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Sep 3

Yifeng Liu
Beilinson-Bloch conjecture and arithmetic inner product formula

In this talk, we study the Chow group of the motive associated to a tempered global L-packet $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is -1. We show that, under some restrictions on the ramification of $\pi$, if the central derivative $L'(1/2,\pi)$ is nonvanishing, then the $\pi$-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and L-functions. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain $\pi$-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,\pi)$ and local doubling zeta integrals. This is a joint work with Chao Li.

Sep 10

Yifeng Liu
The joints problem for varieties

We generalize the Guth-Katz joints theorem from lines to varieties. A special case of our result says that N planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture).

Our main innovation is a new way to extend the polynomial method to higher dimensional objects. A simple, yet key step in many applications of the polynomial method is the "vanishing lemma": a single-variable degree-d polynomial has at most d zeros. In this talk, I will explain how we generalize the vanishing lemma to multivariable polynomials, for our application to the joints problem.

Joint work with Jonathan Tidor and Hung-Hsun Hans Yu (https://arxiv.org/abs/2008.01610)

Sep 17

Ziquan Yang
A Crystalline Torelli Theorem for Supersingular K3^[n]-type Varieties

In 1983, Ogus proved that a supersingular K3 surface in characteristic at least 5 is determined up to isomorphism by the Frobenius action and the Poincaré pairing on its second crystalline cohomology. This is an analogue of the classical Torelli theorem for K3's, due to Shapiro and Shafarevich, which says that a complex algebraic K3 surface is determined up to isomorphism by the Hodge structure and the Poinaré pairing on its second singular cohomology. I will explain how to re-interpret Ogus' theorem from a motivic point of view and generalize the stronger form of the theorem to a class of higher dimensional analogues of K3 surfaces, called K3^[n]-type varieties. This is also an analogue of Verbitsky's global Torelli theorem for general irreducible symplectic manifolds. A new feature in Verbitsky's theorem, which did not appear in the classical Torelli theorem for K3's, is the notion of "parallel transport operators". I will explain how to work with this notion in an arithmetic setting.

As an application, I will also present a similar crystalline Torelli theorem for supersingular cubic fourfolds, the Hodge theoretic counterpart of which is a theorem of Voisin.