Dolbeault Complex Research Articles

Monopole harmonics on $\mathbb{CP}^{n-1}$

We find the spectra and eigenfunctions of both ordinary and supersymmetric quantum-mechanical models describing the motion of a charged particle over the \mathbb{CP}^{n-1}ℂℙn−1 manifold in the presence of a background monopole-like gauge field. The states form degenerate SU(n)SU(n) multiplets and their wave functions acquire a very simple form being expressed via homogeneous coordinates. Their relationship to multidimensional orthogonal polynomials of a special kind is discussed. By the well-known isomorphism between the twisted Dolbeault and Dirac complexes, our construction also gives the eigenfunctions and eigenvalues of the Dirac operator on complex projective spaces in a monopole background.

Representation of relative sheaf cohomology

Abstract We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Čech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sections of the sheaf complex, which appears naturally in the theory of Čech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories. We prove a “relative de Rham-type theorem” from the above two viewpoints. It says that, in the case the complex is a soft or fine resolution of a certain sheaf, the cohomology is canonically isomorphic with the relative cohomology of the sheaf. Thus the former provides a handy way of representing the latter. Along the way we develop various theories and establishes canonical isomorphisms among the cohomologies that appear therein. The second viewpoint leads to a generalization of the theory to the case of cohomology of sheaf morphisms. Some special cases together with applications are also indicated. Particularly notable is the application of the Dolbeault complex case to the Sato hyperfunction theory and other problems in algebraic analysis.

Maximally Twisted Eleven-Dimensional Supergravity

We perform the maximal twist of eleven-dimensional supergravity. This twist is partially topological and exists on manifolds of G_2 times SU(2) holonomy. Our derivation starts with an explicit description of the Batalin–Vilkovisky complex associated to the three-form multiplet in the pure spinor superfield formalism. We then determine the L_infty module structure of the supersymmetry algebra on the component fields. We twist the theory by modifying the differential of the Batalin–Vilkovisky complex to incorporate the action of a scalar supercharge. We find that the resulting free twisted theory is given by the tensor product of the de Rham and Dolbeault complexes of the respective G_2 and SU(2) holonomy manifolds as conjectured by Costello.

Dolbeault-type complexes on $$G_2$$- and $$\mathrm {Spin}(7)$$-manifolds

There are three types of Dolbeault complexes arising from representations of holonomy group on a Riemannian manifold, two of which are dual to each other. Such a complex is elliptic if and only if its generator satisfies an algebraic condition. As applications, we give all Dolbeault complexes on a compact Riemannian manifold with exceptional holonomy. Each cohomology can be described by harmonic forms.

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.

Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

This paper is devoted to the study of the relation between `formal exponential maps,' the Atiyah class, and Kapranov $L_\infty[1]$ algebras associated with dg manifolds in the $C^\infty$ context. Given a dg manifold, we prove that a `formal exponential map' exists if and only if the Atiyah class vanishes. Inspired by Kapranov's construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an $L_\infty[1]$ algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative w.r.t. the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold $(T_X^{0,1}[1],\bar{\partial})$ arising from a complex manifold $X$, we prove that this $L_\infty[1]$ algebra structure is quasi-isomorphic to the standard $L_\infty[1]$ algebra structure on the Dolbeault complex $\Omega^{0,\bullet}(T^{1,0}_X)$.

A Dolbeault Lemma for Temperate Currents

We consider a bounded open Stein subset Ω of a complex Stein manifold X of dimension n. We prove that if f is a current on X of bidegree (p,q+1), ∂ ¯-closed on Ω, we can find a current u on X of bidegree (p,q) which is a solution of the equation ∂ ¯u=f in Ω. In other words, we prove that the Dolbeault complex of temperate currents on Ω (i.e. currents on Ω which extend to currents on X) is concentrated in degree 0. Moreover if f is a current on X=ℂ n of order k, then we can find a solution u which is a current on ℂ n of order k+2n+1.

The Witten deformation of the Dolbeault complex

We introduce a Witten–Novikov type perturbation $$\bar{\partial }_{\bar{\omega }}$$ of the Dolbeault complex of any complex Kahler manifold, defined by a form $$\omega $$ of type (1, 0) with $$\partial \omega =0$$ . We give an explicit description of the associated index density which shows that it exhibits a nontrivial dependence on $$\omega $$ . The heat invariants of lower order are shown to be zero.

English

A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian manifold M. We use invariance theory to show that the perturbed index density is independent of Θ; this result was established previously by J. A. Alvarez Lopez, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on Θ. We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires Θ to be a $$\overline \partial $$ closed 1-form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on Θ even in the setting of Riemann surfaces.

The correspondence formula of Dolbeault complex on pair deformation

Given a holomorphic family of pairs \(\{(X_t,E_t)\}\) where each \(E_t\) is a holomorphic vector bundle over a compact complex manifold \(X_t\), we get a correspondence between the Dolbeault complex of \(E_t\)-valued (p, q)-forms on \(X_t\) and the one of \(E_0\)-valued (p, q)-forms on \(X_0\) for small enough t.

