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Infinite determinants, stable bundles and curvature

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... Using this reformulation of Atiyah and Bott, Donaldson reproved the theorem of Narasimhan-Seshadri via the differential geometric method (see [Don83]), by establishing the equivalence between the existence of the Hermitian-Einstein metric and poly-stability. This equivalence, also 1 known as the Kobayashi-Hitchin correspondence, was extended to higher dimensions by the celebrated works of Donaldson [Don85], Uhlenbeck-Yau [UY86] and Donaldson [Don87]. ...
... A remarkable observation made by Donaldson relates the slope stability of S and the properness of M (K, ·). Since our Theorem 4.1 is ultimately inspired by Donaldson's work, we summarize this result in the following proposition [Don87]. Then there exist constants C 1 , C 2 which depend on (X, H, S, B) such that for ∀s ∈ T B , we have ...
... Assume the same conditions as in Theorem 4.1. For any k ∈ N, there exists a subsequence of H i which converges in C k to a Hermitian metric.A key ingredient of the L ∞ -estimates relies on the understanding of Donaldson's functional[Don85,Don87] for a sequence of Hermitian metrics. Let us first recall the definition of Donaldson's functional. ...
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On any complex smooth projective curve with positive genus, we construct Hilbert bundles that admit Hermitian--Einstein metrics. Our main constructive step is by investigating the arithmetic property of the upper half plane in Bridgeland's definition of stability conditions and its homological countparts. The main analytic ingredient in our proof is a notion called a geometrically well-approximable pair (X,θ)(X,\theta). This notion compares a constant L(X) that can be bounded by the geometric information of the Riemann surface X with a constant L0(θ)L_0(\theta) that depends only on the arithmetic information of the irrational number θ\theta. This notion helps us to apply the Diophantine approximation to Donaldson's functional. We further study the continuous structures, smooth structures, and holomorphic structures on such Hilbert bundles. We hope that this construction can shed some new light on the geometric background of quantum field theory.
... In other words, the trace free part of ΛF (h) is 0. According to Kobayashi [21] and Lübke [27], if (Y, ω Y ) is a smooth projective manifold or more generally a compact Kähler manifold, a holomorphic vector bundle with a Hermitian-Einstein metric is poly-stable with respect to the Kähler form ω Y in the sense of Takemoto [46,47]. Conversely, a stable bundle on a compact Kähler manifold has a Hermitian-Einstein metric, according to the celebrated theorem of Donaldson [8,10,11] and Uhlenbeck-Yau [48]. The correspondence is called Kobayashi-Hitchin correspondence, Hitchin-Kobayashi correspondence, or Donaldson-Uhlenbeck-Yau correspondence. ...
... Proof We just apply the argument of Donaldson [10,11] and Simpson [43] under the Dirichlet condition. We give only an outline. ...
... Therefore, we obtain (13) by using (11). Thus, we obtain (12). ...
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We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying K\"ahler manifolds. We also study the curvature decay of the Hermitian-Einstein metrics. It is useful for the study of the classification of instantons and monopoles on the quotient of 4-dimensional Euclidean space by some types of closed subgroups. We also explain examples of doubly periodic monopoles corresponding to some algebraic data.
... The criterion for [ω 0 ]-(semi)stability obtained through the inequalities given in Eq. (1.30) resembles the notion of stability for a given stability function in the setting of Bridgeland stability [Bri07], [MS17,§4]. From Kobayashi-Hitchin correspondence [Don85,Don87], [UY86], we have the following differential-geometric counterpart of item (5) of the above theorem. ...
... From Kobayashi-Hitchin correspondence [Don85,Don87], [UY86], and item (5) of the last theorem, we obtain the following. ...
... , s r ∈ Z, and all r ∈ Z >0 . From Kobayashi-Hitchin correspondence [Don85,Don87], [UY86], we conclude that E(s 1 , . . . , s r ) is Hermite-Einstein, for all s 1 , . . . ...
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In this paper, we show that the deformed Hermitian Yang-Mills (dHYM) equation on a rational homogeneous variety, equipped with any invariant Kähler metric, always admits a solution. In particular, we describe the Lagrangian phase, with respect to any invariant Kähler metric, of every closed invariant (1, 1)-form in terms of Lie theory. Building on this, we characterize all supercritical and hypercritical homogeneous solutions of the dHYM equation using the Cartan matrix associated with the underlying complex simple Lie algebra. Further, we provide an explicit formula, in terms of Lie theory, for the slope of holomorphic vector bundles over rational homogeneous varieties. Using this formula, we derive a new criterion for slope semistability through restrictions of holomorphic vector bundles to the generators of the associated cone of curves. Moreover , we provide a new characterization, in terms of central charges defined by rational curves, for slope (semi)stability of holomorphic vector bundles over rational homogeneous varieties. As an application of our main results, we describe all supercritical and hyper-critical homogeneous solutions of the dHYM equation on the Fano threefold defined by the Wallach flag manifold P(T P 2). In addition, we introduce a constructive method to obtain non-trivial examples of Hermitian-Einstein metrics on certain holomorphic vector bundles over P(T P 2) from solutions of linear diophantine equations. In this last case, we describe explicitly all associated Hermitian Yang-Mills instantons. We also present some new insights that explore the interplay between intersection theory and number theory.
... This follows for s, t, u ∈ Z + 0 from the Sobolev embedding theorem and Hölder's inequality. The general case is obtained from this by interpolation (15) and duality (16). ...
