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 Advances in MathematicsJournal Prestige (SJR): 3.027 Citation Impact (citeScore): 2Number of Followers: 17      Subscription journal ISSN (Print) 0001-8708 - ISSN (Online) 1090-2082 Published by Elsevier  [3206 journals]
• Moduli of curves on Enriques surfaces
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Ciro Ciliberto, Thomas Dedieu, Concettina Galati, Andreas Leopold Knutsen We compute the number of moduli of all irreducible components of the moduli space of smooth curves on Enriques surfaces. In most cases, the moduli maps to the moduli space of Prym curves are generically injective or dominant. Exceptional behavior is related to existence of Enriques–Fano threefolds and to curves with nodal Prym-canonical model.

• Uniqueness of very weak solutions for a fractional filtration equation
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Gabriele Grillo, Matteo Muratori, Fabio Punzo We prove existence and uniqueness of distributional, bounded solutions to a fractional filtration equation in Rd. With regards to uniqueness, it was shown even for more general equations in [22] that if two bounded solutions u,w of (1.1) satisfy u−w∈L1(Rd×(0,T)), then u=w. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions. For nonnegative initial data, we first show that a minimal solution exists and then that any other solution must coincide with it. A similar procedure is carried out for sign-changing solutions. As a consequence, distributional solutions have locally-finite energy.

• A rigorous derivation of the Hamiltonian structure for the nonlinear
Schrödinger equation
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Dana Mendelson, Andrea R. Nahmod, Nataša Pavlović, Matthew Rosenzweig, Gigliola Staffilani We consider the cubic nonlinear Schrödinger equation (NLS) in any spatial dimension, which is a well-known example of an infinite-dimensional Hamiltonian system. Inspired by the knowledge that the NLS is an effective equation for a system of interacting bosons as the particle number tends to infinity, we provide a derivation of the Hamiltonian structure, which is comprised of both a Hamiltonian functional and a weak symplectic structure, for the nonlinear Schrödinger equation from quantum many-body systems. Our geometric constructions are based on a quantized version of the Poisson structure introduced by Marsden, Morrison and Weinstein [24] for a system describing the evolution of finitely many indistinguishable classical particles.

• Towards a categorical boson-fermion correspondence
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Yin Tian We construct a Heisenberg counterpart of a Clifford categorification. It is a modification of Khovanov's Heisenberg categorification. We relate generators of the Heisenberg category with a complex of generators of the Clifford category. Certain vertex operators associated to the Clifford algebra are lifted to endofunctors of the Fock space categorification.

• An upper bound on the number of perfect quadratic forms
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): W.P.J. van Woerden In a recent publication Roland Bacher showed that the number pd of non-similar perfect d-dimensional quadratic forms satisfies eΩ(d)

• 3-dimensional defect TQFTs and their tricategories
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Nils Carqueville, Catherine Meusburger, Gregor Schaumann We initiate a systematic study of 3-dimensional ‘defect’ topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.

• From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Haonan Zhang In this paper we study the joint convexity/concavity of the trace functionsΨp,q,s(A,B)=Tr(Bq2K⁎ApKBq2)s,p,q,s∈R, where A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of (p,q,s)∈R3 for Ψp,q,s to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of (α,z) for α-z Rényi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψp,0,1/p for 0

• Preprojective algebra structure on skew-group algebras
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Louis–Philippe Thibault We give a class of finite subgroups G

• Mixed type surfaces with bounded Gaussian curvature in three-dimensional
Lorentzian manifolds
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Atsufumi Honda, Kentaro Saji, Keisuke Teramoto A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the behavior of Gaussian curvature at a non-degenerate lightlike point of a mixed type surface. To characterize the boundedness of Gaussian curvature at a non-degenerate lightlike points, we introduce several fundamental invariants along non-degenerate lightlike points, such as the lightlike singular curvature and the lightlike normal curvature. Moreover, using the results by Pelletier and Steller, we obtain the Gauss–Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.

• The moduli space of smooth ample hypersurfaces in Abelian varieties
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Fabrizio Catanese, Yongnam Lee We give a characterization of smooth ample Hypersurfaces in Abelian Varieties and also describe the connected component of the moduli space containing such hypersurfaces of a given polarization type: we show that this component is irreducible and that it consists of these Hypersurfaces, plus the smooth iterated univariate coverings of normal type (of the same polarization type).The above manifolds yield also a connected component of the open set of Teichmüller space consisting of Kähler complex structures.

• Representations of mock theta functions
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Dandan Chen, Liuquan Wang Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters a and b. Specializing the choices of (a,b), we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same (a,b)-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks m(x,q,z) and fa,b,c(x,y,q) introduced by Hickerson and Mortenson, as well as m‾(x,q,z) and f‾a,b,c(x,y,q) introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations.

