Showing papers in "Inventiones Mathematicae in 2015"

On the equivalence of the entropic curvature-dimension condition and bochner's inequality on metric measure spaces

Abstract: We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Emery (via energy and $$\Gamma _2$$ -calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $$L^2$$ -Wasserstein distance.

Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics

Abstract: We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice $$\mathbb Z^d$$ with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions $$d>2$$ ) and prove new optimal estimates for regularized versions of the corrector (in dimensions $$d\ge 2$$ ). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

Global solutions for the gravity water waves system in 2d

Abstract: We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu (Invent Math 177(1):45–135, 2009). We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case (Germain et al., Ann Math 175(2):691–754, 2012; Wu, Invent Math 184(1):125–220, 2011). In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2D.

A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry

Abstract: For \(\phi \) a metric on the anticanonical bundle, \(-K_X\), of a Fano manifold \(X\) we consider the volume of \(X\) $$\begin{aligned} \int _X e^{-\phi }. \end{aligned}$$ In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on \(-K_X\). Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on \(X\), even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando–Mabuchi uniqueness theorem for Kahler–Einstein metrics. A generalization of this theorem to ‘twisted’ Kahler–Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than \(-K_X\), and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kahler–Ricci solitons.

Unirational threefolds with no universal codimension \(2\) cycle

Abstract: We prove that the general quartic double solid with \(k\le 7\) nodes does not admit a Chow theoretic decomposition of the diagonal, (or equivalently has a nontrivial universal \(\mathrm{CH}_0\) group,) and the same holds if we replace in this statement “Chow theoretic” by “cohomological”. In particular, it is not stably rational. We also deduce that the general quartic double solid with seven nodes does not admit a universal codimension \(2\) cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibers of its Jacobian by a family of \(1\)-cycles. This finally implies that its third unramified cohomology group is not universally trivial.

M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra

Abstract: We show that the zeroth cohomology of M. Kontsevich’s graph complex is isomorphic to the Grothendieck–Teichmuller Lie algebra $$\mathfrak {{grt}}_1$$ . The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by $$\mathfrak {{grt}}_1$$ , up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) $$E_n$$ operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for $$\mathfrak {{grt}}_1$$ -elements implies the hexagon equation.

The Tate conjecture for K3 surfaces in odd characteristic

Abstract: We show that the classical Kuga–Satake construction gives rise, away from characteristic $$2$$ , to an open immersion from the moduli of primitively polarized K3 surfaces (of any fixed degree) to a certain regular integral model for a Shimura variety of orthogonal type. This allows us to attach to every polarized K3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety. In turn, this reduces the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic to a version of the Tate conjecture for certain endomorphisms on the attached Kuga–Satake abelian variety, which we prove. As a by-product of our methods, we also show that the moduli stack of primitively polarized K3 surfaces of degree $$2d$$ is quasi-projective and, when $$d$$ is not divisible by $$p^2$$ , is geometrically irreducible in characteristic $$p$$ . We indicate how the same method applies to prove the Tate conjecture for co-dimension $$2$$ cycles on cubic fourfolds.

Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces

Abstract: For the $$d$$ -dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $$H^s(\mathbb R^d)$$ , $$s>s_c:=d/2+1$$ . The borderline case $$s=s_c$$ was a folklore open problem. In this paper we consider the physical dimension $$d=2$$ and show that if we perturb any given smooth initial data in $$H^{s_c}$$ norm, then the corresponding solution can have infinite $$H^{s_c}$$ norm instantaneously at $$t>0$$ . In a companion paper [1] we settle the 3D and more general cases. The constructed solutions are unique and even $$C^{\infty }$$ -smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems.

