligned}$$

(3.13)

Assume that there exists an S-admissible pair, then we call

$$\begin{aligned} \iota (S):=\sup \{\gamma >0: \text {there exists }\psi \text { such that } (\psi , \gamma )\text { is }S\text {-admissible}\} \end{aligned}$$

the Hörmander constant of S.

Since $\psi$ equals to $-\infty$ at S, (3.13) implies that

$$\begin{aligned} |\lambda -\lambda '|\ge 2\epsilon _0, \ \ \forall \ \lambda \ne \lambda ' \in S, \end{aligned}$$

thus S is uniformly discrete. The proof of the Hörmander criterion also implies:

Theorem 3.10

Let S be a discrete set in ${\mathbb {C}}^n$. Assume that $\iota (S)>1$. Then there exists a constant $C>0$ such that for every sequence of complex numbers $a=\{a_\lambda \}_{\lambda \in S}$ with

$$\begin{aligned} |a|^2:=\sum _{\lambda \in S} |a_\lambda |^2 e^{-\pi |\lambda |^2}=1, \end{aligned}$$

there exists $F\in {\mathcal {F}}^2$ such that $F(\lambda )=a_\lambda$ for all $\lambda \in S$ and $||F||^2 \le C$.

Definition 3.4

Let S be a discrete set in ${\mathbb {C}}^n$. We shall define the upper uniform density of S as

$$\begin{aligned} D^{+}(S):= \limsup _{r\rightarrow \infty } \sup _{z_0\in {\mathbb {C}}^n} \frac{n(z_0, r)}{\pi ^nr^{2n}/ n!}, \end{aligned}$$

where $n(z_0, r)$ denotes the number of points in

$$\begin{aligned} B(z_0, r):=\{z\in {\mathbb {C}}: |z-z_0|<r\}. \end{aligned}$$

In case S is a lattice in ${\mathbb {C}}^n$, we know that $D^{+}(S)^{-1}$ is equal to the Lebesgue measure of the torus ${\mathbb {C}}^n/S$. In the one-dimensional case, we also have the following general result.

Theorem 3.11

Let S be a uniformly discrete set in ${\mathbb {C}}$. Then $D^{+}(S) \cdot \iota (S)=1$.

Proof

Since S is uniformly discrete, we have $\iota (S) >0$, a change of variable argument gives

$$\begin{aligned} \iota ((\iota (S)-\varepsilon )^{-1/2}S )=(\iota (S)-\varepsilon )^{-1}\epsilon (S) >1 \end{aligned}$$

for every sufficiently small $\varepsilon >0$. Thus Theorem 3.10 implies that $(\iota (S)-\varepsilon )^{-1/2}S$ is ${\mathcal {F}}^2$ interpolating. Apply the main result in [OS98], we know that

$$\begin{aligned} 1>D^{+}((\iota (S)-\varepsilon )^{-1/2}S) =D^{+}(S)\cdot (\iota (S)-\varepsilon ). \end{aligned}$$

Letting $\varepsilon$ go to zero, we have

$$\begin{aligned} D^{+}(S) \cdot \iota (S) \le 1. \end{aligned}$$

Now it suffices to show that

$$\begin{aligned} \iota (S) \ge D^{+}(S)^{-1}. \end{aligned}$$

Assume that $D^{+}(S)^{-1}=b$, then for every $0<a<\gamma <b$,

$$\begin{aligned} \sup _{z_0\in {\mathbb {C}}} \frac{n(z_0, r)}{\pi r^2} \le \frac{1}{\gamma }, \end{aligned}$$

for all sufficiently large r, which gives

$$\begin{aligned} \int _{B(z_0,r)} i\partial {\overline{\partial }}|z|^2/2=\pi r^2 \ge \gamma \cdot n(z_0, r) \end{aligned}$$

for all $z_0\in {\mathbb {C}}$. Apply the Berndtsson–Ortega construction in [BO95, page 113–114], the above inequality implies that $\iota (S) \ge a$ for every $a<b$. Hence $\iota (S) \ge b=D^{+}(S)^{-1}$. The proof is complete. $\square$

Using results from [BO95, OS98], one may also generalize the above theorem to general weight function $\phi$ with $\phi _{z{\bar{z}}}$ bounded by two positive constants. For general higher-dimensional cases, by [Lin01, Theorem 2], we know that if S is ${\mathcal {F}}^2$ interpolating then $D^{+}(S) \le 1$. However, in general, S may not be ${\mathcal {F}}^2$ interpolating even $D^{+}(S)$ is small enough. Comparing with Theorem 3.10, this means that there exists S with very small upper uniform density whose Hörmander constant is also small.

