## Winter School

## University of Warwick

## 12 - 15 December 2023

We will be hosting a Winter School for students and early career researchers at the University of Warwick. The first talk will start at 1:30pm on Tuesday 12th December and the last talk will finish at 2:30pm on Friday 15th December. Participants can arrive from 12:30pm on Tuesday 12th December.

The school will take place in the Zeeman Building. A campus map is below:

The school will consist of 4 lecture courses covering a range of topics of current research interest in algebraic geometry, delivered by:

Arend Bayer (Edinburgh)

Luca Giovenzana (Sheffield)

Inder Kaur (Glasgow)

Qaasim Shafi (Birmingham)

Schedule

Titles and Abstracts

Luca Giovenzana - A tour of compact Hyperkähler manifolds

1) Definitions and examples: After briefly recalling the notions of K3 surfaces, Kähler manifolds, and Hodge structures, I will introduce Hyperkähler manifolds, present some classical results such as the Beauville-Bogomolov theorem and the BBF quadratic form, and give examples.

2) Torelli Theorem and moduli spaces of Hyperkähler manifolds: In this lecture I wish to cover the basics of deformation theory, state the Bogomolov-Tian-Todorov theorem, the Torelli theorem and show the the relation between the moduli space of Hyperkähler manifolds and the quotient of the period domain.

3) Hyperkähler manifolds as moduli spaces of sheaves on K3 surfaces: In this lecture I will introduce the moduli spaces of stable sheaves and show how these provide examples of Hyperkähler manifolds. Finally, I will present the Hyperkähler tenfold found by O’Grady.

4) Hyperkähler manifolds and cubic fourfolds: Connections between K3 surfaces and cubic fourfolds have been known for a long time. I will present the construction of some of the Hyperkähler tenfolds arising from cubic fourfolds. Time permitting, I will end by presenting some open questions.

Inder Kaur - Ubiquitous techniques from Hodge theory

The classical Torelli theorem for curves states that a smooth, projective curve is determined by its principally polarised Jacobian. In 1968, Mumford and Newstead proved a higher rank version of this i.e a smooth, projective curve is determined by the polarised second intermediate Jacobian of the moduli space of rank 2 stable sheaves with fixed odd degree determinant over the curve. In 2018, with S. Basu and A. Dan we proved an analogue of Mumford-Newstead in the case when the underlying curve is irreducible nodal. The proof of this result will be a guiding target for this lecture series but the emphasis will be on learning the Hodge-thoeretic techniques needed for the proof such as: limit mixed Hodge structures, the local invariant cycle theorem, intermediate Jacobians, and Néron models.

Qaasim Shafi - Enumerative geometry of (log) K3 surfaces

This mini course will be about the enumerative geometry of K3 surfaces and certain log Calabi-Yau (or log K3) surfaces. Rational curves on a K3 surface are related to a certain modular form. Rational curves on a (toric) log K3 surface are related to counts of tropical curves. I will explain both of these results as well as introducing (log) Gromov-Witten theory, a modern framework for curve counting. The end goal will be generalisations of these results to higher genus curves, and in order to motivate the correct generalisation in the log K3 setting I will draw an analogy to the higher genus statement for K3 surfaces, which is the celebrated KKV conjecture.

Accommodation and Travel

We will provide accommodation for all participants from 12th to 15th December. We also have some funding to cover travel costs for UK-based students and postdocs.

If you would like to apply for travel funding, please indicate this on the registration form, and include an estimate of how much travel funding you expect to require. In case we receive more applications than we have funding to cover, travel funding will be allocated on a first-come, first-served basis.

Registration

Registration has now closed for this event.

Funding

This event is taking place as part of the UK Algebraic Geometry Network. The network is funded by a Network Support for the Mathematical Sciences grant from the Isaac Newton Institute, via EPSRC grant EP/V521929/1.