The specialization question for rationality is the following one: assume that very general fibers of a flat proper morphism are rational, does it imply that all fibers are rational? I will talk about recent solution of this question in characteristic zero due to myself and Nicaise, and Kontsevich-Tschinkel. The method relies on a construction of various specialization morphisms for the Grothendieck ring of varieties (stable rationality) and the Burnside ring of varieties (rationality), which in turn rely on the Weak Factorization and Semi-stable Reduction Theorems.

# Past Algebraic Geometry Seminar

In mirror symmetry, the prepotential on the Kahler side has an expansion, the constant term of which is a rational multiple of zeta(3)/(2 pi i)^3 after an integral symplectic transformation. In this talk I will explain the connection between this constant term and the period of a mixed Hodge-Tate structure constructed from the limit MHS at large complex structure limit on the complex side. From Ayoub’s works on nearby cycle functor, there exists an object of Voevodsky’s category of mixed motives such that the mixed Hodge-Tate structure is expected to be a direct summand of the third cohomology of its Hodge realisation. I will present the connections between this constant term and conjecture about how mixed Tate motives sit inside Voevodsky’s category, which will also provide a motivic interpretation to the occurrence of zeta(3) in prepotential.

I will introduce a cohomology theory which combines topological and algebraic concepts. Interpretations of certain cohomology groups will be given. We also generalise the construction of the second Stiefel-Whitney class of a line bundle. As I will explain in my talk, the refined Stiefel-Whitney class of the canonical bundle on certain moduli stacks provides an obstruction for the construction of cohomological Hall algebras.

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a priori) real numbers between 0 and 1, called "jumps". The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions, and generalize Raynaud's description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence which decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

We consider a generalization of degeneracy loci of morphisms between vector bundles based on orbit closures of algebraic groups in their linear representations. Using a certain crepancy condition on the orbit closure we gain some control over the canonical sheaf in a preferred class of examples. This is notably the case for Richardson nilpotent orbits and partially decomposable skew-symmetric three-forms in six variables. We show how these techniques can be applied to construct Calabi-Yau manifolds and Fano varieties of dimension three and four.

This is a joint work with Vladimiro Benedetti, Laurent Manivel and Fabio Tanturri.

Hilbert's Nullstellensatz asserts the existence of a complex point satisfying lying on a given variety, provided there is no (ideal-theoretic) proof to the contrary.

I will describe an analogue for curves (of unbounded degree), with respect to conditions specifying that they lie on a given smooth variety, and have homology class

near a specified ray. In particular, an analogue of the Lefschetz principle (relating large positive characteristic to characteristic zero) becomes available for such questions.

The proof is very close to a theorem of Boucksom-Demailly-Pau-Peternell on moveable curves, but requires a certain sharpening. This is part of a joint project with Itai Ben Yaacov, investigating the logic of the product formula; the algebro-geometric statement is needed for proving the existential closure of $\Cc(t)^{alg}$ in this language.

Given a Fano manifold we will consider two ways of attaching a (usually infinite) collection of polytopes, and a certain combinatorial transformation relating them, to it. The first is via Mirror Symmetry, following a proposal of Coates--Corti--Kasprzyk--Galkin--Golyshev. The second is via symplectic topology, and comes from considering degenerating Lagrangian torus fibrations. We then relate these two collections using the Gross--Siebert program. I will also comment on the situation in higher dimensions, noting particularly that by 'inverting' the second method (degenerating Lagrangian fibrations) we can produce topological constructions of Fano threefolds.

Using tropical geometry and new methods in the theory of Fukaya categories, we explain a mirror symmetry equivalence relating the Fukaya category of a hypersurface and the category of coherent sheaves on the boundary of a toric variety.

I will discuss how \zeta(3) occurs in quantum corrections to the Einstein action, and how this causes \zeta(3) to be seen in the moduli space of CY manifolds. I will also draw attention to the fact that the dependence of the moduli space on \zeta(3) has a p-adic analogue.