Finnur Larusson

EXCISION FOR SIMPLICIAL SHEAVES ON THE STEIN SITE AND GROMOV'S OKA PRINCIPLE

A complex manifold X is said to satisfy the Oka–Grauert property if the inclusion

Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds

{\mathcal O} (S, X)\hookrightarrow {\mathcal C} (S, X)

Holomorphic Flexibility Properties of Compact Complex Surfaces

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. Gromovs 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.

Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds

arusson Abstract. Let Y be an infinite covering space of a projective manifold M in P N of dimension n≥ 2. Let C be the intersection with M of at most n− 1 generic hypersurfaces of degree d in P N. The preimage X of C in Y is a connected submanifold. Letbe the smoothed distance from a fixed point in Y in a metric pulled up from M. LetO�(X) be the Hilbert space of holomorphic functions f on X such that f 2 eis integrable on X, and defineO�(Y) similarly. Our main result is that (under more general hypotheses than described here) the restrictionO�(Y)→O�(X) is an isomorphism for d large enough. This yields new examples of Riemann surfaces and domains of holomorphy in C n with corona. We con- sider the important special case when Y is the unit ball B in C n , and show that for d large enough, every bounded holomorphic function on X extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on B. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from X to B.

Oka properties of groups of holomorphic and algebraic automorphisms of complex affine space

We introduce the notion of a stratified Oka manifold and prove that such a manifold

Applications of a Parametric Oka Principle for Liftings

X

SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION

is strongly dominable in the sense that for every

Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

x\in X

THE MARTIN BOUNDARY ACTION OF GROMOV HYPERBOLIC COVERING GROUPS AND APPLICATIONS TO HARDY CLASSES

, there is a holomorphic map

The Oka principle for holomorphic Legendrian curves in \(\mathbb {C}^{2n+1}\)

f:\C^n\to X