The bounded l 2 curvature conjecture

Abstract: This is the main paper in a sequence in which we give a complete proof of the bounded $$L^2$$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $$L^2$$ -norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, $$L^2$$ bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than $$1+1$$ (based on Strichartz estimates) were obtained in Bahouri and Chemin (Am J Math 121:1337–1777, 1999; IMRN 21:1141–1178, 1999), Tataru (Am J Math 122:349–376, 2000; JAMS 15(2):419–442, 2002), Klainerman and Rodnianski (Duke Math J 117(1):1–124, 2003) and optimized in Klainerman and Rodnianski (Ann Math 161:1143–1193, 2005), Smith and Tataru (Ann Math 162:291–366, 2005), the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. To achieve our goals we recast the Einstein vacuum equations as a quasilinear $$so(3,1)$$ -valued Yang–Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including $$L^2$$ error bounds which is carried out in Szeftel (Parametrix for wave equations on a rough background I: regularity of the phase at initial time, arXiv:1204.1768 , 2012; Parametrix for wave equations on a rough background II: construction of the parametrix and control at initial time, arXiv:1204.1769 , 2012; Parametrix for wave equations on a rough background III: space-time regularity of the phase, arXiv:1204.1770 , 2012; Parametrix for wave equations on a rough background IV: control of the error term, arXiv:1204.1771 , 2012), as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in Szeftel (Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112 , 2013). It is at this level that our problem is critical. Indeed, any known notion of a parametrix relies in an essential way on the eikonal equation, and our space-time possesses, barely, the minimal regularity needed to make sense of its solutions.

Integrable systems, toric degenerations and Okounkov bodies

Abstract: Let \(X\) be a smooth projective variety of dimension \(n\) over \(\mathbb {C}\) equipped with a very ample Hermitian line bundle \(\mathcal {L}\). In the first part of the paper, we show that if there exists a toric degeneration of \(X\) satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from \(X\) to the special fiber \(X_0\) which is a symplectomorphism on an open dense subset \(U\). From this we are then able to construct a completely integrable system on \(X\) in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions \(\{H_1, \ldots , H_n\}\) on \(X\) which are continuous on all of \(X\), smooth on an open dense subset \(U\) of \(X\), and pairwise Poisson-commute on \(U\). Moreover, our integrable system in fact generates a Hamiltonian torus action on \(U\). In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the ‘moment map’ \(\mu = (H_1, \ldots , H_n): X \rightarrow \mathbb {R}^n\) is precisely the Newton-Okounkov body \(\Delta = \Delta (R, v)\) associated to the homogeneous coordinate ring \(R\) of \(X\), and an appropriate choice of a valuation \(v\) on \(R\). Our main technical tools come from algebraic geometry, differential (Kahler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Łojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties \(X\), this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

Expansion of random graphs: new proofs, new results

Abstract: We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let $$\Gamma $$ be a random $$d$$ -regular graph on $$n$$ vertices, and let $$\lambda $$ be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon (Combinatorica 6(2), 83–96, 1986) that a random $$d$$ -regular graph is “almost Ramanujan”, in the following sense: for every $$\varepsilon >0$$ , $$\lambda <2\sqrt{d-1}+\varepsilon $$ asymptotically almost surely. Friedman famously presented a proof of this conjecture in Friedman (Memoirs of the AMS 910, 2008). Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random $$d$$ -regular graph satisfies $$\lambda <2\sqrt{d-1}+1$$ a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For $$d$$ even, a $$d$$ -regular graph on $$n$$ vertices is an $$n$$ -covering space of a bouquet of $$d/2$$ loops. More generally, fixing an arbitrary base graph $$\Omega $$ , we study the spectrum of $$\Gamma $$ , a random $$n$$ -covering of $$\Omega $$ . Let $$\lambda $$ be the largest absolute value of a non-trivial eigenvalue of $$\Gamma $$ . Extending Alon’s conjecture to this more general model, Friedman (Duke Math J 118(1),19–35, 2003) conjectured that for every $$\varepsilon >0,$$ a.a.s. $$\lambda <\rho +\varepsilon $$ , where $$\rho $$ is the spectral radius of the universal cover of $$\Omega $$ . When $$\Omega $$ is regular we get a bound of $$\rho +0.84$$ , and for an arbitrary $$\Omega $$ , we prove a nearly optimal upper bound of $$\sqrt{3}\rho $$ . This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

