Non-constant Holomorphic Functions Research Articles

Conformal geometry of isotropic curves in the complex quadric

Let [Formula: see text] be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in [Formula: see text]. By an isotropic curve, we mean a nonconstant holomorphic map from a Riemann surface into [Formula: see text], null with respect to the conformal structure of [Formula: see text]. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.

Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf

Abstract We prove that a pseudoconvex complex space X of pure dimension n + 1 {n+1} is holomorphically convex provided that its singular set has only compact connected components (e.g., X has isolated singularities) and rank x ⁡ 𝒪 X = n {\operatorname{rank}_{x}\mathscr{O}_{X}=n} for all smooth points x outside a compact set of X. This extends a known result due to Ohsawa asserting that a weakly 1-complete, smooth complex surface is holomorphically convex if it admits a globally defined non-constant holomorphic function. We also revise an example of Markoe concerning non-invariance of holomorphic convexity under normalization.

Clark measures and de Branges–Rovnyak spaces in several variables

Let denote the unit ball of , , and let denote a finite product of , . Given a non-constant holomorphic function , we study the corresponding family , , of Clark measures on the distinguished boundary . We construct a natural unitary operator from the de Branges–Rovnyak space onto the Hardy space . As an application, for and an inner function , we show that the property is directly related to the membership of an appropriate explicit function in .

The $$\partial \overline \partial $$-Bochner Formulas for Holomorphic Mappings between Hermitian Manifolds and Their Applications

In this paper, we derive some \(\partial \overline \partial \)-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, and some rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. non-negative) ℓ-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results [5, 6] proved recently by L. Ni on Kähler manifolds to Hermitian manifolds. We also derive an integral inequality for a holomorphic map between Hermitian manifolds.

Automorphism groups of origami curves

A closed Riemann surface S (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map \(\beta :S \rightarrow E\) with at most one branch value, where E is a genus one Riemann surface. In this case, \((S,\beta )\) is called an origami pair and \(\mathrm{Aut}(S,\beta )\) is the group of conformal automorphisms \(\phi \) of S such that \(\beta =\beta \circ \phi \). Let G be a finite group. It is a known fact that G can be realized as a subgroup of \(\mathrm{Aut}(S,\beta )\) for a suitable origami pair \((S,\beta )\). It is also known that G can be realized as a group of conformal automorphisms of a Riemann surface X of genus \(g \ge 2\) and with quotient orbifold X/G of genus \(\gamma \ge 1\). Given a conformal action of G on a surface X as before, we prove that there is an origami pair \((S,\beta )\), where S has genus g and \(G \cong \mathrm{Aut}(S,\beta )\) such that the actions of \(\mathrm{Aut}(S,\beta )\) on S and that of G on X are topologically equivalent.

Curvature estimate on an open Riemann surface with the induced metric

In this paper, motivated by the result of Osserman and Ru [12], we give a Gauss curvature estimate on an open Riemann surface M whose metric is induced from a nonconstant holomorphic map $$G: M\rightarrow {{\mathbb {P}}}^n({{\mathbb {C}}})$$ .

Periods of holomorphic maps on compact Riemann surfaces and product of spheres

ABSTRACT In this article, we consider non-constant holomorphic maps on Riemann surfaces and product of Riemann spheres, and we give conditions on the maps in order that they have arbitrary large prime numbers as periods. We use Lefschetz fixed point theory and in particular we compute the Lefschetz numbers of period m for large m's.

RC-POSITIVITY, VANISHING THEOREMS AND RIGIDITY OF HOLOMORPHIC MAPS

AbstractLet $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

Flat Bundles Over Some Compact Complex Manifolds

We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete Kahler metric, or are hyperconvex but have no nonconstant holomorphic functions. For any compact Riemannian surface of positive genus, we construct a flat $${\mathbb {P}}^1$$ bundle over it and a Stein domain with real analytic boundary in it whose closure does not have pseudoconvex neighborhood basis. For a compact complex manifold with positive first Betti number, we construct a flat bundle over it such that the total space is hyperconvex but admits no nonconstant holomorphic functions.

Pseudoconcavity of flag domains: the method of supporting cycles

A flag domain of a real from $$G_0$$ of a complex semismiple Lie group G is an open $$G_0$$ -orbit D in a (compact) G-flag manifold. In the usual way one reduces to the case where $$G_0$$ is simple. It is known that if D possesses non-constant holomorphic functions, then it is the product of a compact flag manifold and a Hermitian symmetric bounded domain. This pseudoconvex case is rare in the geography of flag domains. Here it is shown that otherwise, i.e., when $$\mathscr {O}(D)\cong \mathbb {C}$$ , the flag domain D is pseudoconcave. In a rather general setting the degree of the pseudoconcavity is estimated in terms of root invariants. This estimate is explicitly computed for domains in certain Grassmannians.

