Kazhdan-Lusztig Theory and Singular Hodge Theory for Matroids II
Dr. June Huh
Stanford University
9:00am - 10:00am
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Abstract: There is a remarkable parallel between the theory of Coxeter groups (for example the symmetric group or the dihedral group) and matroids (think of your favorite graph or point configuration) from the perspective of combinatorial cohomology theories. I will give an overview of the similarity and report on recent my joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang on singular Hodge theory for combinatorial geometries: https://arxiv.org/abs/2010.06088 ; https://arxiv.org/abs/2002.03341
Hodge Theory, Higgs Bundles on Moduli Spaces of Manifolds and Hyperbolicity II
Dr. Kang Zuo
Johannes Gutenberg Universität Mainz
10:30am - 11:30am
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Abstract:
Given a complex quasi-projective manifold $U$ and a smooth compactification $Y=U+S$, we are particularly interested in the geometry of the log space $(Y, S)$ when $U$ appears as the base of a family $f: V\to U$ parameterizing projective manifolds of non-negative Kodaira-dimension. The powerful theory on variation of Hodge structures (VHS) developed by P. Griffiths shows us a principle of how the variation of the complex structures on the fibers can be mirrored in the global properties of $(Y, S).$
In the first lecture I start to recall two types of graded Higgs bundles. The first one is the so-called system of Hodge bundles $(E,\theta)$ arising from a VHS introduced by C. Simpson in his work on the non-abelian Hodge theory.
The second type of Higgs bundle has been studied in my joint work with E. Viehweg, the so-called deformation Higgs bundle $(F,\tau)$ appearing as the natural extension of the classical Kodaira-Spencer map of $f$ on the cohomologies of the relative tangent sheaf of higher degree.
Based on Kawamata-Viehweg's theorem on the positivity of the relative pluri-differential forms, we construct a non-trivial Higgs map $$ \rho: (F,\tau)\to (E,\theta)\otimes A^{-1},$$ where $(E,\theta)$ is the system of Hodge bundles arising from the geometry origin and $A$ is an ample line bundle on the base. Via $\rho$ Griffiths' curvature formula on the system of Hodge bundles, $(E,\theta)$ can be performed on the deformation Higgs bundle $(F,\tau)$ and we obtain a big sub sheaf in a symmetric power of log differential forms $$\mathcal A\hookrightarrow S^\ell\Omega^1_Y(\log S),$$ which generalizes Griffiths-Schmid's theorem on the strict negativity of horizontal period mappings.
Comparing Geometric and Hodge-Theoretic Compactifications I
Dr. Philip Engel
University of Georgia
1:00pm - 2:00pm
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Abstract: The "classical period domains" D of polarized Hodge structures are those for which Griffiths' transversality condition is automatic. Then D is a Hermitian symmetric domain parameterizing either weight 1 Hodge structures, or weight 2 Hodge structures with h^{2,0}=1. Sometimes, the period mapping identifies a moduli space of varieties with an arithmetic quotient D/Gamma. For instance, in weight 1, we have moduli spaces A_g of principally polarized abelian varieties, and in weight 2, we have moduli spaces F_g of polarized K3 surfaces, or hyperkahler varieties.
Algebro-geometric compactifications of D/Gamma were first studied by Baily-Borel, and later by Ash-Mumford-Rapaport-Tai, who introduced toroidal compactifications. In the weight 2 case, Looijenga simultaneously generalized these to the semitoroidal compactifications. There are infinitely many such, in bijection with rational polyhedral decompositions of cones in certain lattices, called (semi)fans. In weight 1, the cone is the space of positive-definite quadratic forms on \mathbb{R}^g and in weight 2, it is the positive cone of a hyperbolic lattice.
The motivating question of the talks is whether some distinguished (semi)toroidal compactification is a moduli space of generalized geometric objects: "stable abelian varieties" or "stable K3 surfaces."
I will first review the works of Mumford, Namikawa, and Alexeev who studied this question for abelian varieties. There, the so-called "second Voronoi fan" gives a distinguished toroidal compactification. The cones of the fan are the loci of quadratic forms which have fixed combinatorics of Delaunay-Voronoi decomposition. The combinatorics of the "stable abelian varieties" fibering over the corresponding boundary stratum are encoded by this Delaunay-Voronoi decomposition.
I will then discuss recent joint work with Alexeev on how to extend this to "stable K3 surfaces." These moduli spaces were discovered by Kollar-Sheperd-Barron and Alexeev, and parameterize log general type pairs. Since the moduli spaces in question parameterize Calabi-Yau varieties (which are not general type), one must choose canonically a "polarizing divisor" on the objects of the moduli space.
For abelian varieties, this choice was the theta divisor, and for K3 surfaces, we introduce the notion of a "recognizable divisor". Associated to a recognizable divisor is a semifan whose corresponding semitoroidal compactification is the normalization of the KSBA compactification of K3 pairs. For instance, the "rational curve divisor" is the sum of the rational curves in the polarization class, and gives a recognizable divisor for any F_g. The K3 analogue of the Delaunay-Voronoi decompositions are loci where certain polyhedral decompositions of singular integral-affine spheres have fixed combinatorics.