• Coxeter submodular functions and deformations of Coxeter permutahedra
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Federico Ardila, Federico Castillo, Christopher Eur, Alexander Postnikov We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This class of polytopes contains important families such as weight polytopes, signed graphic zonotopes, Coxeter matroids, root cones, and Coxeter associahedra. Our description extends the known correspondence between generalized permutahedra, polymatroids, and submodular functions to any finite reflection group.

• Nested Hilbert schemes on surfaces: Virtual fundamental class
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Amin Gholampour, Artan Sheshmani, Shing-Tung Yau We construct virtual fundamental classes on nested Hilbert schemes of points and curves in complex nonsingular projective surfaces. These classes recover the virtual classes of Seiberg-Witten theory as well as the (reduced) stable theory, and play a crucial role in the reduced Donaldson-Thomas theory of local-surface-threefolds that we study in [15]. We show that certain integrals against the virtual fundamental classes of punctual nested Hilbert schemes are expressed as integrals over the products of the Hilbert scheme of points. We are able to find explicit formulas for some of these integrals by relating them to Carlsson-Okounkov's vertex operator formulas.

• A graphical category for higher modular operads
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Philip Hackney, Marcy Robertson, Donald Yau We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted U, plays a similar role for modular operads that the dendroidal category Ω plays for operads. We carefully study properties of U, including the existence of certain factorization systems. Related structures, such as cyclic operads and stable modular operads, can be similarly treated using categories derived from U.

• Geometrical logarithmic capacitance
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Jie Xiao This paper is devoted to a novel and nontrivial exploration of eight aspects of the geometrical logarithmic capacitance (a very key notion in mathematical physics, quasiconformal geometry and variational calculus) through: (1) identifying with the reduced conformal module; (2) evaluating the minimal log-potential energy; (3) relating to both the volume-radius and the surface-radius; (4) linking with the n-harmonic radius and the log-capacity of the Kevin image of a compact surface; (5) finding the Minkowski inequality and the general variational formula for the log-capacity; (6) pinching the log-isocapacitary inequality from left and solving the left-prescribed problem for the normalized log-capacitary curvature measure; (7) pinching the log-isocapacitary inequality from right and handling the right-prescribed problem for the normalized log-capacitary curvature measure; (8) handling an overdetermination of the n-equilibrium potential of a given convex body via the log-capacitary concavity.

• Extreme points of the set of elements majorised by an integrable function:
Resolution of a problem by Luxemburg and of its noncommutative counterpart

• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): D. Dauitbek, J. Huang, F. Sukochev Let f be an arbitrary integrable function on a finite measure space (X,Σ,ν). We characterise the extreme points of the set Ω(f) of all measurable functions on (X,Σ,ν) majorised by f, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.

• Stability conditions and the A 2 quiver
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Tom Bridgeland, Yu Qiu, Tom Sutherland For each integer n⩾2 we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the A2 quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the A2 singularity.

• Truncated Hecke-Rogers type series
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Chun Wang, Ae Ja Yee The recent work of George Andrews and Mircea Merca on the truncated version of Euler's pentagonal number theorem has opened up a new study on truncated theta series. Since then several papers on the topic have followed. The main purpose of this paper is to generalize the study to Hecke-Rogers type double series, which are associated with some interesting partition functions. Our proofs heavily rely on a formula from the work of Zhi-Guo Liu on the q-partial differential equations and q-series.

• Exponential improvements for superball packing upper bounds
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Ashwin Sah, Mehtaab Sawhney, David Stoner, Yufei Zhao We prove that for all fixed p>2, the translative packing density of unit ℓp-balls in Rn is at most 2(γp+o(1))n with γp

• Three-dimensional normal pseudomanifolds with relatively few edges
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Biplab Basak, Ed Swartz Let Δ be a d-dimensional normal pseudomanifold, d≥3. A relative lower bound for the number of edges in Δ is that g2 of Δ is at least g2 of the link of any vertex. When this inequality is sharp Δ has relatively minimal g2. For example, whenever the one-skeleton of Δ equals the one-skeleton of the star of a vertex, then Δ has relatively minimal g2. Subdividing a facet in such an example also gives a complex with relatively minimal g2. We prove that in dimension three these are the only examples. As an application we determine the combinatorial and topological type of 3-dimensional Δ with relatively minimal g2 whenever Δ has two or fewer singularities. The topological type of any such complex is a pseudocompression body, a pseudomanifold version of a compression body.Complete combinatorial descriptions of Δ with g2(Δ)≤2 are due to Kalai [12] (g2=0), Nevo and Novinsky [13] (g2=1) and Zheng [20] (g2=2). In all three cases Δ is the boundary of a simplicial polytope. Zheng observed that for all d≥0 there are triangulations of Sd⁎RP2 with g2=3. She asked if this is the only nonspherical topology possible for g2(Δ)=3. As another application of relatively minimal g2 we give an affirmative answer when Δ is 3-dimensional.

