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Compactness and rigidity of self-shrinking surfaces Asian J. Math. (IF 0.6) Pub Date : 2023-11-07 Tang-Kai Lee
The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher codimensional case. In this note, we use measure-theoretic techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces
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Spectral convergence in geometric quantization on $K3$ surfaces Asian J. Math. (IF 0.6) Pub Date : 2023-11-07 Kota Hattori
We study the geometric quantization on $K3$ surfaces from the viewpoint of the spectral convergence. We take a special Lagrangian fibrations on the $K3$ surfaces and a family of hyper-Kähler structures tending to large complex structure limit, and show a spectral convergence of the $\overline{\partial}$ Laplacians on the prequantum line bundle to the spectral structure related to the set of Bohr–Sommerfeld
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Terracini locus for three points on a Segre variety Asian J. Math. (IF 0.6) Pub Date : 2023-11-07 Edoardo Ballico, Alessandra Bernardi, Pierpaola Santarsiero
We introduce the notion of $r$-th Terracini locus of a variety and we compute it for at most three points on a Segre variety.
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Ordinary deformations are unobstructed in the cyclotomic limit Asian J. Math. (IF 0.6) Pub Date : 2023-11-07 Ashay Burungale, Laurent Clozel
The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one
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Special termination for log canonical pairs Asian J. Math. (IF 0.6) Pub Date : 2023-11-07 Vladimir Lazić, Joaquín Moraga, Nikolaos Tsakanikas
We prove the special termination for log canonical pairs and its generalisation in the context of generalised pairs.
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Embedding obstructions in $\mathbb{R}^d$ from the Goodwillie–Weiss calculus and Whitney disks Asian J. Math. (IF 0.6) Pub Date : 2023-10-12 Gregory Arone, Vyacheslav Krushkal
Given a finite CW complex $K$, we use a version of the Goodwillie–Weiss tower to formulate an obstruction theory for embedding $K$ into a Euclidean space $\mathbb{R}^d$. For $2$-dimensional complexes in $\mathbb{R}^4$, a geometric analogue is also introduced, based on intersections of Whitney disks and more generally on the intersection theory of Whitney towers developed by Schneiderman and Teichner
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Contracting convex surfaces by mean curvature flow with free boundary on convex barriers Asian J. Math. (IF 0.6) Pub Date : 2023-10-12 Sven Hirsch, Martin Man-Chun Li
We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a
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Moment map for coupled equations of Kähler forms and curvature Asian J. Math. (IF 0.6) Pub Date : 2023-10-12 King Leung Lee
In this paper we introduce two new systems of equations in Kähler geometry: The coupled $\mathrm{p}$ equation and the generalized coupled cscK equation. We motivate the equations from the moment map pictures, prove the uniqueness of solutions and find out the obstructions to the solutions for the second equation. We also point out the connections between the coupled cscK equation, the coupled Kähler
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A criteria for classification of weighted dual graphs of singularities and its application Asian J. Math. (IF 0.6) Pub Date : 2023-10-12 Stephen S.-T. Yau, Qiwei Zhu, Huaiqing Zuo
Let $(V, p)$ be a normal surface singularity. Let $\pi : (M,E) \to (V, p)$ be a minimal good resolution of $V$, such that the irreducible components $E_i$ of $E = \pi^{-1} (p)$ are nonsingular and have only normal crossings. There is a natural weighted dual graph $\Gamma$ associated to $E$. Along with the genera of the $E_i$, $\Gamma$ fully describes the topology and differentiable structure of the
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On $\pi$-divisible $\mathcal{O}$-modules over fields of characteristic $p$ Asian J. Math. (IF 0.6) Pub Date : 2023-06-16 Chuangxun Cheng
In this paper, we construct a Dieudonné theory for $\pi$-divisible $\mathcal{O}$-modules over a perfect field of characteristic $p$. Applying this theory, we prove the existence of slope filtration of $\pi$-divisible $\mathcal{O}$-modules over an integral normal Noetherian base. We also study minimal $\pi$-divisible $\mathcal{O}$-modules over an algebraically closed field of characteristic $p$ and
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Simply and tangentially homotopy equivalent but non-homeomorphic homogeneous manifolds Asian J. Math. (IF 0.6) Pub Date : 2023-06-16 Sadeeb Simon Ottenburger
For each odd integer $r$ greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise nonhomeomorphic $5$-dimensional closed homogeneous spaces with fundamental group isomorphic to $\mathbb{Z}/r$. As an application we construct the first examples of manifolds which possess infinitely many metrics of nonnegative
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Nevanlinna-type theory based on heat diffusion Asian J. Math. (IF 0.6) Pub Date : 2023-06-16 Xianjing Dong
We obtain an analogue of Nevanlinna theory of holomorphic mappings from a complete and stochastically complete Kähler manifold into a complex projective manifold. When certain curvature conditions are imposed, some Nevanlinna-type defect relations based on heat diffusion are established.
