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The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I Publ. math. IHES (IF 6.2) Pub Date : 2024-04-02 Vincent Colin, Paolo Ghiggini, Ko Honda
Given an open book decomposition \((S,\mathfrak{h} )\) adapted to a closed, oriented 3-manifold \(M\), we define a chain map \(\Phi \) from a certain Heegaard Floer chain complex associated to \((S,\mathfrak{h} )\) to a certain embedded contact homology chain complex associated to \((S,\mathfrak{h} )\), as defined in (Colin et al. in Geom. Topol., 2024), and prove that it induces an isomorphism on
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The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II Publ. math. IHES (IF 6.2) Pub Date : 2024-04-02
Abstract This paper is the sequel to (Colin et al. in Publ. Math. Inst. Hautes Études Sci., 2024), and is devoted to proving some of the technical parts of the HF=ECH isomorphism.
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The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus Publ. math. IHES (IF 6.2) Pub Date : 2024-04-02 Vincent Colin, Paolo Ghiggini, Ko Honda
Given a closed oriented 3-manifold \(M\), we establish an isomorphism between the Heegaard Floer homology group \(HF^{+} (-M)\) and the embedded contact homology group \(ECH(M)\). Starting from an open book decomposition \((S,\mathfrak{h} )\) of \(M\), we construct a chain map \(\Phi ^{+}\) from a Heegaard Floer chain complex associated to \((S,\mathfrak{h} )\) to an embedded contact homology chain
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An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity Publ. math. IHES (IF 6.2) Pub Date : 2024-02-21 Camillo De Lellis, Stefano Nardulli, Simone Steinbrüchel
We consider integral area-minimizing 2-dimensional currents \(T\) in \(U\subset \mathbf {R}^{2+n}\) with \(\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]\), where \(Q\in \mathbf {N} \setminus \{0\}\) and \(\Gamma \) is sufficiently smooth. We prove that, if \(q\in \Gamma \) is a point where the density of \(T\) is strictly below \(\frac{Q+1}{2}\), then the current is regular at \(q\). The regularity
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Ancient solutions and translators of Lagrangian mean curvature flow Publ. math. IHES (IF 6.2) Pub Date : 2024-01-15
Abstract Suppose that ℳ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in \(\mathbf {C} ^{n}\) . We show that if ℳ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then ℳ is a translator. In particular in \(\mathbf {C} ^{2}\) , all almost calibrated, exact, ancient solutions of Lagrangian
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A priori bounds for GIETs, affine shadows and rigidity of foliations in genus two Publ. math. IHES (IF 6.2) Pub Date : 2023-10-31 Selim Ghazouani, Corinna Ulcigrai
We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman’s theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that, if a smooth, orientable foliation with non-degenerate (Morse) singularities on a closed surface of genus two is minimal, then, under a full measure condition for
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Martin-Löf identity types in C-systems Publ. math. IHES (IF 6.2) Pub Date : 2023-07-31 Vladimir Voevodsky
This paper continues a series of papers that develop a new approach to syntax and semantics of dependent type theories. Here we study the interpretation of the rules of the identity types in the intensional Martin-Löf type theories on the C-systems that arise from universe categories. In the first part of the paper we develop constructions that produce interpretations of these rules from certain structures
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Globally $\pmb{+}$ -regular varieties and the minimal model program for threefolds in mixed characteristic Publ. math. IHES (IF 6.2) Pub Date : 2023-05-10 Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, Jakub Witaszek
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global \(F\)-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita’s conjecture to mixed characteristic.
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The p-widths of a surface Publ. math. IHES (IF 6.2) Pub Date : 2023-05-04 Otis Chodosh, Christos Mantoulidis
The \(p\)-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the \(p\)-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality
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Stable homotopy groups of spheres: from dimension 0 to 90 Publ. math. IHES (IF 6.2) Pub Date : 2023-05-04 Daniel C. Isaksen, Guozhen Wang, Zhouli Xu
Using techniques in motivic homotopy theory, especially the theorem of Gheorghe, the second and the third author on the isomorphism between motivic Adams spectral sequence for \(C\tau \) and the algebraic Novikov spectral sequence for \(BP_{*}\), we compute the classical and motivic stable homotopy groups of spheres from dimension 0 to 90, except for some carefully enumerated uncertainties.
