On finite regular and holomorphic mappings

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 study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.

With C3 /C * as motivation, in this paper we study somewhat more general affine surfaces. Let X be a normal algebraic surface defined over C with at worst quotient singularities. We call X a logarithmic Q-homology plane if H~(X; Q) = (0) for all i > 0. If X is smooth, we simply say that X is a Q-homology plane. Similarly, if H~(X; Z) = (0) for all i > 0, we call X a logarithmic homology plane or simply a homology plane if X is smooth. Let X be a logarithmic Q-homology plane. Then there exists a projective surface W and a reduced effective divisor D such that W D = X , W is smooth along D and D has simple normal crossings. By [10, Lemma 2.2] we know:

Let X be a smooth variety over a field k and D an effective divisor whose support has simple normal crossings. We construct an explicit cycle map from the r-th Nisnevich motivic complex of the pair (X,D) to a shift of the r-th relative Milnor K-sheaf of (X,D). We show that this map induces an isomorphism for all i greater or equal the dimension of X between the motivic Nisnevich cohomology of (X,D) in bidegree (i+r,r) and the i-th Nisnevich cohomology of the r-th relative Minor K-sheaf of (X,D). This generalizes the well-known isomorphism in the case D=0. We use this to prove a certain Zariski descent property for the motivic cohomology of the pair (\A^1_k, (m+1){0}).

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 .

We study Bergman-harmonic maps between balls 8 : Bn ! BN extending of class either C2 orM1 to the boundary of Bn. For every holomorphic (anti-holomorphic) map 8 : Bn ! BN extending smoothly to the boundary and every smooth homotopy H : 8 ' 9 we prove a Lichnerowicz-type (cf. [28]) result, i.e., we show that E✏ (9) # E✏ (8) + O(✏−n+1). When 8 is proper, Bergman-harmonic, and C2 up to the boundary, the boundary values map % : S2n−1 ! S2N−1 is shown to satisfy a compatibility system similar to the tangential Cauchy-Riemann equations on S2n−1 (and satisfied by the boundary values of any proper holomorphic map). For every weakly Bergman-harmonic map 8 2 W1(Bn,BN ) admitting Sobolev boundary values % 2 M1(S2n−1,BN ) in the sense of [6], the boundary values % are shown to be a weakly subelliptic harmonic map of (S2n−1, ⌘) into (BN , h), provided that 8−1rh stays bounded at the boundary of Bn and % has vanishing weak normal derivatives.

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

Applications holomorphes propres d'un domaine quasi-disqué sur un domaine disqué

We define semistable generalized parabolic Hitchin pairs (GPH) on a disjoint union [Formula: see text] of integral smooth curves and construct their moduli spaces. We define a Hitchin map on the moduli space of GPH and show that it is a proper map. We construct moduli spaces of semistable Hitchin pairs on a reducible projective curve [Formula: see text]. When [Formula: see text] is the normalization of [Formula: see text], we give a birational morphism [Formula: see text] from the moduli space [Formula: see text] of good GPH on [Formula: see text] to the moduli space [Formula: see text] of Hitchin pairs on [Formula: see text] and show that the Hitchin map on [Formula: see text] induces a proper Hitchin map on [Formula: see text]. We determine the fibers of the Hitchin maps. We study the relationship between representations of the (topological) fundamental group of [Formula: see text] and Higgs bundles on [Formula: see text]. We show that if all the irreducible components of [Formula: see text] are smooth, then the Hitchin map is defined on the entire moduli space [Formula: see text].

The goal of this article is to prove the Sum of Squares Conjecture for real polynomials \(r(z,\bar{z})\) on \(\mathbb {C}^3\) with diagonal coefficient matrix. This conjecture describes the possible values for the rank of \(r(z,\bar{z}) \left\Vert {z} \right\Vert ^2\) under the hypothesis that \(r(z,\bar{z})\left\Vert {z} \right\Vert ^2=\left\Vert {h(z)} \right\Vert ^2\) for some holomorphic polynomial mapping h. Our approach is to connect this problem to the degree estimates problem for proper holomorphic monomial mappings from the unit ball in \(\mathbb {C}^2\) to the unit ball in \(\mathbb {C}^k\). D’Angelo, Kos, and Riehl proved the sharp degree estimates theorem in this setting, and we give a new proof using techniques from commutative algebra. We then complete the proof of the Sum of Squares Conjecture in this case using similar algebraic techniques.