• Operator means of probability measures
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Fumio Hiai, Yongdo Lim Let P be the complete metric space consisting of positive invertible operators on an infinite-dimensional Hilbert space with the Thompson metric. We introduce the notion of operator means of probability measures on P, in parallel with Kubo and Ando's definition of two-variable operator means, and show that every operator mean is contractive for the ∞-Wasserstein distance. By means of a fixed point method we consider deformation of such operator means, and show that the deformation of any operator mean becomes again an operator mean in our sense. Based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures.

• S ( R ) +with+polynomial+symbols&rft.title=Advances+in+Mathematics&rft.issn=0001-8708&rft.date=&rft.volume=">Spectrum of composition operators on S ( R ) with polynomial symbols
• Abstract: Publication date: 13 May 2020Source: Advances in Mathematics, Volume 365Author(s): Carmen Fernández, Antonio Galbis, Enrique Jordá We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum reduces to {0}, while the spectrum of any non mean ergodic composition operator with a polynomial always contains the closed unit disc except perhaps the origin. We obtain a complete description of the spectrum of the composition operator with a quadratic polynomial or a cubic polynomial with positive leading coefficient.

• Path developments and tail asymptotics of signature for pure rough paths
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Horatio Boedihardjo, Xi Geng, Nikolaos P. Souris Solutions to linear controlled differential equations can be expressed in terms of global iterated path integrals along the driving path. This collection of iterated integrals encodes essentially all information about the underlying path. While upper bounds for iterated path integrals are well known, lower bounds are much less understood, and it is known only relatively recently that some types of asymptotics for the n-th order iterated integral can be used to recover some intrinsic quantitative properties of the path, such as the length for C1 paths.In the present paper, we investigate the simplest type of rough paths (the rough path analogue of line segments), and establish uniform upper and lower estimates for the tail asymptotics of iterated integrals in terms of the local variation of the underlying path. Our methodology, which we believe is new for this problem, involves developing paths into complex semisimple Lie algebras and using the associated representation theory to study spectral properties of Lie polynomials under the Lie algebraic development.

• On the classification of homogeneous affine tube domains with large
automorphism groups in arbitrary dimensions
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Alexandr Medvedev, Gerd Schmalz, Vladimir Ezhov We classify tube domains in Cn+1 (n≥1) with affinely homogeneous base of their boundary and a.) with positive definite Levi form and b.) with Lorentzian type Levi form and affine isotropy of dimension at least (n−2)(n−3)2.

• Hahn polynomials on polyhedra and quantum integrability
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Plamen Iliev, Yuan Xu Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in Rd, which include hexagons in R2 and truncated tetrahedrons in R3, are defined and studied. The polynomials are given explicitly in terms of the classical one-dimensional Hahn polynomials. They are also characterized as common eigenfunctions of a family of commuting partial difference operators. These operators provide symmetries for a system that can be regarded as a discrete extension of the generic quantum superintegrable system on the d-sphere. Moreover, the discrete system is proved to possess all essential properties of the continuous system. In particular, the symmetry operators for the discrete Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of orthogonal polynomials, and an explicit set of 2d−1 generators for the symmetry algebra is constructed. Furthermore, other discrete quantum superintegrable systems, which extend the quantum harmonic oscillator, are obtained by considering appropriate limits of the parameters.

• Logarithmic potentials and quasiconformal flows on the Heisenberg group
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Alex D. Austin Let H be the sub-Riemannian Heisenberg group. That H supports a rich family of quasiconformal mappings was demonstrated by Korányi and Reimann using the so-called flow method. Here we supply further evidence of the flexible nature of this family, constructing quasiconformal mappings with extreme behavior on small sets. More precisely, we establish criteria to determine when a given logarithmic potential Λ on H is such that there exists a quasiconformal mapping of H with Jacobian comparable to e2Λ (so that the Jacobian is zero or infinity at the same points as e2Λ). When Λ is continuous and meets the criteria, we show the canonical (sub-Riemannian) metric g0 and the weighted metric g=eΛg0 generate bi-Lipschitz equivalent distance functions. These results rest on an extension to the theory of quasiconformal flows on H and constructions that adapt the iterative method of Bonk, Heinonen, and Saksman.

• Regular mappings and non-existence of bi-Lipschitz embeddings for slit
carpets
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Guy C. David, Sylvester Eriksson-Bique We prove that the “slit carpet” studied by Merenkov does not admit a bi-Lipschitz embedding into any uniformly convex Banach space. In particular, this includes any space Rn, but also spaces such as Lp for p∈(1,∞). This resolves Question 8 in the 1997 list by Heinonen and Semmes.

