Speaker
Askold Khovansky, University of Toronto
Abstract
Schubert developed enumerative geometry. In particular he dened cer-
tain subvarieties in Grassmannians, now called Schubert cells, and computed the number of intersection points of several cells of appropriate dimensions moved to a general position by linear transformations. Later (by solving the 15-th Hilbert problem) a rigorous justication for Schubert calculus was established. One can reduce Schubert calculus to the cohomology rings of Grassmannians.
In the talk I will present a new point of view on the ring of conditions (see [1], [2]) of the group (C*)^n. I will dene cells (special subvarieties) in (C*)^n and will compute the number of intersection points of several cells of appropriate dimensions moved to a general position by actions of the group (C*)^n. The computation is based only on BKK (Bernstein, Koushnirenko, Khovanskii) Theorem and is elementary. It allows us to dene a ring, whose elements are a linear combination of cells, and completely describe it using the theory of mixed volumes of convex polyhedra and its description in terms of tropical geometry.
Without changing the ring one can include in it all algebraic subvarieties of (C*)^n and show that the ring is isomorphic to the ring of conditions of the group (C*)^n. This foundational part is based on the Good Compactication Theorem [1] for the group (C*)^n and on properties of cohomology rings of smooth projective toric varieties. If time permits, I will sketch my elementary proof [3] of the Good Compactication Theorem for the group (C*)^n.
References
1. C. De Concini and C. Procesi, Complete symmetric varieties. II. Intersection theory”, Algebraic groups and related topics (Kyoto/Nagoya
1983), Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam 1985, pp. 481-513.
2. B.Ya. Kazarnovskii, A.G. Khovanskii, A.I. Esterov. Newton polytopes and tropical geometry”. Math. Surveys 76:1, (2021) 91-175.
3. A. Khovanskii. Newton polyhedra and good compactification theorem.” ArMJ, V. 7, (2021) 135-157.