Holomorphic Bundles Research Articles

Warped AdS 6 × S 2 in Type IIB supergravity I: local solutions

We investigate the existence of solutions with 16 residual supersymmetries to Type IIB supergravity on a space-time of the formc AdS 6×S 2 warped over a two-dimensional Riemann surface Σ. The SO(2, 5) × SO(3) isometry extends to invariance under the exceptional Lie superalgebra F (4). In the present paper, we construct the general Ansatz compatible with these symmetries, derive the corresponding reduced BPS equations, and obtain their complete local solution in terms of two locally holomorphic functions $$ {\mathcal{A}}_{\pm } $$ on Σ, subject to certain positivity and regularity conditions. Globally, ( $$ {\mathcal{A}}_{+} $$ , $$ {\mathcal{A}}_{-} $$ ) are allowed to be multiple-valued on Σ and be holomorphic sections of a holomorphic bundle over Σ with structure group contained in $$ \mathrm{S}\mathrm{U}\left(1,1\right)\times \mathbb{C} $$ . Globally regular solutions are expected to provide the near-horizon geometry of (p, q) 5-brane and 7-brane webs which are holographic duals to five-dimensional conformal field theories. A preliminary analysis of the positivity and regularity conditions will be presented here, leaving the construction of globally regular solutions to a subsequent paper.

Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties

Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite \'etale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$, ramified only over the singularities of $X$, such that the \'etale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every \'etale cover of $Y_{\mathrm{reg}}$ extends to an \'etale cover of $Y$. As first major application, we show that every flat holomorphic bundle defined on $Y_{\mathrm{reg}}$ extends to a flat bundle that is defined on all of $Y$. As a consequence, we generalise a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an Abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarised endomorphisms.

Compact homogeneous locally conformally Kähler manifolds

In this paper we show as main results two structure theorems of a compact homogeneous locally conformally Kahler (or shortly l.c.K.) manifold, a holomorphic structure theorem asserting that it has a structure of holomorphic principal fiber bundle over a flag manifold with fiber a $1$-dimensional complex torus, and a metric structure theorem asserting that it is necessarily of Vaisman type. We also discuss and determine l.c.K. reductive Lie groups and compact locally homogeneous l.c.K. manifolds of reductive Lie groups.

ON DE RHAM AND DOLBEAULT COHOMOLOGY OF SOLVMANIFOLDS

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 g of G. In this paper we construct, up to a finite group, a new Lie algebra eg whose cohomology is isomorphic to the de Rham cohomology of G/Γ by using a modification of G associated with an algebraic sub-torus of the Zariski-closure of the image of the adjoint representation. This technique includes the construction due to Guan and developed by the first two authors. In this paper, we also give a Dolbeault version of such technique for complex solvmanifolds, i.e., for solvmanifolds endowed with an invariant complex structure. We construct a finite-dimensional cochain complex which computes the Dolbeault cohomology of a complex solvmanifold G/Γ with holomorphic Mostow bundle and we give a construction of a new Lie algebra $$ \overset{\smile }{\mathfrak{g}} $$ with a complex structure whose cohomology is isomorphic to the Dolbeault cohomology of G/Γ.

Semi-stability for Holomorphic Bundles with Global Sections

Semi-stability for Holomorphic Bundles with Global Sections

Hermitian metrics on F-manifolds

An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metrics on $F$-manifolds from $tt^*$ geometry. It clarifies the logic between several notions. It also introduces a new {\it canonical} hermitian metric. Near irreducible points it makes the manifold almost hyperbolic. This holds for the singularity case and will hopefully lead to applications there.

Balanced metrics on twisted Higgs bundles

A twisted Higgs bundle on a Kähler manifold X is a pair $$(E,\phi )$$ consisting of a holomorphic vector bundle E and a holomorphic bundle morphism $$\phi :M \otimes E \rightarrow E$$ for some holomorphic vector bundle M. Such objects were first considered by Hitchin when X is a curve and M is the tangent bundle of X, and also by Simpson for higher dimensional base. The Hitchin–Kobayashi correspondence for such pairs states that $$(E,\phi )$$ is polystable if and only if E admits a hermitian metric solving the Hitchin equation. This correspondence is a powerful tool to decide whether there exists a solution of the equation, but it provides little information as to the actual solution. In this paper we study a quantization of this problem that is expressed in terms of finite dimensional data and balanced metrics that give approximate solutions to the Hitchin equation. Motivation for this study comes from work of Donagi–Wijnholt (JHEP 05:068, 2013) concerning balanced metrics for the Vafa–Witten equations.

Complex geometry of moment-angle manifolds

Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F which is equivariant with respect to the (C*)^m-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi-Eckmann manifolds, and their deformations. We construct transversely Kaehler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that any Kaehler submanifold (or, more generally, a Fujiki class C subvariety) in such a moment-angle manifold is contained in a leaf of the foliation F. For a generic moment-angle manifold Z in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension. This implies, in particular, that the algebraic dimension of Z is zero.

Inverse of Abelian integrals and ramified Riemann domains

We deal with the Levi problem (Hartogs’ inverse problem) for ramified Riemann domains by introducing a positive scalar function $$\rho (a, X)$$ for a complex manifold X with a global frame of the holomorphic cotangent bundle by closed Abelian differentials, which is an analogue of Hartogs’ radius. We obtain some geometric conditions in terms of $$\rho (a, X)$$ which imply the validity of the Levi problem for finitely sheeted ramified Riemann domains over $${\mathbf {C}}^n$$ . On the course, we give a new proof of the Behnke–Stein Theorem.

