Friday short talks: location Center Hall, Room 119
Soheil Memarian
Title: Hyperbolicity Properties of Non-Compact Complex Ball Quotients
Abstract: Let X be a complex ball quotient associated with a nonuniform neat lattice in PU(n,1). We study positivity properties of the cotangent bundle of X and deduce that if the cusps of X are sufficiently deep, then every subvariety of X is of general type. Moreover, we show that the canonical volume of subvarieties of X is controlled by the systole of X.
Jon Kim
Title: Moduli of (b, c)-weighted stable marked cubic surfaces
Abstract: We will construct many KSBA compactifications of cubic surfaces with a marked line by computing the full wall-and-chamber decomposition for KSBA compactifications where we asymmetrically weight a marked line with weight b, and the other 26 lines uniformly with weight c. This generalizes the recent work done by Schock where the lines are weighted uniformly.
Ben Church
Title: Unirational surfaces in positive characteristic
Abstract: In characteristic zero, Castelnuovo proved that every unirational surface is rational. In positive characteristic, this fails dramatically: there exist many non-rational, often even general-type, surfaces that are nevertheless unirational. In 1977, Shioda conjectured that such phenomena are completely explained by the Galois representation—i.e., a simply-connected surface is unirational if and only if it is supersingular. In this talk, I will present a counterexample to this conjecture. The construction requires a new obstruction, inspired by ideas from the study of the hyperbolicity of complex varieties
Feiyang Lin
Title: Tautological bundles on Quot schemes
Abstract: On the Quot scheme parametrizing quotients of a trivial vector bundle on P^1, there are tautological vector bundles that arise naturally from the moduli problem. I will explain some results towards understanding their cohomology. This is joint work with Ajay Gautam and Shubham Sinha.
Ying Wang
Title: Calabi-Yau under non-Archimedean lens
Abstract: We discuss how to study geometric and topological properties of Calabi--Yau varieties from a non-Archimedean perspective, using Berkovich spaces.
Shubham Saha
Title: A Strong Franchetta Property for the universal moduli of bundles on hyperelliptic curves
Abstract: We define a strong Franchetta property for families and show that different versions of the universal moduli functors of bundles on hyperelliptic curves (slope-stable, Higgs stable, and stack of all bundles) satisfy the strong Franchetta property for rank two. Consequently, we can compute the rational Chow rings in several instances of these moduli functors in low genus. This is based on ongoing joint work with Soumik Ghosh.
Shend Zhjeqi
Title: Applications of Hodge modules to the geometry of stacks
Abstract: In this talk we will discuss recent progress on applications of Hodge modules to geometric properties of stacks.
We will focus on explicit examples to showcase the meaning and motivation for our results.
Everything discussed is joint work with Sebastian Casalaina-Martin.
Ting Gong
Title: A Torelli theorem for moduli of twisted sheaves
Abstract: The Torelli theorem for moduli of vector bundles says that if we have two smooth projective curves with isomorphic moduli space of vector bundles of fixed rank and degree, then the two curves are isomorphic over an algebraically closed field. In our setting, taking an arbitrary field, the moduli space of vector bundles descend to that of twisted vector bundles, and we show that curves, along with a Brauer class, can be recovered from the moduli of twisted vector bundles. However, the proof is much more subtle, especially in the small rank and degree cases, where we used the stacky structure, and identified the Torelli theorem as an association of Clifford algebras. This is work in progress with Soham Ghosh, Max Lieblich and Arnab Roy.
Ignacio Rojas
Title: A Pseudostable Analogue of the Multiple Cover Formula
Abstract: The Gromov-Witten invariants of \(\mathcal{O}(-1)\oplus \mathcal{O}(-1)\) were computed by Faber and Pandharipande using Atiyah-Bott localization, which expresses them in terms of Hodge integrals. In 1991, Schubert introduced an alternative compactification of \(M_{g,n}\) by allowing cuspidal singularities which arose after contracting elliptic tails. This gives rise to pseudo-stable curves. The aim of this work is to extend the multiple cover formula to this pseudo-stable setting. After applying localization, the invariants are computed as Hodge integrals on moduli spaces of pseudo stable curves. Based on work of Cavalieri and Williams (2024) and using a formula for pseudo-stable lambda classes described by Cavalieri, Gallegos, Ross, Van Over, and Wise (2022), we aim to find a pseudo-stable version of the multiple cover formula.
Lillian McPherson
Title: The algebra of symmetric tensors for ruled surfaces
Abstract: The algebra of symmetric tensors for a smooth complex projective variety X was defined in 2024 by Beauville and Liu as $H^0 (X, S^\bullet T_X)$. This graded $\mathbb{C}$-algebra is defined analogously to the cotangent ring, $H^0 (X, S^\bullet \Omega^1_X)$, studied starting with Sakai's work in 1978. While the algebra of symmetric tensors has only begun to be studied recently compared to the cotangent ring, its computation produces many interesting algebras and can give information about the projective variety, such as the positivity of its cotangent bundle $T^\ast X$. This talk will focus on the computation of the algebra of symmetric tensors for ruled surfaces of genus $\geq 2$. Expanding on the example given in Beauville and Liu's paper, we will see that the ruled surfaces case produces several interesting algebras despite being trivial for ruled surfaces defined by a general stable vector bundle.
Alex Wang
Title: Rational points on varieties and the Brauer--Manin obstruction
Abstract: A central topic of study in arithmetic geometry investigates the structure of rational points on algebraic varieties. In this talk, we discuss the Brauer--Manin obstruction, a method in which varieties can fail to have rational points despite having points over all completions, and survey my work in this area.
