Warwick Algebraic Geometry Seminar

Abstracts

Roberto Pignatelli (Trento) - Rigid compact complex surfaces that are not infinitesimally rigid

A complex manifold is rigid if every small deformation of its complex structure is trivial. The usual argument for proving the rigidity of a complex manifold is by a well known "standard" cohomological criterium. Morrow and Kodaira posed in 1971 the problem of constructing a rigid manifold that does not satisfy it.
I will present a new criterium for rigidity of a manifold of dimension 2 that is more general than the standard one. As an application, I will produce a family of examples satisfying our criterium and not the classical one, so answering the above question.
This is a joint work with I. Bauer.

Adam Gyenge (Oxford) - Hilbert and Quot schemes of simple surface singularities

The Hilbert schemes of points on the affine complex plane has the structure of a Nakajima quiver variety. For a finite subgroup G of SL(2, C), I will discuss the construction of the Hilbert scheme of n points on the Kleinian singularity C^2/G as a Nakajima quiver variety for the framed McKay quiver of G with a specific non-generic stability parameter. I will also present a formula for the generating series collecting the Euler numbers of these varieties, a specific case of which was proved recently by Nakajima. Given enough time, I will explain the analogous problem for certain Quot schemes of C^2/G. (Joint work with Alastair Craw, Soren Gammelgaard and Balazs Szendroi).

Tyler Kelly (Birmingham) - A few aspects of hypergeometric functions in geometry and arithmetic

Hypergeometric functions are special functions that go back all the way to Euler and come all the way to Mirror Symmetry, Hodge theory, and Gromov-Witten Theory. I plan to give a (very biased) stroll through some of their contexts/computations in algebraic geometry and Hodge theory, and then explain the analogue of hypergeometric functions over finite fields. This will then give us a way to organise point counts on certain varieties over finite fields. This talk will involve joint work with C Doran, A Salerno, S Sperber, U Whitcher, and J Voight.

Will Donovan (Yau MSC, Tsinghua University) - Windows on the Pfaffian-Grassmannian correspondence

The Pfaffian-Grassmannian correspondence has been a key example in the development of Homological Projective Duality: it concerns certain pairs of non-birational Calabi-Yau threefolds which share a mirror partner, and can be proved to be derived equivalent. Physically, such an equivalence is associated to B-brane transport along a path in a mirror symmetry moduli space, and is dependent on the homotopy class of that path: I give a mathematical implementation of this dependency, in terms of mutations of an exceptional collection on the relevant Grassmannian. This follows a physical analysis of Hori and Eager-Hori-Knapp-Romo, and builds on work with Addington and Segal.

Anthony Várilly-Alvarado (Rice University) - Quasi-hyperbolicity via explicit symmetric differentials

A surface X is algebraically quasi-hyperbolic if it contains finitely many curves of genus 0 or 1. In 2006, Bogomolov and de Oliveira used asymptotic computations to show that sufficiently nodal surfaces of high degree in projective three-space carry symmetric differentials, and they used this to prove quasi-hyperbolicity of these surfaces. We explain how a granular analysis of their ideas, combined with computational tools and insights, yield explicit results for the existence of symmetric differentials, and we show how these results can be used to give constraints on the locus of rational curves on surfaces like the Barth Decic, Buechi's surface, and certain complete intersections of general type, including the surface parametrizing perfect cuboids. This is joint work with Nils Bruin and Jordan Thomas.

Michel van Garrel (Warwick) - Prelog Chow rings

In this joint work with Christian Böhning and Hans-Christian Graf von Bothmer, we explore Chow rings in the setting of log geometry, leading to the construction of prelog Chow rings of normal crossings varieties with smooth components. For a strictly semistable degeneration, the prelog Chow ring of the central fiber admits a specialization morphism from the Chow group of the generic fiber. After introducing the definition, I will describe examples illustration the construction.

Enrica Mazzon (MPI Bonn) - Tropical affine manifolds from mirror symmetry to Berkovich geometry

Mirror symmetry is a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, it suggests that certain geometrical objects (complex Calabi-Yau manifolds) should come in pairs, in the sense that each of them has a mirror partner and the two share interesting geometrical properties. In this talk, I will introduce some notions relating mirror symmetry to tropical geometry, inspired by the work of Kontsevich-Soibelman and Gross-Siebert. In particular, I will focus on the construction of a so-called “tropical affine manifold” using methods of non-archimedean geometry, and the guiding example will be the case of K3 surfaces and some hyper-Kähler varieties. This is based on joint work with Morgan Brown and a work in progress with Léonard Pille-Schneider.

Jean-Louis Colliot-Thélène (CNRS et Université Paris-Saclay) - On the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field

Let X be the product of a smooth projective curve C and a smooth projective surface S over a finite field F. Assume the Chow group of zero-cycles on S is just Z over any algebraically closed field extension of F (example : Enriques surface). We give a simple condition on C and S which ensures that the integral Tate conjecture holds for 1-cycles on X. An equivalent formulation is a vanishing result for unramified cohomology of degree 3. This generalizes a result of A. Pirutka (2016). It is a joint work with Federico Scavia (UBC, Vancouver).

Ana-Maria Castravet (Versailles) - Exceptional collections on moduli spaces of pointed stable rational curves

I will report on joint work with Jenia Tevelev answering a question of Orlov. We prove that the Grothendieck-Knudsen moduli spaces of pointed stable rational curves with n markings admit full, exceptional collections which are invariant under the action of the symmetric group S_n permuting the markings. In particular, a consequence is that the K-group with integer coefficients is a permutation S_n-lattice.

Hans-Christian Graf von Bothmer (Hamburg) - Rigid, not infinitesimally rigid surfaces with ample canonical bundle

It was a long standing problem of Morrow and Kodaira wether there are compact complex manifolds X with Def(X) a non reduced point. The first examples answering this question in the affirmative were given by Bauer and Pignatelli in 2018. As explained by Roberto Pignatelli in this seminar some weeks ago, these are certain surfaces of general type that have nodal canonical models. These canonical models are rigid AND infinitesimally rigid, while their desingularizations are still rigid, but not infinitesimally rigid anymore. One can therefore ask, if this situation is typical for rigid, not infinitesimally rigid surfaces of general type, or if it is possible to have examples with smooth canonical models. We answer this question also in the affirmative by constructing such a surface X via line arrangements and abelian covers. This construction was inspired by Vakil's version of „Murphy’s law in algebraic geometry“. (This is joint work with Christian Böhning and Roberto Pignatelli.)

Victor Batyrev (Tübingen) - Fine interior of lattice polytopes: MMP and Mirror Symmetry

The notion "Fine interior" of a lattice polytope P was introduced by Miles Reid in his famous lectures on canonical singularities. A non-degenerated hypersurface in a d-dimensional algebraic torus T is birational to a Calabi-Yau if and only if the Fine interior of its Newton polytope is a single lattice point. This is the starting point for my talk with the ambitious goal to explain my solutions to MMP and Mirror Symmetry for toric hypersurfaces.