Showing papers in "Duke Mathematical Journal in 2019"

On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties

Abstract: In this paper we investigate the geometry of the orbit space of the closure of the subscheme parameterizing smooth Kahler–Einstein Fano manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski-open condition, and we establish the uniqueness of the Gromov–Hausdorff limit for a punctured flat family of Kahler–Einstein Fano manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of Q-Gorenstein smoothable, K-semistable Q-Fano varieties, and we verify various necessary properties to guarantee that it is a good moduli space.

The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures

Abstract: Let g0,…,gk:N→D be 1-bounded multiplicative functions, and let h0,…,hk∈Z be shifts. We consider correlation sequences f:N→Z of the form f(a):=m→∞1logωm∑xm/ωm≤n≤xmg0(n+ah0)⋯gk(n+ahk)n, where 1≤ωm≤xm are numbers going to infinity as m→∞ and is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences f are the uniform limit of periodic sequences fi. Furthermore, if the multiplicative function g0⋯gk “weakly pretends” to be a Dirichlet character χ, the periodic functions fi can be chosen to be χ-isotypic in the sense that fi(ab)=fi(a)χ(b) whenever b is coprime to the periods of fi and χ, while if g0⋯gk does not weakly pretend to be any Dirichlet character, then f must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three and of the Mobius function of length up to four.

Stability and invariant random subgroups

Abstract: Consider Sym(n) endowed with the normalized Hamming metric dn. A finitely generated group Γ is P-stable if every almost homomorphism ρnk:Γ→Sym(nk) (i.e., for every g,h∈Γ, lim k→∞dnk(ρnk(gh),ρnk(g)ρnk(h))=0) is close to an actual homomorphism φnk:Γ→Sym(nk). Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and Paunescu showed the same for abelian groups and raised many questions, especially about the P-stability of amenable groups. We develop P-stability in general and, in particular, for amenable groups. Our main tool is the theory of invariant random subgroups, which enables us to give a characterization of P-stability among amenable groups and to deduce the stability and instability of various families of amenable groups.

K-stability of cubic threefolds

Abstract: We prove that the K-moduli space of cubic threefolds is identical to their geometric invariant theory (GIT) moduli. More precisely, the K-semistability, K-polystability, and K-stability of cubic threefolds coincide with the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kahler–Einstein (KE) metrics and provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension 3 of the volumes of Kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of 3-dimensional canonical and terminal singularities, which was established during the study of the explicit 3-dimensional minimal model program.

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Abstract: In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton–Okounkov bodies for Grassmannians. We consider the Grassmannian X=Grn−k(Cn), as well as the mirror dual Landau–Ginzburg model (Xˇ∘,W:Xˇ∘→C), where Xˇ∘ is the complement of a particular anticanonical divisor in a Langlands dual Grassmannian Xˇ=Grk((Cn)∗) and the superpotential W has a simple expression in terms of Plucker coordinates. Grassmannians simultaneously have the structure of an A-cluster variety and an X-cluster variety; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps. Given a plabic graph or, more generally, a cluster seed G, we consider two associated coordinate systems: a network or X-cluster chart ΦG:(C∗)k(n−k)→X∘ and a Plucker cluster or A-cluster chart ΦG∨:(C∗)k(n−k)→Xˇ∘. Here X∘ and Xˇ∘ are the open positroid varieties in X and Xˇ, respectively. To each X-cluster chart ΦG and ample boundary divisor D in X∖X∘, we associate a Newton–Okounkov body ΔG(D) in Rk(n−k), which is defined as the convex hull of rational points; these points are obtained from the multidegrees of leading terms of the Laurent polynomials ΦG∗(f) for f on X with poles bounded by some multiple of D. On the other hand, using the A-cluster chart ΦG∨ on the mirror side, we obtain a set of rational polytopes—described in terms of inequalities—by writing the superpotential W as a Laurent polynomial in the A-cluster coordinates and then tropicalizing. Our first main result is that the Newton–Okounkov bodies ΔG(D) and the polytopes obtained by tropicalization on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of ) these Newton–Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton–Okounkov bodies, in the case in which the cluster seed G corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians.

