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Finnur Larusson

Publications and source records attributed to Finnur Larusson.

At least 37 records · Page 2Linked to original sources
arXiv2017

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

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ of continuous maps from $M$ to the sphere $\mathbb S^{4n-1}$. If $M$ has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of $\mathscr L(M,\mathbb C^{2n+1})$ in terms of the homotopy groups of $\mathbb S^{4n-1}$. It follows that $\mathscr L(M,\mathbb C^{2n+1})$ is $(4n-3)$-connected.

math.DG↗
arXiv2017

Homotopy principles for equivariant isomorphisms

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $ϕ$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $ϕ$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $Φ\colon X\to Y$, inducing $ϕ$, which is $G$-biholomorphic on the reduced fibres of the quotient mappings, then $Φ$ is homotopic, through $G$-diffeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. The second result roughly says that if we have a $G$-homeomorphism $Φ\colon X\to Y$ which induces a continuous family of $G$-equivariant biholomorphisms of the fibres $p_X^{-1}(q)$ and $p_Y^{-1}(ϕ(q))$ for $q\in Q_X$ and if $X$ satisfies an auxiliary property (which holds for most $X$), then $Φ$ is homotopic, through $G$-homeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. Our results improve upon earlier work of the authors and use new ideas and techniques.

math.CV↗
arXiv2017

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 $\mathbb C^n$, $n\geq 1$, such that the convex hull of $Y$ is all of $\mathbb C^n$. Let $\mathscr O_*(X, Y)$ be the space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic $1$-form $θ$ on $X$, not identically zero, and let $π:\mathscr O_*(X,Y) \to H^1(X,\mathbb C^n)$ 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 Forstneric and Larusson in 2016.

math.CV↗
arXiv2016

An equivariant parametric Oka principle for bundles of homogeneous spaces

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein space $X$, such that the fibre of $E$ is a homogeneous space of the fibre of $\mathscr G$, with the complexification $K^\mathbb C$ of a compact real Lie group $K$ acting on $X$, $\mathscr G$, and $E$. Our main result is that the inclusion of the space of $K^\mathbb C$-equivariant holomorphic sections of $E$ over $X$ into the space of $K$-equivariant continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch's classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.

math.CV↗
arXiv2016

Algebraic subellipticity and dominability of blow-ups of affine spaces

Little is known about the behaviour of the Oka property of a complex manifold with respect to blowing up a submanifold. A manifold is of Class $\mathscr A$ if it is the complement of an algebraic subvariety of codimension at least $2$ in an algebraic manifold that is Zariski-locally isomorphic to $\mathbb C^n$. A manifold of Class $\mathscr A$ is algebraically subelliptic and hence Oka, and a manifold of Class $\mathscr A$ blown up at finitely many points is of Class $\mathscr A$. Our main result is that a manifold of Class $\mathscr A$ blown up along an arbitrary algebraic submanifold (not necessarily connected) is algebraically subelliptic. For algebraic manifolds in general, we prove that strong algebraic dominability, a weakening of algebraic subellipticity, is preserved by an arbitrary blow-up with a smooth centre. We use the main result to confirm a prediction of Forster's famous conjecture that every open Riemann surface may be properly holomorphically embedded into $\mathbb C^2$.

math.AG↗
arXiv2016

The parametric h-principle for minimal surfaces in $\mathbb R^n$ and null curves in $\mathbb C^n$

Let $M$ be an open Riemann surface. It was proved by Alarcón and Forstnerič (arXiv:1408.5315) that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. In this paper, we prove the following much stronger result in this direction: for any $n\geq 3$, the inclusion $ι$ of the space of real parts of nonflat null holomorphic immersions $M\to\mathbb C^n$ into the space of nonflat conformal minimal immersions $M\to \mathbb R^n$ satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. We prove analogous results for several other related maps, and we describe the homotopy type of the space of all holomorphic immersions $M\to\mathbb C^n$. For an open Riemann surface $M$ of finite topological type, we obtain optimal results by showing that $ι$ and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences.

math.DG↗
arXiv2016

Sufficient Conditions for Holomorphic Linearisation

Let $G$ be a reductive complex Lie group acting holomorphically on $X={\mathbb C}^n$. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ${\mathbb C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $Φ\colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $Q_X$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $Φ$ as above. Then $Φ$ induces a biholomorphism $ϕ\colon Q_X\to Q_V$ which is stratified, i.e., the stratum of $Q_X$ with a given label is sent isomorphically to the stratum of $Q_V$ with the same label. The counterexamples to the Linearisation Problem construct an action of $G$ such that $Q_X$ is not stratified biholomorphic to any $Q_V$. Our main theorem shows that, for most $X$, a stratified biholomorphism of $Q_X$ to some $Q_V$ is sufficient for linearisation. In fact, we do not have to assume that $X$ is biholomorphic to ${\mathbb C}^n$, only that $X$ is a Stein manifold.

