Connected Complex Manifold Research Articles

Strongly Jordan property and free actions of non-abelian free groups

AbstractLet $X$ be a connected complex manifold and let $Z$ be a compact complex subspace of $X$. Assume that ${\rm Aut}(Z)$ is strongly Jordan. In this paper, we show that the automorphism group ${\rm Aut}(X,\, Z)$ of all biholomorphisms of $X$ preserving $Z$ is strongly Jordan. A similar result has been proved by Meng et al. for a compact Kähler submanifold $Z$ of $X$ instead of a compact complex subspace $Z$ of $X$. In addition, we also show some rigidity result for free actions of large groups on complex manifolds.

Picard groups of certain compact complex parallelizable manifolds and related spaces

Let G be a complex simply connected semisimple Lie group and let Γ be a torsionless uniform irreducible lattice in G. Then Γ﹨G is a compact complex non-Kähler manifold whose tangent bundle is holomorphically trivial. In this note we compute the Picard group of Γ﹨G when rank(G)≥3. When rank(G)<3, we determine the group Pic0(Γ﹨G)⊂Pic(Γ﹨G) of topologically trivial holomorphic line bundles. When rank(G)≥2, we also show that Pic0(PΓ) is isomorphic to Pic0(Y) where PΓ is a Γ﹨G-bundle associated to a principal G-bundle over a compact connected complex manifold Y, and, when rank(G)≥3, we show that Pic(Y)→Pic(PΓ) is injective with finite cokernel.

Fujiki class 𝒞 and holomorphic geometric structures

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class [Formula: see text], whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class [Formula: see text] and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.

Nonnegative Hermitian vector bundles and Chern numbers

We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically immersed into complex tori, some of which improve several classical results.

Automorphism group of principal bundles, Levi reduction and invariant connections

Let M be a compact connected complex manifold and G a connected reductive complex affine algebraic group. Let $$E_G$$ be a holomorphic principal G–bundle over M and $$T\, \subset \, G$$ a torus containing the connected component of the center of G. Let N (respectively, C) be the normalizer (respectively, centralizer) of T in G, and let W be the Weyl group N / C for T. We prove that there is a natural bijective correspondence between the following two: The composition of maps $$E'_C\, {\mathop {\longrightarrow }\limits ^{\psi }}\, E_W \, {\mathop {\longrightarrow }\limits ^{\phi }}\, M$$ defines a principal N–bundle on M. This principal N–bundle $$E_N$$ is a reduction in structure group of $$E_G$$ to N. Given a complex connection $$\nabla $$ on $$E_G$$ , we give a necessary and sufficient condition for $$\nabla $$ to be induced by a connection on $$E_N$$ . This criterion relates Hermitian–Einstein connections on $$E_G$$ and $$E'_C$$ in a very precise manner.

On the normal bundle of Levi-flat real hypersurfaces

Let $X$ be a connected complex manifold of dimension $\geq 3$ and $M$ a smooth compact Levi-flat real hypersurface in $X$. We show that the normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature along the leaves. This generalizes a result obtained by Brunella.

Fundamental group and analytic disks

Let $W$ be a domain in a connected complex manifold $M$ and $w_0\in W$. Let ${\mathcal A}_{w_0}(W,M)$ be the space of all continuous mappings of a closed unit disk $\overline D$ into $M$ that are holomorphic on the interior of $\overline D$, $f(\partial\mathbb D)\subset W$ and $f(1)=w_0$. On the homotopic equivalence classes $\eta_1(W,M,w_0)$ of ${\mathcal A}_{w_0}(W,M)$ we introduce a binary operation $\star$ so that $\eta_1(W,M,w_0)$ becomes a semigroup and the natural mappings $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ and $\delta_1:\,\eta_1(W,M,w_0)\to\pi_2(M,W,w_0)$ are homomorphisms. \par We show that if $W$ is a complement of an analytic variety in $M$ and if $S=\delta_1(\eta_1(W,M,w_0))$, then $S\cap S^{-1}=\{e\}$ and any element $a\in\pi_2(M,W,w_0)$ can be represented as $a=bc^{-1}=d^{-1}g$, where $b,c,d,g\in S$. \par Let ${\mathcal R}_{w_0}(W,M)$ be the space of all continuous mappings of $\overline D$ into $M$ such that $f(\partial{\mathbb D})\subset W$ and $f(1)=w_0$. We describe its open dense subset ${\mathcal R}^{\pm}_{w_0}(W,M)$ such that any connected component of ${\mathcal R}^{\pm}_{w_0}(W,M)$ contains at most one connected component of ${\mathcal A}_{w_0}(W,M)$.