Kähler currents and null loci

Abstract: We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein–Lazarsfeld–Mustaţa–Nakamaye–Popa. As an application, we show that finite time non-collapsing singularities of the Kahler–Ricci flow on compact Kahler manifolds always form along analytic subvarieties, thus answering a question of Feldman–Ilmanen–Knopf and Campana. We also extend the second author’s results about noncollapsing degenerations of Ricci-flat Kahler metrics on Calabi–Yau manifolds to the nonalgebraic case.

Homoclinic groups, IE groups, and expansive algebraic actions

Abstract: We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of $$p$$ -expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is $$1$$ -expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on $$X$$ , it is shown that the entropy of the action is equal to the entropy of the induced action on the Pontryagin dual of the homoclinic group, the homoclinic group is a dense subgroup of the IE group, the homoclinic group is nontrivial exactly when the action has positive entropy, and the homoclinic group is dense in $$X$$ exactly when the action has completely positive entropy.

Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space

Abstract: We prove Homological Mirror Symmetry for a smooth $$d$$ -dimensional Calabi–Yau hypersurface in projective space, for any $$d \ge 3$$ (for example, $$d=3$$ is the quintic threefold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the ‘ $$d$$ -dimensional pair of pants’; the introduction of the ‘relative Fukaya category’, and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an ‘orbifold’ Fukaya category); a Morse–Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.

Geometry of canonical bases and mirror symmetry

Abstract: A decorated surface S is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group over Q. A pair (G, S) gives rise to a moduli spaceAG,S , closely related to the moduli space of G-local systems on S. It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). So a set AG,S(Z ) of its integral tropical points is defined. We introduce a rational positive function W on the space AG,S , called the potential. Its tropicalisation is a function W t : AG,S(Z ) → Z. The condition W t ≥ 0 defines a subset of positive integral tropical points AG,S(Zt ). For G = SL2, we recover the set of positive integralA-laminations on S from Fock and Goncharov (Publ Math IHES 103:1–212, 2006). We prove that when S is a disc with n special points on the boundary, the set AG,S(Zt ) parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Asterisque 101–102:208– 229, 1983; Ginzburg,1995; Mirkovic and Vilonen, Ann Math (2) 166(1):95– 143, 2007; Beilinson and Drinfeld, Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51, 2004) they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group GL : (Vλ1 ⊗ . . .⊗ Vλn )G L . (1) A. Goncharov (B) · L. Shen Mathematics Department, Yale University, New Haven, CT 06520, USA e-mail: alexander.goncharov@yale.edu

A two-sided estimate for the Gaussian noise stability deficit

Abstract: The Gaussian noise-stability of a set $$A \subset {\mathbb R}^n$$ is defined by $$ \begin{aligned} {\mathcal {S}}_\rho (A) = {\mathbb P}\left( X \in A ~ \& ~ Y \in A \right) \end{aligned}$$ where $$X,Y$$ are standard jointly Gaussian vectors satisfying $${\mathbb E}[X_i Y_j] = \delta _{ij} \rho $$ . Borell’s inequality states that for all $$0 < \rho < 1$$ , among all sets $$A \subset {\mathbb R}^n$$ with a given Gaussian measure, the quantity $${\mathcal {S}}_\rho (A)$$ is maximized when $$A$$ is a half-space. We give a novel short proof of this fact, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set $$A$$ and its corresponding half-space $$H$$ (namely the distance between the two centroids), we show that the deficit $${\mathcal {S}}_\rho (H) - {\mathcal {S}}_\rho (A)$$ can be controlled from both below and above by essentially the same function of the distance, up to logarithmic factors. As a consequence, we also establish the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variation distance as a metric. In the limit $$\rho \rightarrow 1$$ , we obtain an improved dimension-free robustness bound for the Gaussian isoperimetric inequality. Our estimates are also valid for a generalized version of stability where more than two correlated vectors are considered.

