Heidelberg Algebraic Geometry Seminar

Time: Fridays, 11:15 - 12:15+questions
Location: Seminarroom 10, Im Neuenheimer Feld 205

Yota Maeda: On the Kodaira dimension of even-dimensional ball quotients 

In this talk, I will prove that even-dimensional ball quotients are of general type when either the dimension or the absolute value of the odd discriminant of the underlying imaginary quadratic field is sufficiently large. This result is a unitary analogue of the works of Tai, Freitag, and Mumford for Siegel modular varieties, and of Kondo, Gritsenko, Hulek, Sankaran, and Ma for orthogonal modular varieties. The key ingredients of our proof are twofold: first, we extend Prasad's volume formula to a form applicable to general arithmetic subgroups, and use it to estimate the dimension of spaces of automorphic forms; second, we construct suitable cusp forms using Arthur's multiplicity formula in the theory of automorphic representations. This work is based on joint work with Horinaga and Yamauchi, and partly with Ohara. 

FALL 2024 ARCHIVE

List of abstracts:

Arkadij Bojko: Borisov-Joyce and Oh-Thomas developed the theory for counting sheaves on Calabi-Yau fourfolds. My work focuses on proving wall-crossing in this setting. It is desirable that the end result can have many concrete applications to existing conjectures. For this purpose, I introduce a new structure - formal families of vertex algebras. Apart from being a natural extension of the vertex algebras introduced by Joyce, they allow us to wall-cross with insertions instead of the full virtual fundamental classes.  Many fundamental hurdles needed to be overcome to prove wall-crossing in this setting. They included constructing Calabi-Yau four obstruction theories on (enhanced) master spaces and showing that the invariants counting semistable torsion-free sheaves are well-defined. Towards the end of the talk, I will use the complete package to address existing conjectures with applications to 3-fold DT/PT correspondences.

Denis Nesterov:  The talk will be about a stability condition interpolating between Hilbert schemes of points and Fulton-MacPherson compactifications with a view towards enumerative geometry.  

Robert Crumplin: An interesting question in enumerative geometry is how to count curves in a variety X with fixed tangencies to a divisor D. One approach to this is with ``twisted stable maps to root stacks" as introduced by Abramovich-Vistoli and Cadman. I will explain how the corresponding moduli spaces are all controlled by a ``universal moduli space" parameterising line bundle-section pairs. Furthermore, in the case D is smooth I will give a complete description of this space in terms of combinatorial data and show how this can be used to deduce results in orbifold Gromov--Witten theory.

Simon Schirren: Higgs bundles over curves play an important role in recent developments in algebraic geometry, such as the P=W conjecture. We focus instead on Higgs sheaves on a surface, which can be seen as spectral sheaves on a Calabi-Yau threefold. Their enumerative theory and the resulting virtual cycle giving SL(n)-Vafa-Witten invariants has been defined by Tanaka-Thomas. In the rank-2 case, we discuss a different approach to define this cycle, which may also lead to construct new invariants. 

Jan-Willem van Ittersum: There are several instances where Gromov-Witten invariants can be expressed in terms of (quasi)Jacobi forms. In other examples in enumerative geometry, one also encounters mock modular forms or evenq-analogues of multiple zeta values. We explain the origin and properties of these series and provide examples of their occurrences as generating series of geometrical invariants. 

Tomoyoshi Ibukiyama: A theory of differential operators on automorphic forms which preserve automorphy after restrictions of the domains have a long history and turned out to include a nice theory of special functions. We will give rough outline on results since 1990 (partly a joint work with D. Zagier) and explain several arithmetic applications.

Qaasim Shafi:  An old theorem, due to Mikhalkin, says that the number of rational plane curves of degree d through 3d-1 points is equal to a count of tropical curves (combinatorial objects which are more amenable to computations). There are two natural directions for generalising this result: extending to higher genus curves and allowing for more general conditions than passing through points. I’ll discuss a generalisation which does both, as well as recent work connecting it to mirror symmetry for log Calabi-Yau surfaces. This is joint work with Patrick Kennedy-Hunt and Ajith Urundolil Kumaran.