Convexity Properties of Intersections of Decreasing Sequences of $q$-Complete Domains in Complex Spaces

We construct a decreasing sequence of 3-complete open subsets in \mathbb C^5 such that the interior of their intersection is not 3-complete. We also prove that, for every q\geq 2 , there exists a normal Stein space X with only one isolated singularity and a decreasing sequence of open sets that are 2 -complete, but the interior of their intersection is not q -complete with corners. In the concave case we show that, for every integer n &gt; 1 , there exists a connected complex manifold M of dimension n such that M is an increasing union of 1 -concave open subsets and M is not weakly (n{-}1) -concave.

Kähler quantization and entanglement

For a very ample line bundle L on a compact connected complex manifold X, with a real structure, we discuss entanglement properties of certain sequences of vectors in tensor products of spaces of holomorphic sections of powers of L.

Some results of quantum correction on compact Calabi–Yau manifolds of dimension 3 and 4

It is shown that for compact Calabi–Yau threefolds, “no strong quantum correction” is equivalent to the condition that, with respect to the Hodge metric, the image Φ(T) of the Teichmüller space T under the period map Φ is an open submanifold W of the same dimension as T. In this paper, we extend this result to the case of dimension 4. We also obtained similar results for compact generalized Calabi–Yau manifolds in dimension 3 and 4, under the assumption that their Teichmüller space T exists and is a simply connected complex manifold.

Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain [Formula: see text] and any connected complex manifold [Formula: see text], the space [Formula: see text] contains a dense holomorphic disc. Our second result states that [Formula: see text] is an Oka manifold if and only if for any Stein space [Formula: see text] there exists a dense entire curve in every path component of [Formula: see text]. In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain [Formula: see text], any fixed-point-free automorphism of [Formula: see text] and any connected complex manifold [Formula: see text], there exists a universal map [Formula: see text]. We also characterize Oka manifolds by the existence of universal maps.

Equivariant bundles and connections

Let X be a connected complex manifold equipped with a holomorphic action of a complex Lie group G. We investigate conditions under which a principal bundle on X admits a G-equivariance structure.

A criterion for a degree-one holomorphic map to be a biholomorphism

Let X and Y be compact connected complex manifolds of the same dimension with . We prove that any surjective holomorphic map of degree one from X to Y is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that . We show that this condition is actually automatically satisfied.

Effective Transitive Actions of the Unitary Group on Quotients of Hopf Manifolds

In our article of 2002 joint with N. Kruzhilin we showed that every connected complex manifold of dimension $n\ge 2$ that admits an effective transitive action by holomorphic transformations of the unitary group ${\rm U}_n$ is biholomorphic to the quotient of a Hopf manifold by the action of ${\mathbb Z}_m$ for some integer $m$ satisfying $(n,m)=1$. In this note, we complement the above result with an explicit description of all effective transitive actions of ${\rm U}_n$ on such quotients, which provides an answer to a 10-year old question.

Complex manifolds with maximal torus actions

Abstract In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples ( Δ , 𝔥 , G ) {(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in 𝔤 {\mathfrak{g}} , complex vector subspace 𝔥 {\mathfrak{h}} of 𝔤 ℂ {\mathfrak{g}^{\mathbb{C}}} and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate ℂ n {\mathbb{C}^{n}} -actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.

Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two

AbstractIf X is a connected complex manifold with dX = 2 that admits a (connected) Lie group G acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification Ĝ of G on X except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of X.

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.

Superstable manifolds of invariant circles and codimension-one Böttcher functions

AbstractLet $f: X~\dashrightarrow ~X$ be a dominant meromorphic self-map, where $X$ is a compact, connected complex manifold of dimension $n\gt 1$. Suppose that there is an embedded copy of ${ \mathbb{P} }^{1} $ that is invariant under $f$, with $f$ holomorphic and transversally superattracting with degree $a$ in some neighborhood. Suppose that $f$ restricted to this line is given by $z\mapsto {z}^{b} $, with resulting invariant circle $S$. We prove that if $a\geq b$, then the local stable manifold ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition $a\geq b$ cannot be relaxed without adding additional hypotheses by presenting two examples with $a\lt b$ for which ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is not real analytic in the neighborhood of any point.

A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations

We prove that a transversely holomorphic foliation, which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not a zero measure subset. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold is finite provided that the set of periodic orbits is not a zero measure subset.

Completely integrable torus actions on complex manifolds with fixed points

We show that if a holomorphic n-dimensional compact torus action on a compact connected complex manifold of complex dimension n has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.