Witten’s top Chern class via cosection localization

Abstract: For a Landau–Ginzburg space \(([\mathbb {C}^n/G],W)\), we construct Witten’s top Chern class as an algebraic cycle using cosection localized virtual cycles in the case where all sectors are narrow, verify all axioms of this class, and derive an explicit formula for it in the free case. We prove that this construction is equivalent to the constructions of Polishchuk–Vaintrob, Chiodo, and Fan–Jarvis–Ruan.

Moments and distribution of central $$L$$ L -values of quadratic twists of elliptic curves

Abstract: We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real) moments and a one-sided central limit theorem holds. We illustrate our method for the family of quadratic twists of an elliptic curve, obtaining sharp upper bounds for all moments below the first. We also establish a one sided central limit theorem supporting a conjecture of Keating and Snaith. Our work leads to a conjecture on the distribution of the order of the Tate-Shafarevich group for rank zero quadratic twists of an elliptic curve, and establishes the upper bound part of this conjecture (assuming the Birch-Swinnerton-Dyer conjecture).

Effectiveness of Demailly’s strong openness conjecture and related problems

Abstract: In this article, stimulated by the effectiveness in Berndtsson’s solution of the openness conjecture and continuing our solution of Demailly’s strong openness conjecture, we discuss conditions to guarantee the effectiveness of the conjecture and establish such an effectiveness result. We explicitly point out a lower semicontinuity property of plurisubharmonic functions with a multiplier, which is implicitly contained in Guan and Zhou (2014, arXiv:1401.7158). We also obtain optimal effectiveness of the conjectures of Demailly-Kollar and Jonsson-Mustata respectively.

Decay of correlations for normally hyperbolic trapping

Abstract: We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows, classical flows in molecular dynamics, and null geodesic flows for black holes metrics. The decay of correlations is a consequence of the existence of resonance free strips for Green’s functions (cut-off resolvents) and polynomial bounds on the growth of those functions in the semiclassical parameter.

Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

Abstract: In this paper, we associate an invariant $$\alpha _{x}(L)$$ to an algebraic point $$x$$ on an algebraic variety $$X$$ with an ample line bundle $$L$$ . The invariant $$\alpha $$ measures how well $$x$$ can be approximated by rational points on $$X$$ , with respect to the height function associated to $$L$$ . We show that this invariant is closely related to the Seshadri constant $$\epsilon _{x}(L)$$ measuring local positivity of $$L$$ at $$x$$ , and in particular that Roth’s theorem on $$\mathbb {P}^1$$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

On hyperboundedness and spectrum of Markov operators

Abstract: Consider an ergodic Markov operator \(M\) reversible with respect to a probability measure \(\mu \) on a general measurable space. It is shown that if \(M\) is bounded from \(\mathbb {L}^2(\mu )\) to \(\mathbb {L}^p(\mu )\), where \(p>2\), then it admits a spectral gap. This result answers positively a conjecture raised by Hoegh-Krohn and Simon (J. Funct. Anal. 9:121–80, 1972) in the more restricted semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee et al. (Proceedings of the 2012 ACM Symposium on Theory of Computing, 1131–1140, ACM, New York, 2012). It provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general. In addition, the usual Cheeger inequality is extended to the higher eigenvalues in the compact Riemannian setting and the exponential behaviors of the small eigenvalues of Witten Laplacians at small temperature are recovered.

The asymptotics of ECH capacities

Abstract: In a previous paper, the second author used embedded contact homology (ECH) of contact three-manifolds to define “ECH capacities” of four-dimensional symplectic manifolds. In the present paper we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This follows from a more general theorem relating the volume of a contact three-manifold to the asymptotics of the amount of symplectic action needed to represent certain classes in ECH. The latter theorem was used by the first and second authors to show that every contact form on a closed three-manifold has at least two embedded Reeb orbits.

