# Amherst 2019

AGNES Spring 2019 was held Friday, March 22 through Sunday, March 24 at UMass Amherst.

**Videos of talks:**

Andrei Caldararu (U. Wisconsin) *Computing a categorical Gromov-Witten invariant.* videos: pre-talk , talk . slides: pre-talk , talk

In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-infinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical nature of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other settings like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.

In my talk I shall describe recent joint work with Junwu Tu on giving a new definition of Costello's invariants. The main improvement is that the new approach involves no choices, so in particular the new invariants are amenable to explicit computer calculations. If time allows I will explain how to compute the invariants at g=1, n=1, for elliptic curve targets. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.

Alessio Corti (Imperial College London) *Hyperelliptic integrals and mirrors of the Johnson-Kollár del Pezzo surfaces (work with Giulia Gugiatti). *videos: pretalk , talk

For all k>0 integer, we consider the regularised I-function of the family of del Pezzo surfaces of degree 8k+4 in P(2,2k+1,2k+1, 4k+1). We show that this function, which is of hypergeometric type, is a period of an explicit pencil of curves. Thus the pencil is a candidate LG mirror of the family of del Pezzo surfaces. The main feature of these surfaces, which makes the mirror construction especially interesting, is that the anticanonical system is empty: because of this, our mirrors are not covered by any other construction.

Laure Flapan (Northeastern U.) *Chow motives, L-functions, and powers of algebraic Hecke characters*. video: talk

The Langlands and Fontaine-Mazur conjectures in number theory describe when an automorphic representation f arises geometrically, meaning that there is a smooth projective variety X, or more generally a Chow motive M in the cohomology of X, such that there is an equality of L-functions L(M,s)=L(f,s). We explicitly describe how to produce such a variety X and Chow motive M in the case of powers of certain automorphic representations, called algebraic Hecke characters. This is joint work with J. Lang.

Kieran O'Grady (Rome 1) *The period map for quartic surfaces*. video: talk

Joint work with Radu Laza. The period map from the GIT moduli space of quartic surfaces to the Baily-Borel compactification of the relevant locally symmetric variety is birational, but far from regular, with an intricate indeterminacy set. According to Looijenga, the complexity of the period map is related to the combinatorial complexity of the hyperplane arrangement in the relevant bounded symmetric domain corresponding to hyperelliptic and unigonal degree 4 polarized K3 surfaces. We have formulated precise conjectures on the behaviour of the period map in this and similar cases such as double EPW sextics and hyperelliptic K3 surfaces which are double covers of a smooth quadric. The conjectures are motivated by Borcherds-like relations, and are formulated in analogy with the Hassett-Keel program for M_g. In the case of hyperelliptic K3's we prove that the conjectures hold, by identifying our Hassett-Keel-Looijenga program with a VGIT, and carrying out the (non trivial) GIT analysis.

Jessica Sidman (Mt. Holyoke College) *Matroid varieties*. video: talk

Each point x in G(k,n) corresponds to a k x n matrix A_x which gives rise to a matroid M_x on the columns of A_x. Gelfand, Goresky, MacPherson, and Serganova showed that the sets {y in G(k,n) | M_y = M_x} form a stratification of G(k,n) with many beautiful properties. However, results of Mnev and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. In this talk we will focus on constructions giving defining equations of the matroid variety V_x given by the closure of the stratum associated to M_x. If M_x is a positroid, then Knutson, Lam, and Speyer have shown that the ideal of V_x is generated by Plucker coordinates, but this is not true for arbitrary matroids. We describe how the Grassmann-Cayley algebra may be used to generate non-trivial equations in the ideal of V_x when the geometry of M_x is sufficiently rich. This work is joint with Will Traves and Ashley Wheeler.

Bernd Siebert (UT Austin) *Periods and analyticity of toric degenerations. *videos: pre-talk , talk

Toric degenerations provide a flexible technique for studying all kinds of phenomena related to mirror symmetry. The talk reports on a strengthening of a 2014 paper with Helge Ruddat concerning period integrals in toric degenerations constructed by wall structures. The progress concerns the appearance of the complex Ronkin function in local mirror symmetry, cohomological formulations of the period integral, finite determinacy of toroidal singularities and the analyticity of the formal canonical toric degenerations constructed jointly with Mark Gross in 2007. Another application is the engineering of families of K3 surfaces with prescribed Picard rank.

Burt Totaro (UCLA) *Bott vanishing for algebraic surfaces*. videos: pre-talk , talk

Bott proved a strong vanishing theorem for sheaf cohomology on projective space. The statement does not hold for most varieties, and we survey what is known. We find new varieties that satisfy Bott vanishing, building on our knowledge of moduli spaces of K3 surfaces.

