Abstract. 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.
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 deÿned for the ÿrst time here, which allow us to develop holomorphic homotopy theory. More speciÿcally, 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 ÿbrations in suitable model structures. We are naturally led to a simplicial, rather than a topological, approach, which is a novelty in analysis. c 2004 Elsevier B.V. All rights reserved.MSC: Primary: 32Q28; secondary: 18F10; 18F20; 18G30; 18G55; 32E10; 55U35 1. Introduction. This paper, like its predecessor [10], is about model structures in complex analysis. Model structures are good for many things, but here we view them primarily as a tool for studying lifting and extension properties of holomorphic maps, such as those given by the Oka Principle. More precisely, model structures provide a framework for investigating two classes of holomorphic maps such that the ÿrst has the right lifting property with respect to the second and the second has the left lifting property with respect to the ÿrst in the absence of topological obstructions. (It is more natural, actually, to consider homotopy lifting properties rather than plain lifting properties.) We seek to make the maps in the ÿrst class into ÿbrations and those in the second class into coÿbrations, with weak equivalences being understood in the topological sense. The machinery of abstract homotopy theory can then be applied.
Let M be an open Riemann surface. It was proved by Alarcón and Forstnerič [1] that every conformal minimal immersion M → R 3 is isotopic to the real part of a holomorphic null curve M → C 3 . In this paper, we prove the following much stronger result in this direction: for any n ≥ 3, the inclusion ι : ℜN * (M, C n ) ֒→ M * (M, R n ) of the space of real parts of nonflat null holomorphic immersions M → C n into the space of nonflat conformal minimal immersions M → R n satisfies the parametric h-principle with approximation (see Theorem 4.1). In particular, ι is a weak homotopy equivalence (see Theorem 1.1). We prove analogous results for several other related maps (see Theorems 1.2 and 5.6 and Corollary 1.3), and we describe the rough shape of the space of all holomorphic immersions M → C n (Theorem 1.4). 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 (see Corollary 6.2 and Remark 6.3). We use the standard notation O(M, Y ) for the space of all holomorphic maps from M to a complex manifold Y with the compact-open topology.
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 a‰ne 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 a‰ne 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.The first-named author was supported in part by the Natural Sciences and Engineering Research Council of Canada.Brought to you by | The University of York Authenticated Download Date | 7/7/15 7:53 AM
Abstract. 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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