arXiv research

We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group ${\mathbb C}^*$ associated to a compact connected Riemann surface.

Paper generalizes Mochizuki's theorem to stable λ-flat bundles and explores applications to moduli spaces.

Paper studies compactifications of Higgs bundles and self-duality equations.

Let $M= G/Γ$ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space ${\cal C} ({\frak g})$ of invariant complex structures on $M$, the Dolbeault cohomology of $M$ is isomorphic to the one of the differential bigraded algebra ass…

The paper studies the local structure of a moduli space for a specific string theory system.

We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the compactification. We also show that these Calabi-Yau metrics admit a polyhomogeneous…

It is shown that any compact Kähler manifold $M$ gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is ag…

Study transverse Dolbeault cohomology for almost complex structures.

Constructs a Dolbeault-Hilbert complex for spaces with isolated singular points.

Constructs Lagrangian correspondences for Higgs bundles and holomorphic connections.

The loop space LP_1 of the Riemann sphere is an infinite dimensional complex manifold consisting of maps (loops) from S^1 to P_1 in some fixed C^k or Sobolev W^{k,p} space. In this paper we compute the Dolbeault cohomology groups H^{0,1}(LP_1).

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

Local vanishing theorems for complex spaces with smooth boundaries.

Formula connects complex forms on deforming spaces.

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

It has long been known that differential forms on complex manifolds can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for quaternionic manifolds using the fact that the cotangent space is isomorphic to a quaternionic vector space. …

Study of harmonic oscillators on singular geometries using supersymmetry.

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

Researchers calculate exact moduli for type II flux backgrounds using spectral sequences.

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…

Study shows Dolbeault cohomology is unchanged by complex Lie group actions.

Computational techniques calculate dimensions of complex structures.

Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.

The paper characterizes when the $\partial \overline{\partial}$-lemma holds for twistor spaces.

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.

Study on Kähler manifolds proves weak decompositions and relates harmonic forms.

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…

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.

Study Dolbeault cohomology on complex manifolds with torus action.

The paper develops a deformation theory for Dolbeault cohomology classes.

A new Witten deformation modifies Dolbeault complex properties.

The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.

The paper extends Bott-Chern Laplacian definition and explores its properties on almost Hermitian 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.

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 …

Complex foliation cohomology is shown to be simple.

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…