... Thus ||ξ(t)|| : Σ → [0, ∞) are positive functions satisfying ∆ξ(t) ≤ ||R(t)|| at points where ξ(t) = 0. An argument of Donaldson [16] (see [40] Prop 2.1) using the mean-value property of harmonic functions shows that this implies an estimate ...
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The symplectic vortex equations admit a variational description as global minimum of the Yang-Mills-Higgs functional. We study its negative gradient flow on holomorphic pairs (A,u) where A is a connection on a principal G-bundle P over a closed Riemann surface Σ\Sigma and u:PXu: P \rightarrow X is an equivariant map into a K\"ahler Hamiltonian G-manifold. The connection A induces a holomorphic structure on the K\"ahler fibration P×GXP\times_G X and we require that u descends to a holomorphic section of this fibration. We prove a Lojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the W1,2×W2,2W^{1,2}\times W^{2,2}-topology when X is equivariantly convex at infinity with proper moment map, X is holomorphically aspherical and its K\"ahler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang-Mills-Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet's Kobayashi-Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment-weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.
... Hermitian Yang Mills metrics are in bijection to stable vector bundles via the Kobayashi-Hitchin correspondence thanks to the works of Donaldson for algebraic surfaces [9] and manifolds [10], Uhlenbeck Yau for Kähler manifolds [22], Buchdahl [5] for surfaces, and Li Yau [15] for Hermitian manifolds. In particular, uniqueness theorems of that theory yield that there is at most one projectively flat metric on a given compact complex manifold; we should emphasize here that, since a globally conformal metric of a projectively flat metric is again projectively flat, we understand uniqueness modulo global conformal transformations of the metric. ...
... where G is a group as described in (10). In particular, we only consider those G such that the quotient is compact. ...
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We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-K\"ahler metrics on Hopf manifolds, explicitly characterized by Vaisman. Finally, we review the properties of zero projectively flat metrics. As applications, we refine a list of possible projectively flat metrics by Li, Yau, and Zheng; moreover we prove that projectively flat astheno-K\"ahler metrics are in fact K\"ahler and globally conformally flat.
... Motivation. The Donaldson-Uhlenbeck-Yau theorem asserts that for compact Kähler manifolds, there is a correspondence between irreducible Hermitian Yang-Mills connections on the vector bundle E and stable holomorphic structures on E cf. [17,49]. Thus, this allows one to translate the differential geometric problem of constructing Hermitian Yang-Mills connections to an algebraic geometry problem. ...
... Our goal in this paper however is simply to construct explicitly such solutions so we shall not address the problem if our solutions are exhaustive. We should point out that one might not always be able to find Hermitian Yang-Mills connections [17,49]. Since C 0 = 0 only for the trivial line bundle, we already know that ψ = 0 is the only solution to (4.13) which gives a globally well-defined SU(3)-instanton. ...
... gluings, deformations, smoothings, adiabatic constructions), where the common feature is to start with a solution of a given geometric PDE on a given geometric object that we perturb, trying then to solve the same PDE on the perturbed object. A specific feature for HYM connections and cscK metrics is that the solutions of the geometric PDE correspond to zeros of a moment map [4,19,20], and are related to a Geometric Invariant Theory (GIT) stability notion. ...
... Remark 4 We should point out that the method we use in the semistable situation is similar to the one developed in [12]. While in Leung's work, the perturbation is on the equations, in the present work the perturbation is on the geometry of the manifold, leading to extra complications in the arguments, see Remark 12. Finally, we focus on a very special case where our results apply, namely when the fibres are reduced to a point, that is when X = B. ...
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We investigate hermitian Yang–Mills connections on pullback bundles with respect to adiabatic classes on the total space of holomorphic submersions with connected fibres. Under some technical assumptions on the graded object of a Jordan–Hölder filtration, we obtain a necessary and sufficient criterion for when the pullback of a strictly semistable vector bundle will carry an hermitian Yang–Mills connection, in terms of intersection numbers on the base of the submersion. Together with the classical Donaldson–Uhlenbeck–Yau correspondence, we deduce that the pullback of a stable (resp. unstable) bundle remains stable (resp. unstable) for adiabatic classes, and settle the semi-stable case.
... Note that Theorem 1.1 is a Riemannian counterpart of the Hitchin-Kobayashi correspondence in Kähler geometry [Do1,Do2,UY]. Concerning Theorem 1.1, there are a few previous works in the presence of extra assumptions which we should mention. As a postscript to Hitchin's paper [Hi] on self-duality equations, borrowing ideas from harmonic maps, Donaldson [Do3] proved that any vector bundle over a compact Riemannian surface equipped with a semi-simple flat connection must admit harmonic metrics. ...
... Our Theorem 1.2 extends the main result in [BK1](see also [BM] for the case of quasi-regular Sasakian manifolds) to the nonvanishing Chern classes case. Note these are Sasakian analogues of the Corlette-Donaldson-Hitchin-Simpson correspondence built upon [Co1,Do3,Hi,Si1,Si3], dating back to Narasimhan-Seshadri [NS], Donadlson [Do1,Do2] and Uhlenbeck-Yau [UY]. Readers are also refereed to [BK2,Ka,KM] for recent works concerning related topics. ...
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Given a flat vector bundle over a compact Riemannian manifold, Corlette and Donaldson proved that it admits harmonic metrics if and only if it is semi-simple. In this paper, we extend this equivalence to arbitrary vector bundles without any additional hypothesis, the result can be viewed as a Riemannian Hitchin-Kobayashi correspondence. Furthermore, we also prove an equivalence of categories in Sasakian geometry, relating the category of projective flat complex vector bundles to the category of Higgs bundles.
