Dolbeault Cohomology Research Articles

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

Holomorphic Laplacian on the Lie ball and the Penrose transform

We prove that any holomorphic function f on the Lie ball of even dimension satisfying Δf=0 is obtained uniquely by the higher-dimensional Penrose transform of a Dolbeault cohomology for a twisted line bundle of a certain domain of the Grassmannian of isotropic subspaces. To overcome the difficulties arising from our setting that the line bundle parameter is outside the good range, we use some techniques from algebraic representation theory.

O razreshimosti lineynykh differentsial'nykh operatorov na vektornykh rassloeniyakh nad mnogoobraziem

Necessary and sufficient condition is established for the closedness of the rangeor surjectivity of a differential operator acting on smooth sections of vector bundles. For connectednoncompact manifolds it is shown that these conditions are derived from the regularityconditions and the unique continuation property of solutions. An application of these results toelliptic operators (more precisely, to operators with a surjective principal symbol) with analyticcoefficients, to second-order elliptic operators on line bundles with a real leading part, and to theHodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively, Dolbeault)cohomology group on a connected noncompact smooth (respectively, complex-analytic)manifold vanishes. For elliptic operators, we prove that solvability in smooth sections impliessolvability in generalized sections.

On the Sobolev Dolbeault cohomology of a domain with pseudoconcave boundaries

On the Sobolev Dolbeault cohomology of a domain with pseudoconcave boundaries

Almost complex structures, transverse complex structures, and transverse Dolbeault cohomology

We define a transverse Dolbeault cohomology associated to any almost complex structure j on a smooth manifold M . This we do by extending the notion of transverse complex structure and by introducing a natural j -stable involutive limit distribution with such a transverse complex structure. We relate this transverse Dolbeault cohomology to the generalized Dolbeault cohomology of (M,j) introduced by Cirici and Wilson in 2001, showing that the (p,0) cohomology spaces coincide. This study of transversality leads us to suggest a notion of minimally non-integrable almost complex structure.

Superconformal quantum mechanics and growth of sheaf cohomology

We give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kähler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of K points on ℂ2, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as K → ∞. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics.

Levi-flat CR structures on compact Lie groups

Pittie (Proc Indian Acad Sci Math Sci 98:117-152, 1988) proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We generalized Pittie’s result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups. The main tools we used was a version of the Leray–Hirsch theorem for CR principal bundles and the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups (Charbonnel and Khalgui in J Lie Theory 14:165-198, 2004) .

Sato hyperfunctions via relative Dolbeault cohomology

The relative Dolbeault cohomology which naturally comes up in the theory of Čech–Dolbeault cohomology turns out to be canonically isomorphic with the local (relative) cohomology of Grothendieck and Sato so that it provides a handy way of representing the latter. In this paper we use this cohomology to give simple explicit expressions of Sato hyperfunctions, some fundamental operations on them and related local duality theorems. This approach also yields a new insight into the theory of hyperfunctions and leads to a number of further results and applications. As one of such, we give an explicit embedding morphism of Schwartz distributions into the space of hyperfunctions.

Holomorphic Koszul–Brylinski homology via Dolbeault cohomology

We use the Dolbeault cohomology to investigate the Koszul-Brylinski homology on holomorphic Poisson manifolds. We obtain the Leray-Hirsch theorem for Hochschild homology and the Mayer-Vietoris sequence, K\"{u}nneth theorem for holomorphic Koszul-Brylinski homology. In particular, we show some relations of holomorphic Koszul-Brylinski homologies around a blow-up transformation for the general case (\emph{not necessarily compact}) by our previous works on the Dolbeault cohomology.

Formality of the Dolbeault complex and deformations of holomorphic Poisson manifolds

The purpose of this paper is to study the properties of holomorphic Poisson manifolds (M,π) under the assumption of ∂∂¯–lemma or ∂π∂¯–lemma. Under these assumptions, we show that the Koszul–Brylinski homology can be recovered by the Dolbeault cohomology, and prove that the DGLA (AM•,•,∂¯,[−,−]∂π) is formal. Furthermore, we discuss the Maurer–Cartan elements of (AM•,•[[t]],∂¯,[−,−]∂π) which induce the deformations of complex structure of M.

Maximally non-integrable almost complex structures: an h-principle and cohomological properties

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an h-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension 2nge 10 (respectively ge 6) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed 4-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds).

Twisted basic Dolbeault cohomology on transverse Kähler foliations

In this paper, we study the twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem on a transverse Kähler foliation. And we give some properties for \(\Delta _\kappa \)-harmonic forms and prove the Kodaira–Serre-type duality and Dolbeault isomorphism for the twisted basic Dolbeault cohomology.

FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICS

Formulation of the problem. Precision studies of the Higgs boson, supersymmetric particles, the magnetic moment of the muon, electric dipole moment of the electron, flavor anomalies demonstrate the deviation beyond Standard Model. They are connected with a new understanding of quantum field theory through the unification of gravity with particle physics in the framework of string theory - the powerful instrument, which has changed the theory picture. The article is devoted to the study of new physics through these two components. First, we considered particle physics in terms of the latest experimental data and then moved on to the mathematical apparatus of string theory. Materials and methods. The N = 2 Yang-Mills theory is the heterotic string analog determined in ten-dimensional space: four usual space-time coordinates and six extra dimensions, known as Calabi-Yau manifold in weighted projective space. We studied the Calabi-Yau manifold in terms of both differential forms and reflexive polyhedrа to extract the elementary particle information. For further work with Calabi-Yau manifolds, differential forms for calculation of cohomology groups and reflexive polyhedrа for calculation of Hodge numbers were used. We used two definitions of general properties of toric varieties: hypersurfaces in terms of differential forms and projective space in terms of reflexive polyhedral. Then we investigated lattice polyhedra ∆ which gives rise to families of Calabi-Yau hypersurfaces in weighted projective space, P∆. Such polyhedra admit a combinatorial characterization and are called reflexive polyhedra. Results. The comparison of two approaches to the description of Calabi-Yau manifold as a complex manifold and as weighted projective space led us to the conclusion about the equivalence of these two treatments in the context of calculation of the Euler characteristic. As Euler’s characteristic for elementary particle physics is the number of generations of quarks and leptons, the selection of Calabi-Yau manifolds with appropriate topological properties is one of the urgent problems of modern physics. It is necessary to stress that the important result of our paper is the coincidence of the value of the Euler characteristic, found in terms of Dolbeault cohomology and terms of reflexive polyhedral. The obtained information about topological invariants is necessary for predicting the number of generations in particle physics. Conclusions. Although a unified theory of all interactions has not yet been found, however, certain aspects related to the interpretation of the unified theory of all interactions in terms of modern mathematics give their significant results. Therefore, the use and development of the apparatus of algebraic geometry for finding topological invariants that have the value of observables in physics is an urgent task.

From β to η: a new cohomology for deformed Sasaki-Einstein manifolds

We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups {H}_{overline{partial}}^{left(p,0right)}(k) graded by their charge under the Reeb vector. We then introduce a new cohomology, η-cohomology, which is defined by a CR structure and a holomorphic function f with non-vanishing η ≡ df. It is the natural cohomology associated to a class of supersymmetric type IIB flux backgrounds that generalise the notion of a Sasaki-Einstein manifold. These geometries are dual to finite deformations of the 4d mathcal{N} = 1 SCFTs described by conventional Sasaki-Einstein manifolds. As such, they are associated to Calabi-Yau algebras with a deformed superpotential. We show how to compute the η-cohomology in terms of the transverse Dolbeault cohomology of the undeformed Sasaki-Einstein space. The gauge-gravity correspondence implies a direct relation between the cyclic homologies of the Calabi-Yau algebra, or equivalently the counting of short multiplets in the deformed SCFT, and the η-cohomology groups. We verify that this relation is satisfied in the case of S5, and use it to predict the reduced cyclic homology groups in the case of deformations of regular Sasaki-Einstein spaces. The corresponding Calabi-Yau algebras describe non-commutative deformations of ℙ2, ℙ1× ℙ1 and the del Pezzo surfaces.

Special Hermitian metrics on Oeljeklaus–Toma manifolds

Oeljeklaus–Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension K $K$ of Q $\mathbb {Q}$ and a subgroup of units U $U$ . We characterize the existence of pluriclosed metrics (also known as strongly Kähler with torsion (SKT) metrics) on any OT manifold X ( K , U ) $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number b 3 $b_3$ and the Dolbeault cohomology group H ∂ ¯ 2 , 1 $H^{2,1}_{\overline{\partial }}$ . Combined with the main result in (Dubickas, Results Math. 76 (2021), 78), these numerical conditions render explicit examples of pluriclosed OT manifolds in arbitrary complex dimension. We prove that in complex dimension 4 and type ( 2 , 2 ) $(2, 2)$ , the existence of a pluriclosed metric on X ( K , U ) $X(K, U)$ is entirely topological, namely, it is equivalent to b 3 = 2 $b_3=2$ . Moreover, we provide an explicit example of an OT manifold of complex dimension 4 carrying a pluriclosed metric. Finally, we show that no OT manifold admits balanced metrics, but all of them carry instead locally conformally balanced metrics.

Extendability and the $$\overline{\partial }$$ operator on the Hartogs triangle

In this paper it is shown that the Hartogs triangle $${\mathbf{T}}$$ in $${\mathbf{C}}^2$$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $${\mathbf{T}}$$ .

Alternative formulations of the twistor double copy

The classical double copy relating exact solutions of biadjoint scalar, gauge and gravity theories continues to receive widespread attention. Recently, a derivation of the exact classical double copy was presented, using ideas from twistor theory, in which spacetime fields are mapped to Cech cohomology classes in twistor space. A puzzle remains, however, in how to interpret the twistor double copy, in that it relies on somehow picking special representatives of each cohomology class. In this paper, we provide two alternative formulations of the twistor double copy using the more widely-used language of Dolbeault cohomology. The first amounts to a rewriting of the Cech approach, whereas the second uses known techniques for discussing spacetime fields in Euclidean signature. The latter approach indeed allows us to identify special cohomology representatives, suggesting that further application of twistor methods in exploring the remit of the double copy may be fruitful.

Elliptic involutive structures on compact Lie groups

Inspired by the work of Chevalley and Eilenberg on the de Rham cohomology on compact Lie groups, we prove that, under certain algebraic and topological conditions, the cohomology associated to left-invariant elliptic, and even hypocomplex, involutive structures on compact Lie groups can be computed by using only Lie algebras, thus reducing the analytical problem to a purely algebraic one. The main tool is the Leray spectral sequence that connects the result obtained Chevalley and Eilenberg to a result by Bott on the Dolbeault cohomology of a homogeneous manifold.

Deformations of Dolbeault cohomology classes

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 deformations by using the power series method. We also prove a simple relation between the existence of deformations and the varying of the dimensions of Dolbeault cohomology. The deformations of (p, q)-forms is shown to be unobstructed under some mild conditions. By analyzing Nakamura’s example of complex parallelizable manifolds, we will see that the deformation theory developed in this work provides precise explanations to the jumping phenomenon of Dolbeault cohomology.

Dolbeault and J-Invariant Cohomologies on Almost Complex Manifolds

In this paper we relate the cohomology of J-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.