Cluster varieties from Legendrian knots

Abstract: Author(s): Shende, V; Treumann, D; Williams, H; Zaslow, E | Abstract: Many interesting spaces-including all positroid strata and wild character varieties- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

The class of Eisenbud–Khimshiashvili–Levine is the local A1-Brouwer degree

Abstract: Given a polynomial function with an isolated zero at the origin, we prove that the local A1-Brouwer degree equals the Eisenbud–Khimshiashvili–Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes, together with associated arithmetic information, by enriching Milnor’s equality between the local degree of the gradient and the number of nodes into which a hypersurface singularity bifurcates to an equality in the Grothendieck–Witt group.

Fourier transform on high-dimensional unitary groups with applications to random tilings

Abstract: A combination of direct and inverse Fourier transforms on the unitary group U(N) identifies normalized characters with probability measures on N-tuples of integers. We develop the N→∞ version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with the law of large numbers and the central limit theorem for global behavior of corresponding random N-tuples. As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon–Okounkov conjecture (which predicts asymptotic Gaussianity and the exact form of the covariance) for a family of non-simply-connected polygons. Another application is a central limit theorem for the U(N) quantum random walk with random initial data.

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Abstract: We consider the problem of constructing solutions to the fractional Yamabe problem which are singular at a given smooth submanifold, for which we establish the classical gluing method of Mazzeo and Pacard (J. Differential Geom., 1996) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional-order ordinary differential equation (ODE). Thus, our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, we first provide a rigorous construction of radial fast-decaying solutions by a blowup argument and a bifurcation method. Then, second, we use conformal geometry to rewrite this nonlocal ODE, giving a hint of what a nonlocal phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrodinger equation with a Hardy-type critical potential. We construct its Green’s function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a 2-dimensional kernel as in the second-order case.

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

Abstract: We consider an n×n linear system of ODEs with an irregular singularity of Poincare rank 1 at z=∞, holomorphically depending on parameter t within a polydisk in Cn centered at t=0, such that the eigenvalues of the leading matrix at z=∞ coalesce along a locus Δ contained in the polydisk, passing through t=0. Namely, z=∞ is a resonant irregular singularity for t∈Δ. We analyze the case when the leading matrix remains diagonalizable at Δ. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon, and monodromy data as t varies in the polydisk, and their limits for t tending to points of Δ. When the system also has a Fuchsian singularity at z=0, we show, under minimal vanishing conditions on the residue matrix at z=0, that isomonodromic deformations can be extended to the whole polydisk (including Δ) in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at the fixed coalescence point t=0. Conversely, when the system is isomonodromic in a small domain not intersecting Δ inside the polydisk, we give certain vanishing conditions on some entries of the Stokes matrices, ensuring that Δ is not a branching locus for the t-continuation of fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius manifolds is explained. An application to Painleve equations is discussed.

Legendrian fronts for affine varieties

Abstract: In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First, we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several new applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed, exact Lagrangian submanifolds. In particular, we prove that the Koras–Russell cubic is Stein deformation-equivalent to C3, and we verify the affine parts of the algebraic mirrors of two Weinstein 4-folds.

Minimal 2-spheres in 3-spheres

Abstract: We prove that any manifold diffeomorphic to S3 and endowed with a generic metric contains at least two embedded minimal 2-spheres. The existence of at least one minimal 2-sphere was obtained by Simon and Smith in 1983. Our approach combines ideas from min–max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in 3-manifolds. We apply our methods to solve a problem posed by S. T. Yau in 1987 on whether the planar 2-spheres are the only minimal spheres in ellipsoids centered about the origin in R4. Finally, considering the example of degenerating ellipsoids, we show that the assumptions in the multiplicity 1 conjecture and the equidistribution of widths conjecture are in a certain sense sharp.

