Algebraic Geometry Seminar

Department of Mathematics
Princeton University

Fall 2016 Lectures and events

Place: Fine Hall 322

Time: Tuesday 4:30-5:30

Date Speaker Title
October 4 Junliang Shen
ETH Zurich
Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories
By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. I will explain a mathematical approach to prove (part of) the HKK Conjecture. Our method is to construct an involution on the derived category and use wall-crossing techniques. The talk is based on joint work with Georg Oberdieck.
October 11 Dawei Chen
Boston College
Compactification of strata of abelian differentials
Many questions about Riemann surfaces are related to study their flat structures induced from abelian differentials. Loci of abelian differentials with prescribed type of zeros form a natural stratification. The geometry of these strata has interesting properties and applications to moduli of complex curves. In this talk we focus on the question of compactifying the strata of abelian differentials from the viewpoints of algebraic geometry, complex analytic geometry, and flat geometry. In particular, we provide a complete description of the strata compactification over the Deligne-Mumford moduli space of pointed stable curves. The upshot is a global residue condition compatible with a full order on the dual graph of a stable curve. This is joint work with Bainbridge, Gendron, Grushevsky and Moeller, based on arXiv:1604.08834.
October 18 Brian Lehmann
Boston College
Convexity in divisor theory
For toric varieties there is a dictionary relating the geometry of divisors to the theory of polytopes. I will discuss how certain aspects of this dictionary can be extended to divisors on arbitrary smooth projective varieties. These results build upon ideas of Khovanskii and Teissier; as in their work, geometric inequalities and convexity theory play an important role. This is joint work with Jian Xiao.
October 25 David Anderson
Ohio State University
Old and new formulas for degeneracy loci
A very old problem asks for the degree of a variety defined by rank conditions on matrices. The story of the modern approach begins in the 1970's, when Kempf and Laksov proved that the degeneracy locus for a map of vector bundles is given by a certain determinant in their Chern classes. Since then, many variations have been studied -- for example, when the vector bundles are equipped with a symplectic or quadratic form, the formulas become Pfaffians. I will describe recent extensions of these results -- beyond determinants and Pfaffians, and beyond ordinary cohomology -- including my joint work with W. Fulton, as well as work of several others.
November 8 Dhruv Ranganathan
Tropical curve counting in superabundant geometries
I will discuss a general framework using Artin fans -- certain logarithmic algebraic stacks -- in which to understand the relationship between logarithmic stable maps and tropical curve counting. These objects provide a flexible tool to study correspondences between algebraic and tropical curves. In particular, we obtain new realization theorems for tropical curves in superabundant settings. After explaining some remarkably cheap consequences of this setup, I will discuss an application, joint with Yoav Len, to the enumerative geometry of elliptic curves on toric surfaces.
November 15 (Special time: 5 pm, same room) Han-Bom Moon
Fordham University
Classical invariant theory and birational geometry of moduli spaces
Invariant theory is a study of the invariant subring of a given ring equipped with a linear group action. Describing the invariant subring was one of the central mathematical problems in the 19th century and many great algebraists such as Cayley, Clebsch, Hilbert, and Weyl had contributed to it. There are many interesting connections between invariant theory and modern birational geometry of moduli spaces. In this talk I will explain some concrete examples including the moduli space of parabolic vector bundles on the projective line and the moduli space of stable rational pointed curves. This talk is based on joint work with Swinarski and Yoo.
December 6 Wenfei Liu
Xiamen University
On Noether's inequality for stable log surfaces
In this talk I report on some recent progress on the geography problem of stable log surfaces. This is about restrictions on their holomorphic invariants, such as the volume K^2 and the geometric genus p_g. Compared to the case of surfaces of general type, a new feature here is that the volume of a stable log surface is not necessarily an integer. Extending the work of Tsunoda and Zhang in the nineties, I will give an optimal lower bound of the volume when the geometric genus is one. Then I will use an example to illustrate that a speculated Noether type inequality for stable log surfaces does not hold in general.
December 13 (joint with symplectic geometry seminar) Hülya Argüz
Log geometric techniques for open invariants in mirror symmetry
We would like to discuss an algebraic-geometric approach to some open invariants arising naturally on the A-model side of mirror symmetry. The talk will start with a smooth overview of the use of logarithmic geometry in the Gross-Siebert program. We then will discuss various illustrations of the use in open invariants, including a description of the symplectic Fukaya category via certain stable logarithmic curves. For this, our main object of study will be the degeneration of elliptic curves, namely the Tate curve. However, the results are expected to generalise to higher dimensional Calabi-Yau manifolds. This is joint work with Bernd Siebert, with general ideas based on discussions of Bernd Siebert and Mohammed Abouzaid.

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