RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n andm, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy- Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

We show that the moduli stack $\mathscr {M}\sb h$ of canonically polarized complex manifolds with Hilbert polynomial $h$ is Brody hyperbolic. Hence if $M\sb h$ denotes the corresponding coarse moduli scheme, and if $U \to M\sb h$ is a quasi-finite morphism, induced by a family, then there are no nonconstant holomorphic maps $\mathbb {C}\to U$.

Let X be a paracompact complex manifold of dimension «J>1. We call X a weakly 1-complete manifold if there exists a C°° plurisubharmonic function (? on X such that for every ceR (real number), <p~1((— oo? c)} is relatively compact in X. cp is called an exhaustion function of X. It is well known that holomorphically convex manifolds are weakly 1-complete. The converse is not true in general (cf. [1]), so one is led to the problem of seeking natural additional conditions which make weakly 1-complete manifolds holomorphically convex. The content of this article is divided into two parts; Section 1 is devoted to prove some properties of weakly 1-complete manifolds which have a nonconstant holomorphic function. In Section 2, first we present an application of the Nakano's vanishing theorem to the Levi problem on projective spaces and hyperquadrics. These results are not new (cf. [3]) but the method will be of some interest. Next, combining the Nakano's vanishing theorem with the result in Section 1 and a well known theorem (due to Bonnet) of differential geometry we obtain the following.

Let X = Ω/Γ be a smooth quotient of a bounded symmetric domain Ω by an arithmetic subgroup Open image in new window. We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a finite etale cover Xg = Ω/Γ(g) of X determined by a subgroup Open image in new window depending only on g, such that for any compact Riemann surface R of genus g and any non-constant holomorphic map f : R → Xg* from R into the Satake-Baily-Borel compactification Xg* of Xg, the image f(R) lies in the boundary ∂Xg: = X*g\Xg. Nadel proved it for g = 0 or 1. Moreover, for any positive integer n and any non-negative integer g≥0, we show that there exists a positive number a(n,g) depending only on n and g with the following property: a principally polarized non-isotrivial n-dimensional abelian variety over a complex function field of genus g does not have a level-N structure for N≥a(n,g). This was proved by Nadel for g = 0 or 1, and by Noguchi for arbitrary g under the additional hypothesis that the abelian variety has non-empty singular fibers.

The Index theorem for quasi-tori

We show that the property of a compact set E in ℂn to satisfy conditions of the type of Leja's famous polynomial condition is invariant under a large class of holomorphic mappings from a neighborhood of the polynomial envelope of E, E, to ℂM (M ≤ N) containing, in particular, all open holomorphic mappings. This yields new examples of sets E in ℂN satisfying the conditions under consideration.

SynopsisThe compact weak topology (kw) on a Banach space is defined as the finest topology that agrees with the weak topology on weakly compact subsets. It appears in a natural manner in the study of certain classes of continuous and holomorphic maps between Banach spaces. In this paper we treat the kw topology and the finest locally convex topology contained in kw, which we call the ckw topology. We prove that kw = ckw if and only if the space is reflexive or Schur, and we derive characterisations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. We also show that ckw is the topology of uniform convergence on (L)-subsets of the dual space. As a consequence, Banach spaces with the reciprocal Dunford–Pettis property are characterised.

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in $\m^2$ we construct a 2-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables.

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 study real bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) in n n complex variables for which r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 is the squared norm of a holomorphic polynomial mapping. Such polynomials are the focus of the Sum of Squares Conjecture, which describes the possible ranks for the squared norm r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 and has important implications for the study of proper holomorphic mappings between balls in complex Euclidean spaces of different dimension. Questions about the possible signatures for r ( z , z ¯ ) r(z,\bar {z}) and the rank of r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 can be reformulated as questions about polynomial ideals. We take this approach and apply purely algebraic tools to obtain constraints on the signature of r r .

Introduction. The holomorphic self-homeomorphisms (automorphisms) of the open unit ball B, in ~L TM have long been known [1] - they are given by certain rational functions which are holomorphic on a neighborhood of Bn and induce a homeomorphism of the boundary, bB,, of the ball. Our first result can be viewed as a local characterization of these automorphisms: For n > 1, a nonconstant holomorphic mapping into ~ which is defined in a neighborhood of a point of bB, and which maps bB, into itself is necessarily an automorphism, or, more precisely, extends to be an automorphism. We apply this to obtain some information on the as yet unsettled question as to whether every proper holomorphic self-mapping of Bn is an automorphism. In particular, we recover (Cor. 1.1) a result of Pelles ([3, 5]). In the second part, we consider holomorphic mappings from polydiscs. According to a classical theorem of Poincar6, there exists no biholomorphism from the polydisc U z in C 2 with the ball B2. We obtain some integral formulas which yield a quantitative explanation of this phenomenon, Finally I wish to acknowledge that the above characterization of automorphisms may have been known to the late Professor L6wner, at least for two complex variables. I want to thank Professors L. Bers and C. Titus for this information on their oral communication with L6wner.

We study the field theory localizing to holomorphic maps from a complex manifold of complex dimension 2 to a toric target (a generalization of A model). Fields are realized as maps to {{(mathbb{C}text{*})}^{N}} where one includes special observables supported on (1,1)-dimensional submanifolds to produce maps to the toric compactification. We study the mirror of this model. It turns out to be a free theory interacting with {{N}_{{{text{comp}}}}} topological strings of type A. Here, {{N}_{{{text{comp}}}}} is the number of compactifying divisors of the toric target. Before the mirror transformation, these strings are vortex (actually, holomortex) strings.

Let M be a complex manifold of dimension n and let £2 be a domain in C" .Let /: M -► Í2 be a holomorphic map which is an isometry for the infinitesimal Kobayashi metric at one point.We given conditions on M and on ÍJ which imply that F must be a biholomorphic map.

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.