Summer Term 2022

Jakub Witaszek (University of Michigan). Quasi-F-splittings. Friday 29th Apr, 1:30-2:30pm. Huxley 140.

Abstract: What allowed for many developments in algebraic geometry and commutative algebra was a discovery of the notion of a Frobenius splitting, which, briefly speaking, detects how pathological positive characteristic Fano and Calabi-Yau varieties can be. Recently, Yobuko introduced a more general concept, a quasi-F-splitting, which captures much more refined arithmetic invariants. In my talk, I will discuss on-going projects in which we develop the theory of quasi-F-splittings in the context of birational geometry and derive applications, for example, to liftability of singularities. This is joint work with Tatsuro Kawakami, Hiromu Tanaka, Teppei Takamatsu, Fuetaro Yobuko, and Shou Yoshikawa.

Richard Thomas (Imperial College). Rank r DT theory from rank 1. Friday 6th May, 1:30-2:30pm. Huxley 140.

Abstract: Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X.
I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture these rank 1 “abelian” invariants are determined by the GW invariants of X.
Along the way we also express rank r DT invariants (and GW invariants) in terms of invariants counting rank 0 sheaves supported on surfaces in X. These invariants are predicted by physicists’ S-duality to be governed by (vector-valued, mock) modular forms.

Qaasim Shafi (Imperial College). Logarithmic Toric Quasimaps. Friday 13th May, 1:30-2:30pm. Huxley 140.

Abstract: Quasimaps provide an alternate curve counting system to Gromov-Witten theory, related by wall-crossing formulas. Relative (or logarithmic) Gromov-Witten theory has proved useful for constructions in mirror symmetry, as well as for determining ordinary Gromov-Witten invariants via the degeneration formula. I’ll discuss how to build a theory of logarithmic quasimaps in the toric case, and why one might want to do so.

Alexei Skorobogatov (Imperial College). Reduction of Kummer surfaces. Friday 20th May, 1:30-2:30pm. Huxley 140.

Abstract: If a Kummer surface over a discretely valued field of characteristic zero has good reduction, then it comes from an abelian surface with good reduction. The converse holds if the residual characteristic is not 2. In a joint work with Chris Lazda we obtain a necessary and sufficient condition for good reduction of Kummer surfaces attached to abelian surfaces with good, non-supersingular reduction in residual characteristic 2. The supersingular case is very interesting but seems to be completely open.

Ignacio Barros Reyes (Universität Paderborn). The Kodaira classification of the moduli of hyperelliptic curves. Friday 27th May, 1:30-2:30pm. Huxley 140.

Abstract: We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces have non Q-factorial singularities forcing us to work with a better birational model provided by Hurwitz spaces. We complete the Kodaira classification. Similarly, we consider the natural finite cover given by ordering the Weierstrass points. In this case, we show that the Kodaira dimension is one when n = 4 and of general type when n ≥ 5. For this, we carry out a singularity analysis of ordered and unordered pointed Hurwitz spaces. We show that the ordered space has canonical singularities and the unordered space has non-canonical singularities. We describe all non-canonical points and show that pluricanonical forms defined on the full regular locus extend to any resolution. Further, we provide a full classification of the structure of the pseudo-effective cone of Cartier divisors for the moduli space of hyperelliptic curves with marked points. We show the cone is non-polyhedral when the number of markings is at least two and polyhedral in the remaining cases. This is all joint work with S. Mullane.

Hélène Esnault (Freie Universität Berlin). Recent developments on rigid local systems. Friday 10th June, 1:30-2:30pm. Huxley 140.

Abstract: We shall review some of the general problems which are unsolved on rigid local systems and arithmetic l-adic local systems. We ‘ll report briefly on a proof (2018 with Michael Groechenig) of Simpson’s integrality conjecture for cohomologically rigid local systems. While all rigid local systems in dimension 1 are cohomologically rigid (1996, Nick Katz), we did not know until very recently of a single example in higher dimension which is rigid but not cohomologically rigid. We’ll present one series of examples in any rank (2022, joint with Johan de Jong and Michael Groechenig).

Please note the unusual room:

Lenny Taelman (University of Amsterdam). Deformations of Calabi–Yau varieties in mixed characteristic. Friday 17th June, 1:30-2:30pm. Huxley 410.

Abstract: (Joint work with Lukas Brantner.) We study deformations of smooth projective varieties with trivial canonical bundle in positive and mixed characteristic. We show that (under suitable hypotheses) these are unobstructed. This is an analogue to the Bogomolov-Tian-Todorov theorem (in characteristic zero), and is inspired by previous results by Ekedahl and Shepherd-Barron. We also show that “ordinary” varieties with trivial canonical bundle admit a preferred “canonical lift” to characteristic zero. This generalizes results of Serre-Tate (for abelian varieties) and Deligne (for K3 surfaces), and is inspired by previous results of Achinger and Zdanowicz. Our proofs rely in an essential way on “derived” deformation theory as developed by Pridham and Lurie.

Soheyla Feyzbakhsh (Imperial College). Moduli spaces of stable objects in the Kuznetsov component of cubic threefolds. Friday 24th June, 1:30-2:30pm. Huxley 140.

Abstract: I will first explain a general criterion that ensures a fractional Calabi-Yau category of dimension less than or equal to 2 admits a unique Serre-invariant stability condition up to the action of the universal cover of GL+(2, R). This result can be applied to a certain triangulated subcategory (called the Kuznetsov component) of the bounded derived category of coherent sheaves on a cubic threefold. As an application, I will show (i) a categorical version of the Torelli theorem holds for cubic threefolds, and (ii) the moduli space of Ulrich bundles of fixed rank r greater than or equal to 2 on a cubic threefold is irreducible. The talk is based on joint work with Laura Pertusi and a group project with A. Bayer, S.V. Beentjes, G. Hein, D. Martinelli, F. Rezaee and B. Schmidt.

Please note that the seminar will be online:
Paul Schwahn (University of Stuttgart). Stability and rigidity of normal homogeneous Einstein manifolds. Friday 1st July, 1:30-2:30pm. Online.

Abstract: The stability of an Einstein metric is decided by the (non-)existence of small eigenvalues of the Lichnerowicz Laplacian on tt-tensors. In the homogeneous setting, harmonic analysis allows us to approach the computation of these eigenvalues. This is easy on symmetric spaces, but considerably more difficult in the non-symmetric case. I review the case of irreducible symmetric spaces of compact type, prove the existence of a non-symmetric stable Einstein metric of positive scalar curvature, and give an outlook on how to investigate the normal homogeneous case. Furthermore, I explore the rigidity and infinitesimal deformability of homogeneous Einstein metrics.