We construct a functorial pushforward homomorphism in geometric Hodge filtered complex cobordism along proper holomorphic maps between arbitrary complex manifolds. This significantly improves previous results on such transfer maps and is a much stronger result than the ones known for differential cobordism of smooth manifolds. This enables us to define and provide a concrete geometric description of Hodge filtered fundamental classes for all proper holomorphic maps. Moreover, we give a geometric description of a cobordism analog of the Abel–Jacobi invariant for nullbordant maps which is mapped to the classical invariant under the Hodge filtered Thom morphism. For the latter we provide a new construction in terms of geometric cycles.

We show that if E is a closed convex set in {mathbb {C}}^n(n>1) contained in a closed halfspace H such that Ecap bH is nonempty and bounded, then the concave domain Omega = {mathbb {C}}^n{setminus } E contains images of proper holomorphic maps f:Xrightarrow {mathbb {C}}^n from any Stein manifold X of dimension <n, with approximation of a given map on closed compact subsets of X. If in addition 2dim X+1le n then f can be chosen an embedding, and if 2dim X=n, then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties.

In this paper we give theorems on the propagation of algebraic dependence for meromorphic mappings, especially under conditions on deficiencies. Let M be a projective algebraic manifold. Let π: X → C m be a finite analytic covering space, that is, X is a reduced irreducible normal complex space and π: X → C m is a proper surjective holomorphic mappings with discrete fibers. We denote by s0 its sheet number and by B its ramification divisor. We first give the definition of algebraic dependence. Set M 2 = M × M. For meromorphic mappings f, g: X → M, we define a meromorphic mapping f × g: X →M 2 by (f × g)(z) = (f (z),g(z)) for z ∈ X — (I (f) ∪ I (g)), where I (f) and I (g) are the loci of the indeterminacy of f and g respectively.

In this chapter we use the Cauchy theory to study functions that are holomorphic on an annulus, and analytic functions with isolated singularities. We describe a classification for isolated singularities. Functions that are holomorphic on an annulus have Laurent series expansions, an analogue of power series expansions for holomorphic functions on discs. Holomorphic functions with a finite number of isolated singularities in a domain can be integrated using the residue theorem, an analogue of the Cauchy integral formula. We discuss local properties of these functions. The study of zeros and poles of meromorphic functions leads to a theorem of Rouché that connects the number of zeros and poles to an integral. The theorem is not only aesthetically pleasing in its own right but also allows us to give alternate proofs of many important results. In the penultimate section of this chapter we illustrate the use of complex function theory in the evaluation of real definite integrals. The appendix discusses Cauchy principal values, a way to integrate functions with certain singularities.KeywordsCauchy TheoryHolomorphic FunctionReal Definite IntegralsLaurent Series ExpansionNonconstant Meromorphic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let <TEX>$B^{n+1}$</TEX> be the unit ball in the complex vector space <TEX>$\mathbb{C}^{n+1}$</TEX> with the standard Hermitian metric. Let <TEX>${\Sigma}^n={\partial}B^{n+1}=S^{2n+1}$</TEX> be the boundary sphere with the induced CR structure. Let f : <TEX>${\Sigma}^n{\hookrightarrow}{\Sigma}^N$</TEX> be a local CR immersion. If N < 3n - 1, the asymptotic vectors of the CR second fundamental form of f at each point form a subspace of the CR(horizontal) tangent space of <TEX>${\Sigma}^n$</TEX> of codimension at most 1. We study the higher order derivatives of this relation, and we show that a linearly full local CR immersion f : <TEX>${\Sigma}^n{\hookrightarrow}{\Sigma}^N$</TEX>, N <TEX>$\leq$</TEX> 3n-2, can only occur when N = n, 2n, or 2n + 1. As a consequence, it gives an extension of the classification of the rational proper holomorphic maps from <TEX>$B^{n+1}$</TEX> to <TEX>$B^{2n+2}$</TEX> by Hamada to the classification of the rational proper holomorphic maps from <TEX>$B^{n+1}$</TEX> to <TEX>$B^{3n+1}$</TEX>.