... It depends on a choice of a polarisation, the variations of which being responsible for wall-crossing phenomena (see [10] for a survey on constructions and variations of moduli spaces of stable bundles). By the Hitchin-Kobayashi correspondence ( [13,16,22,6]), a holomorphic vector bundle over a compact Kähler manifold is slope polystable if and only if it carries an Hermite-Einstein metric, or equivalently an hermitian Yang-Mills connection. While wall-crossing phenomena describe global variations of the moduli spaces from the algebraic point of view, our focus is on the local variations, from the analytic point of view, and inspired by [3]. ...
... We will make the standard identification of a holomorphic vector bundle E with its sheaf of sections, and thus talk about slope stability notions for vector bundles as well. In that case slope stability relates nicely to differential geometry via the Hitchin-Kobayashi correspondence : 16,22,6]). There exists a Hermite-Einstein metric on E with respect to ω if and only if E is polystable with respect to L We will be mostly interested in semi-stable vector bundles. ...
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We consider a sufficiently smooth semi-stable holomorphic vector bundle over a compact K\"ahler manifold. Assuming the automorphism group of its graded object to be abelian, we provide a semialgebraic decomposition of a neighbourhood of the polarisation in the K\"ahler cone into chambers characterising (in)stability. For a path in a stable chamber converging to the initial polarisation, we show that the associated HYM connections converge to an HYM connection on the graded object.
... In each of the two most historically prominent examples, there is a natural analytic counterpart to stability in the shape of a PDE on the space of metrics on a given object. For holomorphic vector bundles, the existence of Hermitian Yang-Mills connections is equivalent to slope stability of the vector bundle-this is the Hitchin-Kobayashi correspondence established by Donaldson and Uhlenbeck-Yau Don87,UY86]. For smooth polarised varieties, the existence of constant scalar curvature Kähler metrics is conjectured by Yau-Tian-Donaldson to be equivalent to K-stability of the polarised variety [Yau93,Tia97,Don02]; this conjecture has been proven in the special case of Kähler-Einstein metrics on Fano manifolds [CDS15,Ber16,Tia97] and much progress has been made in general [BBJ21,Li22]. ...
... Remark 5.3. This argument also shows that Ω agrees with the Kähler metric used by Atiyah-Bott and Donaldson on A(E, h); this has been observed by Donaldson in the case that ω ∈ c 1 (L) for an ample line bundle L using different ideas [Don87,Proposition 14]. ...
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We introduce a natural, geometric approach to constructing moment maps in finite and infinite-dimensional complex geometry. This is applied to two settings: K\"ahler manifolds and holomorphic vector bundles. Our new approach exploits the existence of universal families and the theory of equivariant differential forms. In the setting of K\"ahler manifolds we first give a new, geometric proof of Donaldson-Fujiki's moment map interpretation of the scalar curvature. Associated to arbitrary products of Chern classes of the manifold-namely to a central charge-we then introduce a geometric PDE determining a Z-critical K\"ahler metric, and show that these general equations also satisfy moment map properties. On holomorphic vector bundles, using a similar strategy we give a geometric proof of Atiyah-Bott's moment map interpretation of the Hermitian Yang-Mills condition. We go on to give a new, geometric proof that the PDE determining a Z-critical connection-associated to a choice of central charge on the category of coherent sheaves-can be viewed as a moment map; deformed Hermitian Yang-Mills connections are a special case, in which our work gives a geometric proof of a result of Collins-Yau. Our main assertion is that this is the canonical way of producing moment maps in complex geometry-associated to any geometric problem along with a choice of stability condition-and hence that this accomplishes one of the main steps towards producing PDE counterparts to stability conditions in large generality.
... These are equivalent to D m ǫ = / Dǫ = 0 and the bundle V being holomorphic. A supersymmetric solution is a supersymmetric geometry which also satisfies the Bianchi identity ( [42][43][44]. It was conjectured that these are actually sufficient for existence [45], though it now seems that further notions of stability are needed [46,47]. ...
... 12 The flow for h is known as Donaldson heat flow [43]. This is gauge equivalent to Yang-Mills flow [48], where the connection evolves via ∂ t A = −d † A F , with the right-hand side given by the negative of the gradient of the Yang-Mills functional. ...
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We study the relation between supersymmetry and geometric flows driven by the Bianchi identity for the three-form flux H in heterotic supergravity. We describe how the flow equations can be derived from a functional that appears in a rewriting of the bosonic action in terms of squares of supersymmetry operators. On a complex threefold, the resulting equations match what is known in the mathematics literature as "anomaly flow". We generalise this to seven- and eight-manifolds with G2_2 or Spin(7) structures and discuss examples where the manifold is a torus fibration over a K3 surface. In the latter cases, the flow simplifies to a single scalar equation, with the existence of the supergravity solution implied by the long-time existence and convergence of the flow. We also comment on the α\alpha' expansion and highlight the importance of using the proper connection in the Bianchi identity to ensure that the flow's fixed points satisfy the supergravity equations of motion.
... The celebrated Donaldson-Uhlenbeck-Yau theorem [13,44] confirms the existence of Hermitian-Yang-Mills metrics on stable holomorphic vector bundles over compact Kähler manifolds. Two important corollaries include the polystability of reflexive sheaves under symmetric and exterior powers and tensor products, and the Bogomolov-Gieseker inequality for stable bundles over Kähler manifolds together with a characterization of the equality using projectively flat metrics. ...
