Workshop on geometry at UNC, 2025

Olivia Dumitrescu and I are planning a small geometry workshop at UNC – Chapel Hill, Monday 7th April to Wednesday 9th April. The workshop will be funded by NSF Award#2152130.

 

Travel to UNC: fly to Raleigh/Durham (RDU) airport and take Lyft/Uber/taxi to Chapel Hill. We will reserve hotel rooms for speakers. We can possibly support some other participants too; please email <sawon at email dot unc dot edu> if interested.

 

Proposed schedule of talks:
Titles/Abstracts
Speaker: Hülya Argüz (University of Georgia)
Abstract: The KSBA moduli space, introduced by Kollár-Shepherd-Barron, and Alexeev, is a natural generalization of the moduli space of stable curves to higher dimensions. It parametrizes stable pairs (X,B), where X is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. In the situation when X is a toric variety and B=D+\epsilonC, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that this moduli space is the toric variety defined by the secondary fan. Generally, for a log Calabi-Yau variety (X,D) consisting of a projective variety X with B=D+\epsilonC, where D is an anticanonical divisor and C is an ample divisor, it has been an open question what this moduli space is, and it was conjectured by Hacking-Keel-Yu that it still should be toric (up to passing to a finite cover). In joint work with Alexeev and Bousseau, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory and mirror symmetry.

 

Speaker: Colleen Robles (Duke University)
Abstract: I will give an overview of how Hodge theory is used to study families, and moduli spaces, of smooth projective varieties; how these applications motivate the problem of completing period maps; and current work toward Hodge theoretic generalizations of Mumford et al’s compactifications of locally Hermitian symmetric spaces.

 

Title: Archimedean zeta function and Hodge theory.
Speaker: Ruijie Yang (University of Kansas)
Abstract: Given a holomorphic function f on a complex manifold, Gelfand posed his famous problem in 1954 ICM about meromorphically extending the complex power of f and determining the poles and residues.
The first part is solved in a multiple way, among which the approach of Bernstein in 1971 is one of the origins of the theory of D-modules. The second part, pioneered by Varchenko in 80s, can be approached using Hodge theory, which is seemingly unrelated. After that, a lot of progress was made for isolated singularities.
In this talk, I want to explain how to deal with arbitrary non-isolated singularities using a Hodge-theoretic refinement of D-modules, which builds on the work of Cattani, Kaplan and Schmid on period domains in 70s. As a consequence, several questions and conjectures from 80s-90s as well as recent questions by Mustata and Popa can be resolved. This is based on the joint work with Dougal Davis and Andras Lorincz.

 

Speaker: Michael Schultz (Virginia Tech)
Abstract: Apéry’s original proof of the irrationality of ζ(3) stunned the mathematics community in 1978, and subsequent generations of mathematicians have discovered geometric and modular structures underlying these irrationality proofs that are arguably even more striking. One such well known class of examples are connections to modular pencils of elliptic curves and K3 surfaces and their Picard-Fuchs operators, the latter which exhibit maximally unipotent monodromy at the so-called large complex structure limit of their associated moduli spaces. These objects are respectively mirror dual to anticanonical divisors in certain del Pezzo surfaces and Fano threefolds, and their Picard-Fuchs operators to the A-side connection on small quantum cohomology for these varieties. Although genus zero enumerative invariants for elliptic curves and K3 surfaces are trivial, I will show in this talk how a blend of the perspectives above allows one to compute a modular form of Eisenstein type determined in a simple way from the canonical holomorphic period near the large complex structure limit. The functional inverse is shown in certain cases to determine a generating function of local Gopakumar-Vafa invariants related to the mirror, recovering some known results in the literature. Based on joint work with Andreas Malmendier (arXiv:2403.07349), and work in progress with Andreas Malmendier, Chuck Doran, and Irit Huq-Kuruvilla.

 

Title: The king is dead, long live the king
Speaker: Matthew Ballard (University of South Carolina)
Abstract: Inspired by Beilinson’s well-known result on projective space, King’s conjecture is a very enticing but false guess about the structure of derived categories of toric varieties. It posits every smooth, projective toric variety possesses a full strong exceptional collection of line bundles. We show how, by incorporating birational geometry naturally in the derived category, we get a true statement and give some applications. The talk is based on joint work with Christine Berkesch, Michael K. Brown, Lauren Cranton Heller, Daniel Erman, David Favero, Sheel Ganatra, Andrew Hanlon, and Jesse Huang.

 

Title: Geometry of semi-classical limit of 1D Schrödinger operators and unique quantization as its inverse
Speaker: Motohico Mulase (UC Davis)
Abstract: Every Higgs bundle has a spectral curve, but there is no such spectral curve corresponding to a holomorphic connection on a curve. However, if the connection is an oper, then there is a corresponding unique spectral curve, which appears as a semi-classical limit. This construction is not constructible, but its converse map is, and it goes through identifying the “quantum deformation” of the linear ordinary differential operator corresponding to the starting oper. I will review what is known for the holomorphic case established in joint papers with Olivia Dumitrescu et al. Then I will discuss its expected generalizations and potential applications, based on the work in progress with Raymond Chan and Zach Ibarra.

 

Title: New examples of compact holomorphic symplectic manifolds
Speaker: Ruxandra Moraru (University of Waterloo)
Abstract: A holomorphic symplectic manifold is a complex manifold X together with a closed, non-degenerate holomorphic 2-form \Omega. The top power of \Omega gives a trivialisation of the canonical bundle so that X has trivial first Chern class. In the context of Kähler geometry, such manifolds play a very important role due to the Bogomolov covering theorem, which states that any compact Kähler manifold with vanishing first Chern class has a covering that splits as the product of Calabi–Yau manifolds, complex tori and irreducible holomorphic symplectic manifolds. Among these, the last two are, in fact, compact holomorphic symplectic manifolds. Furthermore, irreducible holomorphic symplectic manifolds correspond to compact hyperkähler manifolds in the Kähler setting. In general, finding compact holomorphic symplectic manifolds is very difficult. In this talk, I will present new examples of compact holomorphic symplectic manifolds. These manifolds correspond to moduli spaces of sheaves on Kodaira surfaces and are non-Kähler. This is work in progress with Tom Baird and Eric Boulter.

 

Speaker: Soham Karwa (Duke University)
Abstract: Period integrals are a fundamental concept in algebraic geometry and number theory. In this talk, we will study the notion of non-archimedean periods as introduced by Kontsevich and Soibelman. We will give an overview of the non-archimedean SYZ program, which is a close analogue of the classical SYZ conjecture in mirror symmetry. Using the non-archimedean SYZ fibration, we will see how non-archimedean periods recover the complex analytic periods for log Calabi-Yau surfaces, verifying a conjecture of Kontsevich and Soibelman. This is joint work with Jonathan Lai.