Title: Excision for simplicial sheaves on the Stein site
and Gromov's Oka Principle
Author: Finnur Larusson
AMS classification numbers: Primary: 32Q28; secondary: 18F10,
18F20, 18G30, 18G55, 32E10, 32H02, 55U35
arXiv:math.CV/0101103
Department of Mathematics
University of Western Ontario
London, Ontario N6A 5B7
Canada
larusson@uwo.ca
ABSTRACT: A complex manifold $X$ satisfies the Oka-Grauert property
if the inclusion $\Cal O(S,X) \hookrightarrow \Cal C(S,X)$ is a weak
equivalence for every Stein manifold $S$, where the spaces of
holomorphic and continuous maps from $S$ to $X$ are given the
compact-open topology. Gromov's Oka principle states that if $X$ has a
spray, then it has the Oka-Grauert property. The purpose of this paper
is to investigate the Oka-Grauert property using homotopical algebra.
We embed the category of complex manifolds into the model category of
simplicial sheaves on the site of Stein manifolds. Our main result is
that the Oka-Grauert property is equivalent to $X$ representing a finite
homotopy sheaf on the Stein site. This expresses the Oka-Grauert
property in purely holomorphic terms, without reference to continuous
maps.