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In this paper, we first prove a complete version of the Donaldson–Uhlenbeck–Yau theorem over normal varieties, including normal Kähler varieties and projective normal varieties with multiple polarizations. In particular, this gives the polystability of reflexive sheaves under symmetric and exterior powers and tensor products. As a consequence of the singular Donaldson–Uhlenbeck–Yau theorem, the complete Hitchin–Kobayashi correspondence over normal varieties smooth in codimension two is built by showing that an admissible Hermitian–Yang–Mills connection defines a polystable reflexive sheaf. Furthermore, it is shown that the Hermitian–Yang–Mills connection gives a lower bound for the discriminants of any Kähler resolutions, which gives a Bogomolov–Gieseker inequality over normal varieties and a characterization of the equality using projectively flat connections. We discuss typical cases including normal surfaces and varieties smooth in codimension two where we could simplify the Bogomolov–Gieseker inequality and endow it with topological meanings. We also prove the Bogomolov–Gieseker inequality for semistable reflexive sheaves and characterize the class of semistable sheaves that satisfy the Bogomolov–Gieseker equality. Finally, we give a new criteria for when a normal Kähler variety with trivial first Chern class is a finite quotient of torus.
... The results build on fundamental works in complex and algebraic geometry, employing tools such as non-pluripolar products for closed positive (1,1)-currents. The KH correspondence, originally established by Donaldson [17] and Uhlenbeck-Yau [36] for compact Kähler manifolds, asserts that a holomorphic vector bundle over a compact Kähler manifold is slope polystable if and only if it admits a Hermitian-Einstein metric. Bando and Siu extended the KH correspondence to reflexive sheaves by defining the Hermitian-Einstein metric on a Zariski open set that satisfies the admissible condition [1]. ...
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In this paper, we introduce the notion of α\alpha-slope stability of sheaves where α\alpha is a big cohomology class. We also define the notion of a T-Hermitian-Einstein metric for sheaves in terms of a closed positive (1,1)-current T. The first main theorem establishes the invariance of these notions under bimeromorphic transformations that are compositions of proper bimeromorphic maps and bimeromorphic morphisms isomorphic in codimension 1. The second main theorem proves the Kobayashi-Hitchin correspondence with respect to the canonical divisor on a projective variety of general type. As a consequence of this correspondence, we show that the Bogomolov-Gieseker inequality holds and the T-Hermitian-Einstein metric provides a characterization of the equality in the inequality on varieties of general type. Additionally, we extend the Bogomolov-Gieseker inequality to non-projective normal varieties with a transcendental big and nef cohomology class.
... We denote by M * EH the moduli space of irreducible EH connections. By the Donaldson-Uhlenbeck-Yau theorem the moduli space M * EH is diffeomorphic to the moduli space of ω-stable holomorphic vector bundles defined by the GIT quotient [31][32] ...
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We describe a class of supersymmetric gauged linear sigma-model, whose target space is the infinite dimensional space of bundles on a Calabi-Yau 3- or 2-fold. This target space can be considered the configuration space of D-branes wrapped around the Calabi-Yau. We propose that this model can be used to define matrix string theory compactifications. In the infrared limit the model flows to a superconformal non-linear sigma-model whose target space is the moduli space of BPS configurations of branes on the compact space, containing the moduli space of semi-stable bundles. We argue that the bulk degrees of freedom decouple in the infrared limit if semi-stability implies stability. We study topological versions of the model on Calabi-Yau 3-folds. The resulting B-model is argued to be equivalent to the holomorphic Chern-Simons theory proposed by Witten. The A-model and half-twisted model define the quantum cohomology ring and the elliptic genus, respectively, of the moduli space of stable bundles on a Calabi-Yau 3-fold.
... Let (X, g, I) be a compact Kähler manifold, let E be a holomorphic vector bundle over X. This functional was introduced in [6, Section 1.2] and [7,§II]. We refrain from a lengthy discussion and only marshal the following three facts, which are used in Section 5. ...
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We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical K\"ahler manifolds: If E\mathscr{E} is a reflexive sheaf over an ACyl K\"ahler manifold, which is asymptotic to a μ\mu-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Yang-Mills metrics (with curvature in Lloc2L^2_{\mathrm{loc}} across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau [UY86] with the geometric regularization scheme introduced by Bando and Siu [BS94].
... It is easy to see that v.A is still integrable. The curvature of v.A is (see page 5 of [6] for instance) ...
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We introduce a vector bundle version of the complex Monge-Ampere equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA positivity) which is necessary for the infinite-dimensional symplectic form to be Kahler. On rank-2 bundles on compact complex surfaces, we prove two consequences of the existence of a "positively curved" solution to this equation - Stability (involving the second Chern character) and a Kobayashi-Lubke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation.
... Theorem 2.3 (Donaldson-Uhlenbeck-Yau[7,8,34], Bando-Siu[3]). A reflexive sheaf F over a compact Kähler manifold (X, ω) admits an admissible Hermitian-Einstein metric if and only if it is polystable.For later purpose we need the followingProposition 2.4 ([3], Proposition 3). ...
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This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable.
... It is an algebro-geometric moduli. Donaldson-Uhlenbeck-Yau Theorem ( [7], [21], [8]) implies that for any holomorphic structure∂ α , the following two conditions are equivalent (also see the presentation in [15,Theorem 8.3]). ...
