Hermitian-Einstein Metrics Research Articles

Instantons on Calabi–Yau cones

The Hermitian Yang–Mills equations on certain vector bundles over Calabi–Yau cones can be reduced to a set of matrix equations; in fact, these are Nahm-type equations. The latter can be analysed further by generalising arguments of Donaldson and Kronheimer used in the study of the original Nahm equations. Starting from certain equivariant connections, we show that the full set of instanton equations reduce, with a unique gauge transformation, to the holomorphicity condition alone.

Sasakian quiver gauge theories and instantons on cones over lens 5-spaces

We consider SU(3)-equivariant dimensional reduction of Yang–Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki–Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki–Einstein structure, which are projective planes. We describe the Higgs branches of these quiver gauge theories as moduli spaces of spherically symmetric instantons which are SU(3)-equivariant solutions to the Hermitian Yang–Mills equations on the associated Calabi–Yau cones, and further compare them to moduli spaces of translationally-invariant instantons on the cones. We provide an explicit unified construction of these moduli spaces as Kähler quotients and show that they have the same cyclic orbifold singularities as the cones over the lens 5-spaces.

(1,1)-Forms Acting on Spinors on Kähler Surfaces

It is known that, for Dirac operators on Riemann surfaces twisted by line bundles with Hermitian-Einstein connections, it is possible to obtain estimates for the first eigenvalue in terms of the topology of the twisting bundle [8]. Attempts to generalize topological estimates for higher rank bundles or higher dimensional manifolds have been so far unsuccessful. In this work, we construct a class of examples, making explicit one problem that must be addressed in attempts to generalize such topological estimates.

Instantons on the exceptional holonomy manifolds of Bryant and Salamon

We give a construction of G2 and Spin(7) instantons on exceptional holonomy manifolds constructed by Bryant and Salamon, by using an ansatz of spherical symmetry coming from the manifolds being the total spaces of rank-4 vector bundles. In the G2 case, we show that, in the asymptotically conical model, the connections are asymptotic to Hermitian Yang–Mills connections on the nearly Kähler S3×S3.

Holomorphic curvatures of twistor spaces

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

Semistability of Principal Bundles on a Kähler Manifold with a Non-Connected Structure Group

We investigate principal $G$-bundles on a compact K\"ahler manifold, where $G$ is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal $G$-bundle $E_G$ admits an Einstein-Hermitian connection if and only if $E_G$ is polystable. We give an equivalent formulation of the (semi)stability condition. A question is to compare this definition with that of math.AG/0506511.

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.

Gluing branes — I

We consider several aspects of holomorphic brane configurations. We recently showed that an important part of the defining data of such a configuration is the gluing morphism, which specifies how the constituents of a configuration are glued together, but is usually assumed to be vanishing. Here we explain the rules for computing spectra and interactions for configurations with non-vanishing gluing VEVs. We further give a detailed discussion of the D-terms for Higgs bundles, spectral covers and ALE fibrations. We highlight a stability criterion that applies to degenerate configurations of the spectral data, and address an apparent discrepancy between the field theory and ALE descriptions. This allows us to show that one gets walls of marginal stability in F -theory even though they are absent in the 11d supergravity description. We also propose a numerical approach for approximating the hermitian-Einstein metric of the Higgs bundle using balanced metrics.

Polystable Parabolic Principal G-Bundles and Hermitian-Einstein Connections

AbstractWe show that there is a bijective correspondence between the polystable parabolic principal G-bundles and solutions of the Hermitian-Einstein equation.

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.

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).

Instantons on the six-sphere and twistors

We consider the six-sphere S6 = G2/SU(3) and its twistor space \documentclass[12pt]{minimal}\begin{document}${\cal Z}= G_2/$\end{document}Z=G2/U(2) associated with the SU(3)-structure on S6. It is shown that a Hermitian Yang-Mills connection (instanton) on a smooth vector bundle over S6 is equivalent to a flat partial connection on a vector bundle over the twistor space \documentclass[12pt]{minimal}\begin{document}${\cal Z}$\end{document}Z. The relation with Tian's tangent instantons on \documentclass[12pt]{minimal}\begin{document}${\mathbb {R}}^7$\end{document}R7 and their twistor description are briefly discussed.

