Algebraic Geometry Seminar

A smooth K3 surface obtained as the blow-up of the quotient of a four-torus by the involution automorphism at all 16 fixed points is called a Kummer surface. Algebraic Kummer surfaces obtained from abelian varieties provide a fascinating arena for string compactification and string dualities as they are not trivial spaces but are sufficiently simple to analyze most of their properties in detail. However, their Picard rank is always bigger or equal 17, and they only provide a geometric description for heterotic string vacua with up to one Wilson line.

In this talk, I give an explicit description of the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. The family of Van Geemen-Sarti partners is a four-parameter family of K3 surfaces with a H + E₇(-1) + E₇(-1) lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type IV. We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions with two Wilson lines. If time allows, I will also describe special six line configurations (such as six lines tangent to a conic, three lines meeting in a point) that correspond to Kummer surfaces of Jacobians of genus-two curves with principal polarization and those associated to (1, 2)-polarized abelian surfaces, as well as their applications in string theory and number theory.

Let G be a reductive group and X be a smooth projective G-variety. In classical geometric invariant theory (GIT), there are stratifications of X that can be used to understand the geometry of the GIT quotients X//G and their dependence on choices. In this talk, after introducing basic theory, I will discuss the moduli of gauged maps, their relation to the Gromov-Witten theory of GIT quotients X//G and work in progress regarding stratifications of the moduli space of gauged maps as well as possible applications to quantum K-theory. This is joint work with D. Halpern-Leistner, P. Solis and C. Woodward.

Understanding the structure of Gromov-Witten invariants of quintic threefolds is an important problem in enumerative geometry which has been studied since the early 90s. Together with Q. Chen, Y. Ruan and A. Sauvaget, we construct new moduli spaces that we call "logarithmic GLSM moduli spaces". One application is toward proving conjectures from physics about higher genus Gromov-Witten invariants of quintic threefolds, such as the holomorphic anomaly equations. Another application, which also was the initial motivation to develop logarithmic GLSM, is toward proving a conjecture of R. Pandharipande, A. Pixton, D. Zvonkine and myself on loci of holomorphic differentials with prescribed zeros. In this talk, I will focus on the second application. Its main actor is the so-called Witten's r-spin class, the analog of the virtual class in the FJRW theory of the A_{r-1}-singularity.

The homological conjectures have been a focus of research in commutative algebra since 1960s. They concern a number of interrelated conjectures concerning homological properties of commutative rings to their internal ring structures. These conjectures had largely been resolved for rings that contain a field, but several remained open in mixed characteristic--until recently Yves Andre proved Hochster’s direct summand conjecture and the existence of big Cohen-Macaulay algebras, which lie in the heart of these homological conjectures. The main new ingredient in the solution is to systematically use the theory of perfectoid spaces, which leads to further developments in the study of mixed characteristic singularities. For example, using integral perfectoid big Cohen-Macaulay algebras, one can define the mixed characteristic analog of rational/F-rational and log terminal/F-regular singularities, and they have applications to study singularities when the characteristic varies (based on recent joint work with Karl Schwede). In this talk, we will give a survey on these results and methods.

In this talk I'll describe how a bicategorical gadget, called the shadow, allows one to extract many interesting algebraic and topological invariants. For example, in algebraic bicategories, one easily recovers group characters, and in certain topological categories, one recovers the Lefschetz number. I'll describe joint work with Kate Ponto generalizing work of Ben-Zvi--Nadler which allows us to simultaneously recover the Lefschetz theorem for DG-algebras due to Lunts and the theory of 2-characters due to Ganter Kapranov, along with many other results. Prerequisites: An appetite for category theory (but I will not assume knowledge of bicategories!). There will be some motivation from stable homotopy theory, but one need only believe in the stable homotopy category, not have knowledge of it.

Shokurov introduced the theory of complements while he investigated log flips of threefold. The theory is further developed by Prokhorov-Shokurov and Birkar. As one of the key steps in the proof of BAB conjecture, Birkar showed a conjecture of Shokurov, i.e., the existence of n-complements for Fano pairs with hyperstandard coefficients. In this talk, I will show that the boundedness of complements holds for pairs with DCC coefficients, and have some additional properties: divisibility, rationality, approximation, and anisotropic. If there's still enough time, I will show some of its applications on the ACC for minimal log discrepancies. This is a joint work with J.Han and V.V.Shokurov.

For a smooth projective variety X containing a smooth irreducible divisor D, the question of counting curves in X with prescribed contact conditions along D is a classical one in enumerative geometry. In more modern approaches to this question, there are two ways to define these counts: as Gromov-Witten invariants of X relative to D, or as Gromov-Witten invariants of the stack X_{D,r} of r-th roots of X along D (for r large). In genus 0, explicit calculations in examples suggested that these two sets of Gromov-Witten invariants are always the same, whether they actually count curves or not. This is proven in full generality by Abramovich-Cadman-Wise. The situation in genus > 0 is not so simple, as an example of D. Maulik showed that the two sets of Gromov-Witten invariants are not equal, even in genus 1. In this talk, we'll explain how these two sets of Gromov-Witten invariants are related in all genera, in full generality. This is based on a joint work with Fenglong You.