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On the stability of linear feedback particle filter Asian J. Math. (IF 0.6) Pub Date : 2023-06-16 Xiuqiong Chen, Stephen S.-T. Yau
In this paper, we study the stability of feedback particle filter (FPF) for linear filtering systems with Gaussian noises. We first provide some local contraction estimates of the exact linear FPF, whose conditional distribution is exactly the posterior distribution of the state as long as their initial values are equal. Then we study the convergence of the linear FPF formed by $N$ particles, and prove
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The Schwarz lemma in Kähler and non-Kähler geometry Asian J. Math. (IF 0.6) Pub Date : 2023-06-16 Kyle Broder
We introduce a new curvature constraint that provides an analog of the real bisectional curvature considered by Yang–Zheng [28] for the Aubin–Yau inequality. A unified perspective of the various forms of the Schwarz lemma is given, leading to novel Schwarz-type inequalities in both the Kähler and Hermitian categories.
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An $\varepsilon$-regularity theorem for line bundle mean curvature flow Asian J. Math. (IF 0.6) Pub Date : 2023-04-27 Xiaoli Han, Hikaru Yamamoto
In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau [7]. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang–Mills metrics on a given Kähler manifold. The goal of this paper is to give an $\varepsilon$-regularity theorem for the line bundle mean curvature flow. To establish the theorem, we provide a scale invariant monotone
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Branched Cauchy–Riemann structures on once-punctured torus bundles Asian J. Math. (IF 0.6) Pub Date : 2023-04-27 Alex Casella
Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realization in Cauchy–Riemann (CR) space. By introducing a new type of 3‑cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle
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A hall of statistical mirrors Asian J. Math. (IF 0.6) Pub Date : 2023-04-27 Gabriel Khan, Jun Zhang
The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher–Rao metric and a pair of torsion-free conjugate connections. In recent work [ZK20], the authors considered parametrized probability distributions as partially-flat statistical manifolds admitting torsion and showed that there is a complex-to-symplectic
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The deformed Hermitian–Yang–Mills equation on the blowup of $\mathbb{P}^n$ Asian J. Math. (IF 0.6) Pub Date : 2023-04-27 Adam Jacob, Norman Sheu
We study the deformed Hermitian–Yang–Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kähler manifolds.
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Contact of circles with surfaces: Answers to a question of Montaldi Asian J. Math. (IF 0.6) Pub Date : 2023-04-13 Peter Giblin, Graham Reeve
We answer a question raised by J. Montaldi in 1986 as to the exact upper bound on the number of circles which can have $5$-point contact with a generic smooth surface $M$ in $\mathbb{R}^3$, at a point of $M$.
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Dirac structures on the space of connections Asian J. Math. (IF 0.6) Pub Date : 2023-04-13 Yuji Hirota, Tosiaki Kori
We shall investigate the Dirac structures on the space of connections over threemanifolds and over four-manifolds. We show that the space of irreducible connections on the trivial $\mathrm{SU}(n)$-bundle over a three-manifold is canonically endowed with a Dirac structure twisted by a $3$-form. We also give a family of Dirac structures twisted by the $3$-form on the space of irreducible connections
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Convergence of Narasimhan–Simha measures on degenerating families of Riemann surfaces Asian J. Math. (IF 0.6) Pub Date : 2023-04-13 Sanal Shivaprasad
Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense
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Differential complexes and Hodge theory on $\log$-symplectic manifolds Asian J. Math. (IF 0.6) Pub Date : 2023-04-13 Ziv Ran
We study certain complexes of differential forms, including ‘reverse de Rham’ complexes, on (real or complex) Poisson manifolds, especially holomorphic $\log$-symplectic ones. We relate these to the degeneracy divisor and rank loci of the Poisson bivector. In some good holomorphic cases we compute the local cohomology of these complexes. In the Kählerian case, we deduce a relation between the multiplicity
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Nowhere vanishing primitive of a symplectic form Asian J. Math. (IF 0.6) Pub Date : 2023-04-13 B. Stratmann
Let $M$ be a manifold with an exact symplectic form $\omega$. Then there is a nowhere vanishing primitive $\beta$ for $\omega$, i.e. $\omega = \mathrm{d} \beta$.
