Abstract:

Let X be a Kähler manifold and let’s consider the total space T*X of the cotangent bundle of X. In many nice cases there exists a natural hyperkähler metric on T*X, called the Feix-Kaledin metric. One of these cases is when X is obtained as a Kähler reduction from C^n. Once we have a hyperkähler metric on T*X we may ask if it is complete. The completeness of this metric turns out to be strongly related to the algebraic geometry of X. Namely, I’m going to derive from this condition that the tangent bundle to X is big and nef. The famous Campana-Peternell conjecture tells us that these two conditions on the tangent bundle should imply that X is a rational homogeneous variety. https://arxiv.org/abs/2007.05773