3.8 Proof of Theorem 1.10

Notice that if $(({\mathbb {Z}}\oplus \frac{1}{2} {\mathbb {Z}})^2, e^{-\pi |t|^2})$ gives a frame in $L^2({\mathbb {R}}^2)$ then $({\mathbb {Z}}^2, e^{-\pi t^2})$ defines a frame in $L^2({\mathbb {R}})$, which is not true by the Balian-Low type theorem. Now it suffices to prove (2) since (2) implies (1) by the remark after Corollary 1.9. Put

$$\begin{aligned} X:={\mathbb {R}}^4/({\mathbb {Z}}\oplus 2{\mathbb {Z}})^2, \end{aligned}$$

then we know that $(X,\omega )$ is of type (1, 4). Since the moduli space of polarized type $(d_1,d_2)$ ($d_1, d_2$ are fixed positive integers) Abelian surfaces is equal to the Siegel upper half-space, to prove (2), by the Hörmander criterion, it suffices to show that the Seshadri constant of a generic polarized type (1, 4) Abelian surface is bigger than two. Since generically a polarized type (1, 4) Abelian surface has Picard number one, by Theorem 6.1 (b) in [Bau99], its Seshadri constant equals

$$\begin{aligned} 8/\sqrt{8+1}=8/3 >2, \end{aligned}$$

where we use the fact that $k=1, l=3$ is the primitive solution of the following Pell’s equation

$$\begin{aligned} l^2-8k^2=1. \end{aligned}$$

The proof is complete.

4 Non transcendental examples

4.1 Gröchenig–Lyubarskii’s example

Let us look at the following lattice in ${\mathbb {R}}^2 \times {\mathbb {R}}^2$ (see [GL20, page 3, (4)]):

$$\begin{aligned} \Lambda =\left\{ \left( e+\frac{1}{2} f,\ \frac{\sqrt{3}}{2}f\right) : e,f \in {\mathbb {Z}}^2 \right\} . \end{aligned}$$

Its symplectic dual is

$$\begin{aligned} \Lambda ^\circ =\left\{ \left( e^*,\ \frac{-1}{\sqrt{3}}e^*+ \frac{2}{\sqrt{3}}f^*\right) : e^*,f^* \in {\mathbb {Z}}^2 \right\} . \end{aligned}$$

Fix $\Omega =i I_n$, where $I_n$ denotes the identity matrix. With the notation in Proposition 1.4, we have

$$\begin{aligned} \Gamma _{\Omega , \Lambda ^\circ }=\left\{ \left( e^*,\ \frac{-1}{\sqrt{3}}e^*+ \frac{2}{\sqrt{3}}f^*\right) : e^*,f^* \in {\mathbb {Z}}[i] \right\} . \end{aligned}$$

Let us estimate the Seshadri constant of

$$\begin{aligned} \omega =\omega _\textrm{euc}:=\frac{i}{2} \sum _{j=1}^2 dz_j \wedge d{\bar{z}}_j \end{aligned}$$

on the complex tori $X:={\mathbb {C}}^2/\Gamma _{\Omega , \Lambda ^\circ }$. Notice that the Riemannian metric induced by $\omega$ is precisely the euclidean metric $|\cdot |$, hence

$$\begin{aligned} \inf _{\lambda \ne \lambda ' \in \Gamma _{\Omega , \Lambda ^\circ }} |\lambda -\lambda '|^2 =\inf _{0 \ne \mu \in \Gamma _{\Omega , \Lambda ^\circ }} |\mu |^2 = |(1, -1/\sqrt{3})|^2 =\frac{4}{3}. \end{aligned}$$

Thus the following ball

$$\begin{aligned} B:=\{|z|<1/\sqrt{3}\} \end{aligned}$$

contains precisely one point in $\Gamma _{\Omega , \Lambda ^\circ }$ and we can think of B as a Kähler ball in X, which gives (see Theorem 3.4) the following Seshadri constant inequality

$$\begin{aligned} \epsilon _0(\omega ) \ge \frac{\pi }{3} >1. \end{aligned}$$

(4.1)

However, from [GL20, page 3, (4)], we know that $(\Lambda , g_\Gamma )$ does not define a frame in $L^2({\mathbb {R}}^2)$. Thus by Proposition 1.4, $\Gamma _{\Omega , \Lambda ^\circ }$ is not a set of interpolation for ${\mathcal {F}}^2$. To summarize, we obtain:

Theorem 4.1

There exists a lattice in ${\mathbb {C}}^2$ whose Seshadri constant is bigger than one but it is not a set of interpolation for ${\mathcal {F}}^2$.

Remark

By the Hörmander criterion, we know every lattice in ${\mathbb {C}}^2$ with Seshadri constant bigger than two is a set of interpolation for ${\mathcal {F}}^2$. In the above example, one may further prove that

$$\begin{aligned} \frac{4}{3}\ge \epsilon _0(\omega ) \ge \frac{\pi }{3}. \end{aligned}$$

(4.2)

In fact the complex line ${\mathbb {C}}(0, 2/\sqrt{3})$ covers a subtorus, say

$$\begin{aligned} V\simeq {\mathbb {C}} / (2/\sqrt{3}){\mathbb {Z}}[i], \end{aligned}$$

of X, thus [Tos18, Theorem 4.6] implies that

$$\begin{aligned} (2/\sqrt{3})^2=\int _V \omega \ge \epsilon _0(\omega ), \end{aligned}$$

from which (4.2) follows.