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Topological convexity in complex surfaces Asian J. Math. (IF 0.6) Pub Date : 2023-04-13 Robert E. Gompf
We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded $3$-manifolds in complex surfaces. Topologically pseudoconvex (TPC) $3$-manifolds behave similarly to their smooth analogues, cutting out open domains of holomorphy (Stein surfaces), but they are much more common. We provide tools for constructing TPC embeddings, and show that every closed, oriented
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Parabolic Higgs bundles, $tt^\ast$ connections and opers Asian J. Math. (IF 0.6) Pub Date : 2023-03-24 Murad Alim, Florian Beck, Laura Fredrickson
The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi–Yau manifolds. This setup enables us to apply techniques from mirror symmetry. For example, the corresponding
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Invariance of plurigenera and Chow-type lemma Asian J. Math. (IF 0.6) Pub Date : 2023-03-24 Sheng Rao, I-Hsun Tsai
This paper answers a question of Demailly whether a smooth family of nonsingular projective varieties admits the deformation invariance of plurigenera affirmatively, and proves this more generally for a flat family of varieties with only canonical singularities and uncountable ones therein being of general type and also two Chow-type lemmata on the structure of a family of projective complex analytic
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On the stability of homogeneous Einstein manifolds Asian J. Math. (IF 0.6) Pub Date : 2023-03-24 Jorge Lauret
Let $g$ be a $G$-invariant Einstein metric on a compact homogeneous space $M = G/K$. We use a formula for the Lichnerowicz Laplacian of $g$ at $G$-invariant $TT$-tensors to study the stability type of $g$ as a critical point of the scalar curvature function. The case when $g$ is naturally reductive is studied in special detail.
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On the global moduli of Calabi–Yau threefolds Asian J. Math. (IF 0.6) Pub Date : 2023-03-24 Ron Donagi, Mark Macerato, Eric Sharpe
In this note we initiate a program to obtain global descriptions of Calabi–Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle. We do this here for several Calabi–Yau’s obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16]
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Non-existence of negative weight derivations of the local $1\textrm{-st}$ Hessian algebras of singularities Asian J. Math. (IF 0.6) Pub Date : 2023-03-06 Shuanghe Fan, Stephen S.-T. Yau, Huaiqing Zuo
In our previous work, we proposed a conjecture about the non-existence of negative weight derivations of the $k\textrm{-st}$ Tjurina algebras of weighted homogeneous hypersurface singularities. In this paper, we verify this conjecture for three dimensional fewnomial singularities.
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On the Iwasawa invariants of non-cotorsion Selmer groups Asian J. Math. (IF 0.6) Pub Date : 2023-03-06 Sören Kleine
We study the variation of Iwasawa invariants of Selmer groups and fine Selmer groups of abelian varieties over $\mathbb{Z}_p$-extensions of a fixed number field $K$. It is shown that the $\lambda$-invariants can be unbounded if the $\Lambda$-coranks of the Selmer groups (respectively fine Selmer groups) vary. In contrast, the classical Iwasawa $\lambda$-invariants of $\mathbb{Z}_p$-extensions are expected
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Intermediate curvatures and highly connected manifolds Asian J. Math. (IF 0.6) Pub Date : 2023-03-06 Diarmuid Crowley, David J. Wraith
We show that after forming a connected sum with a homotopy sphere, all $(2j-1)$-connected $2j$-parallelisable manifolds in dimension $4j+1, j \geq 2$, can be equipped with Riemannian metrics of $2$-positive Ricci curvature. The condition of $2$-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues of the Ricci tensor is positive at every point. This result is a counterpart
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Moduli space of irregular singular parabolic connections of generic ramified type on a smooth projective curve Asian J. Math. (IF 0.6) Pub Date : 2023-01-30 Michi-Aki Inaba
We give an algebraic construction of the moduli space of irregular singular connections of generic ramified type on a smooth projective curve. We prove that the moduli space is smooth and give its dimension. Under the assumption that the exponent of ramified type is generic, we give an algebraic symplectic form on the moduli space.