Purity for the Brauer group

Abstract: A purity conjecture due to Grothendieck and Auslander–Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension ≥2. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic (0,p). We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover.

On the rationality problem for quadric bundles

Abstract: We classify all positive integers n and r such that (stably) nonrational complex r-fold quadric bundles over rational n-folds exist. We show in particular that, for any n and r, a wide class of smooth r-fold quadric bundles over PCn are not stably rational if r∈[2n−1−1,2n−2]. In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thelene and Pirutka which avoids universally CH0-trivial resolutions of singularities.

Interior C2 regularity of convex solutions to prescribing scalar curvature equations

Abstract: We establish interior C2 estimates for convex solutions of the scalar curvature equation and the σ2-Hessian equation. We also prove interior curvature estimates for isometrically immersed hypersurfaces (Mn,g)⊂Rn+1 with positive scalar curvature. These estimates are consequences of interior estimates for these equations obtained under a weakened condition.

Congruences of $5$-secant conics and the rationality of some admissible cubic fourfolds

Abstract: The works of Hassett and Kuznetsov identify countably many divisors Cd in the open subset of P55=P(H0(OP5(3))) parameterizing all cubic fourfolds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family C14. We use congruences of 5-secant conics to prove rationality for the first three of the families Cd, corresponding to d=14,26,38 in Hassett’s notation.

A gradient estimate for nonlocal minimal graphs

Abstract: We consider the class of measurable functions defined in all of Rn that give rise to a nonlocal minimal graph over a ball of Rn. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known results, leads to the C∞ regularity of the function in the ball. While the smoothness of nonlocal minimal graphs was known for n=1,2—but without a quantitative bound—in higher dimensions only their continuity had been established. To prove the gradient bound, we show that the normal to a nonlocal minimal graph is a supersolution of a truncated fractional Jacobi operator, for which we prove a weak Harnack inequality. To this end, we establish a new universal fractional Sobolev inequality on nonlocal minimal surfaces. Our estimate provides an extension to the fractional setting of the celebrated gradient bounds of Finn and of Bombieri, De Giorgi, and Miranda for solutions of the classical mean curvature equation.

Twisted moments of $L$-functions and spectral reciprocity

Abstract: We establish a reciprocity formula that expresses the fourth moment of automorphic L-functions of level q twisted by the lth Hecke eigenvalue as the fourth moment of automorphic L-functions of level l twisted by the qth Hecke eigenvalue. Direct corollaries include subconvexity bounds for L-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic L-functions.

Weak subconvexity without a Ramanujan hypothesis

Abstract: We describe a new method to obtain weak subconvexity bounds for L-functions with mild hypotheses on the size of the Dirichlet coefficients. We verify these hypotheses for all automorphic L-functions and (with mild restrictions) the Rankin–Selberg L-functions attached to two automorphic representations. The proof relies on a new unconditional log-free zero density estimate for Rankin–Selberg L-functions.

Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient

Abstract: We study local and global properties of positive solutions of $-{\Delta}u=u^p]{\left |{ abla u}\right |}^q$ in a domain ${\Omega}$ of ${\mathbb R}^N$, in the range $1

On the polynomial Szemerédi theorem in finite fields

Abstract: Let P1,…,Pm∈Z[y] be any linearly independent polynomials with zero constant term. We show that there exists γ>0 such that any subset of Fq of size at least q1−γ contains a nontrivial polynomial progression x,x+P1(y),…,x+Pm(y), provided that the characteristic of Fq is large enough.

Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

Abstract: Let π traverse a sequence of cuspidal automorphic representations of GL2 with large prime level, unramified central character, and bounded infinity type. For G∈{GL1,PGL2}, let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint L-function of π, with polynomial dependence upon the conductor of the twist. We show that H(GL1) implies H(PGL2). In geometric terms, H(PGL2) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL1), to the special case in which the relevant sequence of measures is tested against an Eisenstein series.

