Deformation in the large of some complex manifolds, II
Fabrizio Catanese, Paola Frediani
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23 pages, to appear in the AMS Series 'Contemporary Mathematics',in the Proceedings of the 10th anniversary (2004) Conference for the Mathematics Institute at East China Normal University
The compact complex manifolds considered in this article are principal torus bundles over a torus. We consider the Kodaira Spencer map of the complete Appell Humbert family (introduced by the first author in Part I) and are able to show that we obtain in this way a connected component of the space of complex structures each time that the base dimension is two, the fibre dimension is one, and a suitable topological condition is verified.
We investigate when the tangent bundle of a projective manifold has a non-trivial first order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.
A new proof of the assertion, that any global kaehler deformation of a flag manifold F with b2=1 is biholomorphic to F. Essential use is made of rational connected properties of F
We consider (small) algebraic deformations of germs of real-algebraic CR submanifolds in complex space and study the biholomorphic equivalence problem for such deformations. We show that two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are holomorphically equivalent if and only if they are algebraically equivalent.
Why withdrawn: Main theorem can only be proved, if the flag manifold F with b2=1 is additionally hermitian symmetric.Mistake made at the following place: If F is not hermitian symmetric, certain vector fields are not left invariant
The main result of this paper is to study the local deformations of Calabi-Yau $\partial\bar{\partial}$-manifold that are co-polarised by the Gauduchon metric by considering the subfamily of co-polarised fibres by the class of Aeppli/De Rham-Gauduchon cohomology of Gauduchon metric given at the beginning on the central fibre. In the latter part, we prove that the $p$-SKT $h$-$\partial\bar{\partial}$-property is deformation open by constructing and studying a new notion called $hp$-Hermitian symplectic ($hp$-HS) form.
Let M be a compact hyperkaehler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in $H^2(M)$ defines a divisor $D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W.
We review the relations between compact complex manifolds carrying various types of Hermitian metrics (Kähler, balanced or {\it strongly Gauduchon}) and those satisfying the $\partial\bar\partial$-lemma or the degeneration at $E_1$ of the Frölicher spectral sequence, as well as the behaviour of these properties under holomorphic deformations. The emphasis will be placed on the notion of {\it strongly Gauduchon} (sG) manifolds that we introduced recently in the study of deformation limits of projective and Moishezon manifolds. Various examples of sG and non-sG manifolds are exhibited while a ra…
Let $X$ be a compact Kähler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X'$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X, Ω^2) \neq 0$. Using this, we recover a recent theorem of Catanese and Demleitner, which states that a rigid smooth quotient of a complex torus is always projective. We also produce many examples of non-algebraic flat Kähler manifolds with vanishing first Betti number.
We extend the notion of essential deformations from the case of the Iwasawa manifold, for which they were introduced recently by the first-named author, to the general case of page-$1$-$\partial\bar\partial$-manifolds that were jointly introduced very recently by all three authors. We go on to obtain an analogue of the unobstructedness theorem of Bogomolov, Tian and Todorov for page-$1$-$\partial\bar\partial$-manifolds. As applications of this discussion, we study the small deformations of certain Nakamura solvmanifolds and reinterpret the cases of the Iwasawa manifold and its $5$-dimensional …
We determine the successive pages of the Frölicher spectral sequence of the Iwasawa manifold and some of its small deformations, providing new examples and counterexamples on its properties, including the behaviour under small deformations.
In this paper, we introduce the notions of $p$-Hermitian-symplectic and $p$-pluriclosed compact complex manifolds as generalisations for an arbitrary positive integer $p$ not exceeding the complex dimension of the manifold of the standard notions of Hermitian-symplectic and SKT manifolds that correspond to the case $p=1$. We then notice that these two properties are equivalent on $\partial\bar{\partial}$-manifolds and go on to prove that in (smooth) complex analytic families of $\partial\bar{\partial}$-manifolds, they are deformation open. Concerning closedness results, we prove that the cones…
Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-Kähler) Hermitian metric $ω$, we introduce the concept of deformations of $X$ that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1}(X,\,\C)\subset H^{2n-2}(X,\,\C)$ and show that the resulting theory of balanced co-polarised deformations is a natural extension of the classical theory of Kähler polarised deformations in the context of Calabi-Yau or even holomorphic symplectic compact complex manifolds. The c…
We investigate the behaviour of the Oka property with respect to deformations of compact complex manifolds. We show that in a family of compact complex manifolds, the set of Oka fibres corresponds to a G-delta subset of the base. We give a necessary and sufficient condition for the limit fibre of a sequence of Oka fibres to be Oka in terms of a new uniform Oka property. We show that if the fibres are tori, then the projection is an Oka map. Finally, we consider holomorphic submersions with noncompact fibres.
We first give a deformation theory of integrable distributions of codimension 1. We define a parametrization of families of smooth hypersurfaces near a Levi flat hypersurface L such that the Levi flat deformations are given by the solutions of the Maurer-Cartan equation in a DGLA associated to the Levi foliation. We say that L is infinitesimally rigid if the tangent cone at the origin to the moduli space of Levi flat deformations of L is trivial. We prove the infinitesimal rigidity of compact transversally parallelisable Levi flat hypersurfaces in compact complex manifolds and give sufficient …
We define a complex whose cohomology group of order 1 contains the infinitesimal deformations of a Levi flat structure on a smooth manifold. In the case of real analytic Levi flat structures, this cohomology group is the product of the d-bar cohomology group of order 1 of tangent vector fields to the Levi structure and the cohomology group of order 1 of the associated DGLA.
We show that if $\{M_t\}_{t\in Δ}$ is a polarized family of compact Kähler manifolds over the open unit disk $Δ$, if $N$ is a Riemannian manifold of nonpositive complexified sectional curvature, and if $\{ϕ_t:M_t\to N\}_{t\in Δ}$ is a smooth family of pluriharmonic maps, then the Dirichlet energy $E(ϕ_t)$ is a subharmonic function of $t\inΔ$.
In this paper, we prove an existence theorem of a local moduli space for geometric structures in a very general setting. Then to show the interest of this result, we apply it to the case of sasakian and Sasaki-Einstein structures.