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Geometry of polarised varieties Publ. math. IHES (IF 6.2) Pub Date : 2023-02-13 Caucher Birkar
In this paper, we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we study birational boundedness of linear systems. We show that if \(X\) is a projective variety of dimension \(d\) with \(\epsilon \)-lc singularities for \(\epsilon >0\), and if \(N\) is a nef and big Weil divisor on \(X\) such that \(N-K_{X}\) is pseudo-effective
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On the Severi problem in arbitrary characteristic Publ. math. IHES (IF 6.2) Pub Date : 2022-12-01 Karl Christ, Xiang He, Ilya Tyomkin
In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction
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Two dimensional neighborhoods of elliptic curves: analytic classification in the torsion case Publ. math. IHES (IF 6.2) Pub Date : 2022-07-11 Frank Loray, Frédéric Touzet, Sergei M. Voronin
We investigate the analytic classification of two dimensional neighborhoods of an elliptic curve with torsion normal bundle. We provide the complete analytic classification for those neighborhoods in the simplest formal class and we indicate how to generalize this construction to general torsion case.
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Smith–Treumann theory and the linkage principle Publ. math. IHES (IF 6.2) Pub Date : 2022-06-20 Simon Riche, Geordie Williamson
We apply Treumann’s “Smith theory for sheaves” in the context of the Iwahori–Whittaker model of the Satake category. We deduce two results in the representation theory of reductive algebraic groups over fields of positive characteristic: (a) a geometric proof of the linkage principle; (b) a character formula for tilting modules in terms of the \(\ell \)-canonical basis, valid in all blocks and in all
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Langevin dynamic for the 2D Yang–Mills measure Publ. math. IHES (IF 6.2) Pub Date : 2022-06-07 Ajay Chandra, Ilya Chevyrev, Martin Hairer, Hao Shen
We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions. To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties
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Hecke correspondences for smooth moduli spaces of sheaves Publ. math. IHES (IF 6.2) Pub Date : 2022-05-11 Andrei Neguţ
We define functors on the derived category of the moduli space ℳ of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain commutation relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the \(K\)-theory of the moduli space ℳ, thus generalizing the action studied
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On the divergence of Birkhoff Normal Forms Publ. math. IHES (IF 6.2) Pub Date : 2022-04-27 Raphaël Krikorian
It is well known that a real analytic symplectic diffeomorphism of the \(2d\)-dimensional disk (\(d\geq 1\)) admitting the origin as a non-resonant elliptic fixed point can be formally conjugated to its Birkhoff Normal Form, a formal power series defining a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This
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The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case Publ. math. IHES (IF 6.2) Pub Date : 2022-01-27 Raphaël Beuzart-Plessis, Pierre-Henri Chaudouard, Michał Zydor
In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups \(U_{n}\times U_{n+1}\) in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain cuspidal data, called ∗-regular, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on
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Abelian surfaces over totally real fields are potentially modular Publ. math. IHES (IF 6.2) Pub Date : 2021-11-29 George Boxer, Frank Calegari, Toby Gee, Vincent Pilloni
We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces \(A\) over \({\mathbf {Q}}\) with \(\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}\)
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Symmetric power functoriality for holomorphic modular forms Publ. math. IHES (IF 6.2) Pub Date : 2021-10-15 James Newton, Jack A. Thorne
Let \(f\) be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting \(\operatorname{Sym}^{n} f\) for every \(n \geq 1\). We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over \(\mathbf{Q}\).