• Stability of geometric flows of closed forms
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Lucio Bedulli, Luigi Vezzoni We prove a general result about the stability of geometric flows of “closed” sections of vector bundles on compact manifolds. Our theorem allows us to prove a stability result for the modified Laplacian coflow in G2-geometry introduced by Grigorian in [9] and for the balanced flow introduced by the authors in [2].

• C P 2 &rft.title=Advances+in+Mathematics&rft.issn=0001-8708&rft.date=&rft.volume=">The Leray transform: Factorization, dual CR structures, and model
hypersurfaces in C P 2
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): David E. Barrett, Luke D. Edholm We compute the exact norms of the Leray transforms for a family Sβ of unbounded hypersurfaces in two complex dimensions. The Sβ generalize the Heisenberg group, and provide local projective approximations to any smooth, strongly C-convex hypersurface S to two orders of tangency. This work is then examined in the context of projective dual CR-structures and the corresponding pair of canonical dual Hardy spaces associated to S, leading to a universal description of the Leray transform and a factorization of the transform through orthogonal projection onto the conjugate dual Hardy space.

• Linear response for random dynamical systems
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Wael Bahsoun, Marks Ruziboev, Benoît Saussol We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to Pε' For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to ε; moreover, we obtain a linear response formula. Our results cover random maps whose transfer operator does not necessarily admit a spectral gap. We apply our results to iid compositions, with respect to various distributions Pε, of uniformly expanding circle maps, Gauss-Rényi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.

• Functional Brunn-Minkowski inequalities induced by polarity
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): S. Artstein-Avidan, D.I. Florentin, A. Segal We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geometric inf-convolution arises as the pull back of the second order reversing transform on geometric convex functions (called either polarity transform or A-transform). These are, up to linear terms, the only order reversing isomorphisms on the class of geometric convex functions. We prove that the integral of this new geometric convolution of two functions is bounded from below by the harmonic average of the individual integrals. Our inequality implies the Brunn-Minkowski inequality, as well as some other, new, inequalities for volumes of bodies. Our inequalities are intimately connected with Busemann's convexity theorem, a new variant of which we prove for 1-convex hulls and log-concave densities.

• Equivariant localization and completion in cyclic homology and derived
loop spaces
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Harrison Chen We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups X/G in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack X/G and its classical z-fixed point stack π0(Xz)/Gz become equivalent after completion along a semisimple parameter [z]∈G//G, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes Perf(X/G). We then prove an analogue of the Atiyah-Segal completion theorem in the setting of periodic cyclic homology, where the completion of the periodic cyclic homology of Perf(X/G) at the identity [e]∈G//G is identified with a 2-periodic version of the derived de Rham cohomology of X/G. Together, these results identify the completed periodic cyclic homology of a stack X/G over a parameter [z]∈G//G with the 2-periodic derived de Rham cohomology of its z-fixed points.

• Dirichlet p-Laplacian eigenvalues and Cheeger constants on
symmetric graphs
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Bobo Hua, Lili Wang In this paper, we study eigenvalues and eigenfunctions of p-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of p-Laplacian, as p→1, we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.

• The dual complex of log Calabi–Yau pairs on Mori fibre spaces
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Mirko Mauri In this paper we show that the dual complex of a dlt log Calabi–Yau pair (Y,Δ) on a Mori fibre space π:Y→Z is a finite quotient of a sphere, provided that either the Picard number of Y or the dimension of Z is ≤2. This is a partial answer to Question 4 in [12].

• A bound for Castelnuovo-Mumford regularity by double point divisors
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Sijong Kwak, Jinhyung Park Let X⊆Pr be a non-degenerate smooth projective variety of dimension n, codimension e, and degree d defined over an algebraically closed field of characteristic zero. In this paper, we first show that reg(OX)≤d−e, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that reg(OX) is not even bounded above by any polynomial function of d when X is not smooth. For a normality bound in the smooth case, we establish that reg(X)≤n(d−2)+1, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that reg(X)≤d−1+m, where m is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projections, we also obtain a slightly better bound for reg(X) under suitable assumptions.

• The Hochschild cohomology ring of a global quotient orbifold
• Abstract: Publication date: 15 April 2020Source: Advances in Mathematics, Volume 364Author(s): Cris Negron, Travis Schedler We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields TXpoly on X, and is generated as a TXpoly-algebra by the sum of the determinants det⁡(NXg) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne–Mumford stacks in general. We discuss, in the case of a symplectic group action on a symplectic variety X, relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, we employ compatible trivializations of the determinants det⁡(NXg), which requires (for the cup product) a nontrivial normalization missing in previous literature.

• Corrigendum to “On divisors of modular forms” [Adv. Math. 329
(2018) 541–554]
• Abstract: Publication date: Available online 5 August 2019Source: Advances in MathematicsAuthor(s): Kathrin Bringmann, Ben Kane, Steffen Löbrich, Ken Ono, Larry Rolen

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