STABILITY AND HERMITIAN–EINSTEIN METRICS FOR VECTOR BUNDLES ON FRAMED MANIFOLDS

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford–Takemoto and Hermitian–Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that KX⊗ [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization KX⊗ [D] coincides with the degree with respect to the complete Kähler–Einstein metric gX\Don X\D. For stable holomorphic vector bundles, we prove the existence of a Hermitian–Einstein metric with respect to gX\Dand also the uniqueness in an adapted sense.

Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections

We provide nontrivial examples of solutions to the system of coupled equations introduced by M. Garcia-Fernandez for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kahler metric and Hermitian-Einstein metric.

A construction of Spin(7)-instantons

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphic involution fixing only the singular points. On the other hand, there are higher-dimensional analogues of anti-self-dual instantons in four dimensions on Spin(7)-manifolds, which are called Spin(7)-instantons. They are minimizers of the Yang-Mills action, and the Spin(7)-instanton equation together with a gauge fixing condition forms an elliptic system. In this article, we construct Spin(7)-instantons on the examples of compact Spin(7)-manifolds above, starting with Hermitian-Einstein connections on the Calabi-Yau four-orbifolds and ALE spaces. Under some assumptions on the Hermitian-Einstein connections, we glue them together to obtain Spin(7)-instantons on the compact Spin(7)-manifolds. We also give a simple example of our construction.

Hermitian–Einstein connections on polystable orthogonal and symplectic parabolic Higgs bundles

Let X be a smooth complex projective curve and S⊂X a finite subset. We show that an orthogonal or symplectic parabolic Higgs bundle on X with parabolic structure over S admits a Hermitian–Einstein connection if and only if it is polystable.

Numerical Hermitian Yang-Mills connections and Kähler cone substructure

We further develop the numerical algorithm for computing the gauge connection of slope-stable holomorphic vector bundles on Calabi-Yau manifolds. In particular, recent work on the generalized Donaldson algorithm is extended to bundles with Kahler cone substructure on manifolds with h^{1,1}>1. Since the computation depends only on a one-dimensional ray in the Kahler moduli space, it can probe slope-stability regardless of the size of h^{1,1}. Suitably normalized error measures are introduced to quantitatively compare results for different directions in Kahler moduli space. A significantly improved numerical integration procedure based on adaptive refinements is described and implemented. Finally, an efficient numerical check is proposed for determining whether or not a vector bundle is slope-stable without computing its full connection.

ON THE ALGEBRAIC HOLONOMY OF STABLE PRINCIPAL BUNDLES

Let EG be a stable principal G–bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over ℂ. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group to H. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction (which means that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group), then EH is unique in the following sense: For any other minimal reduction of structure group (H′, EH′) of EG to some reductive subgroup H′, there is some element g ∈ G such that H′ = g-1Hg and EH′ = EHg. As an application, we show the following: Let M be a simply connected, irreducible smooth complex projective variety of dimension n such that the Picard number of M is one. If the canonical line bundle KM is ample, then the algebraic holonomy of the holomorphic tangent bundle T1, 0M is GL (n, ℂ). If [Formula: see text] is ample, the rank of the Picard group of M is one, the biholomorphic automorphism group of M is finite, and M admits a Kähler–Einstein metric, then the algebraic holonomy of T1, 0M is GL (n, ℂ). These answer some questions posed in V. Balaji and J. Kollár, Publ. Res. Inst. Math. Sci.44 (2008) 183–211.

Determinant line bundle on moduli space of parabolic bundles

In Biswas and Raghavendra (Proc Indian Acad Sci (Math Sci) 103:41–71, 1993; Asian J Math 2:303–324, 1998), a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line bundle on a moduli space of usual vector bundles over a covering curve. The Hermitian structure on the parabolic determinant bundle was taken to be the pullback of the Quillen metric on the determinant line bundle on the moduli space of usual vector bundles. Here a direct construction of the Hermitian structure is given. For that we need to establish a version of the correspondence between the stable parabolic bundles and the Hermitian–Einstein connections in the context of conical metrics. Also, a recently obtained parabolic analog of Faltings’ criterion of semistability plays a crucial role.

Hermitian-Einstein connections on polystable parabolic principal Higgs bundles

Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincare-type metric on X - D, on polystable parabolic principal Higgs bundles with parabolic structure over D, satisfying certain conditions on its restriction to D.