4.2 Complex lattices

We call $\Gamma$ a complex lattice if

$$\begin{aligned} i \Gamma = \Gamma . \end{aligned}$$

One may verify the following:

Proposition 4.2

For a lattice $\Gamma$ in ${\mathbb {C}}^n$, the followings are equivalent

• (1) $\Gamma$ is a complex lattice;

• (2) $\Gamma ={\mathbb {Z}}[i] \{\gamma _1,\ldots , \gamma _n\}$ for some $\gamma _j\in {\mathbb {C}}^n$;

• (3) $\Gamma =A {\mathbb {Z}}[i]^n$ for some $A\in GL(n, {\mathbb {C}})$;

• (4) $X:={\mathbb {C}}^n/\Gamma$ is biholomorphic to ${\mathbb {C}}^n/{\mathbb {Z}}[i]^n$.

Now we can prove a generalization of (4.2).

Theorem 4.3

Assume that $\Gamma =A {\mathbb {Z}}[i]^n$ is a complex lattice. Then

$$\begin{aligned} m(\Gamma ) \ge \epsilon _0(\omega )\ge \max \left\{ \frac{\pi }{4} m(\Gamma ), \ e_{min}(A)\right\} , \end{aligned}$$

where

$$\begin{aligned} e_{min}(A):=\inf _{z\in {\mathbb {C}}^n, \, |z|=1} |Az|^2. \end{aligned}$$

Proof

Choose $z=A^{-1}w$ as the new variable, one may assume that $A=I_n$. Then

$$\begin{aligned} \omega =\frac{i}{2} \sum _{j=1}^n dw_j \wedge d{\bar{w}}_j \ge e_{min}(A) \cdot \omega _0, \ \ \ \omega _0:=\frac{i}{2} \sum _{j=1}^n dz_j \wedge d{\bar{z}}_j. \end{aligned}$$

Denote by $\psi _0(z)$ the Green function on ${\mathbb {C}}/{\mathbb {Z}}[i]$ satisfying

$$\begin{aligned} i\pi dz \wedge d{\bar{z}} +i\partial {\overline{\partial }}\psi _0= i\partial {\overline{\partial }}\log |z|^2. \end{aligned}$$

then we know that

$$\begin{aligned} \psi := e_{min}(A) \cdot \max \{\psi _0(z_1),\cdots , \psi _0(z_n)\} \end{aligned}$$

satisfies $2\pi \omega +i\partial {\overline{\partial }}\psi \ge 0$ and has order one log pole at $\Gamma$. Thus we know that $\epsilon _0(\omega )\ge e_{min}(A)$, together with Theorem 3.4, it gives the lower bound of the Seshadri constant that we need. To prove the upper bound, it suffices to choose a subtorus

$$\begin{aligned} V_\gamma :={\mathbb {C}}/ {\mathbb {Z}}[i]\gamma , \ \ 0\ne \gamma \in \Gamma , \end{aligned}$$

then [Tos18, Theorem 4.6] gives

$$\begin{aligned} \epsilon _0(\omega ) \le \int _{V_\gamma } \omega =|\gamma |^2. \end{aligned}$$

Take the infimum over all $0\ne \gamma \in \Gamma$, the upper bound follows. $\square$

Remark

In case

$$\begin{aligned} \Gamma = a_1 {\mathbb {Z}}[i] \times \cdots \times a_n {\mathbb {Z}}[i], \ \ a_j >0, \end{aligned}$$

we have

$$\begin{aligned} e_{min}(A)=m(\Gamma ) = \min \{a^2_1,\ldots , a^2_n\}, \end{aligned}$$

thus the above theorem gives

$$\begin{aligned} \epsilon _0(\omega )=\min \{a^2_1,\ldots , a^2_n\}. \end{aligned}$$

4.3 Seshadri sequence and Gröchenig’s result

In this section, we shall rephrase the main result of Gröchenig in [Gro11] in terms of the Seshadri constant. The main idea is to consider a sequence of extensions, more precisely, let

$$\begin{aligned} \{0\}=X_0\subset X_1\cdots \subset X_k=X:={\mathbb {C}}^n/\Gamma , \ \ n_k:=\dim _{{\mathbb {C}}} X_k, \ \ k\ge 1, \end{aligned}$$

(4.3)

be an increasing sequence of complex Lie subgroups of X. We shall introduce the Seshadri constant $\epsilon _j$, $1\le j\le k$, for extension from $X_{j-1}$ to $X_{j}$. Let

$$\begin{aligned} \pi _j: E_j \rightarrow X_j \end{aligned}$$

be the covering map, where $E_j$ is an $n_j$ dimensional complex subspace of ${\mathbb {C}}^n$. Let

$$\begin{aligned} E_j= E_{j-1}\oplus F_j, \end{aligned}$$

be the orthogonal decomposition with respect to the Euclidean metric $\omega$. Then

$$\begin{aligned} \Gamma _j:=F_j \cap \pi _j^{-1}(X_{j-1}) \end{aligned}$$

define a lattice in $F_j$. Put

$$\begin{aligned} X^\bot _{j-1}:=F_j/ \Gamma _j, \end{aligned}$$

(in general, $X^\bot _{j-1}$ is not a subtorus of $X_j$). Denote by

$$\begin{aligned} \epsilon _j:= \epsilon _0(\omega ; X^\bot _{j-1}) \end{aligned}$$

(4.4)

the Seshadri constant at the origin of $X^\bot _{j-1}$ with respect to $\omega$.