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Comparing the Carathéodory pseudo-distance and the Kähler–Einstein distance on complete reinhardt domains Asian J. Math. (IF 0.6) Pub Date : 2023-01-30 Gunhee Cho
We show that on a certain class of bounded, complete Reinhardt domains in $\mathbb{C}^n$ that enjoy a lot of symmetries, the Carathéodory pseudo-distance and the geodesic distance of the complete Kähler–Einstein metric with Ricci curvature $-1$ are different.
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$6$-dimensional FJRW theories of the simple–elliptic singularities Asian J. Math. (IF 0.6) Pub Date : 2023-01-30 Alexey Basalaev
We give explicitly in the closed formulae the genus zero primary potentials of the three $6$-dimensional FJRW theories of the simple–elliptic singularity $\tilde{E}_7$ with the non–maximal symmetry groups. For each of these FJRW theories we establish the CY/LG correspondence to the Gromov–Witten theory of the elliptic orbifold $[\mathcal{E} / (\mathbb{Z}/2\mathbb{Z})]$ — the orbifold quotient of the
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Even and odd instanton bundles on Fano threefolds Asian J. Math. (IF 0.6) Pub Date : 2023-01-30 Vincenzo Antonelli, Gianfranco Casnati, Ozhan Genc
We define non–ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles introduced in [14]. We determine a lower bound for the quantum number of a non–ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_X \geq 2$ or $i_X = 1$ and $\mathrm{rk} \:
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Slope equality of plane curve fibrations and its application to Durfee’s conjecture Asian J. Math. (IF 0.6) Pub Date : 2023-01-30 Makoto Enokizono
We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a “strong” Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee’s strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.
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Ricci-flat graphs with maximum degree at most $4$ Asian J. Math. (IF 0.6) Pub Date : 2022-10-24 Shuliang Bai, Linyuan Lu, Shing-Tung Yau
A graph is called Ricci-flat if its Ricci curvatures vanish on all edges, here the definition of Ricci curvature on graphs was given by Lin–Lu–Yau [7]. The authors in [8] and [3] obtained a complete characterization for all Ricci-flat graphs with girth at least five. In this paper, we completely determined all Ricci-flat graphs with maximum degree at most $4$.
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Gauss–Kronecker curvature and equisingularity at infinity of definable families Asian J. Math. (IF 0.6) Pub Date : 2022-10-24 Nicolas Dutertre, Vincent Grandjean
Assume given a polynomially bounded $o$-minimal structure expanding the real numbers. $Let (T_s)_{s \in \mathbb{R}}$ be a definable family of $C^2$-hypersurfaces of $\mathbb{R}^n$. Upon defining the notion of generalized critical value for such a family, we show that the functions $s \to {\lvert K \rvert} (s)$ and $s \to K(s)$, respectively the total absolute Gauss–Kronecker and total Gauss–Kronecker
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Representations and modules of Rota–Baxter algebras Asian J. Math. (IF 0.6) Pub Date : 2022-10-24 Li Guo, Zongzhu Lin
We give a general study of representation and module theory of Rota–Baxter algebras. Regular-singular decompositions of Rota–Baxter algebras and Rota–Baxter modules are obtained under the condition of quasi-idempotency. Representations of a Rota–Baxter algebra are shown to be equivalent to the representations of the ring of Rota–Baxter operators whose categorical properties are obtained and explicit
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Generating functions for Ohno type sums of finite and symmetric multiple zeta-star values Asian J. Math. (IF 0.6) Pub Date : 2022-10-24 Minoru Hirose, Hideki Murahara, Shingo Saito
Ohno’s relation states that a certain sum, which we call an Ohno type sum, of multiple zeta values remains unchanged if we replace the base index by its dual index. In view of Oyama’s theorem concerning Ohno type sums of finite and symmetric multiple zeta values, Kaneko looked at Ohno type sums of finite and symmetric multiple zeta-star values and made a conjecture on the generating function for a
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A new proof for global rigidity of vertex scaling on polyhedral surfaces Asian J. Math. (IF 0.6) Pub Date : 2022-10-24 Xu Xu, Chao Zheng
