Standard Poisson structures on Bott-Samelson varieties : degenerations and Frobenius splittings

Abstract

In this thesis, connections between several areas of mathematics, namely that of degenerations of algebraic varieties, tropicalizations of birational transfor mations, Poisson geometry, and Frobenius splittings, were investigated through the examples of Bott-Samelson varieties associated to complex semisimple Lie groups. Bott-Samelson varieties are examples of strongly uniformly rational varieties, which, by definition, are varieties admitting finite Zariski open covers by affine spaces. In the first part of the thesis, for a strongly uniformly rational variety Z and a finite atlas A on Z consisting of coordinate charts parametrized by the affine space, a general theory on degenerations of the pair (Z,A), which are degenerations of Z by rescaling the coordinates in the charts ofA, was developed, and such degenerations were described in terms of the tropicalizations and the initial forms of the birational coordinate transformations between the charts ofA. In particular, a complete invariant of such a degeneration, which is an integral vector called the rescaling weight, was introduced, and it was shown that the rescaling weights of all degenerations (resp. all toric degenerations) of (Z,A) form a cone called the degeneration cone (resp. the toric degeneration cone) of (Z,A). As an example of the general theory, for a Bott-Samelson variety Zu, where u is a sequence of simple roots of a complex semisimple Lie group G, and for a natural finite affine atlas Au on Zu, all degenerations of (Zu,Au) were constructed, up to equivalences, in Lie theoretical terms, and the degeneration cone of (Zu,Au) was shown to be isomorphic to the set of integral points in a closed polyhedral cone Cu explicitly described using the sequence u, with the toric degeneration cone isomorphic to the set of integral points in the interior Cu of Cu. Poisson geometry was brought out in the second part of the thesis, and it was shown that the degeneration cone (resp. the toric degeneration cone) of (Zu,Au) is isomorphic to the degeneration cone (resp. the log-canonical degeneration cone), as introduced by A. Alekseev and I. Davydenkova, of a certain standard Poisson structure on Zu restricted to one particular coordinate chart of Au. All the degenerations of (Zu,Au) were then shown to be degenerations of compact complex Poisson manifolds. In the case of toric degenerations, the standard Poisson structure on Zu was shown to degenerate to a log-canonical Poisson structure on the central fiber which must be a Bott tower as defined by M. Grossberg and Y. Karshon. In the last part of the thesis, a general theory on Frobenius splittings on smooth T-Poisson varieties via Poisson T-Pfaffians was introduced, where T is an algebraic torus, and it was shown that for such splittings, compatibly split sub varieties must be Poisson sub-varieties. As an application, the standard Poisson structure on Bott-Samelson varieties of an algebraic semisimple Lie group G were used to construct a Frobenius splitting on the variety G/U, where U is the uniradical of a Borel subgroup of G.