This afternoon session was held during week 2 of the 2025 Summer Research Institute in Algebraic Geometry, July 21-25, 2025 at Colorado State University in Fort Collins, Colorado. 

Organizers: Arend Bayer, Alex Perry, and Giulia Saccà

Speakers 

• Benjamin Bakker (University of Illinois Chicago) 

• Agnieszka Bodzenta (University of Warsaw)

• Daniel Halpern-Leistner (Cornell University) 

• Chunyi Li (University of Warwick)

• Eyal Markman (University of Massachusetts Amherst)

• Amnon Neeman (Australian National University)

• Kieran O'Grady (Sapienza University of Rome)

• Tony Pantev (University of Pennsylvania)

• Alexander Perry (University of Michigan)

• Laura Pertusi (University of Milan) 

• Evgeny Shinder (University of Sheffield) 

• Yukinobu Toda (Kavli IPMU, University of Tokyo)

Schedule 

Monday, July 21 (YATES 104)

1:30-2:20: Daniel Halpern-Leistner

2:50-3:40: Benjamin Bakker

4:10-5:00: Alex Perry

Tuesday, July 22 (YATES 104) 

1:30-2:20: Chunyi Li

2:50-3:40: Amnon Neeman

4:10-5:00: Laura Pertusi

Thursday, July 24 (LCS ballroom)

1:30-2:20: Kieran O'Grady

2:50-3:40: Eyal Markman

4:10-5:00: Tony Pantev

Friday, July 25 (YATES 104)

1:30-2:20: Agnieszka Bodzenta

2:50-3:40: Yukinobu Toda

4:10-5:00: Evgeny Shinder

Titles and abstracts

Agnieszka Bodzenta: Periodic semiorthogonal decompositions

Abstract: I will recall the notion of a semiorthogonal decomposition and will introduce periodic semiorthogonal decompositions. I will explain their relation to spherical functors. I will then give examples of periodic decompositions. Two will be related to threefold flopping contractions and one to effective Cartier divisors. The talk is based on joint work with A. Bondal and W. Donovan.

Benjamin Bakker: Baily--Borel compactifications of period images and the b-semiampleness conjecture

Abstract: We address two questions related to the semiampleness of line bundles arising from Hodge theory.  First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which a natural line bundle extends amply.  This generalizes the Baily--Borel compactification of a Shimura variety, and for instance produces Baily--Borel type compactifications of moduli spaces of Calabi--Yau varieties.  We prove more generally that the Hodge bundle of a Calabi--Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture of Prokhorov--Shokurov.  The semiampleness results crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Kollár on the geometry of minimal lc centers to verify the extra conditions.  This is joint work with S. Filipazzi, M. Mauri, and J. Tsimerman.

Daniel Halpern-Leistner:  Moduli spaces of semistable objects in dg-categories.

Abstract: The moduli of vector bundles on a smooth curve is a beautiful illustration of many of the ideas and techniques that arise in algebraic moduli theory. In the past few years, a general theory of stability, $\Theta$-stratifications, and good moduli spaces for algebraic stacks has generalized many aspects of this moduli problem. However, questions of boundedness do not yet have a general answer -- this tends to be one of the trickiest parts of applying the machinery. I will discuss a new approach to boundedness for a vast generalization of the moduli of vector bundles on a curve: moduli spaces of objects in a dg-category that are semistable with respect to a Bridgeland stability condition. This involves reformulating the notion of a Bridgeland stability condition in a way that is especially convenient for moduli theory.

Chunyi Li: Bridgeland stability conditions with a real reduction

Abstract: I will survey some results on wall-crossing phenomena in the moduli spaces of stable sheaves on surfaces via Bridgeland stability conditions, as well as the construction of stability conditions on threefolds. I will then discuss a related concept, termed a reduced stability condition, which offers a new perspective on these results and may provide a pathway toward generalizing them to higher-dimensional varieties.

Eyal Markman: Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds

Abstract: In 1977 Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian varieties are said to be of Weil type and these Hodge classes are known as Weil classes.

The connected components of the moduli space of polarized abelian varieties A of Weil type have three discrete invariants, dim(A), the imaginary quadratic number field K, and the discriminant. The latter is the coset in Q^*/Nm(K^*) of the determinant of a natural Hermitian form.

We prove that the Weil classes are algebraic for all abelian sixfold of Weil type of discriminant -1, for all imaginary quadratic number fields. The algebraicity of the Weil classes follows for all abelian fourfolds of Weil type (for all discriminants and all imaginary quadratic number fields), by a degeneration argument of C. Schoen. The Hodge conjecture for abelian fourfolds is known to follow from the above result. 