-Nice triples and the Grothendieck–Serre conjecture concerning principal G-bundles over reductive group schemes

Abstract: The main result of this article is to reduce a proof of the conjecture to a statement about principal bundles on affine line over a regular local scheme. This reduction is obtained via a theory of nice triples, which goes back to the ideas of Voevodsky. As an application, an unpublished result due to Gabber is proved.

Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume

Abstract: To a complex projective structure Σ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms ‖ϕΣ‖∞ and ‖ϕΣ‖2 of the quadratic differential ϕΣ of Σ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well-known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil–Petersson gradient flow of renormalized volume on the space CC(N) of convex cocompact hyperbolic structures on a compact manifold N with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of DN, the double of N.

Asymptotics of Chebyshev polynomials, II: DCT subsets of ${\mathbb{R}}$

Abstract: We prove Szego[double acute]-Widom asymptotics for the Chebyshev polynomials of a compact subset of ℝ which is regular for potential theory and obeys the Parreau-Widom and DCT conditions.

Nonabelian Cohen-Lenstra Moments

Abstract: In this article, we give a conjecture for the average number of unramified G-extensions of a quadratic field for any finite group G. The Cohen–Lenstra heuristics are the specialization of our conjecture to the case in which G is abelian of odd order. We prove a theorem toward the function field analogue of our conjecture and give additional motivations for the conjecture, including the construction of a lifting invariant for the unramified G-extensions that takes the same number of values as the predicted average and an argument using the Malle–Bhargava principle. We note that, for even |G|, corrections for the roots of unity in Q are required, which cannot be seen when G is abelian.

Noncommutative boundaries and the ideal structure of reduced crossed products

Abstract: A C∗-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C∗-algebra. In this paper we characterize this property for unital C∗-dynamical systems over discrete groups. To every C∗-dynamical system we associate a “twisted” partial C∗-dynamical system that encodes much of the structure of the action. This system can often be “untwisted,” for example, when the algebra is commutative or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtain relatively simple necessary and sufficient conditions for the ideal separation property. A key idea is a notion of noncommutative boundary for a C∗-dynamical system that generalizes Furstenberg’s notion of topological boundary for a group.

A tropical motivic Fubini theorem with applications to Donaldson–Thomas theory

Abstract: We present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semialgebraic sets. As applications, we prove a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and give a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Le Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson–Thomas theory.

Simple closed curves, finite covers of surfaces, and power subgroups of Out(Fn)

Abstract: We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set O⊂Fn which is contained in the union of finitely many Aut(Fn)-orbits, we construct finite-index normal subgroups of Fn whose first rational homology is not spanned by powers of elements of O. These examples answer questions of Farb and Hensel, Kent, Looijenga, and Marche. We also show that the quotient of Out(Fn) by the subgroup generated by kth powers of transvections often contains infinite-order elements, strengthening a result of Bridson and Vogtmann that it is often infinite. Finally, for any set O⊂Fn which is contained in the union of finitely many Aut(Fn)-orbits, we construct integral linear representations of free groups that have infinite image and that map all elements of O to torsion elements.

Symplectically knotted codimension-zero embeddings of domains in R4

Abstract: We show that many toric domains X in R4 admit symplectic embeddings ϕ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes ϕ(X) to X. For instance X can be taken equal to a polydisk P(1,1) or to any convex toric domain that both is contained in P(1,1) and properly contains a ball B4(1); by contrast a result of McDuff shows that B4(1) (or indeed any 4-dimensional ellipsoid) cannot have this property. The embeddings are constructed based on recent advances in symplectic embeddings of ellipsoids, though in some cases a more elementary construction is possible. The fact that the embeddings are knotted is proved using filtered positive S1-equivariant symplectic homology.