Equivariant completions of affine space

• 主讲人：Ivan Arzhantsev（HSE University）
• 举办方： Beijing-Moscow Mathematics Colloquium
• 时间： 2023-12-01 17:00 - 2023-12-01 18:00
• 地点： online

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Abstract: In this talk we survey recent results on open embeddings of the complex affine space A^n into a complete algebraic variety X such that the action of the vector group G on A^n by translations extends to an action of G on X. We begin with Hassett-Tschinkel correspondence describing equivariant embeddings of A^n into projective spaces and give its generalization for embeddings into projective hypersurfaces. We prove that non-degenerate projective hypersurfaces admitting such an embedding are in bijection with Gorenstein local algebras. Moreover, such an embedding into a projective hypersurface is unique if and only if the hypersurface is non-degenerate.Further we deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Supported by the Russian Science Foundation grant 23-21-00472 

 

Bio: Ivan Arzhantsev graduated from the Faculty of Mechanics and Mathematics of Moscow State Lomonosov University in 1995. Since 2014, he is a Dean of the Faculty of Computer Science at the HSE University in Moscow and a Professor of the Big Data and Information Retrieval School at the same faculty. In 2021, he initiated the creation of the laboratory on Algebraic Transformation Groups at the HSE University and since that time he has been the head of this laboratory. Arzhantsev is a Deputy head of the Dissertation Coucil on Mathematics at the HSE University and a Tenured Professsor at Independent University of Moscow. 

 

Area of scientific interests: Algebraic Transformation Groups, Affine Algebraic Geometry, Invariant Theory, Toric Varieties, Cox Rings. 
 

Arzhantsev's main results concern automorphisms of algebraic varieties, infinite transitivity of group actions, embeddings of homogeneous spaces and equivariant competions, locally nilpotent derivations of graded algebras and other actual areas of modern Algebra and Geometry. His results are published in Duke Mathematical Journal, Advances in Mathematics, Journal of the London Mathematical Society, and many other prestigious journals.  He co-authored the monography "Cox Rings" published in 2015 in Cambridge Studies in Advanced Mathematics. 

In 2006, he was awarded the Academiae Europaeae Prize for Young Russian Scientists, and in 2008 he got the Pierre Deligne’s Grant based on 2004 Balzan Prize in Mathematics. He regularly supervises grants from the Russian Science Foundation and other institutions.