Sam Johnston: Mirror symmetry for Fano varieties is typically stated as a duality between the symplectic geometry of a Fano variety and the complex geometry of a Landau-Ginzburg model, realized as a pair (X,W) with X a quasi-projective variety and W a regular function on X. The pair (X,W) itself is expected to reflect a pair on the Fano side, namely a decomposition Y into a disjoint union of an affine open subset and an anticanonical divisor D, thought of as mirror to W. We will discuss upcoming work which shows how the intrinsic mirror construction of Gross and Siebert may be used to construct mirrors to pairs (Y,D) assuming milder conditions on the singularities of D than typically required for the intrinsic mirror construction. In the setting when Y\D is an affine cluster variety, we will describe how these LG models encode toric degenerations of Y. As an example, we consider Y = Gr(k,n), D a particular choice of anticanonical divisor with affine cluster variety compliment and give an explicit description of the intrinsic LG model in terms of Plucker coordinates on Gr(n-k,n), recovering mirrors constructed and investigated by Marsh-Rietsch and Rietsch-Williams. 

Nikolas Kuhn: The Joyce-Song wall-crossing formulas for Donaldson-Thomas invariants of Calabi-Yau threefolds have proven to be a crucial and versatile tool. In the presence of a torus action, there are interesting threefold geometries in which the Calabi-Yau condition only holds up to an equivariant twist - examples include Vafa-Witten invariants, local curves and surfaces and the threefold vertex. In these cases, invariants are defined using localization, and Joyce-Song's theory doesn't apply. I will explain how ideas from Joyce's recent work on wall-crossing in abelian categories can be used to prove wall-crossing in this situation, and which difficulties arise.  This is joint work with Henry Liu and Felix Thimm.

Weisheng Wang: Let Y be an Enriques surface and let A be anAzumaya  algebra corresponding to the non-trivial Brauer class. LetM  be the moduli space of stable twisted sheaves on Enriques surfaces with fixed twistedChern  character. The virtual dimension of M is N. We show that the virtual Euler characteristic of M only depends on N, more precisely, it is 0 when N isodd and it equals to two times the Euler characteristic of Hilbert scheme of N/2 points when N is even.

Brandon Williams: The NL-cone of an orthogonal modular variety is the cone spanned by effective linear combinations of irreducible components of Noether--Lefschetz divisors. By the work of Bruinier--Moeller it is known (under some conditions) that the NL-cone is rational polyhedral. I will discuss the problem of computing that cone explicitly, i.e. writing it as the convex hull of a finite set of boundary divisors. This is joint work with I. Barros, P. Beri and L. Flapan.

Rahul Pandharipande: I will explain developments in the study of cycles on the moduli space of abelian varieties with connections to the moduli space of curves, the cohomology of the Lagrangian Grassmannian, modular forms, and the quantum cohomology of the Hilbert scheme of points of the plane.

Sabrina Pauli: We study the problem of counting rational curves of fixed degree on a toric del Pezzo surface subject to point conditions. Over algebraically closed fields, this count is invariant under the choice of point conditions. Over non-algebraically closed fields, however, the invariance fails. For real numbers, Welschinger's groundbreaking work introduced a signed count of real curves that restores invariance.

 Building on this, Levine and Kass-Levine-Solomon-Wickelgren have developed curve counts over arbitrary fields that not only generalize Welschinger's signed counts and classical counts over algebraically closed fields, but also encode much richer arithmetic information. 

 In this talk I will survey these different approaches to counting rational curves with point conditions and discuss a recent joint result with A. Jaramillo Puentes, H. Markwig, and F. Röhrle. We establish a tropical correspondence theorem for curve counts over arbitrary fields, identifying the count of algebraic curves with point conditions with a weighted count of their tropical counterparts with point conditions. The latter are combinatorial objects and there are several purely combinatorial methods to find all tropical curves with point conditions.