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Symmetric power functoriality for holomorphic modular forms, II Publ. math. IHES (IF 6.2) Pub Date : 2021-10-11 James Newton, Jack A. Thorne
Let \(f\) be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of the symmetric power lifting \(\operatorname{Sym}^{n} f\) for every \(n \geq 1\).
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Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry Publ. math. IHES (IF 6.2) Pub Date : 2021-07-21 Piermarco Cannarsa, Wei Cheng, Albert Fathi
If \(U:[0,+\infty [\times M\) is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ where \(M\) is a not necessarily compact manifold, and \(H\) is a Tonelli Hamiltonian, we prove the set \(\Sigma (U)\), of points in \(]0,+\infty [\times M\) where \(U\) is not differentiable, is locally contractible. Moreover, we study the
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Stability conditions in families Publ. math. IHES (IF 6.2) Pub Date : 2021-05-17 Arend Bayer, Martí Lahoz, Emanuele Macrì, Howard Nuer, Alexander Perry, Paolo Stellari
We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show
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The action of Young subgroups on the partition complex Publ. math. IHES (IF 6.2) Pub Date : 2021-03-25 Gregory Z. Arone, D. Lukas B. Brantner
We study the restrictions, the strict fixed points, and the strict quotients of the partition complex \(|\Pi_{n}|\), which is the \(\Sigma_{n}\)-space attached to the poset of proper nontrivial partitions of the set \(\{1,\ldots,n\}\). We express the space of fixed points \(|\Pi_{n}|^{G}\) in terms of subgroup posets for general \(G\subset \Sigma_{n}\) and prove a formula for the restriction of \(|\Pi_{n}|\)
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Stationary characters on lattices of semisimple Lie groups Publ. math. IHES (IF 6.2) Pub Date : 2021-03-02 Rémi Boutonnet, Cyril Houdayer
We show that stationary characters on irreducible lattices \(\Gamma < G\) of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice \(\Gamma < G\), the left regular
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Tightness of Liouville first passage percolation for γ ∈ ( 0 , 2 ) $\gamma \in (0,2)$ Publ. math. IHES (IF 6.2) Pub Date : 2020-11-04 Jian Ding, Julien Dubédat, Alexander Dunlap, Hugo Falconet
We study Liouville first passage percolation metrics associated to a Gaussian free field \(h\) mollified by the two-dimensional heat kernel \(p_{t}\) in the bulk, and related star-scale invariant metrics. For \(\gamma \in (0,2)\) and \(\xi = \frac{\gamma }{d_{\gamma }}\), where \(d_{\gamma }\) is the Liouville quantum gravity dimension defined in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934,
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Deviations of ergodic sums for toral translations II. Boxes Publ. math. IHES (IF 6.2) Pub Date : 2020-10-19 Dmitry Dolgopyat, Bassam Fayad
We study the Kronecker sequence \(\{n\alpha \}_{n\leq N}\) on the torus \({\mathbf {T}}^{d}\) when \(\alpha \) is uniformly distributed on \({\mathbf {T}}^{d}\). We show that the discrepancy of the number of visits of this sequence to a random box, normalized by \(\ln ^{d} N\), converges as \(N\to \infty \) to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the
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Generic regularity of free boundaries for the obstacle problem Publ. math. IHES (IF 6.2) Pub Date : 2020-07-02 Alessio Figalli, Xavier Ros-Oton, Joaquim Serra
The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in \(\mathbf {R}^{n}\). By classical results of Caffarelli, the free boundary is \(C^{\infty }\) outside a set of singular points. Explicit examples show that the singular set could be in general \((n-1)\)-dimensional—that is, as large as the regular set. Our main result establishes that, generically
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Explicit spectral gaps for random covers of Riemann surfaces Publ. math. IHES (IF 6.2) Pub Date : 2020-06-25 Michael Magee, Frédéric Naud
We introduce a permutation model for random degree \(n\) covers \(X_{n}\) of a non-elementary convex-cocompact hyperbolic surface \(X=\Gamma \backslash \mathbf {H}\). Let \(\delta \) be the Hausdorff dimension of the limit set of \(\Gamma \). We say that a resonance of \(X_{n}\) is new if it is not a resonance of \(X\), and similarly define new eigenvalues of the Laplacian. We prove that for any \(\epsilon
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Complexity of parabolic systems Publ. math. IHES (IF 6.2) Pub Date : 2020-05-15 Tobias Holck Colding, William P. Minicozzi
We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. We use this in a major application of our new methods to give the first general bounds on generic singularities of surfaces