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Let X be a closed 66-dimensional manifold with a half-closed SU(3)SU(3)-structure. On the product manifold X×S1X\times S^{1}, with respect to the product G2G_{2}-structure and on a pullback vector bundle from X, we show that any G2G_{2}-instanton is equivalent to a Hermitian Yang-Mills connection on X via a "broken gauge". This result reveals the topological type of the moduli of G2G_{2}-instantons on X×S1X\times S^{1}. In dimension 8, similar result holds for moduli of Spin(7)Spin(7)-instantons. A generalization and an example are given.
... A Hermitian metric H on a holomorphic vector bundle is called PHYM if its Chern connection A H is PHYM. The celebrated Donaldson-Uhlenbeck-Yau Theorem [3,4,11] asserts that a holomorphic vector bundle E on a compact Kähler manifold admits a PHYM metric if and only if it is µ-polystable; moreover, any two PHYM metrics are related by an automorphism of E and by multiplication with a conformal factor. If H is a PHYM metric, then the connection A • on PU(E, H) induced by A H is Hermitian Yang-Mills (HYM), that is, it satisfies F 0,2 A • = 0 and iΛF A • = 0; it depends only on the conformal class of H. Conversely, any HYM connection A • on PU(E, H) can be lifted to a PHYM connection A; any two choices of lifts lead to isomorphic holomorphic vector bundles E and conformal metrics H. ...
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We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on μ\mu-polystable holomorphic bundles over Pn1\mathbf{P}^{n-1}.
... We start the proof by noticing that we could take a partition of unity {θ p } of the base manifold B subordinate to an open cover {U p } and construct the Hermitian-Einstein metric on each trivialization of the holomorphic vector bundle E over each cover. It suffices to consider the cone chart U , which intersects with the divisor D, since far from the divisor all the arguments are the same to [3,[14][15][16]48]. The proof is divided in several steps. ...
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Over a compact K\"ahler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a K\"ahler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature K\"ahler metrics with conic singularities: existence result under small deformations of K\"ahler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivisation of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian-Einstein metric on this bundle enjoys a regularity property along the divisor on the base.
... with a real constant λ h , where √ −1Λ g is the contraction with ω, F h is the curvature form of the Chern connection of (E, h) and id E is the identity endomorphism of E. The relationship of the notions of g-stability and g-Hermitian-Einstein metrics is classical: An irreducible holomorphic vector bundle on X admits a g-Hermitian-Einstein metric if and only if it is gstable, cf. Narasimhan/Seshadri [25], S. Kobayashi [17], Lübke [20], Donaldson [9], [10], [11] and Uhlenbeck/Yau [36], [37]. ...
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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 K_X \otimes [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization K_X \otimes [D] coincides with the degree with respect to the complete K\"ahler-Einstein metric g_{X \setminus D} on X \setminus D. For stable holomorphic vector bundles, we prove the existence of a Hermitian-Einstein metric with respect to g_{X \setminus D} and also the uniqueness in an adapted sense.
... Motivated by the above important results and in analogy with Hitchin-Kobayashi correspondence for holomorphic vector bundles [Don87,UY86,LY87,LT95], it was conjectured, e.g. [Tia97,Don02b]), that the relations of the existence problem with some (to be understood) algebrogeometric notion of stability is indeed the crucial aspect to completely characterize, in purely algebro-geometric terms, the existence of KE metrics on Fano manifolds (and, even more generally, for constant scalar curvature Kähler metrics). ...
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We present classical and recent results on K\"ahler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course "K\"ahler-Einstein metrics" given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.
... The fundamental result, that is generically known as the Hitchin-Kobayashi correspondence, states the equivalence between the Mumford-Takemoto stability of a vector bundle-an algebraic notion-and the existence of Hermite-Yang-Mills metrics on it, which can be realised as critical points of the Donaldson and Yang-Mills functionals (see [15,17] for details). The Hitchin-Kobayashi correspondence was first established in the pionnering works of Donaldson [10,11,12], Kobayashi [15] and Uhlenbeck-Yau [21] for holomorphic vector bundles over compact Kähler manifolds, and has also been extended to more general contexts, including holomorphic vector bundles over arbitrary compact complex manifolds [17], and reflexive sheaves over compact Kähler manifolds [3]. In fact, Kobayashi [15] also noted that for holomorphic bundles on a compact Kähler manifold, the 2010 Mathematics Subject Classification. ...
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We define a functional J(h){\cal J}(h) for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact K\"ahler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles \cite{Kobayashi}, and study some of its basic properties. We show that J(h){\cal J}(h) is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the K\"ahler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating J(h){\cal J}(h) and another functional I(h){\cal I}(h), closely related to the Yang-Mills-Higgs functional \cite{Bradlow-Wilkin, Wentworth}, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of J(h){\cal J}(h), which is expressed as a certain L2L^{2}-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of J(h){\cal J}(h) if and only if the corresponding Hitchin--Simpson mean curvature is parallel with respect to the Hitchin--Simpson connection.
... The famous Donaldson-Uhlenbeck-Yau theorem reveals the deep relationship between the stability of a holomorphic vector bundle and the existence of Hermitian-Einstein metrics (cf. [21], [6], [7], [8], [25]). In Section 2, we introduce the notions of the nonlinear semistability (stability) and the nonlinear polystablity associated to a triple (X , M, L) and discuss the relationships between the existence of geodesic-Einstein metrics on L and these stabilities. ...
