Showing papers in "Inventiones Mathematicae in 2018"

Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures

Abstract: We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger–Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger–Kakutani and Kantorovich–Wasserstein distances.

Quantitative estimates of propagation of chaos for stochastic systems with $$W^{-1,\infty }$$ W - 1 , ∞ kernels

Abstract: We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. We have to develop for this new laws of large numbers at the exponential scale. But our result only requires very weak regularity on the interaction kernel in the negative Sobolev space $$\dot{W}^{-1,\infty }$$ , thus including the Biot–Savart law and the point vortices dynamics for the 2d incompressible Navier–Stokes.

Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations

Abstract: We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are:

Time quasi-periodic gravity water waves in finite depth

Abstract: We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions—namely periodic and even in the space variable x—of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations—the highest order x-derivative appears in the nonlinear term but not in the linearization at the origin—and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash–Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.

Modularity lifting beyond the Taylor–Wiles method

Abstract: We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions—one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on the automorphic side, applies to automorphic forms on the group $$\mathrm {GL}(n)$$ over a general number field; it is contingent on a conjecture which, in particular, predicts the existence of Galois representations associated to torsion classes in the cohomology of the associated locally symmetric space. We show that if this conjecture holds, then our main theorem implies the following: if E is an elliptic curve over an arbitrary number field, then E is potentially automorphic and satisfies the Sato–Tate conjecture. In addition, we also prove some unconditional results. For example, in the setting of $$\mathrm {GL}(2)$$ over $$\mathbf {Q}$$ , we identify certain minimal global deformation rings with the Hecke algebras acting on spaces of p-adic Katz modular forms of weight 1. Such algebras may well contain p-torsion. Moreover, we also completely solve the problem (for p odd) of determining the multiplicity of an irreducible modular representation $$\overline{\rho }$$ in the Jacobian $$J_1(N)$$ , where N is the minimal level such that $$\overline{\rho }$$ arises in weight two.

Canonical bases arising from quantum symmetric pairs

Abstract: We develop a general theory of canonical bases for quantum symmetric pairs $$({\mathbf{U}}, {\mathbf{U}}^\imath )$$ with parameters of arbitrary finite type. We construct new canonical bases for the finite-dimensional simple $${\mathbf{U}}$$ -modules and their tensor products regarded as $${\mathbf{U}}^\imath $$ -modules. We also construct a canonical basis for the modified form of the $$\imath $$ quantum group $${\mathbf{U}}^\imath $$ . To that end, we establish several new structural results on quantum symmetric pairs, such as bilinear forms, braid group actions, integral forms, Levi subalgebras (of real rank one), and integrality of the intertwiners.

Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity

Abstract: We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple plane wave solutions, starting from smooth initial data, is stable under perturbations of the initial data that break the plane symmetry. Moreover, we provide a sharp asymptotic description of the singularity formation. The new feature of our work is that the perturbed solutions are allowed to have small but non-zero vorticity, even at the location of the shock. Thus, our results provide the first constructive description of the vorticity near a singularity formed from compression. Specifically, the vorticity remains uniformly bounded, while the vorticity divided by the density exhibits even more regular behavior: the ratio remains uniformly Lipschitz relative to the standard Cartesian coordinates. To control the vorticity, we rely on a coalition of new geometric and analytic insights that complement the ones used by Christodoulou in his groundbreaking, sharp proof of shock formation in vorticity-free regions. In particular, we rely on a new formulation of the compressible Euler equations (derived in a companion article) exhibiting remarkable structures. To derive estimates, we construct an eikonal function adapted to the acoustic characteristics (which correspond to sound wave propagation) and a related set of geometric coordinates and differential operators. Thanks to the remarkable structure of the equations, the same set of coordinates and differential operators can be used to analyze the vorticity, whose characteristics are transversal to the acoustic characteristics. In particular, our work provides the first constructive description of shock formation without symmetry assumptions in a system with multiple speeds.

