Andrei Caldararu. (Wisconsin)Computing a categorical Gromov-Witten invariant.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)Hyperelliptic integrals and mirrors of the Johnson-Kollár del Pezzo surfaces (work with Giulia Gugiatti).
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)Chow motives, L-functions, and powers of algebraic Hecke characters.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. (Rome 1)Kieran O'Grady. The period map for quartic surfaces. 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)Matroid Varieties.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.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.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.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. Titles and abstracts for the poster session See the pdf file below. |

Amherst 2019 >