Chenyang Xu (MIT) *K-moduli of Fano varieties*. video: talk

K-stability of Fano varieties was originally defined to capture the existence of a Kahler-Einstein metric. One of the deepest applications of K-stability in algebraic geometry is that it conjecturally provides a well behaved moduli space, which we call the K-moduli. I will discuss some recent progress toward the conjecture and also the remaining open part.

**Posters:**

Shamil Asgarli (Brown) *Regular birational automorphisms of the plane. *pdf

Given a field k, we say that a birational automorphism of P 2 is regular if it restricts to a bijection on the k-rational points. When k is a finite field, such an automorphism induces a permutation on those k-points. Can we recover every abstract permutation on the k-points in this way? Using a geometric construction provided by Cantat in 2009, we give a positive answer

to the above question when the characteristic is odd. Using a family of homaloidal linear systems found by Ruffini in 1877, we also prove that the subgroup of such automorphisms is of infinite index in the Cremona group and infinitely-generated as an abstract group. This is a work in progress joint with Kuan-Wen Lai, Masahiro Nakahara, and Susanna Zimmermann.

Nathan Chen (Stony Brook) *Degree of irrationality of very general abelian surfaces. *pdf

The degree of irrationality of a projective variety X is defined to be the smallest degree rational dominant map to a projective space of the same dimension. For abelian surfaces, Yoshihara computed this invariant in specific cases, while Stapleton gave a sublinear upper bound for very general polarized abelian surfaces (A, L) of degree d. Somewhat surprisingly, we show that the degree of irrationality of a very general polarized abelian surface is uniformly bounded above by 4, independently of the degree of the polarization. This result disproves part of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld and Ullery.

Shanna Dobson (California State University, Los Angeles) *The Lubin-Tate Tower at Infinite Level as a Perfectoid Space and a Derived Categorical Hodge-Tate Period Map.*

Scholze reformulated Rapoport and Viehmann’s construction and reformulation of towers of moduli spaces as local Shimura varieties, in hopes that the cohomology of the moduli space of Shimura varieties provides a realization of the Local Langlands correspondences. We review the Lubin-Tate tower as a Perfectoid Space, and then consider a derived categorical analogue of the Hodge-Tate period map.

Lian Duan (UMass Amherst) *Transverse lines to surfaces over finite fields*. pdf

Given a smooth projective hypersurface X defined over an algebraically closed field k, a classical theorem of Bertini implies that X ∩ L is smooth for a general line L defined over k. The same result in fact holds for any infinite field k. However, when k is a finite field, it is possible that X ∩ L is singular for every L defined over k. One approach to remedy the original Bertini theorem in the case of finite fields is to investigate how large q=|k| should be with respect to the invariants of the variety X (such as its degree d) so that X admits a favorable k-line. When X is a reflexive curve, the first author proved that there is a k-transverse line to X provided that q ≥ d − 1. In this recent work, we generalize this result and show that when X is a smooth reflexive surface of degree d satisfying q ≥ 1.8d, there exists a k-transverse line to X. This is a joint work with Shamil Asgarli and Kuan-Wen Lai.

Iulia Gheorghita (Boston College)* Effective divisors in the projectivized Hodge bundle.* pdf

We compute the class of the closure of the locus of canonical divisors in the projectivization of the Hodge bundle over M_g which have a zero at a Weierstrass point. We also show that the strata of canonical and bicanonical divisors with a double zero span extremal rays of the respective pseudoeffective cones.

Ross Goluboff (Boston College) *Genus six curves, K3 surfaces, and stable pairs. *pdf

A general smooth curve of genus six lies on a quintic del Pezzo surface. Artebani and Kondo have constructed a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. In this poster, I will describe a smooth Deligne-Mumford stack parametrizing certain stable surface-curve pairs that essentially resolves this map.

Brian Hwang (Cornell) *Toric degenerations, limit linear series, and quivers in Bruhat–Tits buildings.*

We show how quivers (directed graphs) in Bruhat–Tits buildings give rise to degenerations of Grassmannians. These are flat families whose generic fiber is a Grassmannian and whose special fiber is a certain quiver Grassmannian attached to the diagram. It turns out that these can be understood as moduli spaces of limit linear series and provide an easy way to unify a number of such degenerations that arise in algebraic geometry, such as local models of Shimura varieties, linked Grassmannians, the Mumford degeneration, and Mustafin varieties. In fact, it turns out that there are some non-obvious isomorphisms between such degenerations that arise from seemingly unrelated quivers, such as one with a single cycle, and one with multiple cycles. As an illustration, we construct some toric degenerations and point out some interesting connections between these embedded quivers, the toric data of the special fiber, and associated combinatorial diagrams.