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In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.
... But if we want to be faithful to the finite-dimensional Kempf-Ness theorem, we would want the infinite-dimensional symplectic form to arise as the curvature of an infinitedimensional hermitian holomorphic line bundle. Indeed, the correct infinite dimensional line bundle often turns out to be a Quillen determinant bundle (see [9] for instance). ...
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We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the generalised Monge-Amp`ere equation is conditioned on a conjecture from algebraic geometry. In addition, we prove that for small values of the perturbation parameters, some of these equations have solutions.
... The main result we wish to describe is the Hitchin-Kobayashi correspondence. It was proved first for Riemann surfaces by Narasimhan and Seshadri [35], then for projective surfaces by Donaldson [11], and then for compact Kähler manifolds in arbitrary dimension by Uhlenbeck and Yau [44] (and independently, for smooth projective varieties in arbitrary dimension by Donaldson [12]). ...
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In this survey article we describe moduli spaces of simple, stable, and semistable sheaves on K3 surfaces, following the work of Mukai, O'Grady, Huybrechts, Yoshioka, and others. We also describe some recent developments, including applications to the study of Chow rings of K3 surfaces, determination of the ample and nef cones of irreducible holomorphic symplectic manifolds, and moduli spaces of Bridgeland stable complexes of sheaves.
... The Donaldson-Uhlernbeck-Yau theorem states that holomorphic vector bundles admit Hermitian-Einstein metrics if they are stable. It was proved by Narasimhan and Seshadri in [26] for compact Riemann surface case, by Donaldson in [10,11] for algebraic manifolds and by Uhlenbeck and Yau in [28,29] for general compact Kähler manifolds. The inverse problem that a holomorphic bundle admitting such a metric must be poly-stable( i.e. a direct sum of stable bundles with the same slope) was solved by Kobayashi [17] and Lübke [23] independently. ...
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In this paper, we consider the existence of approximate Hermitian-Einstein structure and the semi-stability on Higgs bundles over compact Gauduchon manifolds. By using the continuity method, we show that they are equivalent.
... Our definition builds on Simpson's definition of the Donaldson Lagrangian in [Sim88,p.883] and was first used in [Don87a,Don85]. Let Ψ : R × R → [0, ∞) be given by ...
Preprint
We introduce the notion of a Nakajima bundle representation. Given a labelled quiver and a variety or manifold X, such a representation involves an assignment of a complex vector bundle on X to each node of the doubled quiver; to the edges, we assign sections of, and connections on, associated twisted bundles. We for the most part restrict attention in our development to algebraic curves or Riemann surfaces. Our construction simultaneously generalizes ordinary Nakajima quiver representations on the one hand and quiver bundles on the other hand. These representations admit gauge-theoretic characterizations, analogous to the ADHM equations in the original work of Nakajima, allowing for the construction of these generalized quiver varieties using a reduction procedure with moment maps. We study the deformation theory of Nakajima bundle representations, prove a Hitchin-Kobayashi correspondence between such representations and stable quiver bundles, examine the natural torus action on the resulting moduli varieties, and comment on scenarios where the variety is hyperk\"ahler. Finally, we produce concrete examples that recover known and new moduli spaces.
... In general, detecting harmonic metrics on vector bundles is a nonlinear system generalization of solving the Laplace equation and obstruction emerges in a natural way. A basic issue of harmonic metrics is to investigate its existence theory, there is a Riemannian Kobayashi-Hitchin correspondence for harmonic metrics analogous to the celebrated Do2,UY] in complex geometry, which was well established by Corlette [Co] and Donaldson [Do3] for flat vector bundles in the late 1980s, by in full generality for arbitrary vector bundles until very recently. ...
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In this paper, we prove two Liouville theorems for harmonic metrics on complex flat line bundles on gradient steady Ricci solitons and gradient shrinking K\"{a}hler-Ricci solitons, which imply that they arise from fundamental group representations into S1S^1.
... [29]). Since a slope-polystable holomorphic vector bundle is in particular slope-semistable, from Kobayashi-Hitchin correspondence [20,21,48], it follows that E does not admit a HYM connection. On the other hand, since F, G ∈ L m 1 (ω 0 ), it follows that ...
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We construct the first explicit non-trivial example of deformed Hermitian Yang–Mills (dHYM) connection on a higher rank slope-unstable holomorphic vector bundle over a Fano threefold. Additionally, we provide a sufficient algebraic condition in terms of central charges for the existence of dHYM connections on Whitney sum of holomorphic line bundles over rational homogeneous varieties. As a consequence, we obtain several new examples of dHYM connections on higher rank holomorphic vector bundles.
... A Hermitian metric satisfying the HE equation will be called a HE metric and a holomorphic vector bundle will be called indecomposable if it is not a direct sum of serval holomorphic vector bundles. Thanks to a series of works including those of Narasimhan-Seshadri [19], Kobayashi [8], Lübke [15], Metha-Ramanathan [16], Donaldson [3,4] and Uhlenbeck-Yau [24], the celebrated Kobayashi-Hitchin correspondence (also called Hitchin-Kobayashi correspondence or Donaldson-Uhlenbeck-Yau correspondence) states that for an indecomposable holomorphic vector bundle, the existence and uniqueness of HE metrics is equivalent to the Mumford-Takemoto stability. It is a landmark in complex geometry and has been generalized in many different directions, the large majority of which themselves became very important results in complex geometry. ...
