Matroids and the Integral Hodge Conjecture

This is an algebraic geometry learning seminar (2026 Fall) on matroids, degenerations of abelian varieties, and their applications to the integral Hodge conjecture, stable rationality, and complex structures on S⁶.

References

1. [EFS] Philip Engel, Olivier de Gaay Fortman, and Stefan Schreieder, Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction.
2. [EFS2] Philip Engel, Olivier de Gaay Fortman, and Stefan Schreieder, Matroids and the integral Hodge conjecture for abelian varieties.
3. [E] Philip Engel, Complex structures on S⁶.

Schedule

• September 18: Introductions (Speaker: James Hotchkiss)
• September 25:
  (§2 [EFS]) Backgrounds on degenerations of abelian varieties (Speaker: Yoonjoo Kim)

  Abstract

  We will learn/review some basic material for our discussion of matroidal degenerations of abelian varieties. This includes principally polarized abelian varieties (ppav), Hodge structures of weight -1, log monodromies, and toroidal compactifications of A_g associated to certain cones B of positive-definite symmetric matrices / Q.

• October 2: No meeting
• October 9:
  (§3 [EFS]) Mumford construction over a polydisk (Speaker: Daigo Ito)

  Abstract

  We will first explain the basic idea of the Mumford construction via equivariant toric extensions of intermediate covers, and then describe the construction over a polydisk. We will also briefly discuss its connection with toroidal compactifications of A_g.

• October 16:
  (§3,4 [EFS]) Matroidal Mumford construction and Segre cubic (Speaker: Zhijia Zhang)

  Abstract

  We will cover the Mumford construction over polydisk and matroidal Mumford construction, as well as how to control the period maps and make the total space smooth. Then we will go through our main example: the degeneration of intermediate jacobians of cubic threefolds, and explain its connection to matroids.

• October 23:
  Transversely shifted matroidal degenerations and Mumford degeneration with nodal singularities (Speaker: Anh Duc Vo)

  Abstract

  Starting from the Mumford construction, we explain how regular matroids and shifted hyperplane arrangements produce degenerations of abelian varieties with nodal singularities and smooth total space. Our goal is to explain the geometric ingredients behind the existence of projective nodal extensions for families with matroidal monodromy (Theorem 1.1).

• October 30:
  Complex Structure on the Six-Sphere (Speaker: Hechen Hu)

  Abstract

  In this talk I'll discuss the recent proof that S^6 has a complex structure and a non-constant meromorphic function. After reviewing the classical fact that only S^2 and S^6 can have a complex structure, I'll assume the existence of X \to P^1 with X a homological six-sphere and discuss some constraints on the family, e.g. types of singular fibers, numerical bounds on irreducible components of the fiber. I'll describe how the geometric construction (explained by Engel) fits the constraint and several other variations on the construction.

• November 6: TBD
• November 13: TBD
• November 20: TBD
• November 27: Thanksgiving
• December 4: TBD
• December 11: TBD