math.CV↗
arXiv2015

Eight lectures on Oka manifolds

Over the past decade, the class of Oka manifolds has emerged from Gromov's seminal work on the Oka principle. Roughly speaking, Oka manifolds are complex manifolds that are the target of "many" holomorphic maps from affine spaces. They are "dual" to Stein manifolds and "opposite" to Kobayashi-hyperbolic manifolds. The prototypical examples are complex homogeneous spaces, but there are many other examples: there are many ways to construct new Oka manifolds from old. The class of Oka manifolds has good formal properties, partly explained by a close connection with abstract homotopy theory. These notes were prepared for lectures given at the Institute of Mathematics of the Chinese Academy of Sciences in Beijing in May 2014. They are meant to give an accessible introduction, not to all of Oka theory, but more specifically to Oka manifolds, how they arise and what we know about them.

math.CV↗
arXiv2013

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

The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at once, in a way that depends continuously on $f$ and leaves $f$ fixed if it is holomorphic to begin with. In other words, is $\scrO(S,X)$ a deformation retract of $\scrC(S,X)$? We prove that it is if $S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points; in particular, if $S$ is affine algebraic. The only property of $X$ used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that $X$ is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces.

math.CV↗
arXiv2013

An Oka principle for equivariant isomorphisms

Let $G$ be a reductive complex Lie group acting holomorphically on normal Stein spaces $X$ and $Y$, which are locally $G$-biholomorphic over a common categorical quotient $Q$. When is there a global $G$-biholomorphism $X\to Y$? If the actions of $G$ on $X$ and $Y$ are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that $X$ and $Y$ are $G$-biholomorphic if $X$ is $K$-contractible, where $K$ is a maximal compact subgroup of $G$, or if $X$ and $Y$ are smooth and there is a $G$-diffeomorphism $ψ:X\to Y$ over $Q$, which is holomorphic when restricted to each fibre of the quotient map $X\to Q$. We prove a similar theorem when $ψ$ is only a $G$-homeomorphism, but with an assumption about its action on $G$-finite functions. When $G$ is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of $G$-biholomorphisms from $X$ to $Y$ over $Q$. This sheaf can be badly singular, even for a low-dimensional representation of $\mathrm{SL}_2(\C)$. Our work is in part motivated by the linearisation problem for actions on $\C^n$. It follows from one of our main results that a holomorphic $G$-action on $\C^n$, which is locally $G$-biholomorphic over a common quotient to a generic linear action, is linearisable.

math.CV↗
arXiv2013

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.

math.CV↗
arXiv2012

Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into CxC*

Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map $X \to \C\times\C^*$ from a finitely connected planar domain $X$ without isolated boundary points, a stronger Oka property holds, namely that the map is homotopic to a proper holomorphic embedding. Here we show that every continuous map from a finitely connected planar domain, possibly with punctures, into $\C\times\C^*$ is homotopic to a proper immersion that identifies at most countably many pairs of distinct points, and in most cases, only finitely many pairs. By examining situations in which the immersion is injective, we obtain a strong Oka property for embeddings of some classes of planar domains with isolated boundary points. It is not yet clear whether a strong Oka property for embeddings holds in general when the domain has isolated boundary points. We conclude with some observations on the existence of a null-homotopic proper holomorphic embedding $\C^* \to \C\times\C^*$.

math.CV↗
arXiv2012

Plurisubharmonic subextensions as envelopes of disc functionals

We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain $W$ in a Stein manifold to a larger domain $X$ under suitable conditions on $W$ and $X$. We introduce a related equivalence relation on the space of analytic discs in $X$ with boundary in $W$. The quotient, if it is Hausdorff, is a complex manifold with a local biholomorphism to $X$. We use our disc formula to generalise Kiselman's minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.

math.CV↗
arXiv2011

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.

math.CV↗
arXiv2010

Survey of Oka theory

Oka theory has its roots in the classical Oka principle in complex analysis. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. Following a brief review of Stein manifolds, we discuss the recently introduced category of Oka manifolds and Oka maps. We consider geometric sufficient conditions for being Oka, the most important of which is ellipticity, introduced by Gromov. We explain how Oka manifolds and maps naturally fit into an abstract homotopy-theoretic framework. We describe recent applications and some key open problems. This article is a much expanded version of the lecture given by the first-named author at the conference RAFROT 2010 in Rincon, Puerto Rico, on March 22, 2010, and of a brief survey article by the second-named author in Notices Amer. Math. Soc., January 2010.

math.CV↗