Stability in the homology of congruence subgroups

Abstract: The homology groups of many natural sequences of groups $$\{G_n\}_{n=1}^{\infty }$$ (e.g. general linear groups, mapping class groups, etc.) stabilize as $$n \rightarrow \infty $$ . Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense as representations. They called this phenomena representation stability. We prove that the homology groups of congruence subgroups of $${{\mathrm{GL}}}_n(R)$$ (for almost any reasonable ring $$R$$ ) satisfy a strong version of representation stability that we call central stability. The definition of central stability is very different from Church-Farb’s definition of representation stability (it is defined via a universal property), but we prove that it implies representation stability. Our main tool is a new machine for proving central stability that is analogous to the classical homological stability machine.

Epipelagic L-packets and rectifying characters

Abstract: We provide an explicit construction of the local Langlands correspondence for general tamely-ramified reductive p-adic groups and a class of wildly ramified Langlands parameters. Furthermore, we verify that our construction satisfies many expected properties of such a correspondence. More precisely, we show that each $$L$$ -packet we construct admits a parameterization in terms of the Langlands dual group, contains a unique generic element for a fixed Whittaker datum, satisfies the formal degree conjecture, is compatible with central and cocentral characters, provides a stable virtual character, and satisfies the expected endoscopic character identities. Moreover, we show that in the case of $$\mathrm{{GL}}_n$$ , our construction coincides with the established local Langlands correspondence. Our techniques provide a general approach to the construction of the local Langlands correspondence for tamely-ramified groups and regular supercuspidal parameters.

Bogomolov's inequality for Higgs sheaves in positive characteristic

Abstract: We prove Bogomolov’s inequality for Higgs sheaves on varieties in positive characteristic $$p$$ that can be lifted modulo $$p^2$$ . This implies the Miyaoka–Yau inequality on surfaces of non-negative Kodaira dimension liftable modulo $$p^2$$ . This result is a strong version of Shepherd-Barron’s conjecture. Our inequality also gives the first algebraic proof of Bogomolov’s inequality for Higgs sheaves in characteristic zero, solving the problem posed by Narasimhan.

Hyperbolicity of generic high-degree hypersurfaces in complex projective space

Abstract: We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet differentials on the complex projective space to a hypersurface. The other is slanted vector fields of low vertical pole order on the vertical jet space of the universal hypersurface. We also present a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, (ii) nonexistence of nontrivial sets of entire functions satisfying certain polynomial equations with slowly varying coefficients, and (iii) Second Main Theorems for jet differentials and slowly moving targets.

Universal Gauss-Thakur sums and L-series ∗†

Abstract: We prove that the maximal abelian extension tamely ramified at infinity of the rational function field over \(\mathbb {F}_q\) is generated by the values at the points in the algebraic closure of \(\mathbb {F}_q\) of the higher derivatives of the so-called Anderson and Thakur function \(\omega .\) We deduce a similar property for the special values of the higher derivatives of a new kind of \(L\)-series introduced by the second author.

Countable abelian group actions and hyperfinite equivalence relations

Abstract: An equivalence relation E on a standard Borel space is hyperfinite if E is the increasing union of countably many Borel equivalence relations $$E_n$$ where all $$E_n$$ -equivalence classs are finite. In this article we establish the following theorem: if a countable abelian group acts on a standard Borel space in a Borel manner then the orbit equivalence relation is hyperfinite. The proof uses constructions and analysis of Borel marker sets and regions in the space $$2^{{\mathbb {Z}}^{<\omega }}.$$ This technique is also applied to a problem of finding Borel chromatic numbers for invariant Borel subspaces of $$2^{{\mathbb {Z}}^n}$$ .