Vector Fields and Cohomology of G/B

Abstract

The topic I will discuss today is one which arose from a question which I believe Professor Matsushima originally asked: namely, if one is given a holomorphic vector field V on a projective manifold X, is it true that X has no nontrivial holomorphic p-forms if p > dimC zero (V)? Alan Howard answered this question affirmatively in [H] and later, D. Lieberman and I discovered other relationships between zeros of holomorphic vector fields and topology. Perhaps the most interesting of these is that if one has a holomorphic vector field V on a compact Kaehler manifold X with isolated zeros, then the whole cohomology ring of X can be calculated on the zeros of V. Although holomorphic vector fields with isolated zeros are not abundant, they do exist on a fundamental class of spaces, namely the algebraic homogeneous spaces. In the one example that has been carefully analyzed, the Grassmannians, the calculation of the cohomology ring on the zeros of V gives a new insight on the connection between Schubert calculus and the theory of symmetric functions [C].