Holomorphic Vector Bundles Research Articles

The geometry of supersymmetric partition functions

We consider supersymmetric field theories on compact manifolds $ \mathcal{M} $ and obtain constraints on the parameter dependence of their partition functions $ {Z_{\mathcal{M}}} $ . Our primary focus is the dependence of $ {Z_{\mathcal{M}}} $ on the geometry of $ \mathcal{M} $ , as well as background gauge fields that couple to continuous flavor symmetries. For $ \mathcal{N} $ = 1 theories with a U(1) R symmetry in four dimensions, $ \mathcal{M} $ must be a complex manifold with a Hermitian metric. We find that $ {Z_{\mathcal{M}}} $ is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over $ \mathcal{M} $ and $ {Z_{\mathcal{M}}} $ is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional $ \mathcal{N} $ = 2 theories with a U(1) R symmetry, where the necessary geometric structure on $ \mathcal{M} $ is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that $ {Z_{\mathcal{M}}} $ is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S 3 × S 1 or S 2 × S 1, which are related to supersymmetric indices, and manifolds diffeomorphic to S 3 (squashed spheres). In examples where $ {Z_{\mathcal{M}}} $ has been calculated explicitly, our results explain many of its observed properties.

The limit of the Yang–Mills flow on semi-stable bundles

Abstract By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kähler manifold X, the Yang–Mills flow converges away from an analytic singular set. If E is semi-stable, then the limiting metric is Hermitian-Einstein and will decompose the limiting bundle E ∞ into a direct sum of stable bundles. Bando and Siu prove E ∞ can be extended to a reflexive sheaf E ^ ∞ ${\hat{E}_\infty }$ on all of X. In this paper, we construct an isomorphism between E ^ ∞ ${\hat{E}_\infty }$ and the double dual of the stable quotients of the graded Seshadri filtration Gr s ( E ) * * ${{\rm Gr}^{\hspace*{0.56905pt}s}(E)^{**}}$ of E.

Infinitely divisible metrics and curvature inequalities for operators in the Cowen-Douglas class

The curvature $\mathcal K_T(w)$ of a contraction $T$ in the Cowen-Douglas class $B_1(\mathbb D)$ is bounded above by the curvature $\mathcal K_{S^*}(w)$ of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this note, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle $E_T$ corresponding to the operator $T$ in the Cowen-Douglas class $B_1(\mathbb D)$ which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the class $B_1(\Omega)$, for a bounded domain $\Omega$ in $\mathbb C^m$.

Deformation quantization with separation of variables of an endomorphism bundle

Given a holomorphic Hermitian vector bundle E and a star-product with separation of variables on a pseudo-Kähler manifold, we construct a star product on the sections of the endomorphism bundle of the dual bundle E∗ which also has the appropriately generalized property of separation of variables. For this star product we prove a generalization of Gammelgaard’s graph-theoretic formula.

Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on $ \mathbb{C}{{\mathbb{P}}^1} $

Consider the moduli space of parabolic Higgs bundles (E, Φ) of rank two on ℂℙ1 such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by \( \left( {E,\varPhi } \right)\mapsto \left( {E,-\varPhi } \right) \). We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.

Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit

Abstract In the following article we study the limiting properties of the Yang–Mills flow associated to a holomorphic vector bundle E over an arbitrary compact Kähler manifold (X,ω). In particular we show that the flow is determined at infinity by the holomorphic structure of E. Namely, if we fix an integrable unitary reference connection A 0 defining the holomorphic structure, then the Yang–Mills flow with initial condition A 0 converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle E ∞, which is isomorphic to the associated graded object of the Harder–Narasimhan–Seshadri filtration of (E,A 0). Moreover, E ∞ extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of one and two complex dimensions and proves the general case of a conjecture of Bando and Siu.

Monad constructions of omalous bundles

We consider a particular class of holomorphic vector bundles relevant for supersymmetric string theory, called omalous, over nonsingular projective varieties. We use monads to construct examples of such bundles over 3-fold hypersurfaces in P4, complete intersection Calabi–Yau manifolds in Pk, blow-ups of P2 at n distinct points, and products Pm×Pn.

Solutions of Strominger System from Unitary Representations of Cocompact Lattices of $${{\rm SL}(2, \mathbb{C})}$$ SL ( 2 , C )

Given an irreducible unitary representation of a cocompact lattice of \({{\rm SL}(2, \mathbb{C})}\), we explicitly write down a solution of the Strominger system of equations. These solutions satisfy the equation of motion, and the underlying holomorphic vector bundles are stable.

The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let $$\mathcal{O }$$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $$(X,\omega ).$$ We define a functional $$\mathcal{C }:\mathcal{O } \rightarrow \mathbb{R }$$ for each differential form $$\beta $$ of middle degree satisfying $$\beta \wedge \omega = 0$$ and an exactness condition. If the exactness condition does not hold, $$\mathcal{C }$$ is defined on the universal cover of $$\mathcal{O }.$$ A particular instance of $$\mathcal{C }$$ recovers the Calabi homomorphism. If $$\beta $$ is the imaginary part of a holomorphic volume form, the critical points of $$\mathcal{C }$$ are special Lagrangian submanifolds. We present evidence that $$\mathcal{C }$$ is related by mirror symmetry to a functional introduced by Donaldson to study Einstein–Hermitian metrics on holomorphic vector bundles. In particular, we show that $$\mathcal{C }$$ is convex on an open subspace $$\mathcal{O }^+ \subset \mathcal{O }.$$ As a prerequisite, we define a Riemannian metric on $$\mathcal{O }^+$$ and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.

Localization of Atiyah classes

We construct the Atiyah classes of holomorphic vector bundles using (1,0)-connections and developing a Chern–Weil type theory, allowing us to effectively compare Chern and Atiyah forms. Combining this point of view with the Čech–Dolbeault cohomology, we prove several results about vanishing and localization of Atiyah classes, and give some applications. In particular, we prove a Bott type vanishing theorem for (not necessarily involutive) holomorphic distributions. As an example we also present an explicit computation of the residue of a singular distribution on the normal bundle of an invariant submanifold that arises from the Camacho–Sad type localization.

Equivariant vector bundles and logarithmic connections on toric varieties

Let X be a smooth complete complex toric variety such that the boundary is a simple normal crossing divisor, and let E be a holomorphic vector bundle on X. We prove that the following three statements are equivalent:•The holomorphic vector bundle E admits an equivariant structure.•The holomorphic vector bundle E admits an integrable logarithmic connection singular over D.•The holomorphic vector bundle E admits a logarithmic connection singular over D. We show that an equivariant vector bundle on X has a tautological integrable logarithmic connection singular over D. This is used in computing the Chern classes of the equivariant vector bundles on X. We also prove a version of the above result for holomorphic vector bundles on log parallelizable G-pairs (X,D), where G is a simply connected complex affine algebraic group.

The Yang-Mills equations over Klein surfaces

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic point of view, these Lagrangian quotients are connected sets of real points inside a complex moduli variety endowed with a real structure; when the rank and the degree are coprime, they are in fact the connected components of the fixed-point set of the real structure. This presentation as a quotient enables us to generalize the methods of Atiyah and Bott to a setting with involutions, and compute the mod 2 Poincare polynomials of these moduli spaces in the coprime case. We also compute the mod 2 Poincare series of moduli stacks of all real and quaternionic vector bundles of a fixed topological type. As an application of our computations, we give new examples of maximal real algebraic varieties.

Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

Let \(X\) be a compact connected Riemann surface and \(G\) a connected reductive complex affine algebraic group. Given a holomorphic principal \(G\)-bundle \(E_G\) over \(X\), we construct a \(C^\infty \) Hermitian structure on \(E_G\) together with a \(1\)-parameter family of \(C^\infty \) automorphisms \(\{F_t\}_{t\in \mathbb R }\) of the principal \(G\)-bundle \(E_G\) with the following property: Let \(\nabla ^t\) be the connection on \(E_G\) corresponding to the Hermitian structure and the new holomorphic structure on \(E_G\) constructed using \(F_t\) from the original holomorphic structure. As \(t\rightarrow -\infty \), the connection \(\nabla ^t\) converges in \(C^\infty \) Frechet topology to the connection on \(E_G\) given by the Hermitian–Einstein connection on the polystable principal bundle associated to \(E_G\). In particular, as \(t\rightarrow -\infty \), the curvature of \(\nabla ^t\) converges in \(C^\infty \) Frechet topology to the curvature of the connection on \(E_G\) given by the Hermitian–Einstein connection on the polystable principal bundle associated to \(E_G\). The family \(\{F_t\}_{t\in \mathbb R }\) is constructed by generalizing the method of [6]. Given a holomorphic vector bundle \(E\) on \(X\), in [6] a \(1\)-parameter family of \(C^\infty \) automorphisms of \(E\) is constructed such that as \(t\rightarrow -\infty \), the curvature converges, in \(C^0\) topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.

