Mathematics – Algebraic Geometry
Scientific paper
1998-04-07
Mathematics
Algebraic Geometry
Scientific paper
In this article we study how the birational geometry of a normal projective variety $X$ is influenced by a normal subvariety $A \subset X.$ One of the most basic examples in this context is provided by the following situation. Let $f:X\to Y$ 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 $$ \kappa(X)\le \kappa(A)+\dim Y. $$ This article grew out of the realization that this result should be true with $\dim Y$ replaced by the codimension $\cod_X A$ 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 $\cod_A (A\cap\sing X)\ge 2$. 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.
Peternell Thomas
Schneider Michael
Sommese Andrew J.
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