The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo [17], who proved the local rigidity by establishing a variational principle and conjectured the global rigidity. Luo’s conjecture was solved by Bobenko–Pinkall–Springborn [3], who also introduced the vertex scaling for piecewise hyperbolic metrics and proved its global rigidity. Bobenko–Pinkall–Spingborn’s
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Closed $\mathrm{G}_2$-structures with a transitive reductive group of automorphisms Asian J. Math. (IF 0.6) Pub Date : 2022-10-24 Fabio Podestà, Alberto Raffero
We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel $\mathrm{G}_2$-structure and admitting a transitive reductive group $\mathrm{G}$ of automorphisms. In particular, we show that the center of $\mathrm{G}_2$ is one-dimensional and the manifold is the Riemannian product of a flat factor and a non-compact homogeneous six-dimensional manifold endowed
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On the analytic classification of irreducible plane curve singularities Asian J. Math. (IF 0.6) Pub Date : 2022-07-06 Eduardo Casas-Alvero
We present new results regarding which Puiseux coefficients the analytic type of a complex irreducible plane curve singularity depends on.
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Hodge filtration and Hodge ideals for $\mathbb{Q}$-divisors with weighted homogeneous isolated singularities or convenient non-degenerate singularities Asian J. Math. (IF 0.6) Pub Date : 2022-07-06 Mingyi Zhang
We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $\mathcal{O}_X (*Z) f^{1-\alpha}$ associated to the effective $\mathbb{Q}$-divisor $D = \alpha \cdot Z$, where $0 \lt \alpha \leq 1$ and $Z = (f = 0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the
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Deformations of CR maps and applications Asian J. Math. (IF 0.6) Pub Date : 2022-07-06 Giuseppe Della Sala, Bernhard Lamel, Michael Reiter
We study the deformation theory of CR maps in the positive codimensional case. In particular, we study structural properties of the mapping locus $E$ of (germs of nondegenerate) holomorphic maps $H: (M, p) \to M^\prime$ between generic real submanifolds $M \subset \mathbb{C}^N$ and $ M^\prime \subset \mathbb{C}^{N^\prime}$, defined to be the set of points $p^\prime \in M^\prime$ which admit such a
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Moduli of curves of genus one with twisted fields Asian J. Math. (IF 0.6) Pub Date : 2022-07-06 Yi Hu, Jingchen Niu
We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil–Zinger’s desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone
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Finsler perturbation with nondense geodesics with irrational directions Asian J. Math. (IF 0.6) Pub Date : 2022-07-06 Dmitri Burago, Dong Chen
We show that given any Liouville direction and flat Finsler torus, one can make a $C^\infty$‑small perturbation on an arbitrarily small disc to get a nondense geodesic in the given direction.
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Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators $\mathbb{I}_n$ Asian J. Math. (IF 0.6) Pub Date : 2022-07-06 V. V. Bavula, V. Bekkert, V. Futorny
For the algebra $\mathbb{I}_n = K {\langle x_1, \dotsc, x_n, \partial_1, \dotsc, \partial_n, \int_1, \dotsc, \int_n \rangle}$ of polynomial integrodifferential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $\mathbb{I}_n$‑modules is given. It is proven that the category of weight $\mathbb{I}_n$‑modules is semisimple. An explicit
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Meromorphic connections, determinant line bundles and the Tyurin parametrization Asian J. Math. (IF 0.6) Pub Date : 2022-04-25 Indranil Biswas, Jacques Hurtubise
We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the “sheaf of holomorphic connections” (the sheaf of holomorphic splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic
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Metrics and compactifications of Teichmüller spaces of flat tori Asian J. Math. (IF 0.6) Pub Date : 2022-04-25 Mark Greenfield, Lizhen Ji
Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R}) / \mathrm{SO}(n)$ with the Teichmüller space of flat $n$‑tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichmüller theory and symmetric spaces. We define and study analogs of the Thurston, Teichmüller, and Weil–Petersson metrics. We show the Teichmüller metric
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On strong exceptional collections of line bundles of maximal length on fano toric Deligne–Mumford stacks Asian J. Math. (IF 0.6) Pub Date : 2022-04-25 Lev Borisov, Chengxi Wang