The Hodge realization of the polylogarithm on the product of multiplicative groups

The purpose of this article is to describe explicitly the polylogarithm class in absolute Hodge cohomology of a product of multiplicative groups, in terms of the Bloch-Wigner-Ramakrishnan polylogarithm functions. We will use the logarithmic Dolbeault complex defined by Burgos to calculate the corresponding absolute Hodge cohomology groups.

Enhanced Laplace transform and holomorphic Paley-Wiener-type theorems

Starting from a remark about the computation of Kashiwara–Schapira’s enhanced Laplace transform by using the Dolbeault complex of enhanced distributions, we explain how to obtain explicit holomorphic Paley–Wiener-type theorems. As an example, we get back some classical theorems due to Polya and Méril as limits of tempered Laplace-isomorphisms. In particular, we show how contour integrations naturally appear in this framework.

Transverse Kähler structures on central foliations of complex manifolds

For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. If there exists a transverse Kahler structure on such a foliation, then we obtain a nice differential graded algebra which is quasi-isomorphic to the de Rham complex and a nice differential bi-graded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kahler manifolds. Moreover, under certain additional condition, we can develop Morgan’s theory of mixed Hodge structures as similar to the study on smooth algebraic varieties.

The Neumann Problem for the k-Cauchy–Fueter Complex over k-Pseudoconvex Domains in $$\mathbb {R}^4$$ R 4 and the $$L^2$$ L 2 Estimate

The k-Cauchy–Fueter operator and complex are quaternionic counterparts of the Cauchy–Riemann operator and the Dolbeault complex in the theory of several complex variables, respectively. To develop the function theory of several quaternionic variables, we need to solve the non-homogeneous k-Cauchy–Fueter equation over a domain under the compatibility condition, which naturally leads to a Neumann problem. The method of solving the $$\overline{\partial }$$ -Neumann problem in the theory of several complex variables is applied to this Neumann problem. We introduce notions of k-plurisubharmonic functions and k-pseudoconvex domains, establish the $$L^2$$ estimate and solve the Neumann problem over k-pseudoconvex domains in $$\mathbb {R}^4$$ . Namely, we get a vanishing theorem for the first cohomology group of the k-Cauchy–Fueter complex over such domains.

The bar{partial }-equation on a non-reduced analytic space

Let X be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of overline{partial }-equation on X and prove a Dolbeault–Grothendieck lemma. We obtain fine sheaves mathcal {A}_X^q of (0, q)-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf mathscr {O}_X. Our construction is based on intrinsic semi-global Koppelman formulas on X.

Nahm transformation for parabolic Higgs bundles on the projective line—Case of non-semisimple residues

We extend our earlier construction of Nahm transformation for parabolic Higgs bundles on the projective line to solutions with not necessarily semisimple residues and show that it determines a holomorphic mapping on corresponding moduli spaces. The construction relies on suitable elementary modifications of the logarithmic Dolbeault complex.

Explicit Serre duality on complex spaces

In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof, of Serre duality on any reduced pure n-dimensional paracompact complex space X. At the core of the paper is the introduction of certain fine sheaves BXn,q of currents on X of bidegree (n,q), such that the Dolbeault complex (BXn,•,∂¯) becomes, in a certain sense, a dualizing complex. In particular, if X is Cohen–Macaulay then (BXn,•,∂¯) is an explicit fine resolution of the Grothendieck dualizing sheaf.

The Dolbeault dga of a formal neighborhood

Inspired by a work of Kapranov (Kap99), 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 com- plex analytic approach, as in the case of usual complex manifolds and their Dolbeault complexes. Moreover, our the Dolbeault complex as a differential graded algebra can be associated with a dg-category according to Block (Blo10). We show this dg-category is a dg-enhancement of the bounded derived category over the formal neighborhood under the assumption that the submanifold is compact. This generalizes a similar result of Block in the case of usual complex manifolds. CONTENTS

On the Hodge-type decomposition and cohomology groups of k-Cauchy–Fueter complexes over domains in the quaternionic space

The k -Cauchy–Fueter operator D 0 ( k ) on one dimensional quaternionic space H is the Euclidean version of spin k / 2 massless field operator on the Minkowski space in physics. The k -Cauchy–Fueter equation for k ≥ 2 is overdetermined and its compatibility condition is given by the k -Cauchy–Fueter complex. In quaternionic analysis, these complexes play the role of Dolbeault complex in several complex variables. We prove that a natural boundary value problem associated to this complex is regular. Then by using the theory of regular boundary value problems, we show the Hodge-type orthogonal decomposition, and the fact that the non-homogeneous k -Cauchy–Fueter equation D 0 ( k ) u = f on a smooth domain Ω in H is solvable if and only if f satisfies the compatibility condition and is orthogonal to the set ℋ ( k ) 1 ( Ω ) of Hodge-type elements. This set is isomorphic to the first cohomology group of the k -Cauchy–Fueter complex over Ω , which is finite dimensional, while the second cohomology group is always trivial.

Dolbeault–Massey triple products of low degree

Let A = ( A • , • , ∂ ¯ A ) be a differential bigraded algebra. We characterize non-vanishing Dolbeault–Massey triple products of low degree (see Theorem 3.1 , Theorem 3.2 ). We give some applications for the Dolbeault complex on a compact complex manifold.