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Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations Publ. math. IHES (IF 6.2) Pub Date : 2020-05-14 Helge Ruddat, Bernd Siebert
We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber. One application is a proof that the mirror map for the canonical formal families
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Discrete series multiplicities for classical groups over Z$\mathbf {Z}$ and level 1 algebraic cusp forms Publ. math. IHES (IF 6.2) Pub Date : 2020-03-05 Gaëtan Chenevier, Olivier Taïbi
The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group \(G\) over \(\mathbf {Z}\), and provide numerical applications in absolute rank \(\leq 8\). Second, we prove a classification result for the level one cuspidal algebraic automorphic representations
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Riemannian hyperbolization Publ. math. IHES (IF 6.2) Pub Date : 2020-02-28 Pedro Ontaneda
The strict hyperbolization process of Charney and Davis produces a large and rich class of negatively curved spaces (in the geodesic sense). This process is based on an earlier version introduced by Gromov and later studied by Davis and Januszkiewicz. If M is a manifold its Charney-Davis strict hyperbolization is also a manifold, but the negatively curved metric obtained is very far from being Riemannian
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Quasimap wall-crossings and mirror symmetry Publ. math. IHES (IF 6.2) Pub Date : 2020-02-07 Ionuţ Ciocan-Fontanine, Bumsig Kim
We state a wall-crossing formula for the virtual classes of \({\varepsilon }\)-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus \(g\) descendant Gromov-Witten potential and the genus \(g\)\({\varepsilon }\)-quasimap descendant potential
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E2$E_{2}$ -cells and mapping class groups Publ. math. IHES (IF 6.2) Pub Date : 2019-06-17 Søren Galatius, Alexander Kupers, Oscar Randal-Williams
We prove a new kind of stabilisation result, “secondary homological stability,” for the homology of mapping class groups of orientable surfaces with one boundary component. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of \(E_{2}\)-algebras, which have no \(E_{2}\)-cells below a certain vanishing line.
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The period-index problem for real surfaces Publ. math. IHES (IF 6.2) Pub Date : 2019-05-28 Olivier Benoist
We study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that it is always the case if the surface has no real points (more generally, if the class vanishes in restriction to the real points of the locus where it is well-defined), and give a necessary and sufficient condition for unramified classes. As an application
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Separation for the stationary Prandtl equation Publ. math. IHES (IF 6.2) Pub Date : 2019-09-05 Anne-Laure Dalibard, Nader Masmoudi
In this paper, we prove that separation occurs for the stationary Prandtl equation, in the case of adverse pressure gradient, for a large class of boundary data at \(x=0\). We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at \(x=0\), there exists \(x^{*}>0\) such that \(\partial _{y} u_{|y=0}(x) \sim C \sqrt{x^{*} -x}\) as \(x\to x^{*}\)
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A local model for the trianguline variety and applications Publ. math. IHES (IF 6.2) Pub Date : 2019-08-22 Christophe Breuil, Eugen Hellmann, Benjamin Schraen
We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck’s simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial
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Covariantly functorial wrapped Floer theory on Liouville sectors Publ. math. IHES (IF 6.2) Pub Date : 2019-08-23 Sheel Ganatra, John Pardon, Vivek Shende
We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid’s generation criterion
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Foliations with positive slopes and birational stability of orbifold cotangent bundles Publ. math. IHES (IF 6.2) Pub Date : 2019-04-18 Frédéric Campana, Mihai Păun
Let \(X\) be a smooth connected projective manifold, together with an snc orbifold divisor \(\Delta \), such that the pair \((X, \Delta )\) is log-canonical. If \(K_{X}+\Delta \) is pseudo-effective, we show, among other things, that any quotient of its orbifold cotangent bundle has a pseudo-effective determinant. This improves considerably our previous result (Campana and Păun in Ann. Inst. Fourier
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Joinings of higher rank torus actions on homogeneous spaces Publ. math. IHES (IF 6.2) Pub Date : 2019-02-14 Manfred Einsiedler, Elon Lindenstrauss
We show that joinings of higher rank torus actions on \(S\)-arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.