Complexes of differential forms associated with a normalized manifold over the algebra of dual numbers

We construct some complexes of differential forms on a smooth manifold MnD over the algebra of dual numbers D on the base of a decomposition of the tensor product TMnD⊗ℝD into the Whitney sum of two subbundles. It is shown that these complexes can be obtained as restrictions of some complexes of holomorphic (D-smooth) forms defined on the tangent bundle TMnD. For holomorphic fiber bundles over MnD, we introduce complexes of D-valued forms holomorphic along the fibers and express in terms of cohomology classes of such complexes the obstructions to existence of holomorphic connections in holomorphic principal bundles.

A Kobayashi–Hitchin correspondence for I±-holomorphic bundles

In this paper, we introduce the notions of α-Hermitian–Einstein metric and α-stability for I±-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi–Hitchin correspondence for I±-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include generalized holomorphic bundles on generalized Kähler manifolds. We also show that α-stability of a vector bundle, in this sense, can depend on the parameter α.

A generalized Schrödinger equation via a complex Lagrangian of electrodynamics

In this paper we give a generalized form of the Schrödinger equation in the relativistic case, which contains a generalization of the Klein-Gordon equation. By complex Legendre transformation, the complex Lagrangian of electrodynamics produces a complex relativistic Hamiltonian H of electrodynamics, on the holomorphic cotangent bundle T′* M. By a special quantization process, a relativistic time dependent Schrödinger equation, in the adapted frames of (T′* M, H) is obtained. This generalized Schrödinger equation can be expressed with respect to the Laplace operator of the complex Hamilton space (T′*M, H). Finally, under some additional conditions on the proper time s of the complex space-time M and the time parameter t along the quantum state, by the method of separation of variables, we obtain two classes of solutions for the Schrödinger equation, one for the weakly gravitational complex curved space M, and the second in the complex space-time with Schwarzschild metric.

G2–instantons over twisted connected sums

We introduce a method to construct $G_2$-instantons over compact $G_2$-manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing $G_2$-instantons obtained from holomorphic bundles over the building blocks via the first named author's work [SE11]. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface.

Holomorphic Lagrangian branes correspond to perverse sheaves

Let X be a compact complex manifold, $D_c^b(X)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk(T^*X)$ be the Fukaya category of $T^*X$. A Lagrangian brane in $Fuk(T^*X)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^*X_{\mathbb{C}}$, the holomorphic cotangent bundle of $X$. We prove that under the quasi-equivalence between $D^b_c(X)$ and $DFuk(T^*X)$ established in [NaZa09] and [Nad09], holomorphic Lagrangian branes with appropriate grading correspond to perverse sheaves.

Analytic cycles, Bott–Chern forms, and singular sets for the Yang–Mills flow on Kähler manifolds

It is shown that the singular set for the Yang–Mills flow on unstable holomorphic vector bundles over compact Kähler manifolds is completely determined by the Harder–Narasimhan–Seshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves. We derive a singular Bott–Chern formula relating the second Chern form of a smooth metric on the bundle to the Chern current of an admissible metric on the associated graded sheaf. This is used to show that the multiplicities of the top dimensional bubbling locus defined via the Yang–Mills density agree with the corresponding multiplicities for the Harder–Narasimhan–Seshadri filtration. The set theoretic equality of singular sets is a consequence.

Massive And Massless Gauge Fields Formed by Flat Connections

The Yang - Mills type massive and massless gauge theories are interpreted in the geometrical frame of holomorphic principal bundles on a complex 2 - manifold. It is seen in this formalism that, the component (1,1) of the curvature of this connection appears because of flat connections generated by holomorphic structure although connection is flat. Thus it is possible to write a Lagrangian for a Yang - Mills theory including massive and massless gauge fields. However, the mass matrix of a massive gauge field on such a bundle isn’t nilpotent and this field is generated by a noncommutative flat connection on the same bundle, then the structure group of this bundle is non - Abelian complex Lie group. However, if the gauge field is massless, then this is generated by commutative flat connection, and so the structure group of the bundle is Abelian complex Lie group. Also one sees that the second Chern number or topological charge is proportional to the total volume of the base manifold for each massless and massive gauge theories and Abelian (massless) gauge theories are indeed the theories of the Kahler potential on the complex projective space $\mathbb {C}P^{2}$ .

A maximum principle for Hermitian (and other) metrics

We consider homomorphisms of hermitian holomorphic Hilbert bundles. Assuming the homomorphism decreases curvature, we prove that its pointwise norm is plurisubharmonic.

Holomorphic Poisson Cohomology

Abstract Holomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology is obtained by twisting the Dolbeault @-operator by the holomorphic Poisson bivector field. Therefore, the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, Kähler manifolds, and nilmanifolds with abelian complex structures or parallelizable complex structures.

Semidirect products and invariant connections

Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via biholomorphisms. We give an explicit description of the isomorphism classes of $G$-equivariant almost holomorphic hermitian principal bundles on $N$. Under the assumption that there is a central subgroup $Z= \text{U}(1)$ of $K(S)$ that acts on $\text{Lie}(N)$ as multiplication through a single nontrivial character, we give an explicit description of the isomorphism classes of $G$-equivariant holomorphic hermitian principal bundles on $N$.

On the torsion of the first direct image of a locally free sheaf

Let π:M→B be a proper holomorphic submersion between complex manifolds and ℰ a holomorphic bundle on M. We study and describe explicitly the torsion subsheaf Tors(R 1 π * (ℰ)) of the first direct image R 1 π * (ℰ) under the assumption R 0 π * (ℰ)=0. We give two applications of our results. The first concerns the locus of points in the base of a generically versal family of complex surfaces where the family is non-versal. The second application is a vanishing result for H 0 (Tors(R 1 π * (ℰ))) in a concrete situation related to our program to prove existence of curves on class VII surfaces.