Definition 4.1

We call (4.3) an admissible sequence of X if

$$\begin{aligned} \epsilon _j > n_j-n_{j-1}, \ \ \forall \ 1\le j\le n. \end{aligned}$$

X is said to be Seshadri admissible if it possesses an admissible sequence.

Theorem 4.4

Assume that X is Seshadri admissible, then $\Gamma$ is a set of interpolation in ${\mathcal {F}}^2$.

Proof

Let $\pi : {\mathbb {C}}^n \rightarrow X$ be the covering map. Put

$$\begin{aligned} {\mathcal {F}}^2_j:=\left\{ F \ \text {holomorphic on}\ \pi ^{-1}(X_j): \int _{\pi ^{-1}(X_j)} |F|^2 e^{-\pi |z|^2} <\infty \right\} . \end{aligned}$$

It suffices to prove that each element f in ${\mathcal {F}}^2_{j-1}$ extends to an element F in ${\mathcal {F}}^2_j$ with $||F||\le C_j ||f||$. Since $\pi ^{-1}(X_j)$ is a disjoint union of translates of $E_j$ and

$$\begin{aligned} \pi ^{-1}(X_{j-1}) \cap E_j=\pi _j^{-1}(X_{j-1}), \end{aligned}$$

the above extension problem reduces to the extension from ${\mathcal {F}}^2_{j-1}|_{\pi _j^{-1}(X_{j-1})}$ to ${\mathcal {F}}^2_j|_{E_j}$. Apply the Hörmander method, it suffices to construct an $\omega$ plurisubharmonic function with order $(n_j-n_{j-1})$ log pole along $\pi _j^{-1}(X_{j-1})$. Now the assumption $\epsilon _j > n_j-n_{j-1}$ gives an $\omega$ plurisubharmonic $\psi _j$ with order $n_j-n_{j-1}$ log pole at the origin of $F_j$, the pull back of $\psi _j$ along the natural projection

$$\begin{aligned} E_j \rightarrow F_j \end{aligned}$$

gives the function that we need. $\square$

Now we shall show how to use the above theorem to give a new proof of [Gro11, Theorem 9] on the Gabor frame property for $(\Lambda , g_\Omega )$. The setup for [Gro11, Theorem 9] is the following:

$$\begin{aligned} \Omega =iI_n, \ \ \Gamma _{\Omega , \Lambda ^\circ } \ \text {is a complex lattice}. \end{aligned}$$

Based on Proposition 1.4, we shall prove a similar result with a weaker assumption, i.e. we shall only assume that $\Gamma _{\Omega , \Lambda ^\circ }$ is a complex lattice. Let us write

$$\begin{aligned} \Gamma _{\Omega , \Lambda ^\circ }= A {\mathbb {Z}}[i]^n, \end{aligned}$$

where $A\in GL(n, {\mathbb {C}})$. By the Iwasawa decomposition (see [Bum13, Proposition 26.1]), we have

$$\begin{aligned} A=US, \end{aligned}$$

where U is unitary and S is lower triangular with positive eigenvalues $\lambda _j$ (U, S are uniquely determined by A, $\lambda _j^{-1}$ is equal to $\gamma _j$ in [Gro11, Theorem 9]). Since the Euclidean metric $\omega$ is unitary invariant, one may assume that

$$\begin{aligned} \Gamma _{\Omega , \Lambda ^\circ }=S {\mathbb {Z}}[i]^n. \end{aligned}$$

Put

$$\begin{aligned} E_j=\{z\in {\mathbb {C}}^n: z_1=\cdots =z_{n-j}=0\}, \ \ 1\le j\le n, \end{aligned}$$

and

$$\begin{aligned} X_j=\pi (E_j). \end{aligned}$$

Then we have

$$\begin{aligned} F_j \simeq {\mathbb {C}}, \ \ \ \Gamma _j\simeq \lambda _{n-j+1} {\mathbb {Z}}[i], \end{aligned}$$

and

$$\begin{aligned} \epsilon _j=\lambda _{n-j+1}^2, \ \ 1\le j\le n. \end{aligned}$$

Since $n_j=j$, we have $n_j-n_{j-1}=1$. Hence if

$$\begin{aligned} \lambda _{n-j+1} >1, \ \ 1\le j\le n, \end{aligned}$$

then X is Seshadri admissible. Thus Theorem 4.4 implies the following slight generalization of Gröchenig’s result (notice again that $\gamma _j =\lambda _j^{-1}$):

Theorem 4.5

(Theorem 9 in [Gro11]). Let $\Gamma _{\Omega , \Lambda ^\circ }$ be a complex lattice. With the notation above, assume that $\lambda _j>1$ for all $1\le j\le n$. Then $\Gamma _{\Omega , \Lambda ^\circ }$ is set of interpolation for ${\mathcal {F}}^2$ (and equivalently $(\Lambda , g_\Omega )$ defines a frame in $L^2({\mathbb {R}}^n)$).

