arXiv research

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 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.

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 …

Researchers describe how special conic bundles deform into double solids.

Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.

Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as tra…

New examples of deformed Hermitian-Yang-Mills connections found.

In this note we give a simplified proof of a recent result of X.X. Chen, which together with work of G. Szekelyhidi implies that on a sufficiently small deformation of a polarized constant scalar curvature Kahler manifold the K-energy has a lower bound.

Study on symplectic structures and their deformations.

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic …

Any given surface of revolution embedded in Euclidean three-space can always be perturbed by arbitrarily small ambient isotopies as to admit highly nontrivial vector fields inducing infinitesimal deformations. For this matter Morse Theory is used, clarifying and giving a general response of a problem started with an id…

Defines a new metric on Fano Kaehler-Ricci solitons.

In this paper we study the deformations of bihamiltonian PDEs of hydrodynamic type with one dependent variable. The reason we study such deformations is that the deformed systems maintain an infinite number of commuting integrals of motion up to a certain order in the deformation parameter. This fact suggests that thes…

We will consider locally conformally balanced manifolds. We prove that a locally conformally balanced condition is not stable under a small deformation. We prove that locally conformally balanced condition is stable under any proper modification. We prove that symmetric products of the Kodaira surface can be resolve to…

In this paper,we obtain two results on closed Reimainnian manifold $M\times [0,T]$.When $T$ is small enough,to any prescribed scalar curvature, the existence and uniqueness of metrics are obtained on the volume element preserving deformation.When $T$ is large and the given scalar curvature is small enough,the same resu…

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…

In these notes we give a shortened and more direct proof of Goto's generalized Kaehler stability theorem stating that if (J_1,J_2) is a generalized kaehler structure for which J_2 is determined by a nowhere vanishing closed form, then small deformations of J_1 can be coupled with small deformations of J_2 so that the p…

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…

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 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…

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

Extends Kollár's result to fibered Calabi-Yau varieties with cohomological assumption.

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…

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 …

Let $S$ be a torus with a hyperbolic metric admitting one puncture or cone singularity. We describe which infinitesimal deformations of $S$ lengthen (or shrink) all closed geodesics. We also study how the answer degenerates when $S$ becomes Euclidean, i.e. very small.

Stability of SKT metrics under deformations on complex manifolds.

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…

Introduces new holomorphic contact structures and proves unobstructedness theorems.

Develops deformation theory for symplectic foliations using $L_\infty$-algebras.

In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectra…

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…

The paper studies deformations of symplectic forms and Lagrangian submanifolds.

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 paper examines conditions for Kähler-Einstein metrics on deformations of Fano manifolds.

We describe the space of Poisson bivectors near a log-symplectic structure up to small diffeomorphisms.

Study extends Hausdorff dimension Hessian results to new hyperconvex representations.

Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form on M is closed, we study deformations of a compact coassociative submanifold N wi…

We analyze the horizon and geodesic structure of a class of 4D off--diagonal metrics with deformed spherical symmetries, which are exact solutions of the vacuum Einstein equations with anholonomic variables. The maximal analytic extension of the ellipsoid type metrics are constructed and the Penrose diagrams are analyz…

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 consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $π$: any first-order deformation changes either one of those angles or the conformal …

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

Let $X$ be a negatively curved symmetric space and $Γ$ a non-cocompact lattice in $\rm{Isom}(X)$. We show that small, parabolic-preserving deformations of $Γ$ into the isometry group of any negatively curved symmetric space containing $X$ remain discrete and faithful (the cocompact case is due to Guichard). This applie…

The study explores deformations of standard locally homogeneous spaces.

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…

Let $X$ be a compact quotient of the product of the real Heisenberg group $H_{4m+1}$ of dimension $4m+1$ and the 3-dimensional real Euclidean space $\bR^3$. A left invariant hypercomplex structure on $H_{4m+1}\times \bR^3$ descends onto the compact quotient $X$. The space $X$ is a hyperholomorphic fibration of 4-tori o…

We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with transversely Riemannian foliations. We use thi…

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

Paper finds surface groups can deform in reductive symmetric spaces.