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Topological Hochschild homology and integral p $p$ -adic Hodge theory Publ. math. IHES (IF 6.2) Pub Date : 2019-04-17 Bhargav Bhatt, Matthew Morrow, Peter Scholze
In mixed characteristic and in equal characteristic \(p\) we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic \(K\)-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex \(A\Omega\) constructed in our previous work, and in equal characteristic \(p\) to crystalline cohomology
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Categorical actions on unipotent representations of finite unitary groups Publ. math. IHES (IF 6.2) Pub Date : 2019-03-08 O. Dudas, M. Varagnolo, E. Vasserot
Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincides with
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Fourier interpolation on the real line Publ. math. IHES (IF 6.2) Pub Date : 2018-09-17 Danylo Radchenko, Maryna Viazovska
In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set \(\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}\). The functions in the interpolating basis are constructed in a closed form as an integral transform of
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Integral p $p$ -adic Hodge theory Publ. math. IHES (IF 6.2) Pub Date : 2019-01-16 Bhargav Bhatt, Matthew Morrow, Peter Scholze
We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of \(\mathbf {C}_{p}\). It takes values in a mixed-characteristic analogue of Dieudonné modules, which was previously defined by Fargues as a version of Breuil–Kisin modules. Notably, this cohomology theory specializes to all other known \(p\)-adic cohomology theories, such as crystalline, de Rham and
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Measure concentration and the weak Pinsker property Publ. math. IHES (IF 6.2) Pub Date : 2018-02-15 Tim Austin
Let \((X,\mu)\) be a standard probability space. An automorphism \(T\) of \((X,\mu)\) has the weak Pinsker property if for every \(\varepsilon > 0\) it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less than \(\varepsilon \). This property was introduced by Thouvenot, who asked whether it holds for all ergodic automorphisms. This paper proves that it does
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Integral models of Shimura varieties with parahoric level structure Publ. math. IHES (IF 6.2) Pub Date : 2018-04-30 M. Kisin, G. Pappas
For a prime \(p > 2\), we construct integral models over \(p\) for Shimura varieties with parahoric level structure, attached to Shimura data \((G,X)\) of abelian type, such that \(G\) splits over a tamely ramified extension of \({\mathbf {Q}}_{\,p}\). The local structure of these integral models is related to certain “local models”, which are defined group theoretically. Under some additional assumptions
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La conjecture du facteur direct Publ. math. IHES (IF 6.2) Pub Date : 2017-12-07 Yves André
M. Hochster a conjecturé que pour toute extension finie \(S\) d’un anneau commutatif régulier \(R\), la suite exacte de \(R\)-modules \(0\to R \to S \to S/R\to0\) est scindée. En nous appuyant sur sa réduction au cas d’un anneau local régulier \(R\) complet non ramifié d’inégale caractéristique, nous proposons une démonstration de cette conjecture dans le contexte de la théorie perfectoïde de P. Scholze
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Le lemme d’Abhyankar perfectoide Publ. math. IHES (IF 6.2) Pub Date : 2017-12-07 Yves André
Nous étendons le théorème de presque-pureté de Faltings-Scholze-Kedlaya-Liu sur les extensions étales finies d’algèbres perfectoïdes au cas des extensions ramifiées, sans restriction sur le lieu de ramification. Nous déduisons cette version perfectoïde du lemme d’Abhyankar du théorème de presque-pureté, par un passage à la limite mettant en jeu des versions perfectoïdes du théorème d’extension de Riemann
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Invariant and stationary measures for the action on Moduli space Publ. math. IHES (IF 6.2) Pub Date : 2018-04-17 Alex Eskin, Maryam Mirzakhani
We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.
