arXiv research

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…

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…

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

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

We calculate the second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator associated to high powers of a Hermitian line bundle with non-degenerate curvature, using the method of formal power series developed by Ma and Marinescu.

Researchers compute the index of a specific operator on contact manifolds.

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.

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.

Proves the Hodge conjecture for complex projective manifolds.

Researchers find spectral gaps in quantum flag manifolds using twisted operators.

We study the compact Hermitian spin surfaces with positive conformal scalar curvature on which the first eigenvalue of the Dolbeault operator of the spin structure is the smallest possible. We prove that such a surface is either a ruled surface or a Hopf surface. We give a complete classification of the ruled surfaces …

We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohom…

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

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormand…

Study complex structures with perturbed differential operators to compute curvature-like operators and obtain vanishing results.

We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator $D_t$ on each regular level set $C_t$ of a fixed Morse function defining this cobordism. We show that as we approac…

We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, …

Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it giv…

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Che…

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. …

Noncommutative Kähler structures were recently introduced as an algebraic framework for studying noncommutative complex geometry on quantum homogeneous spaces. In this paper, we introduce the notion of a \emph{compact quantum homogeneous Kähler space} which gives a natural set of compatibility conditions between covari…

The article studies cohomology on complex manifolds and proves vanishing theorems.

We review the notion of relative Dolbeault cohomology and prove that it is canonically isomorphic with the local (relative) cohomology of A. Grothendieck and M. Sato with coefficients in the sheaf of holomorphic forms. We deal with this cohomology from two viewpoints. One is the Cech theoretical approach, which is conv…

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 $L^2$ Hilbert complexes on complex manifolds.

The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator …

Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.

Study transverse Dolbeault cohomology for almost complex structures.

With respect to the Dirac operator and the conformally invariant Laplacian, an explicit description of the inverse Penrose transform on Riemannian twistor spaces is given. A Dolbeault representative of cohomology on the twistor space is constructed from a solution of the field equation on the base manifold.

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

M. F. Atiyah proved that the index of a transversally elliptic operator relative to a free action can be computed by using indices of elliptic operators on the orbit manifold. In this paper, we derive an explicit formula for the transversal indices on S^1-bundles over complex projective spaces. Using this explicit form…

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.…

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corre…

Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.

Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.

A new Witten deformation modifies Dolbeault complex properties.

Quantum systems on coadjoint orbits yield spectra matching Dolbeault and de Rham indices.

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

The spectrum of the Laplace-Dolbeault operator for any line bundle with parallel curvature on a flat complex torus is computed. The Ray-Singer analytic torsion is then deduced, generalizing thus Bost's result for ample line bundles and Ray-Singer's ones for flat bundles, of which we a geometric interpretation is given.

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of t…

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 introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-…

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…