Algebra and Geometry Seminar

During the last few decades, much progress has been made in classification of complex algebraic varieties. From the viewpoint of Minimal Model Program, projective manifolds should be classified according to "sign" of their canonical class $K_X$. It is then natural to ask how far we can lift the positivity or the negativity of $K_X$ to the cotangent bundle. In the positive case, Miyaoka showed that if $K_X$ is pseudoeffective, then the cotangent bundle is generically nef, that is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. If moreover $K_X$ is nef, he also showed that the second Chern class of $X$ has non-negative intersection numbers with ample divisors in this case. We are interested in the negative case. We show that if $−K_X$ is nef, then the tangent bundle is generically nef, and the second Chern class of $X$ has the same positivity as before. We also investigate under which conditions the postivities would be strict.