Spring Term 2023

Jenia Tevelev (University of Massachusetts, Amherst). Semi-orthogonal decompositions of moduli spaces. Friday 13th Jan, 1:30-2:30pm. Huxley 140.

Abstract: Let C be a smooth projective curve of genus g at least 2 and let N be the moduli space of stable rank 2 vector bundles on C of odd degree. It is a Fano variety of dimension 3g-3, Picard number 1 and index 2. We construct a semi-orthogonal decomposition of the bounded derived category of N conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has two blocks for each i-th symmetric power of C for i = 0,…,g−2 and one block for the (g − 1)-st symmetric power. Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows. Joint work with Sebastian Torres.

Please note the unusual time and room:
Dennis Sullivan (Stony Brook University). The third exterior power of a d-dimensional vector space is interesting. Friday 13th Jan, 3pm. Huxley 311.

Abstract: There are “roughly“ (d choose 3 )-d ^2 parameters to determine the “nature” of an element in this exterior space . This number of moduli is positive starting in dimension at least nine . For dimensions 6, 7 and 8 the structures are finite and important in various areas of mathematics. Finally there is a general result relating these structures precisely to the homology intersection rings of all possible closed oriented three manifolds.

André Belotto da Silva (Institut de Mathématiques de Jussieu-Paris Rive Gauche). Partial desingularization. Friday 20th Jan, 1:30-2:30pm. Huxley 140.

Abstract: Can we understand the nature of the singularities that have to be admitted after a blow-up sequence that preserves the normal crossings locus of an algebraic (or complex-analytic) variety X? For example, every surface can be transformed by blowings-up preserving normal crossings to a surface with at most additional Whitney umbrella singularities. We will discuss general conjectures, and solutions for X of dimension up to four. The techniques involve circulant matrices, elementary Galois theory and Newton-Puiseux expansion in several variables. Work in collaboration with Edward Bierstone and Ramon Ronzon Lavie.

Fabio Bernasconi (EPFL). On the properness of the moduli space of stable surfaces over Z[1/30]. Friday 27th Jan, 1:30-2:30pm. Huxley 140.

Abstract: The moduli functor M_{n,v} of stable varieties of dimension n is a higher-dimensional generalization proposed by Kollár and Shepherd-Barron to find a geometric compactification of moduli spaces of varieties with ample canonical bundle, similar to the Deligne–Mumford compactification for smooth curves. In the past decades, work of various birational geometers showed that M_{n,v} is a proper DM stack of finite type, admitting a coarse projective moduli space over the complex numbers.
In positive and mixed characteristic many basic questions on moduli of stable varieties are still open. Recent progress on the MMP allowed to show that M_{2,v} exists as a separated Artin stack of finite type over Z[1/30]. In this talk, I will report on a joint work in progress with E. Arvidsson and Zs. Patakfalvi where, assuming the existence of semi-stable reduction, we conclude that M_{2,v} is proper. To achieve this, we give a geometric characterisation of the failure of the S_3-condition for 3-dimensional log canonical singularities.

Ana-Maria Castravet (Université Paris-Saclay, Versailles). Higher Fano manifolds. Friday 3rd Feb, 1:30-2:30pm. Huxley 140.

Abstract: Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been some effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds.
In this talk, I will discuss higher Fano manifolds which are defined in terms of positivity of higher Chern characters. After a brief survey of what is currently known, I will present recent joint work with Carolina Araujo, Roya Beheshti, Kelly Jabbusch, Svetlana Makarova, Enrica Mazzon and Nivedita Viswanathan, regarding toric higher Fano manifolds. I will explain a strategy towards proving that projective spaces are the only higher Fano manifolds among smooth projective toric varieties.

Toby Stafford (University of Manchester). Invariant holonomic systems for symmetric spaces. Friday 10th Feb, 1:30-2:30pm. Huxley 140.

Abstract: Fix a complex reductive Lie group G with Lie algebra g and let V be a symmetric space over g with ring of differential operators D(V). A fundamental class of D(V)-modules consists of the admissible modules (these are natural analogues of highest weight g-modules). In this lecture I will describe the structure of some important admissible modules. In particular, when V=g these results reduce to give Harish-Chandra’s regularity theorem for G-equivariant eigendistributions and imply results of Hotta and Kashiwara on invariant holonomic systems. If I have time I will describe extensions of these results to the more general polar g-representations. A key technique is relate (the admissible module over) invariant differential operators on V to (highest weight modules over) Cherednik algebras. This research is joint with Bellamy, Levasseur and Nevins.

