Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Classifying complex algebraic manifolds with rather positive tangent bundles.

There are many ways to define related, non-equivalent notions of a "positively curved" complex manifold. It is generally interesting to see how these notions relate, find criteria for a manifold to be positively curved, or describe consequences of having positive curvature, in one sense or the other of the words "positive" and "curvature". Many different results throughout differential, complex, and algebraic geometry seem to encourage the vague intuition that positively curved varieties are better behaved than their flat or hyperbolic counterparts. Two main objects used by complex algebraic geometers to measure the curvature of a variety are the tangent bundle, and its determinant, the so-called anticanonical bundle - K_X. There are different notions of positivity for these objects, such as (from the stricter to the looser) amplitude, strict nefness, and nefness. In the first part of this talk, we define these notions, discuss classical examples and interplay between them. We also present classical results and open questions about varieties whose tangent bundle or anticanonical bundle that is ample, strictly nef, or nef. In the second part of the talk, we investigate the same positivity properties for intermediate exterior powers of the tangent bundle. We present partial classification results for varieties X such that the 3rd or 4th exterior power of T_X is strictly nef. We also prove an unexpected relationship between the positivity of the (dim X - 1)th exterior power of T_X and that of - K_X. If time permits, I will explain the role that large families of rational curves deforming in X play both results.