Vanishing theorems for the Calabi curvature operator on Kähler manifolds

Speaker: James Stanfield
Affiliation: University of Münster (Germany)

Abstract

Important questions in Riemannian geometry ask: "What can be said about the underlying topology of a Riemannian manifold given some positivity condition on the curvature?" In Kähler geometry, Bochner gave a very interesting answer in the 1940s: namely, if the Ricci curvature of a Kähler metric is k-positive (i.e., the sum of the first k eigenvalues of the Ricci tensor is positive), then there are no holomorphic p-forms for p between k and n.

In this talk, we discuss a different kind of curvature operator in Kähler geometry, which we call the Calabi curvature operator, first considered by Calabi–Vesentini in the 1960s. It is an endomorphism defined on type (2,0) bivectors that arises naturally from the symmetries of Kähler curvature tensors.

We show that n-dimensional Kähler manifolds must be rational cohomology projective spaces if their Calabi curvature operator is n/2-positive. This result is sharp in even dimensions, because the complex quadric is n/2-nonnegative but has nth Betti number equal to 2. We also classify Kähler manifolds with n/2-nonnegative Calabi curvature operator.

This is joint work with Kyle Broder, Jan Nienhaus, Peter Petersen, and Matthias Wink.