The homotopy theory of equivalence relations
We give a detailed exposition of the homotopy theory of equivalence relations, perhaps the simplest nontrivial example of a model structure.
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arXiv subjects
Finnur Larusson
Publications and source records attributed to Finnur Larusson.
Affine simplices in Oka manifolds
In this note, we show that the homotopy type of a complex manifold X satisfying the Oka property is captured by holomorphic maps from the affine spaces C^n, n\geq 0, into X. Among such X are all complex Lie groups and their homogeneous spaces. We present generalisations of this result, one of which states that the homotopy type of the space of continuous maps from any smooth manifold to X is given by a simplicial set whose simplices are holomorphic maps into X.
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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 $\mathbb C^n$.
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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.
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Schlicht envelopes of holomorphy and foliations by lines
Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain Y in X an interval domain with respect to F if Y intersects every leaf of F in a nonempty connected set. We show that if X is Stein and if F satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope of holomorphy, which is also an interval domain. This result is a generalization and a global version of a well-known lemma from the mid-1980s. We illustrate the notion of quasiholomorphicity with sufficient conditions, examples, and counterexamples, and present some applications, in particular to a little-studied boundary regularity property of domains called local schlichtness.
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Siciak-Zahariuta extremal functions and polynomial hulls
We use our disc formula for the Siciak-Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs.
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Dessins d'enfants and differential equations
We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d'enfants. We show how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial. We produce a universal annihilating operator for the inverses of a generic polynomial. We classify those plane trees that have a representation by Mobius transformations and those that have a linear representation of dimension at most two. This yields an analogue for trees of Schwarz's classical list, that is, a list of those plane trees whose Riemann-Hilbert problem has a hypergeometric solution of order at most two.
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The Siciak-Zahariuta extremal function as the envelope of disc functionals
We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space, generalizing Lempert's formula for the convex case. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert's formula for this function from the convex case to the connected case.
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Mapping Cylinders and the Oka Principle
We apply concepts and tools from abstract homotopy theory to complex analysis and geometry, continuing our development of the idea that the Oka Principle is about fibrancy in suitable model structures. We explicitly factor a holomorphic map between Stein manifolds through mapping cylinders in three different model structures and use these factorizations to prove implications between ostensibly different Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. Going beyond the Stein case to a study of cofibrant models of arbitrary complex manifolds, using the Jouanolou Trick, we obtain a geometric characterization of an Oka property for a large class of manifolds, extending our result for Stein manifolds. Finally, we prove a converse Oka Principle saying that certain notions of cofibrancy for manifolds are equivalent to being Stein.
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Model Structures and the Oka Principle
We embed the category of complex manifolds into the simplicial category of prestacks on the simplicial site of Stein manifolds, a prestack being a contravariant simplicial functor from the site to the category of simplicial sets. The category of prestacks carries model structures, one of them defined for the first time here, which allow us to develop "holomorphic homotopy theory". More specifically, we use homotopical algebra to study lifting and extension properties of holomorphic maps, such as those given by the Oka Principle. We prove that holomorphic maps satisfy certain versions of the Oka Principle if and only if they are fibrations in suitable model structures. We are naturally led to a simplicial, rather than a topological, approach, which is a novelty in analysis.
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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.
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Plurisubharmonicity of envelopes of disc functionals on manifolds
We show that a disc functional on a complex manifold has a plurisubharmonic envelope if all its pullbacks by holomorphic submersions from domains of holomorphy in affine space do and it is locally bounded above and upper semicontinuous in a certain weak sense. For naturally defined classes of disc functionals on manifolds, this result reduces a property somewhat stronger than having a plurisubharmonic envelope to the affine case. The proof uses a recent Stein neighbourhood construction of Rosay, who proved the plurisubharmonicity of the Poisson envelope on all manifolds. As a consequence, the Riesz envelope and the Lelong envelope are plurisubharmonic on all manifolds; for the former, we make use of new work of Edigarian. The basic theory of the three main classes of disc functionals is thereby extended to all manifolds.
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The third Cauchy-Fantappie formula of Leray
We study the third Cauchy-Fantappie 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 "Probleme 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.
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The Jensen envelope is plurisubharmonic on all manifolds
The Jensen envelope $Jϕ$ of an upper semicontinuous function $ϕ$ on a complex manifold X is defined at $x\in X$ as the infimum of $μ(ϕ)$ over all Jensen measures $μ$ centred at x. The Poisson envelope $Pϕ$ is defined by using only the boundary measures of analytic discs centred at x. One of the main open problems in the theory of disc functionals is whether the Poisson envelope is plurisubharmonic on an arbitrary manifold. This is equivalent to the two envelopes being equal, so plurisubharmonicity of $Jϕ$ is a necessary condition for $Pϕ$ to be plurisubharmonic. We prove that the Jensen envelope is plurisubharmonic, with no assumptions on the manifold X. Hence $Jϕ$ is the largest plurisubharmonic function smaller than $ϕ$. We also show that the Poisson envelope is plurisubharmonic if and only if boundary measures of analytic discs are dense among Jensen measures.
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Holomorphic functions of slow growth on nested covering spaces of compact manifolds
Let Y be an infinite covering space of a projective manifold M in P^N of dimension n geq 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. Let phi be the smoothed distance from a fixed point in Y in a metric pulled up from M. Let O_phi(X) be the Hilbert space of holomorphic functions f on X such that f^2 e^(-phi) is integrable on X, and define O_phi(Y) similarly. Our main result is that (under more general hypotheses than described here) the restriction O_phi(Y) to O_phi(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 consider 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.
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Plurisubharmonic extremal functions, Lelong numbers and coherent ideal sheaves
We introduce a new type of pluricomplex Green function which has a logarithmic pole along a complex subspace A of a complex manifold X. It is the largest negative plurisubharmonic function on X whose Lelong number is at least the Lelong number of log max{|f_1|,...,|f_m|}, where f_1,...,f_m are local generators for the ideal sheaf of A. The pluricomplex Green function with a single logarithmic pole or a finite number of weighted poles is a very special case of our construction. We give several equivalent definitions of this function and study its properties, including boundary behaviour, continuity, and uniqueness. This is based on and extends our previous work on disc functionals and their envelopes.
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