On the direct summand conjecture and its derived variant

Abstract: Andre recently gave a beautiful proof of Hochster’s direct summand conjecture in commutative algebra using perfectoid spaces; his two main results are a generalization of the almost purity theorem (the perfectoid Abhyankar lemma) and a construction of certain faithfully flat extensions of perfectoid algebras where “discriminants” acquire all p-power roots. In this paper, we explain a quicker proof of Hochster’s conjecture that circumvents the perfectoid Abhyankar lemma; instead, we prove and use a quantitative form of Scholze’s Hebbarkeitssatz (the Riemann extension theorem) for perfectoid spaces. The same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de Jong.

Flops and clusters in the homological minimal model programme

Abstract: Suppose that $$f:X\rightarrow \mathrm{Spec}\,R$$ is a minimal model of a complete local Gorenstein 3-fold, where the fibres of f are at most one dimensional, so by Van den Bergh (Duke Math J 122(3):423–455, 2004) there is a noncommutative ring $$\Lambda $$ derived equivalent to X. For any collection of curves above the origin, we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of $$\Lambda $$ is finite dimensional, improving a result of Donovan and Wemyss (Contractions and deformations, arXiv:1511.00406 ). We further show that the mutation functor of Iyama and Wemyss (Invent Math 197(3):521–586, 2014, §6) is functorially isomorphic to the inverse of the Bridgeland–Chen flop functor in the case when the factor of $$\Lambda $$ is finite dimensional. These results then allow us to jump between all the minimal models of $$\mathrm{Spec}\,R$$ in an algorithmic way, without having to compute the geometry at each stage. We call this process the Homological MMP. This has several applications in GIT approaches to derived categories, and also to birational geometry. First, using mutation we are able to compute the full GIT chamber structure by passing to surfaces. We say precisely which chambers give the distinct minimal models, and also say which walls give flops and which do not, enabling us to prove the Craw–Ishii conjecture in this setting. Second, we are able to precisely count the number of minimal models, and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity. Third, we prove a bijective correspondence between maximal modifying R-module generators and minimal models, and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry. This lifts the Auslander–McKay correspondence to dimension three.

A decomposition theorem for singular spaces with trivial canonical class of dimension at most five

Abstract: In this paper we partly extend the Beauville–Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, etale in codimension one, that decomposes as a product of an Abelian variety, and singular analogues of irreducible Calabi–Yau and irreducible holomorphic symplectic varieties.

Algebraic K-theory and descent for blow-ups

Abstract: We prove that algebraic K-theory satisfies ‘pro-descent’ for abstract blow-up squares of noetherian schemes. As an application we derive Weibel’s conjecture on the vanishing of negative K-groups.

Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras

Abstract: Let J be a set of pairs consisting of good $$U'_q(\mathfrak g)$$ -modules and invertible elements in the base field $$\mathbb C(q)$$ . The distribution of poles of normalized R-matrices yields Khovanov–Lauda–Rouquier algebras $$R^J(\beta )$$ for each $$\beta \in \mathsf {Q}^+$$ . We define a functor $$\mathcal F_\beta $$ from the category of graded $$R^J(\beta )$$ -modules to the category of $$U'_q(\mathfrak g)$$ -modules. The functor $$\mathcal F= \bigoplus _{\beta \in \mathsf {Q}^+} \mathcal {F}_\beta $$ sends convolution products of finite-dimensional graded $$R^J(\beta )$$ -modules to tensor products of finite-dimensional $$U'_q(\mathfrak g)$$ -modules. It is exact if $$R^J$$ is of finite type A, D, E. If $$V(\varpi _1)$$ is the fundamental representation of $$U_q'({\widehat{\mathfrak {sl}}_N})$$ of weight $$\varpi _1$$ and $$J=\left\{ \bigl (V(\varpi _1), q^{2i} \bigr ) \mid i \in \mathbb Z \right\} $$ , then $$R^J$$ is the Khovanov–Lauda–Rouquier algebra of type $$A_{\infty }$$ . The corresponding functor $$\mathcal {F}$$ sends a finite-dimensional graded $$R^J$$ -module to a module in $$\mathcal {C}_J$$ , where $$\mathcal {C}_J$$ is the category of finite-dimensional integrable $$U_q'({\widehat{\mathfrak {sl}}_N})$$ -modules M such that every composition factor of M appears as a composition factor of a tensor product of modules of the form $$V(\varpi _1)_{q^{2s}}$$ $$(s \in {\mathbb {Z}})$$ . Focusing on this case, we obtain an abelian rigid graded tensor category $${\mathcal T}_J$$ by localizing the category of finite-dimensional graded $$R^J$$ -modules. The functor $$\mathcal {F}$$ factors through $${\mathcal T}_J$$ . Moreover, the Grothendieck ring of the category $$\mathcal {C}_J$$ is isomorphic to the Grothendieck ring of $${\mathcal T}_J$$ at $$q=1$$ .

