Flexible varieties, images of affine spaces, and ellipticity after Mikhail Gromov

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, one may 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 the joint work with Shulim Kaliman and Mikhail Zaidenberg. The research is supported by the RSF-DST grant 22-41-02019.

About the speaker

Ivan Arzhantsev

Ivan Arzhantsev, Dean of HSE University’s Faculty of Computer Science, Member of the HSE Academic Council.

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https://www.hse.ru/en/staff/arjantsev