Algebraic Geometry Days in Glasgow

We’re excited to announce our “Algebraic Geometry Days in Glasgow”, which will take place on May 8th – 9th 2024 at the University of Glasgow. We will have talks by our guests according to the following schedule: 

On Wednesday, May 8th (Room 116):

• Simon Schirren (Roma Tre) 13:30–14:30

• Giulia Gugiatti (Padova) 14:30–15:30

• Jon Woolf (Liverpool) (as the week’s speaker at the Algebra and Number Theory Seminar – Room 110) 16:00–17:00

On Thursday, May 9th (Room 311B):

• Robert Laterveer (Strasbourg) 11:00–12:00

• Lie Fu (Strasbourg) 13:30–14:30

• Nicholas Williams (Lancaster) 14:30–15:30

• Johannes Krah (Bielefeld) 16:00–17:00

We will confirm titles and abstracts soon, and post them below.

This is meant as a local gathering, so we won’t be able to offer travel or accommodation support to participants. But we hope to see some of you in Glasgow! 

For further information, feel free to email one of the organizers (Timothy De Deyn, Inder Kaur, Emma Lepri, Matthew Pressland, Franco Rota).

Titles and Abstracts:

Simon Schirren:

Virtual cycles are one of the main tools in enumerative geometry. Their existence stems from what is called a “perfect obstruction theory”, which goes back to the famous work of Behrend–Fantechi in the late 90s. We discuss virtual cycles for moduli of Higgs sheaves on a smooth projective surface or equivalently, for torsion sheaves on a Calabi–Yau threefold. If time permits, we’ll have a look at the perfect obstruction theory used in this setting and give an outlook on future projects.  

Giulia Gugiatti: Mirror symmetry and log del Pezzo surfaces outside of the known constructions

Mirror symmetry predicts a correspondence between the complex geometry (the B side) and the symplectic geometry (the A side) of certain pairs of objects. Homological mirror symmetry (HMS) interprets the correspondence as a categorical equivalence. In this talk, I will discuss mirror symmetry for a family of log del Pezzo surfaces falling out of the standard mirror symmetry constructions. Motivated by HMS, I will describe the derived category of the surfaces (their B side), and mention some results on the category encoding the A side of their (Hodge-theoretic) mirrors. As special cases, I will discuss del Pezzo surfaces of degree ≤ 3. This is based on work in progress with Franco Rota.

Robert Laterveer: Who’s afraid of MCK?

In an attempt to explain and unify results on the Chow ring of abelian varieties, results on the Chow ring of K3 surfaces, and the Beauville–Voisin conjecture for 

hyper-Kähler varieties, Shen–Vial have come up with the notion of “multiplicative Chow–Künneth decomposition” (MCK, for short).

I will introduce the notion of MCK, explain the motivation, and leisurely survey what is known and what is not known. Much of this is based on joint work with Lie Fu and Charles Vial, on joint work with Michele Bolognesi, and on joint work in progress with Inder Kaur.

Lie Fu: Categorical polynomial entropy 

In the classical setting of a (discrete) dynamical system, namely a space with a self-map, its entropy measures the complexity of the system and the notion of polynomial entropy provides a refinement allowing us to study “slow” dynamics.  In the categorical setting, Dimitrov, Haiden, Katzarkov, and Kontsevich developed the theory of category entropy for an endofunctor of a triangulated category. In my joint work with Yu-Wei Fan and Genki Ouchi, we defined and studied its natural refinement called categorical polynomial entropy. In this talk, I will present several fundamental properties of this invariant as well as giving many examples which demonstrates its usage. Time permitting, I will also sketch the closely related notion of mass growth rate and its polynomial version. 

Nicholas Williams: Donaldson–Thomas invariants for the Bridgeland–Smith correspondence

Giving a mathematical interpretation of part of the work of Gaiotto, Moore, and Neitzke on theories of class S[A_1], celebrated work of Bridgeland and Smith shows a correspondence between quadratic differentials on Riemann surfaces and stability conditions on certain 3-Calabi–Yau triangulated categories. Part of this correspondence is that finite-length trajectories of the quadratic differential correspond to stable objects of phase 1. Speaking roughly, these stable objects are then counted by an associated Donaldson–Thomas invariant. Work of Iwaki and Kidwai predicts particular values for these Donaldson–Thomas invariants according to the different types of finite-length trajectories, based on the output of topological recursion. We show that the category recently studied by Christ, Haiden, and Qiu produces Donaldson–Thomas invariants matching these predictions. This is joint work with Omar Kidwai.

Johannes Krah: Derived categories of blow-ups of the projective plane 

Let X be a del Pezzo surface. By Bondal–Orlov, D^b(X) admits only standard autoequivalences. Moreover, the braid group action on full exceptional collections up to shifts is transitive (Kuleshov–Orlov). In this talk, we consider blow-ups of the projective plane in 9 or more points. On the one hand, any blow-up of the plane in very general points admits only standard autoequivalences (this is joint work with Xianyu Hu) and on the blow-up of 9 very general points exceptional collections behave similar to the case of del Pezzo surfaces. On the other hand, on the blow-up of 10 general points, we construct a non-full exceptional collection of maximal length. This disproves a conjecture of Kuznetsov and a conjecture of Orlov.