Stefan Kebekus (University of Freiburg). An Albanese construction for Campana’s C-pairs. Friday 17th Feb, 1:30-2:30pm. Huxley 140.

Abstract: Almost twenty years ago, Campana introduced C-pairs to complex geometry. Interpolating between compact and non-compact geometry, C-pairs capture the notion of “fractional positivity” in the “fractional logarithmic tangent bundle”. Today, they are an indispensible tool in the study of hyperbolicity, higher-dimensional birational geometry and several branches of arithmetic geometry. This talk reports on joint work with Erwan Rousseau. Aiming to construct a “C-Albanese variety”, we clarify the notion of a “morphism of C-pairs” and discuss the beginnings of a Nevanlinna theory for “orbifold entire curves”.

Marta Panizzut (Max Planck Institute, Leipzig). Khovanskii bases and toric degenerations of Cox rings. Friday 24th Feb, 1:30-2:30pm. Huxley 140.

Abstract: We study combinatorial properties of toric degenerations of Cox rings of blow-ups of 3-dimensional projective space at points in general positions by computing Khovanskii bases. We focus in particular on Ehrarht-type formulas for the multigraded Hilbert functions of these spaces,as evaluating these functions will amount to counting lattice points on polytopes. From our computations, it follows that the presentation ideal of the Cox ring of the blow-up at seven points is quadratically generated, as conjectured by Lesieutre and Park. The talk is based on recent work with Mara Belotti and will also highlight computational aspects of the project.

Goncalo Tabuada (University of Warwick). Grothendieck classes of quadric hypersurfaces and involution varieties. Friday 3rd March, 1:30-2:30pm. Huxley 140.

Abstract: The Grothendieck ring of varieties, introduced in a letter from Alexander Grothendieck to Jean-Pierre Serre (August 16th 1964), plays an important role in algebraic geometry. However, despite the efforts of several mathematicians, the structure of this ring still remains poorly understood. In this talk, in order to better understand the Grothendieck ring of varieties, I will describe some new structural properties of the Grothendieck classes of quadric hypersurfaces and involution varieties. More specifically, by combining the recent theory of noncommutative motives with the classical theory of motives, I will show that if two quadric hypersurfaces (or, more generally, two involution varieties) have the same Grothendieck class, then they have the same even Clifford algebra and the same signature. As an application, this implies in numerous cases (e.g., when the base field is a local or global field) that two quadric hypersurfaces (or, more generally, two involution varieties) have the same Grothendieck class if and only if they are isomorphic.

Tudor Padurariu (Columbia University). Categorical and K-theoretic Donaldson-Thomas theory of C^3. Friday 10th March, 1:30-2:30pm. Huxley 140.

Abstract: Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of C^3, when the Donaldson-Thomas (DT) invariant of sheaves of zero dimensional support and length d is p(d), the number of plane partitions of d. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies a graded vector space with Euler characteristic equal to the DT invariant. I will talk about two other refinements (categorical and K-theoretic) of DT invariants, focusing on the explicit case of C^3. In particular, we show that the K-theoretic DT invariant for d points on C^3 also equals
p(d). This is joint work with Yukinobu Toda.

Mykola Matviichuk (Imperial College). Poisson brackets on Hilbert schemes of surfaces. Friday 17th March, 1:30-2:30pm. Huxley 140.

Abstract: A classical theorem of Bottacin states that a Poisson bracket on an algebraic surface induces a Poisson bracket on its Hilbert scheme of points. I will discuss the local behavior of this Poisson bracket, e.g. the singularities of the divisors where it drops rank. In good cases, I will explain how to construct a toric degeneration of the Bottacin’s Poisson bracket. Time permitting, I will also discuss the holonomicity condition, defined by Pym and Schedler, that ensures that the Poisson cohomology is a perverse sheaf. This is joint work with Brent Pym and Travis Schedler.

Laura Fredrickson (University of Oregon). The asymptotic geometry of the Hitchin moduli space. Friday 24th March, 1:30-2:30pm. Huxley 140.

Abstract: Hitchin’s equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\”uller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperk\”ahler metric. An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently. I will discuss some recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.