10:20-11:20 Zhao Xiaolei (UCSB)

Non-commutative abelian surfaces and Kummer type hyperkähler manifolds

Abstract: Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkähler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkähler manifolds deformation equivalent to a generalized Kummer variety is not yet available.

In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkähler manifolds of generalized Kummer deformation type. Applications to abelian fourfolds of Weil type will be discussed.

This is joint work in preparation with Arend Bayer, Alex Perry and Laura Pertusi.

13:30-14:30 Zhang Zili （Tongji University）

The P=W identity for cluster varieties

Abstract: The cohomology of even-dimensional full rank Louise-type cluster varieties satisfy the curious hard Lefschetz property. Since curious hard Lefschetz property is rarely observed and strongly related to the P=W identity, we conjecture that there exists a P=W type identity for cluster varieties. In this talk, we will study the cohomology cluster varieties in a more general setting and establish a conjectural P=W type identity. We will also provide various low-dimensional cluster varieties as evidence toward the P=W conjecture for cluster varieties.

14:50-15:50 Zhou Lin（University of Hannover）

Infinite torsion in motivic cohomology and morphic cohomology

Abstract: Motivic cohomology and morphic cohomology naturally generalize the Chow group (cycles modulo rational equivalence) and cycles modulo algebraic equivalence. Since Mumford’s work in the 1970s, questions on the finite generation of Chow and Griffiths groups—such as the finiteness of dimension, the size of their torsion, or their divisibility—have been a central theme in the study of algebraic cycles.

In this talk, I will show how, over an algebraically closed field whose transcendence degree over its prime field is infinite (e.g., complex number field), one can combine Schoen’s injectivity argument with Schreieder’s refined unramified cohomology to construct examples of motivic and morphic cohomology groups with infinitely many torsion elements. Joint work with Theodosis Alexandrou.

16: 00-17:00 Deng Haohua （Dartmouth College）

Baily—Borel compactification: Classical and beyond

I will talk about the Hodge—theoretic perspective of the Baily—Borel compactification. This includes the classical cases, the Green—Griffiths—Laza—Robles approach and the recent breakthrough by Bakker—Filipazzi—Mauri—Tsimerman.