5 Effective interpolation bounds

5.1 Interpolation bounds in terms of the Buser–Sarnak constant

We shall use Theorem 2.9 to prove the following:

Theorem 5.1

Fix a lattice $\Gamma$ in ${\mathbb {C}}^n$. If

$$\begin{aligned} C= \frac{\pi }{4}\cdot \inf _{0 \ne \mu \in \Gamma }|\mu |^2 > n \end{aligned}$$

(5.1)

then every sequence of complex numbers $a=\{a_\gamma \}$ with $\sum _{\gamma \in \Gamma } |a_\gamma |^2 e^{-\pi |\gamma |^2}=1$ extends to a function in ${\mathcal {F}}^2$. Moreover,

$$\begin{aligned} 1-e^{-C} \sum _{k=0}^{n-1} \frac{C^k}{k!}\le \inf _{F\in {\mathcal {F}}^2, \ F(\gamma )=a_\gamma , \ \forall \ \gamma \in \Gamma } ||F||^2 \le \frac{M(C)}{ n! \,e^n }, \end{aligned}$$

where

$$\begin{aligned} M(C):= {\left\{ \begin{array}{ll} (n+1)^{n+1} &{} \text {if} \ \ C\ge n+1,\\ C^{n+1}/(C-n) &{} \text {if} \ \ n<C< n+1. \end{array}\right. } \end{aligned}$$

Proof of the lower bound

Notice that (by an induction on n)

$$\begin{aligned} 1-e^{-C} \sum _{k=0}^{n-1} \frac{C^k}{k!}= \int _{0}^C e^{-t} \frac{t^{n-1}}{(n-1)!}\, dt \end{aligned}$$

and

$$\begin{aligned} \{z\in {\mathbb {C}}^n: \pi |z-\gamma |^2<C\} \cap \{z\in {\mathbb {C}}^n: \pi |z-\gamma '|^2<C\}=\emptyset , \ \ \forall \ \gamma \ne \gamma ' \in \Gamma , \end{aligned}$$

It suffices to show that

$$\begin{aligned} \int _{\{\pi |z-\gamma |^2<C\}} |F(z)|^2e^{-\pi |z|^2} \ge |F(\gamma )|^2 e^{-\pi |\gamma |^2}\int _{0}^C e^{-t} \frac{t^{n-1}}{(n-1)!}\, dt. \end{aligned}$$

(5.2)

Notice that

$$\begin{aligned} \int _{\{\pi |z-\gamma |^2<C\}} |F(z)|^2e^{-\pi |z|^2}=e^{-\pi |\gamma |^2}\int _{\{\pi |w|^2<C\}} |F(w+\gamma )e^{-\pi w^T\gamma }|^2e^{-\pi |w|^2}, \end{aligned}$$

The main observation is that the Taylor expansion

$$\begin{aligned} G(w):=F(w+\gamma )e^{-\pi w^T\gamma }= F(\gamma )+\sum c_\alpha w^\alpha , \end{aligned}$$

is now an orthogonal decomposition, i.e.

$$\begin{aligned} \int _{\{\pi |w|^2<C\}} |G(w)|^2e^{-\pi |w|^2}&=|F(\gamma )|^2 \int _{\{\pi |w|^2<C\}} e^{-\pi |w|^2} \\&\quad +\sum |c_\alpha |^2 \int _{\{\pi |w|^2<C\}} |w^\alpha |^2e^{-\pi |w|^2}, \end{aligned}$$

from which we know that

$$\begin{aligned} \int _{\{\pi |z-\gamma |^2<C\}} |F(z)|^2e^{-\pi |z|^2} \ge |F(\gamma )|^2 e^{-\pi |\gamma |^2}\int _{\{\pi |w|^2<C\}} e^{-\pi |w|^2}. \end{aligned}$$

Now put $t=\pi |\gamma |^2$, we have

$$\begin{aligned} \int _{\{\pi |w|^2<C\}} e^{-\pi |w|^2}= \int _0^C e^{-t}d\, \frac{t^n}{n!} =\int _{0}^C e^{-t} \frac{t^{n-1}}{(n-1)!}\, dt, \end{aligned}$$

which gives (5.2). $\square$

Proof of the upper bound

The main idea is to use Theorem 2.9. The definition of C implies that

$$\begin{aligned} B:=\{z\in {\mathbb {C}}^n: \pi |z|^2<C\} \end{aligned}$$

is embedded ball in $X:={\mathbb {C}}^n /\Gamma$. For

$$\begin{aligned} \frac{\pi }{C}\le \delta < \frac{\pi }{n}, \end{aligned}$$

put (notice that $\{\delta |z|^2 <1\} \subset B$)

$$\begin{aligned} \psi _\delta (z):= {\left\{ \begin{array}{ll} \log (\delta |z|^2)+(1-\delta |z|^2) &{} \ \ \delta |z|^2<1 \\ 0 &{} z\in X\setminus \{\delta |z|^2 <1\}. \end{array}\right. } \end{aligned}$$

Then we have

$$\begin{aligned} i\partial {\overline{\partial }}\psi _\delta \ge -\delta \cdot i\partial {\overline{\partial }}|z|^2. \end{aligned}$$

Denote by $\psi$ the pull back to ${\mathbb {C}}^n$ of $\pi \psi _\delta /\delta$. Apply Theorem 2.9 to $\psi$, we get

$$\begin{aligned} \inf _{F\in {\mathcal {F}}^2, \ F(\gamma )=a_\gamma , \ \forall \ \gamma \in \Gamma } ||F||^2 \le \left( 1-\frac{n\delta }{\pi }\right) ^{-1} \frac{\pi ^n}{n!} e^{-n(1+\log \delta )}. \end{aligned}$$