Saturday & Sunday talks: location Center Hall, Room 101
Title: Realizable classes in Grassmannians
Abstract: Given a class in the cohomology of a projective manifold, one can ask whether the class can be represented by an irreducible subvariety. If the class is represented by an irreducible subvariety, we say that the class is realizable. One can further ask whether the subvariety can be taken to satisfy additional properties such as smooth, nondegenerate, rational, etc. These questions are closely related to central problems in algebraic geometry such as the Hodge Conjecture or the Hartshorne Conjecture. Recently, June Huh and collaborators have made significant progress in understanding realizable classes in products of projective spaces. In this talk, I will discuss joint work with Julius Ross on realizable classes in Grassmannians.
Title: Baily--Borel compactifications of period images and the b-semiampleness conjecture
Abstract: A fundamental tool to study algebraic varieties is given by morphisms to projective space. A line bundle is called semiample if some positive tensor multiple provides a morphism to projective space. Unlike the case of ample line bundles, there are no general numerical criteria to determine whether a line bundle is semiample. In particular, proving the semiampleness of line bundles can be a challenging task. In this talk, we will report on recent developments regarding the semiampleness of line bundles coming from Hodge theory. As a consequence, we will obtain functorial compactifications of general period images, generalizing classic work of Baily--Borel and thus confirming a conjecture of Griffiths. This talk is based on joint work with Bakker, Mauri, and Tsimerman.
Title: An approach to curves in abelian surfaces using Fourier--Mukai and quadratic forms
Abstract: Let $A$ be a complex abelian surface with a primitive polarization of $(1,d)$-type and symmetric polarizing line bundle $L$. By work of Yoshioka, there is an isomorphism between the generalized Kummer variety of $A$, which parametrizes $d$-points that ``sum to $0$'' in the group law, and another hyperk\"ahler variety of Kummer type, which parametrizes certain sheaves in the relative compactified Jacobian of the linear system of $L$. The inverse group law map on $A$ induces an involution on each of these varieties that is compatible with the isomorphism between them. When $d=4$, the fixed locus contains 140 isolated points. On the generalized Kummer, these are easy to describe: they are the choices of four distinct $2$-torsion points in $A$ that sum to $0$. In this talk, we explore what these 140 isolated fixed points are on the other side of the isomorphism, and what this isomorphism tells us about the linear system $L$. This work is joint with Graham McDonald and Peter McDonald.
Title: Higher direct images of dualizing sheaves III.
Abstract: In a series of extremely influential papers that appeared 40 years ago, Kollár showed several surprising properties of higher direct images of dualizing sheaves via projective morphisms. In this talk I will report on recent joint work with Kollár that should be considered a sequel to his original papers (hence the counter in the title). We show that for a flat morphism between varieties with rational singularities even stronger statements are true. In particular, Kollár's splitting theorem from 40 years ago holds for the higher direct images of the relative dualizing sheaf and this has several interesting consequences. For instance it implies that the higher direct images of the structure sheaf are locally free. As an application in a seemingly unrelated direction this leads to the statement that the identity component of the relative Picard scheme of a flat morphism between varieties with rational singularities is a smooth algebraic group scheme.
Title: Cubic fourfolds with many Fourier Mukai partners
Abstract: Two cubic fourfolds are Fourier Mukai partners if there is an equivalence between their Kuznetsov components. It has been suggested by Huybrechts that a pair of Fourier Mukai cubic fourfold partners are birational, however examples of such phenomena are rare. In an attempt to investigate this phenomena, we develop an algorithm for counting the number of Fourier Mukai partners of a given cubic fourfold, and run it on cubics with automorphisms. We prove that a cubic fourfold admitting a symplectic involution has 1120 non-trivial Fourier-Mukai partners, all birational but non-isomorphic. We’ll explain the 1120 in terms of root systems and non-syzygetic pairs of cubic scrolls.
Title: Algebraic geometry of hyper-Kähler varieties via hyperholomorphic bundles
Abstract: Hyper-Kähler varieties are higher-dimensional analogues of K3 surfaces. Addressing many fundamental questions about hyper-Kähler varieties requires constructions of specific algebro-geometric objects. For example, resolving the Bondal–Orlov D-equivalence conjecture involves constructing suitable Fourier–Mukai kernels; approaching the period–index problem requires constructing appropriate vector bundles; attacking the Orlov conjecture for motives requires constructing certain algebraic cycles. In this talk, I will explain how Markman’s hyperholomorphic bundles, which relied heavily on transcendental techniques (metric, twistor lines, etc) of Verbistsky, can be used to produce the desired algebro-geometric constructions in all the directions mentioned above. This is based on joint work with James Hotchkiss, Davesh Maulik, Qizheng Yin, and Ruxuan Zhang.
Title: F-bundles, mirror symmetry and birational invariants
Abstract: There are two main themes in algebraic geometry research: classification and enumeration. In this talk, we will bring them together by applying enumeration to birational classification, more precisely, we will construct new birational invariants from curve counting invariants. I will introduce F-bundles, the spectral decomposition theorem and the equivariant unfolding theorem. I will discuss applications to quantum cohomology, mirror symmetry and birational geometry, in particular, to the proof of irrationality of very general cubic fourfolds. The talk is based on arXiv preprints 2411.02266, 2505.09950, 2508.05105 and 2508.15770.