Abstracts Workshop

Kenny Ascher
Title: Moduli of low degree K3 surfaces
Abstract: The explicit description of low degree K3 surfaces leads to natural compactifications coming from geometric invariant theory (GIT) and Hodge theory. The
relationship between these compactifications for degree two K3 surfaces was studied by
Shah and Looijenga, and revisited by Laza and O’Grady, who also provided a conjecturaldescription for the case of degree four K3 surfaces. I will discuss these results, as well asa verification of this conjectural picture using tools from K-moduli. This is joint workwith Kristin DeVleming and Yuchen Liu.

Laure Flapan
Title: Kodaira dimension of moduli spaces of hyperkähler manifolds
Abstract: We study the geometry of some moduli spaces of polarized hyperkählermanifolds. We use techniques of Gritsenko-Hulek-Sankaran involving the Borcherds modular form to determine a bound on the degree of the polarization after which these moduli
spaces are always of general type. This is joint work with I. Barros, P. Beri, and E. Brakkee.

Christopher Hacon
Title: Recent progress in the Kähler minimal model programAbstract: The minimal model program is an ambituous program that aims to understand the geometry of complex projective varieties (eg. manifolds defined by polynomial equations). In this talk we will discuss some recent results and challenges encounteredtrying to extend the minimal model program to the context of Kähler varieties.

Sándor KovácsTitle: KSB stability is automatic in codimension 3
Abstract: In this talk I will first review KSB/A stability, especially their localversion and then discuss new results, joint with János Kollár, showing that it is enoughto check these conditions, including flatness, up to codimension 2. This implies that we1

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have a very good understanding of this stability condition in general, because local KSBstability is trivial at codimension 1 points, and quite well understood at codimension 2
points, since we have a complete classification of 2-dimensional slc singularities.

Yuchen Liu
Title: On K-moduli of quartic threefolds
Abstract: K-stability provides a powerful tool for constructing compact moduli
spaces, known as K-moduli spaces, for Fano varieties. However, determining the Kmoduli space for specific Fano varieties, such as Fano hypersurfaces, can be a challenging
problem. Previously, the K-moduli space for cubic hypersurfaces was shown to be thesame as the GIT one up to dimension 4. In this talk, I’ll discuss some recent progress on
the K-moduli space of quartic threefolds where K-moduli and GIT differ significantly. We
find a new codimension 3 locus in the K-moduli space that parametrizes (2, 2, 4) complete
intersections in P(15 , 22 ). Moreover, we show that this locus is closed by relating the Kstability of such complete intersections to certain del Pezzo surface pairs. This is joint
work with Hamid Abban, Ivan Cheltsov, Alexander Kasprzyk, and Andrea Petracci.

Junliang Shen
Title: Cohomology of the moduli of vector bundles and the moduli of Higgs bundles
on a Riemann surface
Abstract: The moduli of Higgs bundles on a curve can either be viewed as a variantof the moduli of vector bundle on a curve – a very classical moduli space that has beenstudied for decades, or the non-abelian Dolbeault cohomology of the curve in view of thenon-abelian Hodge theory. In this talk, I will discuss a symmetry of the cohomology of themoduli of Higgs bundles that does not hold for the moduli of vector bundles. I will thenexplain how different viewpoints lead to completely different proofs of this statement.

Shunsuke Takagi
Title: Behavior of multiplier ideals under pure morphisms
Abstract: Z. Zhuang recently proved that the property of being klt type descends
under pure morphisms. In this talk, I will discuss a generalization of his result. Our
key tool is the BCM test ideals associated with a canonical BCM algebra constructed bySchoutens. This talk is based on joint work with Tatsuki Yamaguchi.