Speaker: Sergey Galkin
Title: Cubic forms and related geometries
Date: June 5 (16:20-17:10), June (15:10-16:00), June 7 (10:00-10:50) 2017
Place: Gökova Geometry/Topology Institute
Abstract:
To a single homogeneous cubic polynomial one can associate many spaces of different geometric and topological kinds (Fano manifolds, hyper-kähler manifolds, as well as some manifolds of general type). E.g. for a cubic form f(x1,...,x6) its zeroes in P5 is a Fano fourfold, and there are 3 known constructions of hyper-kähler spaces: 4-folds of Fano-Beauville-Donagi and 8-folds of Lehn-Lehn-Sorger-van Straten are deformations of Hilbert schemes of points on K3 surfaces, and 10-folds of Laza-Sacca-Voisin are deformations of sporadic varieties constructed by O'Grady; much more of similar varieties are yet to be constructed and studied.
These spaces hypothetically should be related by concrete geometric constructions, and as a corollary one should obtain algebraic relations between various invariants of this spaces, such as Euler numbers, Chern numbers, Hodge structures, classes in bordisms and other rings of varieties (such as Grothendieck ring), motives, derived categories of coherent sheaves, mirror duals, ... One such geometric relation between variety of lines and symmetric square of a cubic was carefully studied in works by myself and Shinder, Voisin, Laterveer, and its origin traces back to "secant line" construction of Diophantus. The expression for twisted cubics on a cubic hypersurface is not yet known exactly, however we know its approximate form.
One talk will cover geometry of cubic hypersurfaces and spaces of its rational curves. In another one I will explain definitions and properties of various groups of varieties, as well as operations on some of them (product, grading, differential, symmetric powers, and other operations); also I will explain "decomplexifications" of such groups to groups of topological manifolds (sometimes with orientations and corners). In third talk I will explain various known and conjectural relations, as well as framework of Herbst-Hori-Page to understand some of these relations as phase transitions and mirror symmetry for the respective spaces.