Put

$$\begin{aligned} x:=\frac{n\delta }{\pi }, \end{aligned}$$

then $n/C\le x<1$ and

$$\begin{aligned} \left( 1-\frac{n\delta }{\pi }\right) ^{-1} \frac{\pi ^n}{n!} e^{-n(1+\log \delta )} =\frac{n^n}{(1-x)x^n} \cdot \frac{1}{n!\,e^n}. \end{aligned}$$

Thus the upper bound follows from

$$\begin{aligned} \inf _{n/C\le x<1} \frac{n^n}{(1-x)x^n} =M(C). \end{aligned}$$

The proof of Theorem B is now complete. $\square$

5.2 Interpolation bounds in terms of the Robin constant

In this subsection, we shall generalize Theorem 5.1 to the case that $\epsilon _0(\omega )>n$. The main idea is to consider the following envelope with prescribed singularity

$$\begin{aligned} \psi _a^\pi := \sup \{\psi _0\le 0: \ \ {}&2\pi \omega +i\partial {\overline{\partial }}\psi _0 \ge 0, \ \psi _0 \ \text {has an isolated order} \ a \\&\text {log pole at the origin} \} \end{aligned}$$

on the torus $X={\mathbb {C}}^n/\Gamma$. Denote by $\psi _a$ the pull back to ${\mathbb {C}}^n$ of $\psi _a^\pi$.

Definition 5.1

We call $\psi _a$ the a-envelope function on ${\mathbb {C}}^n$ associated to the lattice $\Gamma$ and

$$\begin{aligned} \rho _a:=\liminf _{z\rightarrow 0} \psi _a(z)-a \log |z|^2. \end{aligned}$$

the a-Robin constant of $\Gamma$.

We have the following generalization of Theorems 5.1 and B.

Theorem 5.2

Assume that $\epsilon :=\epsilon _0(\omega )>n$. Put

$$\begin{aligned} C= \frac{\pi }{4}\cdot \inf _{0 \ne \mu \in \Gamma }|\mu |^2. \end{aligned}$$

(5.3)

Then every sequence of complex numbers $c=\{c_\gamma \}$ with $\sum _{\gamma \in \Gamma } |c_\gamma |^2 e^{-\pi |\gamma |^2}=1$ extends to a function in ${\mathcal {F}}^2$. Moreover,

$$\begin{aligned} 1-e^{-C} \sum _{k=0}^{n-1} \frac{C^k}{k!}\le \inf _{F\in {\mathcal {F}}^2, \ F(\gamma )=c_\gamma , \ \forall \ \gamma \in \Gamma } ||F||^2 \le \inf _{n<a<\epsilon }\left( 1-\frac{n}{a}\right) ^{-1} \frac{\pi ^n}{n!}\, e^{-n\rho _a/a}. \end{aligned}$$

Proof

The proof of the lower bound is the same. For the upper bound, it suffices to apply Theorem 2.9 to $\psi =\psi _a$. $\square$

Remark

In case $n=1$, we know that

$$\begin{aligned} \epsilon =\int _{{\mathbb {C}}/\Gamma } \omega . \end{aligned}$$

Moreover, $\psi _\epsilon ^\pi$ is also well defined. In fact, we have the following

Proposition 5.3

$\psi _\epsilon ^\pi$ is equal to the unique solution, say $\psi$, of

$$\begin{aligned} i\partial {\overline{\partial }}\psi +2\pi \omega =\epsilon \cdot i\partial {\overline{\partial }}\log |z|^2, \ \ \ \ \sup _{X} \psi =0, \end{aligned}$$

on $X={\mathbb {C}}/\Gamma$.

Proof

Since

$$\begin{aligned} \int _X i\partial {\overline{\partial }}\psi +2\pi \omega = \int _X 2\pi \omega =2\pi \epsilon = \int _X\epsilon \cdot i\partial {\overline{\partial }}\log |z|^2, \end{aligned}$$

from the Hodge theory, we know that up to a constant there exists a unique solution $\psi$ such that

$$\begin{aligned} i\partial {\overline{\partial }}\psi +2\pi \omega =\epsilon \cdot i\partial {\overline{\partial }}\log |z|^2. \end{aligned}$$

Thus if we assume further that $\sup _X \psi =0$ then $\psi$ is unique. Moreover, we know that $\psi$ is smooth outside the origin and $\psi -\epsilon \log |z|^2$ is smooth near the origin, thus

$$\begin{aligned} \psi \le \psi _\epsilon ^\pi . \end{aligned}$$

On the other hand, notice that $\psi _\epsilon ^\pi -\psi$ is subharmonic, thus $\psi _\epsilon ^\pi -\psi$ is equal to a constant, say A. Take $z_0$ such that $\psi (z_0)=0$, then

$$\begin{aligned} A=\psi _\epsilon ^\pi (z_0) -\psi (z_0) =\psi _\epsilon ^\pi (z_0) \le 0, \end{aligned}$$

which gives

$$\begin{aligned} \psi \ge \psi _\epsilon ^\pi . \end{aligned}$$

Hence $\psi = \psi _\epsilon ^\pi$. $\square$

Remark

The above Proposition implies that $\psi _\epsilon ^\pi$ is equal to the Arakelov Green function up to a non-zero constant (see [Fal84, page 393 and 417]).

