The Sheffield Algebraic Geometry Seminar - autumn 2024

The mirror of a (family of) K3 surfaces is a (family of) K3 surfaces. But there is more to that: Type II degenerations of a K3 are expected to be mirror to elliptic fibrations on the mirror. I will give a precise definition and interpretation of this expectation and explain compatibility with Dolgachev-Nikulin mirror symmetry for K3 surfaces and the Fano/LG correspondence for (quasi) del Pezzo surfaces. Everything is based on arxiv:2405.12009 and a joint work in progress with Alan Thompson.

Relative Gromov—Witten invariants with only one relative marking are relative invariants with maximal contacts along the unique relative marking. The local-relative correspondence proved by van Garrel—Graber—Ruddat states that genus zero relative invariants with maximal contacts are equal to local Gromov—Witten invariants of a line bundle. Local invariants are usually easier to compute than relative invariants.  However, many computations (e.g. the degeneration formula) involve relative invariants beyond maximal contacts (i.e. with several relative markings). I will explain a generalization of the local-relative correspondence beyond maximal contacts which relates relative invariants with n+1 relative markings to invariants of a P^1-bundle with n relative markings. Repeating this process, they become absolute invariants of toric bundles.

Homological Mirror Symmetry (HMS) predicts a correspondence, expressed as a categorical equivalence, between the complex geometry (the B-side) and the symplectic geometry (the A-side) of suitable pairs of objects. In this talk, motivated by HMS, I will consider certain orbifold del Pezzo surfaces falling out of the standard mirror symmetry constructions. I will describe the derived category of the surfaces (their B-side), and discuss early results about the A-side of their Hodge-theoretic mirrors. This is joint work with Franco Rota.

The group of birational transformations of the projective n-space over a field K is called 'Cremona group of rank n over K'. The structure of such groups depends on the dimension and the field. I will discuss a result about complex Cremona groups of high rank that we obtain using birational geometry of certain surfaces over perfect fields.
More precisely, we describe the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field, proving that if it contains a point of degree 6, then it is not generated by elements of finite order. We then use this result to study Mori fibre spaces over the field of complex numbers and deduce that the Cremona group of rank at least 4 admits any group (of cardinality at most the cardinality of the complex numbers) as a quotient. Moreover, we prove that the 3-torsion in the abelianization of the Cremona group of rank at least 4 is uncountable. This is based on joint work with J. Blanc and E. Yasinsky.

Galois cohomology is a sequence of abelian groups that is associated to a field K and is non-trivial only if the field K is not algebraically closed. Given an algebraic variety over a field K, one could consider its invariants that take values in Galois cohomology of K. Since these invariants would vanish over the algebraic closure of K, this would be especially meaningful if the variety also 'trivializes' over the algebraic closure, i.e. becomes of some 'simple' or 'canonical' form. There are, however, many other cases which are not captured by this scheme, and there is no known universal definition of Galois cohomological invariant.

In my talk, I will present a framework in which one can work with Galois cohomology (of arbitrary fixed degree, with torsion coefficients) and smooth projective varieties on the same footing. In this setting the definition of Galois cohomological invariant of an arbitrary smooth projective variety becomes almost a tautology. In order to define this framework one needs to mix the ideas of Grothendieck motives with the algebro-geometric part of the chromatic homotopy theory, namely, Morava K-theories.
The talk is based on work in progress, partially joint with A.Lavrenov.

The goal of this talk is to introduce some numerical invariants of triangulated categories, and to illustrate them with some examples. These invariants are  categorical entropy (of an endofunctor, due to Dimitrov-Haiden-Katzarkov-Kontsevich) and Serre dimension (a derivative of categorical entropy of the Serre functor). Roughly speaking, dimension of a derived category with respect to an endofunctor measures how iterations of the functor affect the "boundaries" of a complex when applied to some fixed object. For the derived category of coherent sheaves on a smooth projective variety X the Serre dimension is just the dimension of X. However, in general the Serre dimension takes two values (called upper and lower dimension), it can be rational or negative. Time permitting, I will explain how to compute the Serre dimension for a class of triangulated categories, known as derived categories of gentle algebras in representation theory and as Fukaya categories of surface models in symplectic geometry.

The low-energy theory of 4d N=2 supersymmetric gauge theories is encoded in the Seiberg-Witten (SW) curve, which is naturally related to an integrable system. In the presence of Omega-background, this integrable system is given by Painlevé equations. In this talk we will show that the Nekrasov partition functions of 4d SU(2) susy gauge theories on the blowup of spacetime are tau functions, which solve the Painlevé equations in Hirota bilinear form. These solutions can be expressed as expansions in terms of the moduli of the quantum SW curve, and the construction can be applied also to non-Lagrangian theories. Furthermore, these solutions have manifest modular properties that provide a natural non-perturbative completion of the corresponding topological string partition function and directly lead to the BCOV holomorphic anomaly equations of the topological string.

Moduli spaces of K3 surfaces have long been studied, beginning with the proof of the Torelli theorem for K3s in the 70s. The most well-known compactification of this moduli space is still probably the Baily-Borel compactification, whose boundary components consist of Type II and Type III degenerations. Meanwhile, the GIT compactification is able to explicitly describe the elements of its boundary components with equations. In this talk we consider the simplest Type II degenerations, which are called Tyurin degenerations. We explicitly demonstrate a correspondence between the Type II boundary components of the Baily-Borel and GIT compactifications of the moduli space of K3 surfaces of degree 4. Using this, we will classify all Tyurin degenerations of K3 surfaces of degree 4.