Abstract: We define flexible affine algebraic varieties, describe their basic properties, and show that many varieties satisfy the flexibility condition. The group of special automorphisms acts in the regular locus of a flexible variety infinitely transitively, that is, any finite collection of smooth points can be sent to any finite collection of smooth points of the same cardinality. Using flexibility, we show that every non-degenerate toric variety, every homogeneous space of a semisimple group, and every variety covered by affine spaces admits a surjective morphism from an affine space. Applying the ellipticity property introduced by Mikhail Gromov in 1989, we prove that a complete algebraic variety X is an image of an affine space if and only if X is unirational. This result is obtained in a joint work with Shulim Kaliman and Mikhail Zaidenberg.
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.