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Polyakov’s formulation of $2d$ bosonic string theory Publ. math. IHES (IF 6.2) Pub Date : 2019-06-26 Colin Guillarmou,Rémi Rhodes,Vincent Vargas
Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus $\mathbf{g}\geq 2$ and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic string theory \cite{Pol} (also called $2d$ bosonic string theory) and to Liouville
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Flat surfaces and stability structures Publ. math. IHES (IF 6.2) Pub Date : 2017-11-01 F. Haiden,L. Katzarkov,M. Kontsevich
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A viscosity method in the min-max theory of minimal surfaces Publ. math. IHES (IF 6.2) Pub Date : 2017-10-26 Tristan Rivière
We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface \(\Sigma \) into a given closed manifold, we add to the area Lagrangian a term equal to the \(L^{q}\) norm of the second fundamental form of the immersion times a “viscosity” parameter. This relaxation of the area functional satisfies the Palais–Smale condition
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Percolation of random nodal lines Publ. math. IHES (IF 6.2) Pub Date : 2017-09-18 Vincent Beffara,Damien Gayet
We prove a Russo-Seymour-Welsh percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let \(U\) be a smooth connected bounded open set in \(\mathbf{R}^{2}\) and \(\gamma, \gamma '\) two disjoint arcs of positive length in the boundary of \(U\). We prove that there exists a positive constant
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Calabi-Yau manifolds with isolated conical singularities Publ. math. IHES (IF 6.2) Pub Date : 2017-08-25 Hans-Joachim Hein,Song Sun
Let \(X\) be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let \(L\) be an ample line bundle on \(X\). Assume that the pair \((X,L)\) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point \(x \in X\) there exist a Kähler-Einstein Fano manifold \(Z\) and a positive integer \(q\) dividing \(K_{Z}\)
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C*-simplicity and the unique trace property for discrete groups Publ. math. IHES (IF 6.2) Pub Date : 2017-06-28 Emmanuel Breuillard,Mehrdad Kalantar,Matthew Kennedy,Narutaka Ozawa
A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take
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Meromorphic tensor equivalence for Yangians and quantum loop algebras Publ. math. IHES (IF 6.2) Pub Date : 2017-06-28 Sachin Gautam,Valerio Toledano Laredo
Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, and \(Y_{\hbar }(\mathfrak{g})\), \(U_{q}(L\mathfrak{g})\) the corresponding Yangian and quantum loop algebra, with deformation parameters related by \(q=e^{\pi \iota \hbar }\). When \(\hbar \) is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor \(\Gamma \) from the category
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On the hyperbolicity of general hypersurfaces Publ. math. IHES (IF 6.2) Pub Date : 2017-06-27 Damian Brotbek
In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in \(\mathbf {P}^{n}\) are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen
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Double ramification cycles on the moduli spaces of curves Publ. math. IHES (IF 6.2) Pub Date : 2017-05-10 F. Janda,R. Pandharipande,A. Pixton,D. Zvonkine
Curves of genus \(g\) which admit a map to \(\mathbf {P}^{1}\) with specified ramification profile \(\mu\) over \(0\in \mathbf {P}^{1}\) and \(\nu\) over \(\infty\in \mathbf {P}^{1}\) define a double ramification cycle \(\mathsf{DR}_{g}(\mu,\nu)\) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated
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Geometric presentations of Lie groups and their Dehn functions Publ. math. IHES (IF 6.2) Pub Date : 2016-12-20 Yves Cornulier,Romain Tessera
We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local fields, uses and extends Abels’ theory of multiamalgams of graded Lie algebras, in order to provide workable presentations of these