A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces ${mathfrak M}$ $_r,ϵ$ of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space $\overlinel{\matcal M}$ $_0,n$ of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to $\overlinel{\matcal M}$ $_0,n$ , brings up a new tool to study the Kähler topology of $\overlinel{\matcal M}$ $_0,n$ . A wild but precise conjecture on the shape of the Kähler cone of $\overlinel{\matcal M}$ $_0,n$ is given in the end.

We give the definition of the Thom condition and we show that given any germ of complex analytic function f:(X,x)rightarrow ({mathbb {C}},0) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that fin {mathfrak {m}}_{X,x}^2, where {mathfrak {m}}_{X,x} is the maximal ideal of {mathcal {O}}_{X,x}. This result generalizes a well-known theorem of the second named author when X is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A’Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that X has maximal rectified homotopical depth at x and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.

We say that a complex analytic space, X, is an intersection cohomology manifold if and only if the shifted constant sheaf on X is isomorphic to intersection cohomology; with field coefficients, this is quickly seen to be equivalent to X being a homology manifold. Given an analytic function f on an intersection cohomology manifold, we describe a simple relation between V(f) being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of f. We then describe how the Sebastiani–Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Finally, as an easy application, we obtain restrictions on the cohomology of the Milnor fiber of a hypersurface with a special type of one-dimensional critical locus.

This note considers the problem of approximating continuous maps from sets in complex spaces into complex manifolds by holomorphic maps.

In this chapter we shall introduce the very rich geometry of complex and Kähler manifolds. Complex manifolds are manifolds with a geometric structure called a complex structure, which gives every tangent space the structure of a complex vector space. They are defined in §4.1, together with complex submanifolds and holomorphic maps. Section 4.2 discusses tensors on complex manifolds, and their decomposition into components using the complex structure, and §4.3 defines holomorphic vector bundles over a complex manifold.

In this chapter we shall introduce the very rich geometry of complex and Kähler manifolds. Complex manifolds are manifolds with a geometric structure called a complex structure, which gives every tangent space the structure of a complex vector space. They are defined in 5.1, together with complex submanifolds and holomorphic maps. Section 5.2 discusses tensors on complex manifolds, and their decomposition into components using the complex structure, and 5.3 defines holomorphic vector bundles over a complex manifold.

A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, math.CV/0402278; Ann. Inst. Fourier, 55 (2005), 733-751, math.CV/0411048). We also give existence results for holomorphic sections under certain connectivity hypothesis on the fibers. In the final part of the paper we obtain the Oka property for sections of submersions with stratified sprays over Stein spaces.

It is well-known that sums and products of holomorphic functions are holomorphic, and the holomorphic functions on a complex manifold form a commutative algebra over C . The study of complex manifolds using algebras of holomorphic functions upon them is called complex algebraic geometry. The purpose of this paper is to develop an analogue of complex algebraic geometry, in which the complex numbers C are replaced by the quaternions H . The natural quaternionic analogue of a complex manifold is called a hypercomplex manifold. A class of H -valued functions on hypercomplex manifolds will be defined, called q-holomorphic functions, that are analogues of holomorphic functions on complex manifolds. Now, the set of holomorphic functions on a complex manifold is a commutative algebra over C . Therefore one asks: does the set of q-holomorphic functions on a hypercomplex manifold have an analogous algebraic structure, and if so, what is it? We shall show that the q-holomorphic functions on a (noncompact) hypercomplex manifold do indeed possess a rich algebraic structure. To describe it, we shall introduce a theory of quaternionic algebra, which is a quaternionic analogue of real linear algebra. This theory is built on three building blocks: AH-modules, the analogues of vector spaces, AH-morphisms, the analogues of linear maps, and the quaternionic tensor product, the analogue of tensor product of real vector spaces. As far as the author can tell, these ideas seem to be new. They enable us to construct algebraic structures over H as though H were a commutative field. Quaternionic algebra describes the algebraic structure of hypercomplex manifolds in a remarkable way, and it seems to be the natural language of hypercomplex algebraic geometry, the algebraic geometry of hypercomplex manifolds. We believe that quaternionic algebra is worth studying for its own sake. It has many similarities with linear algebra over R or C , which is why the analogies between complex and quaternionic theories work so well, but there are also deep differences, which give quaternionic algebra a flavour all of its own. Quillen [12] has given a sheaf-theoretic interpretation of the ideas of quaternionic algebra, based on a previous version of this paper. He finds a contravariant equivalence between a class of AH-modules and regular sheaves on a real form of CP. Regular sheaves on CP are equivalent to representations of the Kronecker quiver, and out of such a representation Quillen constructs an AH-module. Under Quillen’s equivalence stable AH-modules correspond to regular vector bundles over the real form of CP.

