arXiv research

Study on symplectic structures and their deformations.

Extending the work of G. Székelyhidi and T. Brönnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex …

We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an anal…

Stability of SKT metrics under deformations on complex manifolds.

We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kähler surfaces. This is used to prove that for any scalar-flat Kähler ALE surface, all small deformations of complex structure also admit scalar-flat Kähler ALE metrics. A local moduli space of scalar-flat Kähler ALE metrics is then constr…

We prove that, for some classes of complex nilmanifolds, the Bott-Chern cohomology is completely determined by the Lie algebra associated to the nilmanifold with the induced complex structure. We use these tools to compute the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, com…

The paper studies deformations of astheno-Kähler metrics on complex manifolds.

We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the $\partial\overline\partial$-Lemma, is characterized in terms of the strongly Gauduchon cone and of the first $\partial\overl…

In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which a…

Researchers describe how special conic bundles deform into double solids.

Let $M = Γ\backslash G$ be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result of C. Maclaughlin, H. Pedersen, Y.S. Poon and S. Salamon for 2-step nilmanifolds. We characterize small def…

A notion of geometric formality in the context of Bott-Chern and Aeppli cohomologies on a complex manifold is discussed. In particular, by using Aeppli-Bott-Chern-Massey triple products, it is proved that geometric Aeppli-Bott-Chern formality is not stable under small deformations of the complex structure.

We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several exa…

The paper finds conditions for smooth curves of balanced metrics in Hermitian non-Kähler settings.

We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a $\mathcal{C}^{\infty}$ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result \textit{à la Kodaira-Spencer} for the dimension o…

While small deformations of Kähler manifolds are Kähler too, we prove that the cohomological property to be $\mathcal{C}^\infty$-pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in [Comparing tamed and compatible symp…

Let f be a 1-variable complex polynomial such that f has a singularity at the origin. In the present paper, we show that there exists a deformation of f which has only fold singularities and cusps as singularities of a real polynomial map from the plane to the plane. We then calculate the number of cusps of a deformati…

The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty$ algebra instead. We develop a simplified method for describing this $L_\infty$ algebra a…

Introduces new holomorphic contact structures and proves unobstructedness theorems.

The paper solves a geometric problem related to K3 surfaces and complex-hyperkähler metrics.

Let $X$ be a compact complex manifold with trivial canonical bundle and satisfying the $\partial\bar{\partial}$-Lemma. We show that the Kuranishi space of $X$ is a smooth universal deformation and that small deformations enjoy the same properties as $X$. If, in addition, $X$ admits a complex symplectic form, then the l…

Let $X$ be a complex manifold and $S\hookrightarrow X$ be an embedding of complex submanifold. Assuming that the embedding is $(k-1)$-linearizable or $(k-1)$-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism $F$ from a small neighborhood of the zero section in the normal bundle …

Study extends Hausdorff dimension Hessian results to new hyperconvex representations.

Study on spectral sequence of Iwasawa manifold and its deformations.

Let $(X, J)$ be a compact Kahler manifold with a non-zero holomorphic Poisson structure $β$. If the obstruction space for deformations of generalized complex structures on $(X, J)$ vanishes, we obtain a family of deformations of non-trivial bihermitian structures $(J, J^-_t, h_t)$ on $X$ by using $β$. In addition, if t…

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.

The strip map is a natural map from the arc complex of a bordered hyperbolic surface $S$ to the vector space of infinitesimal deformations of $S$. We prove that the image of the strip map is a convex hypersurface when $S$ is a surface of small complexity: the punctured torus or thrice punctured sphere.

Proves conjecture on deformation invariance of big fundamental groups.

In this paper we provide examples of hypercomplex manifolds which do not carry HKT structure. We also prove that the existence of HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure wit…

Let (X,Ω) be a closed polarized complex manifold, g be an extremal metric on X that represents the Kähler class Ω, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to B)$ of polarized complex deform…

We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any …

Study Witten deformation on noncompact manifolds with bounded geometry.

Study cohomologies of complex manifolds with symplectic forms and their stability.

Study cohomology of Bigolin complex on complex manifolds.

Let $X$ be a compact complex manifold, consider a small deformation $φ: \mathcal{X} \to B$ of $X$, the dimensions of the cohomology groups of tangent sheaf $H^q(X_t,\mathcal{T}_{X_t})$ may vary under this deformation. This paper will study such phenomenons by studying the obstructions to deform a class in $H^q(X,\mathc…

Study on deformations of Bianchi groups into SU(3,1) and SL(4,R).

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

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 Kähler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to exp…

Witten deformation connects manifold spectra to Morse functions.

Let $X$ be a compact complex manifold, consider a small deformation $φ: \mathcal{X} \to B$ of $X$, the dimension of the Dolbeault cohomology groups $H^q(X_t,Ω_{X_t}^p)$ may vary under this defromation. This paper will study such phenomenons by studying the obstructions to deform a class in $H^q(X,Ω_X^p)$ with the param…

Study on special metrics and deformations of solvmanifolds.

Given a holomorphic family of pairs $\{(X_t,E_t)\}$, where each $E_t$ is holomorphic vector bundle over compact complex manifold $X_t$. For small enough $t$, we get a correspondence between the Dolbeault complex of $E_t$-valued $(p,q)$-forms on $X_t$ and the one of $E_0$-valued $(p,q)$-forms on $X_0$.

We prove properties of the Schweitzer complex and its cohomologies.

Uniformizes compact complex manifolds via Anosov representations.

In order to look for a well-behaved counterpart to Dolbeault cohomology in D-complex geometry, we study the de Rham cohomology of an almost D-complex manifold and its subgroups made up of the classes admitting invariant, respectively anti-invariant, representatives with respect to the almost D-complex structure, miming…

Study third order Einstein deformations for Kähler-Einstein metrics on compact manifolds.

In this note we prove that, under a weak condition, small deformations of a compact balanced manifold are also balanced. This condition is satisfied on the twistor space over a compact self-dual four manifold.

Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston's gluing equations. From this, one can recover a representation of th…