Transcendence of period maps
Period domains $D$ can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, however, the period map of a family of algebraic varieties lands in a quotient $D/\Gamma$ by an arithmetic group. In the very special case when $D/\Gamma$ is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization $D\rightarrow D/\Gamma$ is a crucial component of the modern approach to the Andr\'e-Oort conjecture. We prove a version of the Ax-Schanuel conjecture for general period maps $X\rightarrow D/\Gamma$ which says that atypical algebraic relations between $X$ and $D$ are governed by Hodge loci. We will also discuss some geometric and arithmetic applications. This is joint work with J. Tsimerman.
Chern classes of crystals
A vector bundle on a complex manifold has Chern classes in de Rham cohomology. Chern-Weil theory shows that these classes vanish if the bundle admits a flat connection. The analog of a bundle with flat connection in positive characteristic algebraic geometry is an isocrystal in the sense of Grothendieck. In my talk, I will recall what these objects are, and then explain why their Chern classes vanish. The key tool is algebraic K-theory. This is a report on joint work with Jacob Lurie.
Revisiting the Hessian
The Hessian of a plane cubic curve is classically described using partial derivatives or polars. In this talk, we explore another construction of the cubic Hessian (and variants), and relate these constructions to maps between certain moduli spaces of genus one curves with extra structure. We will explain how input from number theory and dynamics also helps to understand such classical objects.
Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic
There are many results characterizing when derived categories of two complex surfaces are equivalent, including theorems of Bridgeland and Maciocia showing that derived equivalent Enriques or bielliptic surfaces must be isomorphic and a theorem of Sosna that a K3 canonically covering an Enriques is not derived equivalent to any varieties other than itself, and that an abelian surface canonically covering a biellptic surface is derived equivalent only to itself and its dual. The proofs of these theorems strongly use Torelli theorems and lattice-theoretic methods which are not available in positive characteristic. In this talk I will discuss how to prove these results over algebraically closed fields of positive characteristic (excluding some low characteristic cases). This work is joint with M. Lieblich, L. Lombardi and S. Tirabassi.
The Prym-Green conjecture for curves of odd genus
A paracanonical curve is a curve together with a torsion bundle of fixed level. The moduli space of such objects is very interesting from several points of view, such as its relationship to Abelian varieties. I will give a complete description of the Betti numbers associated to the homogeneous coordinate ring of such a curve, provided the genus is odd. This is joint work with Gavril Farkas.
Gopakumar-Vafa invariants via vanishing cycles
Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map). Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants. In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi. Conjecturally, these should agree with the invariants as defined by stable maps. I will also explain how to prove the conjectural correspondence in various cases. This is joint work with Yukinobu Toda.
Stability manifold under Fourier-Mukai transforms on an abelian threefold
The notion of Fourier-Mukai transform for abelian varieties was introduced by Mukai in early 1980s. Since then Fourier-Mukai theory turned out to be extremely successful in studying stable sheaves and complexes of them, and also their moduli spaces. I will discuss how the Fourier-Mukai techniques are useful to construct Bridgeland stability conditions on any abelian threefold, and also to study the corresponding stability manifold.