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8th Meeting

BICMR, PKU

Xiaolei Zhao (Northeastern)

Birational geometry of moduli spaces of sheaves and stability conditions

The birational geometry of moduli of sheaves is greatly studied in recent years, after the pioneer work on K3 surfaces by Bayer and Macrì. The key point is to do wall-crossing in the space of Bridgeland stability conditions. In this talk, I will recall the definition and construction of stability conditions, and explain how to carry out this idea of Bayer and Macrì for the projective plane, based on joint work with Chunyi Li. If time permits, I will also discuss some recent progress in higher dimensions.

Chunyi Li (Edinburgh)

Stability conditions on threefolds

The notion of stability condition on a triangulated category has been introduced by Bridgeland around fifteen years ago. The existence of stability conditions on threefolds has become a core problem of this field. The existence of such conditions is equivalent to some generalized version of Bogomolov inequalities, and will imply new bounds on Chern classes of stable sheaves. I will discuss the progress of this problem in recent years and some related open problems.

Junliang Shen (ETH Zürich)

Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories

By physical considerations, Huang, Katz, and Klemm conjectured that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by Jacobi forms. This gives strong structural results for curve counting invariants of elliptic CY 3-folds. I will explain a mathematical approach to proving (part of) the HKK conjecture. Our method is to construct an involution in the derived category and use wall-crossing techniques. As applications, we present new calculations of Gromov-Witten invariants in any genus for several compact CY 3-folds. Finally, we will discuss the connection to the Igusa cusp form conjecture of Oberdieck and Pandharipande, which concerns the enumerative geometry of K3 surfaces. Joint work with Georg Oberdieck.

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