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

For a proper holomorphic map F from a ball B n into another ball B N , we give a criterion when a rational map F is equivalent to a polynomial one.As application, we show that proper rational holomorphic maps from B 2 into B N of degree two are equivalent to polynomial maps, and we also give an example of a rational holomorphic maps of degree 3 that are 'almost' linear but are not equivalent to holomorphic polynomial maps.

We give a new proof of a theorem of Siu and Trautmann on the vanishing of cohomology with support on complex or rigid analytic spaces. We indicate how our methods might be used to prove various comparison theorems in algebraic and analytic geometry.

In this note we show that on any compact subdomain of a K¨ahler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calder´on problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of K¨ahler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calder´on problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.

Let (E, h) be a semipositive hermitian holomorphic line bundle on a compact complex manifold X with dimX>11$$\\end{document}]]>. Assume that for each point x∈X there exists a divisor D∈|E| in the complete linear system determined by E whose complement X\\D is a Stein neighbourhood of x with the density property. Then, the disc bundle Δh(E)={e∈E:|e|h<1} is an Oka manifold while Dh(E)={e∈E:|e|h>1}1\\}$$\\end{document}]]> is a Kobayashi hyperbolic domain. In particular, the zero section of E admits a basis of Oka neighbourhoods {|e|h<c} with c>00$$\\end{document}]]>. We show that this holds if X is a rational homogeneous manifold of dimension >11$$\\end{document}]]>. This class of manifolds includes complex projective spaces, Grassmannians, and flag manifolds. This phenomenon contributes to the heuristic principle that Oka properties are related to metric positivity of complex manifolds.

Related to the concept of p-compact operators with p ? [1,?] introduced by Sinha and Karn [20], this paper deals with the space H? Kp (U, F) of all Banach-valued holomorphic mappings on an open subset U of a complex Banach space E whose ranges are relatively p-compact subsets of F. We characterize such holomorphic mappings as those whose Mujica?s linearisations on the canonical predual of H?(U) are p-compact operators. This fact allows us to make a complete study of them. We show that H? Kp is a surjective Banach ideal of bounded holomorphic mappings which is generated by composition with the ideal of p-compact operators and contains the Banach ideal of all right p-nuclear holomorphic mappings. We also characterize holomorphic mappings with relatively p-compact ranges as those bounded holomorphic mappings which factorize through a quotient space of ?p* or as those whose transposes are quasi p-nuclear operators (respectively, factor through a closed subspace of ?p).

The purpose of the present note is to show that holomorphic fiber bundles with hyperbolic fiber are almost trivial : They are trivial if the base is simply connected, and have a simple structure otherwise. Kobayashi [2],[3] has introduced the notion of a hyperbolic complex manifold.We need not be concerned with the definition here, but only with the following consequence [3], [4]: If M is a hyperbolic complex manifold, p a point of M, and U a neighborhood of /, then there is a <5>0 such that any holomorphic map <f> of the unit disk into M with (f,(0)=p maps the disk {z:|z|<} into U. From this, it follows that, if iv1, , wn are coordinates at/, and / an integer, then there are constants Al depending only on / and the choice of coordinates such that, if-p is any map of the unit disk into M with cf>(0)=p, then \dlwidzl\<Al at z=0.The crux of our argument is the following proposition, which generalizes a result of Kobayashi's that a complex Lie group (of dimension greater than 0) cannot act holomorphically on a hyperbolic manifold. Proposition.Let M and N be complex manifolds with M hyperbolic.Let F=F(w, z) be a holomorphic map of MxN into M. Suppose that, for some z0 e N, the restriction of F to Mx{z0} is a biholomorphic map onto M. Then F is independent of z; i.e., F(w, z)=F(w, z0). Proof.It suffices to prove the proposition for the case that N is a ball, and hence for the case that Nis a disk A, which we take to be the unit disk with z0=0.Without loss of generality, we may assume F(w, z0)=w.Near a point (w0, z0), the map F has a power series expansion of the form F\w, z) = w' + a\w)zl + 0(zl+1).-