The dimension-free structure of nonhomogeneous random matrices

Abstract: Let X be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$\begin{aligned} \mathbf {E}\Vert X\Vert _{S_p} \asymp \mathbf {E}\Bigg [ \Bigg (\sum _i\Bigg (\sum _j X_{ij}^2\Bigg )^{p/2}\Bigg )^{1/p} \Bigg ] \end{aligned}$$ for any $$2\le p\le \infty $$ , where $$S_p$$ denotes the p-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case $$p=\infty $$ , a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on $$\ell _2$$ . Along the way, we obtain optimal dimension-free bounds on the moments $$(\mathbf {E}\Vert X\Vert _{S_p}^p)^{1/p}$$ that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.

Exponential Decay of Correlations for Finite Horizon Sinai Billiard Flows

Abstract: We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semi-group $$\mathcal {L}_t$$ of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions.

Equiangular lines and spherical codes in Euclidean space

Abstract: A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $$\mathbb {R}^n$$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $$\theta $$ and sufficiently large $$n$$ there are at most $$2n-2$$ lines in $$\mathbb {R}^n$$ with common angle $$\theta $$ . Moreover, this bound is achieved if and only if $$\theta = \arccos \frac{1}{3}$$ . Indeed, we show that for all $$\theta e \arccos {\frac{1}{3}}$$ and and sufficiently large n, the number of equiangular lines is at most 1.93n. We also show that for any set of k fixed angles, one can find at most $$O(n^k)$$ lines in $$\mathbb {R}^n$$ having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.

Sharp systolic inequalities for Reeb flows on the three-sphere

Abstract: The systolic ratio of a contact form $$\alpha $$ on the three-sphere is the quantity $$\begin{aligned} \rho _{\mathrm {sys}}(\alpha ) = \frac{T_{\min }(\alpha )^2}{\mathrm {vol}(S^3,\alpha \wedge d\alpha )}, \end{aligned}$$ where $$T_{\min }(\alpha )$$ is the minimal period of closed Reeb orbits on $$(S^3,\alpha )$$ . A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that $$\rho _{\mathrm {sys}}\le 1$$ in a neighbourhood of the space of Zoll contact forms on $$S^3$$ , with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a local middle-dimensional non-squeezing theorem, and a sharp systolic inequality for Finsler metrics on the two-sphere which are close to Zoll ones. Our second main result is that $$\rho _{\mathrm {sys}}$$ is unbounded from above on the space of tight contact forms on $$S^3$$ .

Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers

Abstract: Using perfectoid algebras we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $$p > 0$$ , in the case of a regular ambient ring. We prove several properties about this ideal such as subadditivity. We then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings. The analogous result was shown in equal characteristic by Ein–Lazarsfeld–Smith and Hochster–Huneke.

Gopakumar–Vafa invariants via vanishing cycles

Abstract: In this paper, we propose an ansatz for defining Gopakumar–Vafa invariants of Calabi–Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem–Li, which is itself based on earlier ideas of Hosono–Saito–Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov–Witten theory and Pandharipande–Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem–Li conjectures, where our invariants match the predicted answer. Finally, we give examples where our invariant matches the expected answer in cases where the cycle is non-reduced, non-planar, or non-primitive.