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This paper first investigates solvability of Hermitian-Einstein equation on a Hermitian holomorphic vector bundle on the complement of an arbitrary closed subset in a compact complex manifold. The uniqueness of Hermitian-Einstein metrics on a Zariski open subset in a compact K\"{a}hler manifold was only figured out by Takuro Mochizuki recently, the second part of this paper gives an affirmative answer to a question proposed by Takuro Mochizuki and based on which it leads to an alternative approach to the unique issue. We also prove stability from solvability of Hermitian-Einstein equation, which together with the classical existence result of Carlos Simpson in particular establish a Kobayashi-Hitchin bijective correspondence. The argument is also effective for uniqueness result and Kobayashi-Hitchin bijective correspondence on C\mathbb{C}, as well as on non-K\"{a}hler and semi-stable contexts.
... This requires that the first Pontryagin classes of the tangent bundle of X and the vector bundle V over it are equal. Second, the connection on the holomorphic bundle V should satisfy the hermitian Yang-Mills equation, which holds if and only if V is (poly)stable [51][52][53]. It was conjectured that these are actually sufficient for existence [54], though it now seems that further notions of stability are needed [55,56]. ...
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We study the relation between supersymmetry and geometric flows driven by the Bianchi identity for the three-form flux H in heterotic supergravity. We describe how the flow equations can be derived from a functional that appears in a rewriting of the bosonic action in terms of squares of supersymmetry operators. On a complex threefold, the resulting equations match what is known in the mathematics literature as “anomaly flow”. We generalise this to seven- and eight-manifolds with G22_2 or Spin(7) structures and discuss examples where the manifold is a torus fibration over a K3 surface. In the latter cases, the flow simplifies to a single scalar equation, with the existence of the supergravity solution implied by the long-time existence and convergence of the flow. We also comment on the α′α\alpha ' expansion and highlight the importance of using the proper connection in the Bianchi identity to ensure that the flow’s fixed points satisfy the supergravity equations of motion.
... from Kobayashi-Hitchin correspondence[Don85,Don87],[UY86], we have that F is [ω 0 ]-polystable. From above, we can equip U(E) with a Hermitian structure (Ω, J), such that (a) J| ker(Θ) is given by the lift of the complex structure of X P ; (b) J(Θ 2j−1 ) = −Θ 2j , j = 1, . . . ...
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We construct new examples of t-Gauduchon Ricci-flat metrics, for all t < 1, on compact non-Kähler Calabi-Yau manifolds defined by certain principal torus bundles over rational homogeneous varieties with Picard number ϱ(X) > 1. As an application, we provide a detailed description of new examples of Strominger-Bismut Ricci-flat Hermitian metrics, Lichnerowicz Ricci-flat Hermitian metrics, and balanced Hermitian metrics on principal T 2-bundles over the Fano threefold P(T P 2).
... 此种等价性即被称为 Hitchin-Kobayashi 对应. 该对应最初是由 Narasimhan-Seshadri [25] 、 Donaldson [8,9] 和 Uhlenbeck-丘成桐 [36] 对 Kähler 流形上的全纯丛证明的. 当底流形为 Gauduchon 流形时, Buchdahl [7] 、 李骏 -丘 成桐 [22] 、Lübke-Teleman [17,18] 和 Jacob [15] 证明此时 Hitchin-Kobayashi 对应对 Higgs 丛也是成立 的. 在某些非紧 Kähler 流形上, 将前面所述稳定性换成解析稳定性, Simpson [31] 也得到了上述对 应. 相应的结果可用来研究抛物丛 (抛物 Higgs 丛) 的几何性质. ...
... [Kob87]). Since a slope-polystable holomorphic vector bundle is in particular slope-semistable, from Kobayashi-Hitchin correspondence [Don85,Don87], [UY86], it follows that E does not admit a HYM connection. On the other hand, since F, G ∈ L m 1 (ω 0 ), it follows that ...
Preprint
Full-text available
We construct the first explicit non-trivial example of deformed Hermitian Yang-Mills (dHYM) instanton on a higher rank slope-unstable holomorphic vector bundle over a Fano threefold. Additionally, we provide a sufficient algebraic condition in terms of central charges for the existence of dHYM instantons on Whitney sum of holomorphic line bundles over rational homogeneous varieties. As a consequence, we obtain several new examples of dHYM instantons on higher rank holomorphic vector bundles.
... Here we mention a few important and classical special cases. When φ = 0, the solutions are the Hermitian-Yang-Mills connections which correspond to slope polystable vector bundles by the celebrated Donadlson-Uhlenbeck-Yau theorem ( [4] [21]). When dim R X = 2 i.e. over the Riemann surfaces, this corresponds to the Hitchin equation and the space of solutions correspond to the space of polystable Higgs bundles ( [6]). ...
Preprint
In this paper, we study the analytic properties of solutions to the Vafa-Witten equation over a compact Kaehler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the C\mathbb{C}^* invariant locus of the moduli space is shown to behave similarly as the Hermitian-Yang-Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes' results on rank two solutions over Kaehler surfaces together with a new complex geometric interpretation. The moduli space of SU(2) monopoles and some related examples are also discussed in the final section.
... Since it is the exterior power of a slope-stable bundle, by [HL10, 3.2.10], F is polystable, see also [Don85,Don87,UY86]. On the other hand, from the computation above one deduce that F is simple, since ext 0 (F , F ) = 1, hence it is slope-stable. ...
