arXiv research

A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle $TX$ of a complex manifold $X$ makes the shifted tangent bundle $TX[-1]$ into a Lie algebra object in the derived category $D(X)$. Moreover, he showed that there is an $L_\infty$-algebra structure on the Dolbeault resolution of $TX[-1]$…

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

In this paper we construct an explicit representative for the Grothendieck fundamental class [Z] of a complex submanifold Z of a complex manifold X, under the assumption that Z is the zero locus of a real analytic section of a holomorphic vector bundle E. To this data we associate a super-connection A on the exterior a…

In this paper we construct a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology of an associative algebra with appropriate duality. This is a generalized algebraic version of the main theorem of [AZ] which extends Goldman's results using string topology operations.The main result c…

Study on twisted Dolbeault cohomology in Kähler foliations.

We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we conside…

Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.

Study transverse Dolbeault cohomology for almost complex structures.

We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold structure on Dolbeault cohomology.

Study on Dolbeault complexes for special holonomy manifolds.

This paper extends Dolbeault cohomology and its surrounding theory to arbitrary almost complex manifolds. We define a spectral sequence converging to ordinary cohomology, whose first page is the Dolbeault cohomology, and develop a harmonic theory which injects into Dolbeault cohomology. Lie-theoretic analogues of the t…

Formality of Dolbeault DGAs on complex nilmanifolds restricted to tori.

Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.

We define the relative Dolbeault homology of a complex manifold with currents via a Čech approach and we prove its equivalence with the relative Čech-Dolbeault cohomology as defined in [T. Suwa, Čech-Dolbeault cohomology and the $\overline\partial$-Thom class, {\em Singularities---Niigata---Toyama 2007}, 321--340, Adv.…

Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.

A new Witten deformation modifies Dolbeault complex properties.

In this thesis, we show the existence of a sequence of differential operators starting with with the Dirac operator in k Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,§)\to C^\infty((\R^n)^k,§)$ ($§$ is the spinor module). This operator is the Cauchy-Riemann operato…

Study Dolbeault harmonic forms on Lie group quotients with specific structures.

Let $G$ be a complex Lie group acting on a compact complex Hermitian manifold $M$ by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.

We continue the study the Dolbeault dga of the formal neighborhood of an arbitary closed embedding of complex manifolds previously defined by the author in \cite{DolbeaultDGA}. The special case of the diagonal embedding has been studied in \cite{Diagonal}. We describe the Dolbeault dga explicitly in terms of the formal…

Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.

A d-bar-analogue of differential characters for complex manifolds is introduced and studied using a new theory of homological spark complexes. Many essentially different spark complexes are shown to have isomorphic groups of spark classes. This has many consequences: It leads to an analytic representation of O*-gerbes …

Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.

We consider semi-direct products $\C^{n}\ltimes_φN$ of Lie groups with lattices $Γ$ such that $N$ are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over $G/Γ$ by using the Dolbeaut cohomology of the Lie algebras of the direct …

We study basic Dolbeault cohomology and find new Weitzenböck formulas on a transversely Kähler foliation. We investigate conditions on mean curvature and Ricci curvature that impose restrictions on basic Dolbeault cohomology. For example, we prove that on a transversely Kähler foliation with positive transversal Ricci …

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…

Study complex structure deformations on Lie algebras and Dolbeault cohomology.

We consider the Dolbeault operator of $K^{1/2}$ -- the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface (M,g,J). We prove that the Dolbeault cohomology groups of $K^{1/2}$ vanish if the scalar curvature of g is non-negative and non-identically zero. Moreov…

This paper extends complex Cartan geometry results to noncompact and non-Kähler manifolds.

Inspired by a work of Kapranov, we define the notion of Dolbeault complex of the formal neighborhood of a closed embedding of complex manifolds. This construction allows us to study coherent sheaves over the formal neighborhood via complex analytic approach, as in the case of usual complex manifolds and their Dolbeault…

With respect to the Dolbeault complex over the flat manifold $\C^n$, an explicit description of the inverse correspondence of the twistor correspondence is given.

Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.

In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformation…

Study shows that dimension of Dolbeault harmonic forms is not always equal to B- on certain 4-manifolds.

It is well known that cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result is due to L. Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault…

New findings on complex manifold properties under deformations.

We discuss the known evidence for the conjecture that the Dolbeault cohomology of nilmanifolds with left-invariant complex structure can be computed as Lie-algebra cohomology and also mention some applications.

We use Dirac operator techniques to establish a sharp lower bound for the first eigenvalue of the twisted Dolbeault Laplacian on holomorphic line bundles over compact Kähler manifolds.

The paper improves $L^2$-estimates for Dirac-Dolbeault operators on complex manifolds.

We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a dga…

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $Γ$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/Γ$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$ of $G$. In this paper we construct, up to a finite group, a new Lie algebra $\tilde{\mathfr…

We use Dirac operator techniques to a establish sharp lower bound for the first eigenvalue of the Dolbeault Laplacian twisted by Hermitian-Einstein connections on vector bundles of negative degree over compact Kähler manifolds.

Paper connects cohomologies on almost complex manifolds.

We prove a theorem of Leray-Hirsch type and give an explicit blow-up formula for Dolbeault cohomology on (\emph{not necessarily compact}) complex manifolds. We give applications to strongly $q$-complete manifolds and the $\partial\bar{\partial}$-lemma.

Develops an L^p theory for Dolbeault-Dirac operators on compact Kähler manifolds.

For a Kähler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special atte…

This paper extends geometric structure theory to infinite type structures.

Let ${\cal F}$ be a complex affine Reeb foliation of dimension $1$ on the Hopf manifold ${\Bbb S}^{n+1}\times {\Bbb S}^1$. We prove that its foliated Dolbeault cohomology in degree $1$ is isomorphic to ${\Bbb C}$ by giving an explicit generator.