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

Applications of a parametric Oka principle for liftings

A parametric Oka principle for liftings, recently proved by Forstneric, provides many examples of holomorphic maps that are fibrations in a model structure introduced in previous work of ours. We use this to show that the basic Oka property is equivalent to the parametric Oka property for a large class of manifolds. We introduce new versions of the basic and parametric Oka properties and show, for example, that a complex manifold $X$ has the basic Oka property if and only if every holomorphic map to $X$ from a contractible submanifold of $\mathbb C^n$ extends holomorphically to $\mathbb C^n$.Comment: A few minor improvements in version 2. To appear in a volume in honour of Linda P. Rothschild, Trends in Mathematics series, Birkhause

Deformations of Oka manifolds

We investigate the behaviour of the Oka property with respect to deformations of compact complex manifolds. We show that in a family of compact complex manifolds, the set of Oka fibres corresponds to a G-delta subset of the base. We give a necessary and sufficient condition for the limit fibre of a sequence of Oka fibres to be Oka in terms of a new uniform Oka property. We show that if the fibres are tori, then the projection is an Oka map. Finally, we consider holomorphic submersions with noncompact fibres

Excision for simplicial sheaves on the Stein site and Gromov's Oka principle

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.Comment: Version 3 contains a few very minor improvement

Generalization of a theorem of Gonchar

Let $X, Y$ be two complex manifolds, let $D\subset X,$ $G\subset Y$ be two nonempty open sets, let $A$ (resp. $B$) be an open subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Under a geometric condition on the boundary sets $A$ and $B,$ we show that every function locally bounded, separately continuous on $W,$ continuous on $A\times B,$ and separately holomorphic on $(A\times G) \cup (D\times B)$ "extends" to a function continuous on a "domain of holomorphy" $\hat{W}$ and holomorphic on the interior of $\hat{W}.$Comment: 14 pages, to appear in Arkiv for Matemati

Flexibility properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka-Forstneri\v{c} manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930's, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview article we present 3 classes of properties: 1. density property, 2. flexibility 3. Oka-Forstneri\v{c}. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.Comment: thanks added, minor correction

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

We show that the group of all holomorphic automorphisms of complex affine space C^n, n>1, and several of its subgroups satisfy the parametric Oka property with approximation and with interpolation on discrete sets.Franc Forstnerič and Finnur Lárusso

The Oka principle for holomorphic Legendrian curves in C(2n+1)

Let M be a connected open Riemann surface. We prove that the space L(M,C2n+1) of all holomorphic Legendrian immersions of M to C2n+1, n≥1, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space C(M,S4n−1) of continuous maps from M to the sphere S4n−1. If M has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of L(M,C2n+1) in terms of the homotopy groups of S4n−1. It follows that L(M,C2n+1) is (4n−3)-connected.Franc Forstnerič, Finnur Lárusso

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

Let X be a connected open Riemann surface. Let Y be an Oka domain in the smooth locus of an analytic subvariety of Cn, n ≥ 1, such that the convex hull of Y is all of Cn. Let O∗(X, Y ) be the space of nondegenerate holomorphic maps X → Y. Take a holomorphic 1-form θ on X, not identically zero, and let π : O∗(X, Y ) → H1(X, Cn) send a map g to the cohomology class of gθ. Our main theorem states that π is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on X can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstneriˇc and L´arusson in 2016.Antonio Alarcón, Finnur Lárusso