Holomorphic Deformations Research Articles

Holomorphic support functions for uniformly pseudoconvex hypersurfaces, with an application to CR maps

We construct holomorphic support functions for smooth weakly pseudoconvex hypersurfaces with Levi form of constant rank. These are then applied to show that formal holomorphic curves which are tangential to infinite order to such a hypersurface must be formally contained in its Levi foliation. As a consequence, we obtain a holomorphic deformation theorem for nowhere smooth CR maps into smooth pseudoconvex hypersurfaces with one-dimensional Levi foliation, strengthening a very general result of Lamel and Mir about formal deformations in this particular case.

The Infinitesimal Holomorphically Projective Deformations of Special Parabolically Kahlerian Spaces

We present the most important concepts related to the research: Kahler parabolic space, the holomorphic curve and the projective holomorphic mapping, then define the infinitesimal deformations between Kahler parabolic spaces and the infinitesimally projective holomorphic deformations in Kahler parabolic spaces, and we also find the necessary and sufficient conditions for the Kahler parabolic space to be a non- trivially deformation holomorphic projective. Then we study the infinitesimally projective holomorphic deformation of semi- symmetric and symmetric Kahler parabolic spacesand affine equal Kahler parabolic spaces.

Dolbeault and Bott-Chern formalities: Deformations and [formula omitted]-lemma

It is proved that the properties of being Dolbeault formal and geometrically-Bott-Chern-formal are not closed under holomorphic deformations of the complex structure. Further, we construct a compact complex manifold which satisfies the ∂∂‾-lemma but admits a non vanishing Aeppli-Bott-Chern-Massey product.

Properties of Critical Points of the Dinew-Popovici Energy Functional

Abstract Recently, Dinew and Popovici introduced and studied an energy functional F acting on the metrics in the Aeppli cohomology class of a Hermitian-symplectic metric and showed that in dimension 3 its critical points (if any) are Kähler. In this article we further investigate the critical points of this functional in higher dimensions and under holomorphic deformations. We first prove that being a critical point for F is a closed property under holomorphic deformations. We then show that the existence of a Kähler metric ω in the Aeppli cohomology class is an open property under holomorphic deformations. Furthermore, we consider the case when the (2, 0)-torsion form ρω 2, 0 of ω is ∂-exact and prove that this property is closed under holomorphic deformations. Finally, we give an explicit formula for the differential of F when the (2, 0)-torsion form ρω 2, 0 is ∂-exact.

DEFORMATIONS OF THE VERONESE EMBEDDING AND FINSLER -SPHERES OF CONSTANT CURVATURE

AbstractWe establish a one-to-one correspondence between, on the one hand, Finsler structures on the $2$ -sphere with constant curvature $1$ and all geodesics closed, and on the other hand, Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and whose geodesics are all closed. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $\mathbb {CP}(a_1,a_2)\rightarrow \mathbb {CP}(a_1,(a_1+a_2)/2,a_2)$ of weighted projective spaces provide examples of Finsler $2$ -spheres of constant curvature whose geodesics are all closed.

Transverse Kähler holonomy in Sasaki Geometry andS-Stability

AbstractWe study the transverse Kähler holonomy groups on Sasaki manifolds (M,S) and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti numberb1(M) and the basic Hodge numberh0,2B(S) vanish, thenSis stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure isS-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure isS-stable when dimM≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomyU(n1) ×U(n2)) as well as the fiber join operation preserveS-stability.

Holomorphic Deformations of Balanced Calabi-Yau ∂∂ ¯-Manifolds

Given a compact complex n-fold X satisfying the ∂∂ ¯-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 co-polarised by the balanced class [ω n-1 ]∈H n-1,n-1 (X,ℂ)⊂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 holomorphic symplectic compact complex manifolds. The concept of Weil–Petersson metric still makes sense in this strictly more general, possibly non-Kähler context, while the Local Torelli Theorem still holds.

On the Strominger System and Holomorphic Deformations

We prove that the property of existence of solution to the Strominger system in dimension six is neither open nor closed under holomorphic deformations of the complex structure. These results are obtained both in the case of positive slope parameter as well as in the case of negative slope parameter in the anomaly cancellation equation.

Compact complex manifolds with small Gauduchon cone

This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact K\"ahler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to explore the link between the {\it class} ${\cal C}$ property and the closed positive currents of bidegree $(1,\,1)$ that the manifold supports, a fact leading to the study of semi-continuity properties under deformations of the complex structure of the dual cones of cohomology classes of such currents and of Gauduchon metrics. Our main finding is a new class of compact complex, possibly non-K\"ahler, manifolds defined by the condition that every Gauduchon metric be strongly Gauduchon (sG), or equivalently that the Gauduchon cone be small in a certain sense. We term them sGG manifolds and find numerical characterisations of them in terms of certain relations between various cohomology theories (De Rham, Dolbeault, Bott-Chern, Aeppli). We also produce several concrete examples of nilmanifolds demonstrating the differences between the sGG class and well-established classes of complex manifolds. We conclude that sGG manifolds enjoy good stability properties under deformations and modifications.