Singular foliations with trivial canonical class

Abstract: This paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on complex projective manifolds.

Kirwan surjectivity for quiver varieties

Abstract: For algebraic varieties defined by hyperkahler or, more generally, algebraic symplectic reduction, it is a long-standing question whether the “hyperkahler Kirwan map” on cohomology is surjective. We resolve this question in the affirmative for Nakajima quiver varieties. We also establish similar results for other cohomology theories and for the derived category. Our proofs use only classical topological and geometric arguments.

Holomorphic anomaly equations and the Igusa cusp form conjecture

Abstract: Let S be a K3 surface and let E be an elliptic curve. We solve the reduced Gromov–Witten theory of the Calabi–Yau threefold $$S \times E$$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov–Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov–Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov–Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for elliptic Calabi–Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov–Witten theory of K3 surfaces in primitive classes.

On the nature of the generating series of walks in the quarter plane

Abstract: In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, i.e., we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients.

Punctured holomorphic curves and Lagrangian embeddings

Abstract: We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin’s conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold’s chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions.

Entropy and drift in word hyperbolic groups

Abstract: The fundamental inequality of Guivarc’h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary hyperbolic group which is not virtually free, endowed with a word distance, the fundamental inequality is strict for symmetric measures with finite support, uniformly for measures with a given support. This answers a conjecture of S. Lalley. For admissible measures, this is proved using previous results of Ancona and Blachere–Haissinsky–Mathieu. For non-admissible measures, this follows from a counting result, interesting in its own right: we show that, in any infinite index subgroup, the number of non-distorted points is exponentially small compared to the growth of balls in the whole group. The uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup.

Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions

Abstract: We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as $$ \hat{g}(\xi )= \prod _{j=1}^n (1+2\pi i\delta _j\xi )^{-1} \, e^{-c \xi ^2}$$ for $$\delta _1,\ldots ,\delta _n\in \mathbb {R}, c >0$$ (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density $$>1$$ is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice $$\alpha \mathbb {Z}\times \beta \mathbb {Z}$$ forms a frame, if and only if $$\alpha \beta <1$$ . This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows

Abstract: We prove the weight part of Serre’s conjecture in generic situations for forms of U(3) which are compact at infinity and split at places dividing p as conjectured by Herzig (Duke Math J 149(1):37–116, 2009). We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge–Tate weights (2, 1, 0) for $$K/\mathbb {Q}_p$$ unramified combined with patching techniques. Our results show that the (geometric) Breuil–Mezard conjectures hold for these deformation rings.

Expanding large global solutions of the equations of compressible fluid mechanics

Abstract: Without any symmetry assumptions on the initial data we construct global-in-time unique solutions to the vacuum free boundary three-dimensional isentropic compressible Euler equations when the adiabatic exponent $$\gamma $$ lies in the interval $$(1,\frac{5}{3}]$$ . Our initial data lie sufficiently close to the expanding compactly supported affine motions recently constructed by Sideris and they satisfy the physical vacuum boundary condition.

Two-bubble dynamics for threshold solutions to the wave maps equation

Abstract: We consider the energy-critical wave maps equation $$\mathbb {R}^{1+2} \rightarrow \mathbb {S}^2$$ in the equivariant case, with equivariance degree $$k \ge 2$$ . It is known that initial data of energy $$< 8\pi k$$ and topological degree zero leads to global solutions that scatter in both time directions. We consider the threshold case of energy $$8 \pi k $$ . We prove that the solution is defined for all time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps. The proof combines the classical concentration-compactness techniques of Kenig–Merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation.

Complete intersection varieties with ample cotangent bundles

Abstract: Any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles, of each dimension up to half its own dimension.

Forcing theory for transverse trajectories of surface homeomorphisms

Abstract: This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel’s classification of zero entropy maps of $$\mathbb {S}^2$$ for non-wandering homeomorphisms; we show that if f is a Hamiltonian homeomorphism of the annulus, then the rotation set of f is either a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms, proving a first case of the Franks–Misiurewicz conjecture; we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms.