Algebraic Geometry Archives for Fall 2025 to Spring 2026
Wall crossing for moduli of stable pairs
When: Mon, September 23, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Fanjun Meng (Johns Hopkins University) - https://sites.google.com/u.northwestern.edu/fanjunmeng/home
Abstract: Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.
Motivic Euler characteristics and the transfer
When: Wed, October 2, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Roy Joshua (OSU and UMD) - https://people.math.osu.edu/joshua.1/
Abstract: In the first part of the talk, we will discuss the solution
to a conjecture of Fabien Morel on the motivic
Euler characteristics of certain homogeneous spaces as they relate to
splittings in the motivic
homotopy category. In the second part of the talk, we will discuss
certain applications
of these to computations in Algebraic K-Theory and Brauer groups.
The Cohen-Lenstra moments over function fields
When: Wed, October 9, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Aaron Landesman (Harvard/MIT) - https://people.math.harvard.edu/~landesman/index.html
Abstract: The Cohen-Lenstra heuristics are influential conjectures in arithmetic statistics from 1984 which predict the average number of p-torsion elements in class groups of quadratic fields, for p an odd prime. So far, this average number has only been computed for p = 3. In joint work with Ishan Levy, we verify this prediction for arbitrary p over suitable function fields. The key input to the proof is a computation of the stable homology of Hurwitz spaces associated to dihedral groups.
Elliptic surfaces over an elliptic base
When: Mon, November 4, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: François Greer (MSU) - https://sites.google.com/msu.edu/francoisgreer/
Abstract: Elliptic surfaces are a fairly well understood class of complex projective surfaces. They come with two discrete invariants, $g$ and $d$, which are both nonnegative integers. I will discuss some new results (joint with P. Engel, A. Ward, and Y. Zhang) about the moduli space and Hodge theory of elliptic surfaces with $(g,d)=(1,1)$. While they have Kodaira dimension one, they behave like K3 surfaces in many respects, and they provide an interesting test case for the Hodge Conjecture in dimension 4.
Nowhere vanishing 1-forms on varieties admitting a good minimal model.
When: Wed, November 13, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ben Church (Stanford/Harvard) - https://web.stanford.edu/~bvchurch/about/
Abstract: A theorem of Popa and Schnell shows that if a smooth projective variety X admits a 1-form with no zeros it cannot be of general type. However, one expects far more stringent constraints on the geometry of those X actually admitting nonvanishing 1-forms. If X is not uniruled and assuming the conjectures of MMP, we show that X is birational to an isotrivial fibration over an abelian variety. This partially answers conjectures of Hao--Schreieder, Meng--Popa, and Chen--Church--Hao. The proof involves a decomposition result for families of Calabi-Yau varieties surjecting onto a fixed abelian variety. If X is uniruled, we also give a weak structure theorem that relies on using higher direct image Hodge modules in the method of Popa--Schnell.
Boundedness of singularities and discreteness of local volumes
When: Thu, November 21, 2024 - 3:15pmWhere: MTH 1310
Speaker: Ziquan Zhuang (JHU) -
Abstract: The local volume of a Kawamata log terminal (klt) singularity is an invariant that plays a central role in the local theory of K-stability. By the stable degeneration theorem, every klt singularity has a volume preserving degeneration to a K-semistable Fano cone singularity. I will talk about a joint work with Chenyang Xu on the boundedness of Fano cone singularities when the volume is bounded away from zero. This implies that local volumes only accumulate around zero in any given dimension.
Rational surfaces with a non-arithmetic automorphism group
When: Mon, November 25, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Jennifer Li (Princeton) - https://web.math.princeton.edu/~jl5270/
Abstract: Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur. I will discuss joint work with Sebastián Torres where we give examples of rational surfaces with the same property. Our examples Y are log Calabi-Yau surfaces, i.e., there is a reduced normal crossing divisor D in Y such that KY+D=0.
Formal GAGA for Brauer classes
When: Mon, December 2, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Siddharth Mathur (Universidad Católica de Chile) - https://sites.google.com/view/sidmathur/home
Abstract: Deformation theory studies the variation of geometric data as a variety moves in a family. In this talk, we will introduce some new deformation-theoretic methods and use them to address a question of Grothendieck from the 1960s: how does the Brauer group of a family compare with that of the various thickenings of a special fiber. We will begin by probing a seemingly simpler question concerning the infinitesimal behavior of invertible sheaves and end by using this to answer Grothendieck's question. This is joint work with Andrew Kresch.
Curve Counting for Abelian Surface Fibrations and Modular Forms
When: Wed, December 4, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Stephen Pietromonaco (University of Michigan) - https://sites.google.com/view/spietromonaco-math/home
Abstract: A long-standing prediction of string theory and mirror symmetry is that certain formal generating series of curve-counting invariants are in fact expansions of quasi-modular objects. In this talk I will discuss on-going work with Aaron Pixton aiming to understand this modularity for Calabi-Yau threefolds fibered by Abelian surfaces (of Picard rank 2 or 3). For clarity, I will focus on the explicit example of the banana manifold. We will be interested in the Gromov-Witten (GW) potentials F_{g,k} where we assemble into a generating series the GW invariants of genus g for curve classes of degree k over the base. For the banana manifold, the GW potentials are formal series in 19 variables, which we conjecture to be Siegel-Jacobi forms for the E_{8} lattice, as introduced by Ziegler in the late 80s. We give a collection of evidence for the conjecture, including a proof of the elliptic transformation law.
Motives
When: Tue, December 10, 2024 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Toni Annala (University of Chicago) - https://tannala.com/
Abstract: Algebraic topology is the study of shapes through invariants called cohomology
theories. Algebraic geometry explores the geometry of solution sets of
polynomial equations over commutative rings. Over the past 50 years,
significant progress has been made in bridging these fields, resulting in a
rich theory of cohomology theories in algebraic geometry. To bring unity to
this landscape, Grothendieck introduced the theory of motives, which are
essentially geometric pieces of cohomologies of smooth compact varieties.
Unfortunately, not all cohomology theories (e.g. algebraic de Rham cohomology)
can be fruitfully understood from this perspective, as they behave in a
seemingly pathological fashion on open varieties. In this talk, I explain a new
method of systematically overcoming this, by modifying a cohomology theory in
such a way that its values on smooth projective varieties remain unchanged.
Curves on Varieties and the Degree of Irrationality
When: Wed, March 5, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Junyan Zhao (UMD) - https://sites.google.com/uic.edu/jzhao
Abstract: Inspired by the Noether-Lefschetz theorem, it has been conjectured that the degree of any curve on a (very) general complete intersection variety is divisible by the degree of the variety itself. In this talk, we present a weaker version of this conjecture. Using this result, we confirm a conjecture by Bastianelli--De Poi--Ein--Lazarsfeld--Ullery regarding measures of irrationality. This is joint work with Nathan Chen and Benjamin Church.
The extension of numerically trivial divisor on a family
When: Wed, April 2, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Lingyao Xie (UCSD) - https://sites.google.com/view/lingyaoxie/home
Abstract: Let $f:X\to S$ be a projective morphism of normal varieties. Assume $U$ is an open subset of $S$ and $L_U$ is a divisor on $X_U:=X\times_S U$ such that $L_U\equiv_U 0$. We explore when it is possible to extend $L_U$ to a global $\mathbb{Q}$-divisor $L$ on $X$ such that $L\equiv_S 0$. In particular, we can show that such $L$ always exists after a (weak) semi-stable reduction when $S$ is a curve.
On the other hand, we give an example showing that $L$ may not exist (after any reasonable modification of $f$) if $S$ has dimension $\ge2$, which also gives an $f_U$-nef divisor $M_U$ that cannot extend to an $f$-nef $\mathbb{Q}$-divisor $M$ for any compactification of $f|_U$, even after replacing $X_U$ with any higher birational model.
Surfaces on Calabi-Yau fourfolds
When: Wed, April 9, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Younghan Bae ( University of Michigan) - https://younghbae.github.io
Abstract: Curve counting theory on Calabi-Yau and Fano threefolds has been a central topic in enumerative geometry. For Calabi-Yau fourfolds, DT4 virtual cycles can be defined on the Hilbert scheme of two-dimensional subschemes (or other stable pair type moduli) using the Borisov-Joyce/Oh-Thomas theory. These classes, however, become trivial when the Hodge locus of the surface class has positive codimension. We reduce the theory and prove that the resulting invariants remain deformation invariant along the Hodge locus.
In pursuit of understanding the structure of invariants counting surfaces on Calabi-Yau fourfolds, we turn our attention to the moduli space of stable two-dimensional sheaves. For surfaces with mild singularities, we propose a conjecture that the pushforward of the (reduced) virtual cycle to the Chow variety has modular properties. This is joint work with M. Kool and H. Park.
From abelian schemes to Hitchin systems
When: Mon, April 14, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Junliang Shen (Yale) - https://sites.google.com/view/junliang/home
Abstract: Recent studies of the Hitchin fibration suggest that, from the cohomological perspective, it behaves like an abelian scheme. In this talk, I will discuss this phenomenon and provide supporting evidence. In particular, I will discuss a proof of the motivic decomposition conjecture of Corti-Hanamura for the Hitchin fibration, where the desired algebraic cycles are constructed using a combination of techniques from derived categories, K-theory, and Springer theory. The counter-part of this result for abelian schemes was established by Deninger and Murre over three decades ago. This work is based on joint work with Davesh Maulik and Qizheng Yin.
Strange duality of twisted conformal blocks
When: Wed, April 16, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Swarnava Mukhopadhyay (Tata Institute (TIFR)) - https://mathweb.tifr.res.in/~swarnava/
Abstract: The space of global sections of line bundles on the moduli of principal G-bundles on a curve can be thought as a natural generalization of the space of theta functions on the Jacobian.
Strange duality relates these spaces for pairs of groups. The principal examples being the pair of groups (GL(m),SL(n)), (Sp(2m), Sp(2n)), (G_2, F_4) and many others and the first known case is due to Beauville-Narasimhan-Ramanan for the pair (GL(1), SL(n)). These dualities have their origins in conformal field theory, as dualities between WZW models of conformal blocks as well as in representation theory arising from the duality of the Grassmannians Gr(m,m+n) and Gr(n,m+n).
Motivated by works of Pappas-Rapport, twisted analogs of these non-abelian theta functions have been considered and a Verlinde type formula has also been proved. These spaces are constructed from moduli of non-constant Bruhat-Tits group schemes and generalize the space theta functions on Prym varieties. In this talk we will discuss a twisted analog of strange dualityfor the pair (GL(m), SL(n)) associated to an etale double cover of curves. The untwisted analog for this pair is due to Belkale, Marian-Oprea.
Discriminants and motivic integration
When: Wed, April 23, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Oscar Kivinen (Aalto University ) - https://math.aalto.fi/~kivineo3/
Abstract: Several different invariants can be attached to an isolated hypersurface singularity using motivic integration. In the Euler characteristic limit, these invariants are all related in a straightforward way, but the relationships between the motivic versions are more difficult to understand. We will discuss the case of plane curves in detail, including connections to knot Floer homology, Hilbert schemes, and the Igusa zeta function. Based on joint work with Oblomkov and Wyss.
The Cohomology of Definable Analytic Spaces
When: Wed, May 7, 2025 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Adam Melrod (UMD) - https://math.umd.edu/~abmelrod/
Abstract: The theory of o-minimal geometry has proven to be useful in several areas of algebraic and arithmetic geometry. In the context of Hodge theory, Bakker-Brunebarbe-Tsimerman used this theory to develop a category of definable complex analytic spaces, which leads to a definable GAGA theorem describing how the definable category interpolates the algebraic and complex analytic categories. First, I will introduce these ideas, and then I will discuss some new results, joint with Patrick Brosnan, on the cohomology of coherent sheaves on definable analytic spaces. The first result is that for proper spaces, the cohomology part of Serre's GAGA theorem also holds in the definable setting. The second result is that the cohomology of a definable coherent sheaf on an analytically Stein space is roughly controlled by how far the holomorphic maps on the space are from being definable. I will use this to prove the first cohomology of the structure sheaf of the definable affine complex line has uncountable dimension.
Hilbert scheme of points on threefold
When: Mon, May 12, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ritvkik Ramkumar (Cornell University) - https://sites.google.com/view/ritvikramkumar/home?authuser=0
Abstract: The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will focus on the case where X is a threefold, as there are several open questions regarding its singularities. I will describe the structure of the smooth points of this Hilbert scheme and, time permitting, discuss the structure of the mildly singular points. This is all joint (ongoing) work with Joachim Jelisiejew and Alessio Sammartano.
Boundedness theorems for abelian fibrations
When: Wed, May 14, 2025 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Philip (Engel) - https://philip-engel.github.io/
Abstract: I will report on forthcoming work, joint with Filipazzi, Greer, Mauri, and Svaldi, on boundedness results for abelian fibrations. We will discuss a proof that irreducible Calabi-Yau varieties admitting an abelian fibration are birationally bounded in a fixed dimension; and that Lagrangian fibrations of symplectic varieties, in a fixed dimension, are analytically bounded. Conditional on generalized semiampleness conjectures, this bounds the number of deformation classes of hyperkahler varieties in a fixed dimension, whose second Betti number is at least 5.