Yoonjoo Kim (Stony Brook)* Degeneration of hyperkähler manifolds and Nagai’s conjecture. *pdf

For degenerations of compact hyperkähler manifolds, Nagai’s conjecture predicts the behavior of monodromy on the 2k-th cohomology in terms of monodromy on the second cohomology. Degenerations of hyperkähler manifolds naturally fall into three types, analogous to the well-known situation for K3 surfaces. For type I and III degenerations, the conjecture was established by Kollár-Laza-Saccà-Voisin. Here we settle the conjecture for all currently known examples of compact hyperkähler manifolds, under a certain cohomology assumption on OG10.

Jennifer Li (UMass Amherst) *The Kawamata-Morrison-Totaro Cone Conjecture for log Calabi-Yau surfaces. *pdf

Morrison’s cone conjecture states that for a smooth Calabi-Yau manifold X, the automorphism group of X acts on its effective nef cone with rational polyhedral fundamental domain. Totaro generalized this conjecture to Kawamata log terminal (klt) Calabi-Yau pairs (X,D). In our project, we are studying pairs (X,D) where X is a smooth projective surface and D = D_1 + ... + D_n is a reduced normal crossing divisor on X, such that K_X + D = 0 and D has negative-definite intersection matrix. Results by Gross-Hacking-Keel on mirror symmetry for cusp singularities suggest that we consider the pair (X,D) with a distinguished complex structure for which the mixed Hodge structure on U = X\D is split. The goal of our project is to prove Morrison’s cone conjecture in this special case. We note that this is different from Totaro’s work, because in our project the pair (X,D) is not klt, and we must consider (X,D) with the special complex structure (otherwise the conjecture is false as observed by Totaro). We have shown that the cone conjecture holds when D has at most six components: in these cases the nef cone is rational polyhedral and we give explicit generators for the dual cone. The cases where D has more than six components are current work in progress. Under our conditions, there exists a contraction of (X,D) to a cusp singularity (X',p). Cusp singularities come in mirror dual pairs, and the embedding dimension m of the dual cusp is equal to max(n, 3) where n is the number of components of the boundary divisor D. By studying the nef cone of (X, D), we hope to give a description of the deformation space of the dual cusp, which is not well understood for m greater than six.

Yucheng Liu (Northeastern) *A construction of new Bridgeland stability conditions.*

Motivated by Michael Douglas’ work, Bridgeland constructed a general theory of stability conditions on triangulated categories. The most important question in this theory is the existence of stability conditions. In this paper, I will construct a map from rational stability conditions on D^b(X) to the rational stability conditions on D^b(X x C), where C is an integral smooth curve.

Takumi Murayama (Michigan) *Seshadri constants for vector bundles. *pdf

We introduce Seshadri constants for line bundles in a relative setting. They are a generalization of notions for line bundles and vector bundles respectively due to Demailly and to Beltrametti–Schneider–Sommese and Hacon. We give three applications: (1) A characterization of projective space in terms of the Seshadri constant of the tangent bundle; (2) An identification of new nef classes on self-products of curves; and (3) A generic jet separation statement for direct images of pluricanonical bundles. This is joint work with Mihai Fulger.

Chengxi Wang (Rutgers) *On stringy Euler characteristics of Clifford non-commutative varieties. *pdf

It was shown by Kuznetsov that complete intersections of n generic quadrics in P^{2n−1} are related by Homological Projective Duality to certain non-commutative (Clifford) varieties which are in some sense birational to double covers of P^{n−1} ramified over symmetric determinantal hypersurfaces. Mirror symmetry predicts that the Hodge numbers of the complete intersections of quadrics must coincide with the appropriately defined Hodge numbers of these double covers. We observe that these numbers must be different from the well-known Batyrev stringy Hodge numbers, else the equality fails already at the level of Euler characteristics. We define a natural modification of stringy Hodge numbers for the particular class of Clifford varieties, and prove the corresponding equality of Euler characteristics in arbitrary dimension.

Changwei Zhou (BInghamton) *Effective upper bound of analytic torsion under Arakelov metric*. pdf

Given a choice of metric on the Riemann surface, the regularized determinant of Laplacian (analytic torsion) is defined via the complex power of elliptic operators: det(∆) = exp(−ζ (0)). In this paper we gave an asymptotic effective estimate of analytic torsion under Arakelov metric. In particular, after taking the logarithm it is asymptotically upper bounded by g for g > 1. The construction of a cohomology theory for arithmetic surfaces in Arakelov theory has long been an open problem. In particular, it is not known if h^1(X, L) ≥ 0. We view this as an indirect piece of evidence that if such a cohomology theory exists, the h^1 term may be effectively estimated.

The conference was made possible by support from the NSF, UMass Amherst, Mount Holyoke College, and Amherst College.

Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).