5.3 Faltings’ identity for the Robin constant

Notice that

$$\begin{aligned} \int _X \omega /\epsilon =1, \end{aligned}$$

we know that $[\omega /\epsilon ]$ is the Chern class of the line bundle, $L=[0]$, over X. Choose a metric h on L such that the Chern curvature satisfies

$$\begin{aligned} i\Theta (L, h)=2\pi \omega /\epsilon . \end{aligned}$$

Denote by s the canonical section of [0], then we know that

$$\begin{aligned} i\partial {\overline{\partial }}\log |s|^2_h= i\partial {\overline{\partial }}\log |z|^2- 2\pi \omega /\epsilon , \end{aligned}$$

which implies that

$$\begin{aligned} \epsilon \log |s|^2_h-\sup _X \epsilon \log |s|^2_h= \psi _\epsilon ^\pi . \end{aligned}$$

(5.4)

The pull back of the line bundle [0] to ${\mathbb {C}}$ is a trivial line bundle, thus one may identify the pull back to ${\mathbb {C}}$ of s with a holomorphic function, say $f^s(z)$ on ${\mathbb {C}}$. In case $\Gamma =\textrm{Span}_{{\mathbb {Z}}} \{1, \tau \}$, $\textrm{Im}\, \tau =\epsilon$, (up to a constant) we have

$$\begin{aligned} f^s(z)=\vartheta \left( z+\frac{1}{2}+\frac{\tau }{2}; \tau \right) , \end{aligned}$$

(notice that $\vartheta \left( \frac{1}{2}+\frac{\tau }{2}; \tau \right) =0$) where

$$\begin{aligned} \vartheta (z; \tau )=:\sum _{n\in {\mathbb {Z}}} e^{2\pi i nz} e^{\pi i n^2 \tau } \end{aligned}$$

is known as the Jacobi theta function. The following formula for the Robin constant is based on the Faltings’ theta metric $||\theta ||$ (see [Fal84, page 403, 413 and 416] or the function U below).

Proposition 5.4

Assume further that $\Gamma =\textrm{Span}_{{\mathbb {Z}}} \{1, \tau \}$, $\textrm{Im}\, \tau =\epsilon$. Denote by $\psi _\epsilon$ the pull back to ${\mathbb {C}}$ of $\psi _\epsilon ^\pi$, then

$$\begin{aligned} \frac{\psi _\epsilon }{\epsilon }= \log U- \sup _{{\mathbb {C}}} \log U, \ \ \ U(z):= \big |\vartheta \left( z+\frac{1}{2}+\frac{\tau }{2}; \tau \right) e^{-\pi (\textrm{Im} \, z +\frac{\epsilon }{2})^2/\epsilon }\big |^2, \end{aligned}$$

(5.5)

and the $\epsilon$-Robin constant of $\Lambda$ defined by

$$\begin{aligned} \rho _\epsilon :=\liminf _{z\rightarrow 0} \psi _\epsilon (z)-\epsilon \log |z|^2 \end{aligned}$$

satisfies

$$\begin{aligned} \frac{\rho _\epsilon }{\epsilon }=2 \log (2\pi )+6\log |\eta (\tau )|- \sup _{{\mathbb {C}}} \log U, \end{aligned}$$

(5.6)

where $\eta (\tau )$ is the Dedekind eta function defined by $\eta (\tau ):=e^{\pi i\tau /12} \Pi _{n=1}^\infty (1-e^{2\pi i n \tau })$.

Proof

One may verify that U is $\Gamma$-invariant and

$$\begin{aligned} i\partial {\overline{\partial }}\log U= -2\pi \omega /\epsilon +\sum _{\gamma \in \Gamma } i\partial {\overline{\partial }}\log |z+\gamma |^2, \end{aligned}$$

thus (5.5) follows. To prove (5.6), it suffices to show that

$$\begin{aligned} \lim _{z\rightarrow 0} U(z)/|z|^2 = (2\pi )^2 \cdot |\eta (\tau )|^6 \end{aligned}$$

or equivalently

$$\begin{aligned} \big |\frac{\partial \vartheta }{\partial z} \left( \frac{1}{2}+\frac{\tau }{2}; \tau \right) e^{-\pi \epsilon /4} \big |^2 =(2\pi )^2 \cdot |\eta (\tau )|^6, \end{aligned}$$

which follows from

$$\begin{aligned} \left( \frac{\partial \vartheta }{\partial z} \left( \frac{1}{2}+\frac{\tau }{2}; \tau \right) e^{\pi i \tau /4}\right) ^8=(2\pi )^8 \cdot \eta (\tau )^{24} \end{aligned}$$

(since both sides are cusp forms of degree 12 with the same leading term). $\square$

Remark

Notice that

$$\begin{aligned} \sup _{{\mathbb {C}}} \log U =2\cdot \sup _{z\in {\mathbb {C}}} \log \big |\vartheta \left( z; \tau \right) e^{-\pi (\textrm{Im} \, z)^2/\epsilon }\big | \end{aligned}$$

and

$$\begin{aligned} \sup _{z\in {\mathbb {C}}} \log \big |\vartheta \left( z; \tau \right) e^{-\pi (\textrm{Im} \, z)^2/\epsilon }\big | =\sup _{t\in {\mathbb {R}}} \phi (t)- \pi t^2/\epsilon , \ \ \ \phi (t):=\sup _{\textrm{Im}\, z=t} \log |\vartheta (z; \tau )|. \end{aligned}$$

