1998/04/07 by Thomas Peternell, Michael Schneider, Peternell, Thomas +3
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Geometry and complex manifolds #math.AG #math.CV
paper · pdf · doi:10.48550/arxiv.math/9804034
arxiv created 1998/04/07 · arxiv updated 2009/11/30
In this article we study how the birational geometry of a normal projective variety X is influenced by a normal subvariety A ⊂ X. One of the most basic examples in this context is provided by the following situation. Let f:X→ Y be a surjective holomorphic map with connected fibers between compact connected complex manifolds. It is well known that given a general fiber A of f we have κ(X)≤ κ(A)+dim Y. This article grew out of the realization that this result should be true with dim Y replaced by the codimension \codX A for a pair (X,A) consisting of a normal subvariety A of a compact normal variety X under weak semipositivity conditions on the normal sheaf of A and the weak singularity condition \codA (A∩\sing X)≥ 2. We shall now state our main results in the special case of a submanifold A in a projective manifold X and we also simplify the semipositivity notion.