Gordon Heier

Reduction of manifolds with semi-negative holomorphic sectional curvature

In this note, we continue the investigation of a projective Kähler manifold M of semi-negative holomorphic sectional curvature H. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent truly flat directions of H in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of M. We also prove a splitting theorem for M in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.

Uniformly effective boundedness of Shafarevich Conjecture-type

Abstract The main result of this note is an effective uniform bound for the number of deformation types of certain nonisotrivial families of canonically polarized manifolds. It extends the authors earlier such bound for the classical Shafarevich Conjecture over function fields to the case of higher dimensional fibers, but without the disadvantageous iterated use of Chow or Hilbert varieties that was the core of the proofs in the earlier approaches.

Finite Type and the Effective Nullstellensatz

Improved local and global versions of the effective Nullstellensatz for ideal sheaves on nonsingular complex varieties are obtained, based on a new invariant motivated by the notion of finite type from the theory of several complex variables. Two closely related curve selection theorems for curves with maximal adjusted tangency orders to a given ideal sheaf are established along the way, using normalized blow-ups and integral closures.