Extending holomorphic maps from Stein manifolds into affine toric varieties

A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstneri\v{c}, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular targets. The targets that we study are affine toric varieties, not necessarily normal. We prove that every affine toric variety satisfies a weakening of the interpolation property that is much stronger than the convex interpolation property, but the full interpolation property fails for most affine toric varieties, even for a source as simple as the product of two annuli embedded in $\mathbb{C}^4$.Comment: 14 pages, v2 and v3: minor corrections and clarifications. To appear in Proceedings of the AM

Siciak-Zahariuta extremal functions, analytic discs and polynomial hulls

We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected

Equivariant Oka theory: survey of recent progress.

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle , all over a reduced Stein space X with compatible actions of a reductive complex group on E, , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a notion of a G-manifold being G-Oka

Equivariant Oka theory: survey of recent progress.

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle , all over a reduced Stein space X with compatible actions of a reductive complex group on E, , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a notion of a G-manifold being G-Oka

Holomorphic flexibility properties of compact complex surfaces

We introduce the notion of a stratified Oka manifold and prove that such a manifold $X$ is strongly dominable in the sense that for every $x\in X$, there is a holomorphic map f:\C^n\to X, $n=\dim X$, such that $f(0)=x$ and $f$ is a local biholomorphism at 0. We deduce that every Kummer surface is strongly dominable. We determine which minimal compact complex surfaces of class VII are Oka, assuming the global spherical shell conjecture. We deduce that the Oka property and several weaker holomorphic flexibility properties are in general not closed in families of compact complex manifolds. Finally, we consider the behaviour of the Oka property under blowing up and blowing down.Comment: Version 2: Theorem 11 reformulated and its proof corrected. Minor improvements to the exposition. Version 3: A few minor improvements. To appear in International Mathematics Research Notice

THE THIRD CAUCHY-FANTAPPIÈ FORMULA OF LERAY

Abstract. We study the third Cauchy-Fantappiè formula, an integral representation formula for holomorphic functions on a domain in affine space, presented by Jean Leray in the third paper of his famous series Problème de Cauchy, published in 1959. We show by means of examples that this formula does not hold without some additional conditions, left unmentioned by Leray. We give sufficient conditions and a necessary condition for the formula to hold, and, in the case of a contractible domain, characterize it cohomologically

An extension theorem for holomorphic functions of slow growth on covering spaces of projective manifolds

Abstract. Let X be a projective manifold of dimension n ≥ 2 and Y → X be an infinite covering space. Embed X into projective space by sections of a sufficiently ample line bundle. We prove that any holomorphic function of sufficiently slow growth on the preimage of a transverse intersection of X by a linear subspace of codimension < n extends to Y. The proof uses a Hausdorff duality theorem for L2 cohomology. We also show that every projective manifold has a finite branched covering whose universal covering space is Stein. 1. Introduction. Infinite covering spaces of projective algebraic manifolds form an interesting and natural class of non-compact complex manifolds, whose function theory is still not well understood. The central problem in this area is the conjecture of Shafarevich that the universal covering space of any projective manifold is holomorphicall