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We exhibit examples of slope-stable and modular vector bundles on a hyperk\"ahler manifold of K3[2]^{[2]}-type which move in a 20-dimensional family and study their algebraic properties. These are obtained by performing standard linear algebra constructions on the examples studied by O'Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperk\"ahler.
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In this paper, we introduce a concept of RC-positivity for Hermitian holomorphic vector bundles and prove that, if E is an RC-positive vector bundle over a compact complex manifold X, then for any vector bundle A, there exists a positive integer cA=c(A,E)c_A=c(A,E) such that H0(X,SymEAk)=0H^0(X,\mathrm{Sym}^{\otimes \ell}E^*\otimes A^{\otimes k})=0 for cA(k+1)\ell\geq c_A(k+1) and k0k\geq 0. Moreover, we obtain that, on a compact K\"ahler manifold X, if ΛpTX\Lambda^p T_X is RC-positive for every 1pdimX1\leq p\leq \dim X, then X is projective and rationally connected. As applications, we show that if a compact K\"ahler manifold (X,ω)(X,\omega) has positive holomorphic sectional curvature, then ΛpTX\Lambda^p T_X is RC-positive and Hˉp,0(X)=0H_{\bar\partial}^{p,0}(X)=0 for every 1pdimX1\leq p\leq \dim X, and in particular, we establish that X is a projective and rationally connected manifold, which confirms a conjecture of Yau([57, Problem 47]).
Preprint
Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character \chi of P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by \chi is numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.
Preprint
We investigate higher-dimensional analogues of the bc systems of 2D RCFT. When coupled to gauge fields and Beltrami differentials defining integrable holomorphic structures the bc partition functions can be explicitly evaluated using anomalies and holomorphy. The resulting induced actions generalize the chiral algebras of 2D RCFT to 2n dimensions. Moreover, bc systems in four and six dimensions are closely related to supersymmetric matter. In particular, we show that d=4,N=2d=4, \mathcal{N}=2 hypermultiplets induce a theory of self-dual Yang-Mills fields coupled to self-dual gravity. In this way the bc systems fermionize both the algebraic sector of the WZW4WZW_4 theory, as defined by Losev et. al., and the classical open Nws=2\mathcal{N}_{ws}=2 string.
Article
In this paper, we obtain the Hermitian-Einstein metrics on stable reflexive sheaves over compact Gauduchon manifolds.
Article
In this paper, we study the analytic properties of solutions to the Vafa–Witten equation over a compact Kähler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the C ∗ \mathbb{C}^{*} invariant locus of the moduli space is shown to behave similarly to the Hermitian Yang–Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes’ results on rank two solutions over Kähler surfaces together with a new complex geometric interpretation. The moduli space of SU ⁢ ( 2 ) \mathsf{SU}(2) monopoles and some related examples are also discussed in the final section.
Article
We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of K3[2]^{[2]}-type, which move in a 20-dimensional family and study their algebraic properties. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic four-fold and the Debarre–Voisin hyperkähler manifold.
Article
In this paper, we study the non‐Hermitian Yang–Mills (NHYM) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in Kaledin and Verbitsky ( Selecta Math . (N.S.) 4 (1998) 279–320).
Article
In this paper, we consider the Yang–Mills–Higgs flow for twisted Higgs pairs over Kähler manifolds. We prove that this flow converges to a reflexive twisted Higgs sheaf outside a closed subset of codimension 4, and the limiting twisted Higgs sheaf is isomorphic to the double dual of the graded twisted Higgs sheaves associated to the Harder–Narasimhan–Seshadri filtration of the initial twisted Higgs bundle.
Chapter
We present some results that complement our prequels [27, 28] on holomorphic vector bundles. We apply the method of the Quot-scheme limit of Fubini–Study metrics developed therein to provide a generalisation to the singular case of the result originally obtained by X. W. Wang for the smooth case, which states that the existence of balanced metrics is equivalent to the Gieseker stability of the vector bundle. We also prove that the Bergman 1-parameter subgroups form subgeodesics in the space of Hermitian metrics. This paper also contains a review of techniques developed in [27, 28] and how they correspond to their counterparts developed in the study of the Yau–Tian–Donaldson conjecture.KeywordsFubini-Study metricsBalanced metrics
Article
We relate Berezin–Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including the spectral gap of the Berezin transform and the convergence rate of Donaldson’s iterations toward balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson’s iterations toward balanced metrics on Kähler manifolds with constant scalar curvature.
Article
In this paper, we establish the Kobayashi–Hitchin correspondence, that is, the equivalence of the existence of an Einstein–Hermitian metric and ψ\psi -polystability of a generalized holomorphic vector bundle over a compact generalized Kähler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles, and we obtain ψ\psi -stable Poisson modules over complex surfaces that are not stable in the ordinary sense.
Article
The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a `perfect' functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.
The analysis of elliptic families, I, II
  • J.-M Bismut
  • D S Freed
J.-M. BISMUT AND D. S. FREED, The analysis of elliptic families, I, II, to appear in Commun. Math. Phys.
Classes caractristiques en th3orie d'Arakelov C
  • H Gillwt
  • C Souli
H. GILLWT AND C. SOULI, Classes caractristiques en th3orie d'Arakelov C. R. Acad. Sc. Paris t. 301, S6rie I, No. 9, 1985.
New dimensions in geometry
  • Yu I M Nin
Yu. I. M,NIN, "New dimensions in geometry," in Proceedings of Arbeitstagung, Bonn 1984. Lecture Notes in Math. IIII, Springer-Verlag Berlin, 1985.