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FIELD
Math: CMC
DATE
Jan 30 (Mon), 2023
TIME
16:00 ~ 18:00
PLACE
1423
SPEAKER
Devyatov, Rostislav
HOST
Lee, Junguk
INSTITUTE
KAIST
TITLE
Karpenko’s conjecture on an isomorphism between the Chow ring and the Grothendieck ring of an orthogonal Grassmannian over a non-algebraically closed field
ABSTRACT
For any smooth projective algebraic variety X, one can construct two rings: the Chow ring CH(X), which is generated by the classes of all subvarieties of X, and the Grothendieck ring K(X), which is generated by the classes of all finite-dimensional vector bundles on X. If X is a flag variety of a simple algebraic group over an algebraically closed filed, then there is a well-known connection between these two rings: CH(X) is isomorphic to the associated graded ring GK(X) of K(X) with respect to the topological filtration. For flag varieties over non-algebraically-closed fields, the situation is not so clear. For example, Nikita Karpenko has proved that if X is the maximal orthogonal Grassmannian of a certain 17-dimensional quadratic form over a certain (non-algebraically-closed) field, then the morphism CH(X) -> GK(X) constructed the same way is not an isomorphism. In my talk, based on a work with progess, joint with Sanghoon Baek from KAIST, I will explain how to generalize Karpenko's contruction to any dimension of the form 2^n+1, where n \ge 5. If I have time, I will explain how to explicitly construct a nonzero element of the kernel of the morphism CH(X) -> GK(X).
FILE
556151674560882466_1.pdf