Complex structures of splitting type

We study the six-dimensional solvmanifolds that admit complex structures of splitting type classifying the underlying solvable Lie algebras. In particular, many complex structures of this type exist on the Nakamura manifold X , and they allow us to construct a countable family of compact complex non- \partial\bar{\partial} manifolds X_k , k\in\mathbb Z , that admit a small holomorphic deformation \{(X_{k})_{t}\}_{t\in\Delta_k} satisfying the \partial\bar{\partial} -lemma for any t\in\Delta_k except for the central fibre. Moreover, a study of the existence of special Hermitian metrics is also carried out on six-dimensional solvmanifolds with splitting-type complex structures.

Deformations of some open submanifolds of Calabi–Yau manifolds

Abstract By using L 2 analytic methods we prove some results about holomorphic deformations of some open submanifolds of Calabi–Yau manifolds, which generalize some classical results from compact to noncompact cases.

Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

Let \(M\subset {\mathbb {C}}^N\) and \(M'\subset {\mathbb {C}}^{N'}\) be real-analytic CR submanifolds, with M minimal. We provide a new sufficient condition, that happens to be also essentially necessary, for all sufficiently smooth CR maps \(h:U\rightarrow M'\) defined on a connected open subset of M and of rank larger than a prescribed integer r to be real-analytic on a dense open subset of U. This condition corresponds to the nonexistence of nontrivial holomorphic deformations of germs of real-analytic CR mappings whose rank is larger than r. As a consequence, we obtain several new results about analyticity of CR mappings that, at the same time, generalize and unify a number of previous existing ones.

Cohomological decomposition of compact complex manifolds and holomorphic deformations

Cohomological decomposition of compact complex manifolds and holomorphic deformations

Holomorphic deformations of the Ricci-flat $$\partial \overline \partial $$ -manifolds

We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat $$\partial \overline \partial $$ -manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of the Ricci-flat $$\partial \overline \partial $$ -manifolds.

Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics

We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behavior of the associated Frölicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property are not closed under holomorphic deformation.

Deformation openness and closedness of various classes of compact complex manifolds; Examples

We review the relations between compact complex manifolds carrying various types of Hermitian metrics (K¨ahler, balanced, or strongly Gauduchon) and those satisfying topological properties such as the @ ¯@-lemma, or the degeneration at E1 of the Fr¨olicher spectral sequence. We also review the behaviour of these properties under holomorphic deformations. The emphasis will be placed on the notion of strongly Gauduchon (sG) manifold that we introduced recently in the study of deformation limits of projective and Moishezon manifolds. Besides its expository aspect, the paper presents new results such as exhibiting various examples of sG and non-sG manifolds. These are then used to review and reinterpret from this new standpoint a range of constructions already known in the literature.

Holomorphic arcs on singularities

The study of formal arcs was initiated by Nash in a 1967 preprint. Arc spaces of smooth varieties have a rather transparent structure but difficult problems arise for arcs passing through singularities. The Nash conjecture on the irreducible components of such arc spaces was proved for surfaces by Fernandez de Bobadilla and Pe Pereira and for toric singularities by Ishii and Kollar, but counter examples were found in higher dimensions by Ishii and Kollar, de Fernex and Johnson and Kollar. Here we start the study of holomorphic arcs; these are holomorphic maps from the closed unit disk to a complex analytic space. As one expects, there is not much conceptual difference between the set of formal arcs and the set of holomorphic arcs since every formal arc can be approximated by holomorphic arcs. However, a formal deformation of an arc is a much more local object than a holomorphic deformation. Thus, in many cases, the space of holomorphic arcs has infinitely many connected components while the space of formal arcs always has only finitely many. For a complex analytic space we define two variants—the space of arcs and the space of short arcs—and we study their connected components. For short arcs we obtain complete answers for surface singularities and for isolated quotient singularities in all dimensions.

On the Bott–Chern cohomology and balanced Hermitian nilmanifolds

The Bott–Chern cohomology of six-dimensional nilmanifolds endowed with invariant complex structure is studied with special attention to the cases when balanced or strongly Gauduchon Hermitian metrics exist. We consider complex invariants introduced by Angella and Tomassini and by Schweitzer, which are related to the [Formula: see text]-lemma condition and defined in terms of the Bott–Chern cohomology, and show that the vanishing of some of these invariants is not a closed property under holomorphic deformations. In the balanced case, we determine the spaces that parametrize deformations in type IIB supergravity described by Tseng and Yau in terms of the Bott–Chern cohomology group of bidegree (2, 2).

(0,2) deformations of linear sigma models

We study (0,2) deformations of a (2,2) supersymmetric gauged linear sigma model for a Calabi-Yau hypersurface in a Fano toric variety. In the non-linear sigma model these correspond to some of the holomorphic deformations of the tangent bundle on the hypersurface. Combinatorial formulas are given for the number of these deformations, and we show that these numbers are exchanged by mirror symmetry in a subclass of the models.

Neighborhood of an Embedded J-Holomorphic Disc

We study the deformation space of an embedded J-holomorphic disc D in an almost complex surface (X,J). Every such J is shown to be equivalent to a small deformation of a certain model structure Jβ along D, where β:D→ℂ is a complex valued function whose modulus |β| is a biholomorphic invariant. Furthermore, we find a nonlinear invertible operator mapping the space of all small J-holomorphic deformations of the given J-holomorphic disc onto the space of small holomorphic deformations of the standard disc in ℂ2.