Algebraic Geometry Seminar

Generic smoothness fails to hold for some fibrations in positive characteristic. We study consequences of this failure, in particularly by obtaining a canonical bundle formula relating a fiber with the normalization of its maximal reduced subscheme. This has geometric consequences, including that generic smoothness holds on terminal Mori fiber spaces of relative dimension two in characteristic p at least 11. This is joint work with Zsolt Patakfalvi.

According to SYZ philosophy the same tropical object should admit two ways to be lifted classically: as a complex object, and as a (T-dual) symplectic object. While the tropical-to-complex correspondence is relatively well-studied, tropical-to-symplectic correspondence remains significantly less well-studied up to date. In the talk we'll look at some first instances of tropical- to-symplectic correspondence. As an application of such correspondence in the case of planar tropical curves we'll reprove a theorem of Givental (from about 30 years ago) on Lagrangian embeddings of connected sums of Klein bottles to C^2. For tropical curves in toric 3-folds the resulting Lagrangians turn out to be Waldhausen graph-manifolds. For this case we'll relate the enumerative multiplicity of tropical rational curves to the torsion in the first homology group of the corresponding Lagrangian submanifolds (in full compliance with Mirror Symmetry predictions).

Hyperbolicity is an interesting property of varieties both analytically and algebraically. I will make a recall of several hyperbolicity conjectures and related results, especially hyperbolicity of moduli spaces. Then I will discuss hyperbolicity of base spaces of families of smooth general type varieties, which would imply brody hyperbolicity of some moduli. The main technical tools we used are Hodge modules and Higgs sheaves. If time permitted I will discuss how they could be applied to solve the hyperbolicity problems for families. This is joint work with Mihnea Popa and Behrouz Taji.

The complex Plateau problem has been studied in the context of CR manifolds in works of Rossi (1965) and Harvey and Lawson (1975) where it is proved that every embedded strongly pseudoconvex compact CR manifold of dimension at least 3 is the boundary of a Stein space with isolated singularities. What remains to be determined are intrisic conditions on the CR manifold that guarantee that the Stein filling is smooth. In the talk I will survey this question and discuss some old and recent developments on the problem.

A smooth K3 surface obtained as the blow-up of the quotient of a four-torus by the involution automorphism at all 16 fixed points is called a Kummer surface. Kummer surface need not be algebraic, just as the original torus need not be. However, algebraic Kummer surfaces obtained from abelian varieties provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. In this talk, we give an explicit description for the relation between algebraic Kummer surfaces of Jacobians of genus-two curves with principal polarization and those associated to (1, 2)-polarized abelian surfaces from three different angles: the point of view of 1) the binational geometry of quartic surfaces in P^3 using even-eights, 2) elliptic fibrations on K3 surfaces of Picard-rank 17 over P^1 using Nikulin involutions, 3) theta-functions of genus-two using two-isogeny. Finally, we will explain how these (1,2)-polarized Kummer surfaces naturally allow for an identification of the complex gauge coupling in Seiberg-Witten gauge theory with the axion-dilaton modulus in string theory using an old idea of Sen. (This is joint work with Adrian Clingher.)

I will discuss cohomological obstructions to rationality, descent varieties, and unirationality questions.

We use holomorphic Cayley transformations of quasi-modular forms to realize a correspondence between Fan-Jarvis-Ruan-Witten invariants of Fermat elliptic polynomials and Gromov-Witten invariants of elliptic curves of all genus. Our key observation is that Belorousski-Pandharipande’s relation on \bar{M}_{2,3} implies the genus one invariants on both GW theory and FJRW theory are governed by the Chazy equation. The modularity follows from the Chazy equation and the initial values in FJRW theory can be calculated by cosection localization. This work is joint with J Li and J Zhou.

For a fixed K3 surface, we can study various moduli spaces of sheaves having a given topological type. Under appropriate conditions on the topological data, the moduli spaces are smooth projective varieties. I will discuss a new result about the geometry and arithmetic of these moduli spaces when the K3 surface is defined over an arbitrary field. For any two such moduli spaces of the same dimension, their étale cohomology groups with $\mathbb{Q}_\ell$-coefficients are isomorphic as Galois representations. In particular, when the K3 surface is defined over a finite field, this implies that the moduli spaces have equal zeta functions.

I will discuss new results towards the birational boundedness of low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele Di Certo. Recent work in the minimal model program suggests that pairs with trivial log canonical class should satisfy some boundedness properties. I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are indeed log birationally bounded. This implies birational boundedness of elliptically fibered Calabi-Yau manifolds with a section, in dimension up to 5. If time allows, I will also try to discuss a first approach towards boundedness of rationally connected CY varieties in low dimension (joint with G. Di Cerbo, W. Chen, J. Han and, C. Jiang).

We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric as well as provides a precise list of singular KE ones. To achieve this, the main new ingredient is an estimate in dimension three of the normalized volumes of kawamata log terminal singularities introduced by Chi Li. This is a joint work with Chenyang Xu.

Dual complexes are CW-complexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. They have been finding useful applications for instance in the study of degenerations of projective varieties, mirror symmetry and nonabelian Hodge theory. In particular, Kollár and Xu have asked whether the dual complex of a log Calabi-Yau pair is always a sphere or a finite quotient of a sphere. It is natural to check first if this holds on the end products of minimal model programs. In this talk, we will give a positive answer for Mori fibre spaces of Picard rank two.