Levi’s Problem for Pseudoconvex Homogeneous Manifolds

AbstractSuppose G is a connected complex Lie group and H is a closed complex subgroup. Then there exists a closed complex subgroup J of G containing H such that the fibration π:G/H ⟶ G/J is the holomorphic reduction of G/H; i.e., G/J is holomorphically separable and O(G/H)≅ π*O(G/J). In this paper we prove that if G/H is pseudoconvex, i.e., if G/H admits a continuous plurisubharmonic exhaustion function, then G/J is Stein and J/H has no non-constant holomorphic functions.

Almost complex manifolds with non-degenerate torsion

We show that almost complex manifolds [Formula: see text] of real dimension 4 for which the image of the Nijenhuis tensor forms a non-integrable bundle, called torsion bundle, admit a [Formula: see text]-structure locally, that is, a double absolute parallelism. In this way, the problem of equivalence for such almost complex manifolds can be solved; moreover, the classification of locally homogeneous manifold [Formula: see text] is explicitly given when the Lie algebra of its infinitesimal automorphisms is non-solvable (indeed reductive). It is also shown that the group of the automorphisms of [Formula: see text] is a Lie group of dimension less than or equal to 4, whose isotropy subgroup has at most two elements, and that there are not non-constant holomorphic functions on [Formula: see text].

A long ℂ2 without holomorphic functions

In this paper we construct for every integer $n>1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\mathbb C^n$ (i.e., a long $\mathbb C^n$), but it does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new biholomorphic invariants of a complex manifold $X$, the stable core and the strongly stable core, that are based on the long term behavior of hulls of compact sets with respect to an exhaustion of $X$. We show that every compact polynomially convex set $B\subset \mathbb C^n$ which is the closure of its interior is the strongly stable core of a long $\mathbb C^n$; in particular, biholomorphically nonequivalent sets give rise to nonequivalent long $\mathbb C^n$'s. Furthermore, for any open set $U\subset \mathbb C^n$ there exists a long $\mathbb C^n$ whose stable core is dense in $U$. It follows that for any $n>1$ there is a continuum of pairwise nonequivalent long $\mathbb C^n$'s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long standing open problems.

On the volume growth of Kähler manifolds with nonnegative bisectional curvature

Let $M$ be a complete K\"ahler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth, we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni. There are two essential ingredients in the proof: The Cheeger-Colding theory on Gromov-Hausdorff convergence of manifolds; the three circle theorem for holomorphic functions.

Integral points of bounded degree on affine curves

We generalize Siegel’s theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree $d$ or less over some number field. Generalizing Picard’s theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree $d$ (or less) analytic cover of $\mathbb{C}$.

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f : Y → X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X1 × X2 and Y = Y1 × Y2 of compact connected complex manifolds. When X1 is a Riemann surface of genus ≥ 2, we show that any non-constant holomorphic map F : Y → X is of a special form.

Harmonic maps and stability on locally conformal Kähler manifolds

We study harmonic and pluriharmonic maps on locally conformal Kähler manifolds. We prove that there are no nonconstant holomorphic pluriharmonic maps from a locally conformal Kähler manifold to a Kähler manifold and that any holomorphic harmonic map from a compact locally conformal Kähler manifold to a Kähler manifold is stable.

The Index theorem for quasi-tori

The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the signature of the associated hermitian form. In this article, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert–Morimoto decomposition of quasi-tori as well as the Künneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using L2-methods coupled with the Kazama–Dolbeault isomorphism and Bochner–Kodaira formulas.

Homomorphisms between mapping class groups

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every non-trivial homomorphism $\Map(X) \to \Map(Y)$ is induced by an {\em embedding}, i.e. a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an isomorphism. As an application of our main theorem we obtain that, under the same hypotheses on genus, if $X$ and $Y$ have finite analytic type then every non-constant holomorphic map $\CM(X)\to\CM(Y)$ between the corresponding moduli spaces is a forgetful map. In particular, there are no such holomorphic maps unless $X$ and $Y$ have the same genus and $Y$ has at most as many marked points as $X$.

KMS States and Conformal Measures

From a non-constant holomorphic map on a connected Riemann surface we construct an étale second countable locally compact Hausdorff groupoid whose associated groupoid C*-algebra admits a one-parameter group of automorphisms with the property that its KMS states correspond to conformal measures in the sense of Sullivan. In this way certain quadratic polynomials give rise to quantum statistical models with a phase transition arising from spontaneous symmetry breaking.