Since $\vartheta$ is an even function of z and depends only on $e^{2\pi iz}$, we know that

$$\begin{aligned} \phi (t)=\phi (-t) \end{aligned}$$

is an even convex function of t. Moreover, since $|\vartheta \left( z; \tau \right) e^{-\pi (\textrm{Im} \, z)^2/\epsilon }\big |$ is $\Gamma$ invariant, we have

$$\begin{aligned} \sup _{z\in {\mathbb {C}}} \log \big |\vartheta \left( z; \tau \right) e^{-\pi (\textrm{Im} \, z)^2/\epsilon }\big | =\sup _{0<t<\epsilon } \phi (t)- \pi t^2/\epsilon \end{aligned}$$

and

$$\begin{aligned} \phi (t)=\sup _{0<x<1} \log |\sum _{n\in {\mathbb {Z}}} e^{2\pi i n(x+it)} e^{\pi i n^2 \tau }|. \end{aligned}$$

Notice that

$$\begin{aligned} e^{\phi (t)} \le \sum _{n\in {\mathbb {Z}}} e^{-2\pi nt} e^{-\pi n^2\epsilon } \end{aligned}$$

with identity holds if $\tau =i\epsilon$. By the Poisson summation formula, we have

$$\begin{aligned} \sqrt{\epsilon }\cdot e^{\pi z^2/\epsilon }\sum _{n\in {\mathbb {Z}}} e^{2\pi i nz} e^{-\pi n^2 \epsilon }= \sum _{n\in {\mathbb {Z}}} e^{2\pi i nz/\epsilon } e^{-\pi n^2 /\epsilon }, \end{aligned}$$

take $z=it$, we get

$$\begin{aligned} e^{\phi (t)-\pi t^2 /\epsilon } \le e^{-\pi t^2 /\epsilon }\sum _{n\in {\mathbb {Z}}} e^{-2\pi nt} e^{-\pi n^2\epsilon }=\frac{1}{\sqrt{\epsilon }} \cdot \sum _{n\in {\mathbb {Z}}} e^{-\pi nt/\epsilon } e^{-\pi n^2 /\epsilon }, \end{aligned}$$

which gives

$$\begin{aligned} \sup _{z\in {\mathbb {C}}} \big |\vartheta \left( z; \tau \right) e^{-\pi (\textrm{Im} \, z)^2/\epsilon }\big | \le \frac{1}{\sqrt{\epsilon }} \cdot \sum _{n\in {\mathbb {Z}}} e^{-\pi n^2 /\epsilon }=\sum _{n\in {\mathbb {Z}}} e^{-\pi n^2 \epsilon }, \end{aligned}$$

(for a higher-dimensional generalization of the above argument, see [Pa18, Lemma 8.2]). By (5.6), the above estimate gives the following lower bound for $\rho _\epsilon /\epsilon$.

Theorem 5.5

The $\epsilon$-Robin constant of $\Gamma =\textrm{Span}_{{\mathbb {Z}}} \{1, \tau \}$, $\textrm{Im}\, \tau =\epsilon$ satisfies

$$\begin{aligned} \frac{\rho _\epsilon }{\epsilon }\ge 2 \log (2\pi )+6\log |\eta (\tau )|- 2 \log \sum _{n\in {\mathbb {Z}}} e^{-\pi n^2 \epsilon }, \end{aligned}$$

(5.7)

with identity holds if $\tau = i\epsilon$.

5.4 Proof of Theorem B

Proof of (1) and (2)

Notice that the disc of diameter $\int _{0\ne \lambda \in \Gamma } |\lambda |$ is contained in a fundamental domain of ${\mathbb {C}}/\Gamma$. Hence $|\Lambda |^{-1}=|\Gamma | \ge C$. The frame bounds estimate follows directly from Theorem 5.1 and Theorem 2.7. $\square$

Proof of (3)

For the lower bound, by Theorem 5.2 (let a go to $\epsilon$), it suffices to compute the $\epsilon$-Robin constant $\rho$, where

$$\begin{aligned} \epsilon :=a \, \textrm{Im}\, \tau \end{aligned}$$

is the Seshadri constant. Denote by $\rho '$ the $\textrm{Im}\, \tau /a$-Robin constant of ${\mathbb {C}}/\langle 1, \tau /a\rangle$, then a change of variable argument gives

$$\begin{aligned} \rho =a^2 \rho '. \end{aligned}$$

Thus

$$\begin{aligned} \rho /\epsilon =\rho '/( \textrm{Im}\, \tau /a) \ge 2 \log (2\pi ) +6 \log |\eta (\tau /a)|-2 \log \sum _{n\in {\mathbb {Z}}} e^{-\pi n^2 \textrm{Im}\, \tau /a}, \end{aligned}$$

by Theorem 5.5. Apply Theorems 5.2 and 2.7, we get the lower bound. The upper bound follows directly from Theorems 2.7 and 5.1. $\square$