NTS ABSTRACTFall2020: Difference between revisions
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In this talk, we study the Chow group of the motive associated to a tempered global Lpacket $\pi$ 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 BeilinsonBloch conjecture for Chow groups and Lfunctions. 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.  In this talk, we study the Chow group of the motive associated to a tempered global Lpacket $\pi$ 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 BeilinsonBloch conjecture for Chow groups and Lfunctions. 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.  
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== Sep 17 ==  
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 bgcolor="#F0A0A0" align="center" style="fontsize:125%"  '''Ziquan Yang'''  
  
 bgcolor="#BCD2EE" align="center"  A Crystalline Torelli Theorem for Supersingular K3^[n]type Varieties  
  
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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 reinterpret 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.  
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Revision as of 22:32, 31 August 2020
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Sep 3
Yifeng Liu 
BeilinsonBloch conjecture and arithmetic inner product formula 
In this talk, we study the Chow group of the motive associated to a tempered global Lpacket $\pi$ 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 BeilinsonBloch conjecture for Chow groups and Lfunctions. 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 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 reinterpret 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.