Amnon Neeman: Excellent metrics on triangulated categories

Abstract: There is a recipe for going back and forth, between the bounded derived category of coherent sheaves and the category of perfect complexes. It involves completing with respect to certain metrics. What this talk will focus on is recent work, about conditions on the metrics that allow this back-and-forth passage.

Kieran O'Grady: Moduli of sheaves on hyperkähler manifolds

Abstract: We present some recent results on moduli spaces of sheaves on polarized hyperkähler manifolds of dimension greater than 2. We start by reviewing our model theory, i.e. that of sheaves on K3 surfaces.

Tony Pantev: Birational invariants from nc Hodge theory

Abstract: I will explain how a natural amalgam of classical Hodge theory with the nc Hodge structures arising from Gromov-Witten theory gives rise to new additive invariants of smooth projective varieties called Hodge atoms. Combined with Iritani's blow-up formula, Hodge atoms provide obstructions to birational equivalence. I will discuss applications to classical rationality problems. This is joint work with L. Katzarkov, M. Kontsevich, and T. Y. Yu.

Alex Perry: Deformations of t-structures and stability conditions 

Abstract: A t-structure is a tool for decomposing a derived (or more generally triangulated) category into simpler pieces, while a stability condition is an enhancement of this notion that allows for the formulation of moduli spaces of stable objects. These structures have applications ranging across algebraic geometry, representation theory, and topology, but in general they are difficult to construct. I will discuss a new deformation-theoretic construction method, which in particular leads to stability conditions on very general abelian varieties and hyperkähler varieties of K3^[n] type, as well as a description of such hyperkähler varieties as moduli spaces of stable objects on K3 categories. This is based on joint works with Chunyi Li, Emanuele Macrì, Paolo Stellari, and Xiaolei Zhao.

Laura Pertusi: Non-commutative abelian surfaces and Kummer type hyperkähler manifolds.

Abstract: Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkahler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkahler manifolds deformation equivalent to a generalized Kummer variety is not yet available.

In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkähler manifolds of generalized Kummer deformation type.

This is joint work in preparation with Arend Bayer, Alex Perry and Xiaolei Zhao.

Evgeny Shinder: Canonical semiorthogonal decompositions for G-surfaces

Abstract: We study semiorthogonal decompositions of derived categories of smooth projective varieties and their behavior under birational transformations. Motivated by Kuznetsov’s conjecture on cubic fourfolds and Kontsevich’s vision of canonical decompositions, we introduce the notion of G-atomic theory: canonical, mutation-equivalence classes of G-invariant semiorthogonal decompositions compatible with derived contractions with respect to a group G-action. We prove the existence of such a theory in dimension ≤ 2 for any group G, thereby establishing Kontsevich’s conjecture in this case. This framework refines and extends previous work by Auel–Bernardara and yields a complete birational classification of geometrically rational surfaces over perfect fields in terms of their atoms (canonical building blocks of the derived category). Connections are drawn to the atomic decompositions of quantum cohomology via ongoing work by Katzarkov–Kontsevich–Pantev–Yu. This is joint work in progress with Alexey Elagin and Julia Schneider.

Yukinobu Toda:  Dolbeault geometric Langlands conjecture via limit categories

Abstract: In this talk, I will introduce the notion of limit categories for cotangent stacks of smooth stacks as an effective version of classical limits of the categories of D-modules on them. Using the notion of limit categories, I will propose a precise and tractable formulation of the Dolbeault geometric Langlands conjecture, proposed by Donagi–Pantev as the classical limit of the geometric Langlands correspondence. It states an equivalence between the derived categories of coherent sheaves on moduli stacks of semistable G-Higgs bundles for a reductive group G and the limit category of moduli stacks of G^L-Higgs bundles without a stability condition. I will show the existence of a semiorthogonal decomposition of the limit category into quasi-BPS categories, which (when G=GL_r) categorify BPS invariants on a non-compact Calabi–Yau 3-fold playing an important role in Donaldson-Thomas theory. This semiorthogonal decomposition is interpreted as a Langlands dual to the semiorthogonal decomposition for moduli stacks of semistable Higgs bundles, obtained in our earlier work as a categorical analogue of PBW theorem in cohomological DT theory. It in particular yields a conjectural equivalence between quasi-BPS categories, which gives a categorical version of Hausel-Thaddeus mirror symmetry for Higgs bundles (for any G), This is joint work in progress with Tudor Pădurariu.