Generalized [formula omitted] extension theorem and a conjecture of Ohsawa

In this paper, we determine the optimal constant in the estimate of Ohsawaʼs generalized L2 extension theorem. The result holds for holomorphic vector bundles on a class of complex manifolds including both Stein manifolds and complex projective algebraic manifolds. As an application, we obtain a solution to a related conjecture of Ohsawa.

Balanced metrics on vector bundles and polarized manifolds

We consider a notion of balanced metrics for triples (X, L, E) which depend on a parameter alpha, where X is a smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of alpha, we prove that the limit of a convergent sequence of balanced metrics leads to a Hermitian-Einstein metric on E and a constant scalar curvature Kahler metric in c(1)(L). For special values of alpha, limits of balanced metrics are solutions of a system of coupled equations relating a Hermitian-Einstein metric on E and a Kahler metric in c(1)(L).

A Vanishing Cohomology Theorem of Nakano Type on Projectivized Finsler Bundle

It is well known that a complex Finsler structure F on a holomorphic vector bundle E over a compact Kahler manifold (M;g) denes a bundle-like pseudo-Kahler metric of Sasaki type on the total space of the associated projectivized bundle PE. Then using an argument inspired from the theory of vanishing theorems for complex analytic foliations, we obtain a vanishing cohomology theorem of Nakano's type for base-like forms with values in a fo- liated line bundle over PE. AMS Subject Classication (2000). 53B40; 53B21; 32L20.

Holomorphic connections on filtered bundles over curves

Let X be a compact connected Riemann surface and E_P a holomorphic principal P -bundle over X , where P is a parabolic subgroup of a complex reductive affine algebraic group G . If the Levi bundle associated to E_P admits a holomorphic connection, and the reduction E_P \subset E_P\times^P G is rigid, we prove that E_P admits a holomorphic connection. As an immediate consequence, we obtain a sufficient condition for a filtered holomorphic vector bundle over X to admit a filtration preserving holomorphic connection. Moreover, we state a weaker sufficient condition in the special case of a filtration of length two.

On adding a variable to a Frobenius manifold and generalizations

Let \(\pi :V\rightarrow M\) be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure \((\circ _{M},e_{M},g_{M})\) and typical fiber has the structure of a Frobenius algebra \((\circ _{V},e_{V},g_{V})\). Using a connection \(D\) on the bundle \(\pi : V{\,\rightarrow \,}M\) and a morphism \(\alpha :V\rightarrow TM\), we construct an almost Frobenius structure \((\circ , e_{V},g)\) on the manifold \(V\) and we study when it is Frobenius. In particular, we describe all (real) positive definite Frobenius structures on \(V\) obtained in this way, when \(M\) is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting, we add a real structure \(k_{M}\) on \(M\) and a real structure \(k_{V}\) on the bundle \(\pi : V \rightarrow M\). Using \(k_{M}\), \(k_{V}\) and \(D\) we define a real structure \(k\) on the manifold \(V\). We study when \(k\), together with an almost Frobenius structure \((\circ , e_{V}, g) \), satisfies the tt*- equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and \(tt^{*}\)-geometry.

The Hochschild–Kostant–Rosenberg Isomorphism for Quantized Analytic Cycles

In this article, we provide a detailed account of a construction sketched by Kashiwara in an unpublished manuscript concerning generalized HKR isomorphisms for smooth analytic cycles whose conormal exact sequence splits. It enables us, among other applications, to solve a problem raised recently by Arinkin and C\u{a}ld\u{a}raru about uniqueness of such HKR isomorphisms in the case of the diagonal injection. Using this construction, we also associate with any smooth analytic cycle endowed with an infinitesimal retraction a cycle class which is an obstruction for the cycle to be the vanishing locus of a transverse section of a holomorphic vector bundle.

Holomorphic Banach vector bundles on the maximal ideal space of [formula omitted] and the operator corona problem of Sz.-Nagy

We establish triviality of some holomorphic Banach vector bundles on the maximal ideal space M(H∞) of the Banach algebra H∞ of bounded holomorphic functions on the unit disc D⊂C with pointwise multiplication and supremum norm. We apply the result to the study of the Sz.-Nagy operator corona problem.