We study strong exceptional collections of line bundles on Fano toric Deligne–Mumford stacks $\mathbb{P}_\Sigma$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $\mathbb{P}_\Sigma$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$‑theory group of $\mathbb{P}_\Sigma$
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Spectra related to the length spectrum Asian J. Math. (IF 0.6) Pub Date : 2022-04-25 Conrad Plaut
We extend the Covering Spectrum ($\mathrm{CS}$) of Sormani–Wei to two spectra, called the Extended Covering Spectrum ($\mathrm{ECS}$) and Entourage Spectrum ($\mathrm{ES}$) that are new metric invariants related to the Length Spectrum for Riemannian manifolds. These spectra measure the “size” of certain covering maps called entourage covers that generalize the $\delta$-covers of Sormani–Wei. For Riemannian
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Twisting lemma for $\lambda$-adic modules Asian J. Math. (IF 0.6) Pub Date : 2022-04-25 Sohan Ghosh, Somnath Jha, Sudhanshu Shekhar
A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb{Z}_p[[Γ]]$ with $\Gamma \cong \mathbb{Z}_p$, there exists a continuous character $\theta : \Gamma \to \mathbb{Z}^\times_p$ such that, the $\Gamma^{p^n}$‑Euler characteristic of the twist $M(\theta)$ is finite for every $n$. This twisting lemma has been generalized for the Iwasawa algebra
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Classification of uniformly distributed measures of dimension $1$ in general codimension Asian J. Math. (IF 0.6) Pub Date : 2022-04-25 Paul Laurain, Mircea Petrache
Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $\mathbb{R}^d$ has remained open, except for $d = 1$ and for compactly supported measures in $d = 2$, and for codimension $1$. In this paper we study 1‑dimensional measures in $\mathbb{R}^d$ for all $d$ and classify uniform measures with connected $1$‑dimensional support, which turn out to be homogeneous
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Comparing shapes of high genus surfaces Asian J. Math. (IF 0.6) Pub Date : 2022-04-25 Yanwen Luo
In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer in a fixed homotopy class is achieved by a quasiconformal homeomorphism by the lower
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Segre’s theorem. An analytic proof of a result in differential geometry Asian J. Math. (IF 0.6) Pub Date : 2022-03-14 Karl K. Brustad
We present an analytic approach on how to solve the overdetermined problem $\lvert \nabla u \rvert = f(u), \Delta u = g(u)$, in connected domains $\Omega \subseteq \mathbb{R}^n$.
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Strongly homotopy Lie algebras and deformations of calibrated submanifolds Asian J. Math. (IF 0.6) Pub Date : 2022-03-14 Domenico Fiorenza, Hông Vân Lê, Lorenz Schwachhöfer, Luca Vitagliano
For an element $\Psi$ in the graded vector space $\Omega^\ast (M,TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined as a submanifold $N$ of $M$ such that $\Psi_{\lvert N} \in \Omega^\ast (N,TN)$. The class of $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $\Omega^\ast
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Int-amplified endomorphisms of compact Kähler spaces Asian J. Math. (IF 0.6) Pub Date : 2022-03-14 Guolei Zhong
Let $X$ be a normal compact Kähler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int‑amplified if $f^\ast \xi - \xi = \eta$ for some Kähler classes $\xi$ and $\eta$. First, we show that this definition generalizes the notion in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int‑amplified
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Global perturbation potential function on complete special holonomy manifolds Asian J. Math. (IF 0.6) Pub Date : 2022-03-14 Teng Huang
In this article, we introduce and study the notion of a complete special holonomy manifold $(X,\omega)$ which is given by a global perturbation potential function, i.e., there is a function $f$ on $X$ such that $\omega^\prime = \omega - \mathcal{L}_{\nabla_f} \omega$ is sufficiently small in $L^\infty$-norm. We establish some vanishing theorems on the $L^2$ harmonic forms under some conditions on the
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Higher order moments of generalized quadratic Gauss sums weighted by $L$-functions Asian J. Math. (IF 0.6) Pub Date : 2022-03-14 Nilanjan Bag, Rupam Barman
The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by $L$-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic