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Infinite determinants, stable bundles and curvature
... 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 , . . . ...
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.
... Two fundamental results in the study of stable bundles over compact Kähler and projective algebraic manifolds are the Donaldson-Uhlenbeck-Yau (DUY) theorem and the Mehta-Ramanathan (MR) restriction theorem. The DUY theorem proves the existence of a Hermitian-Einstein (HE) metric on any slope stable vector bundle E over a compact Kähler manifold (Y , ω) [8,9,33]. The associated Chern connections are called Hermitian-Yang-Mills (HYM) connections. ...
... stable) for generic V ∈ P(H 0 (Y , L k )) [10,23]. This result was instrumental in Donaldson's proof of the DUY theorem in higher dimensions [9]. We refer to [17] for a discussion of some other important consequences of the restriction theorem. ...
... As a consequence of Theorem 1.6, we have • E ∞ can be extended to a reflexive sheaf E ∞ over C(V ). Furthermore, for any q > 1; for any global section s, log + |s| 2 ∈ W 1,2 ∩ L ∞ ; • Gr(E ∞ ) * * = Gr(E 0 ) * * ; • Assume n ≥ 3. Then b = C(E 0 ) for any semistable degeneration E of {E t = F} through the deformation to projective cones and E ∞ = Gr(E 0 ) * * Remark 5. 9 • In a sense, under the given assumptions, our results produce a canonical object when the restrictions of semistable bundles to smooth hypersurfaces fail to be semistable. • Using the example E = T P n and V = P n−1 , A ∞ does not have a cone property. ...
In this paper, we first prove a Donaldson–Uhlenbeck–Yau theorem over projective normal varieties smooth in codimension two. As a consequence we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov–Gieseker inequality, along with a precise characterization of the case of equality. This also improves several previously known algebro-geometric results on normalized tautological classes. We study the limiting behavior of semistable bundles over a degenerating family of projective normal varieties. In the case of a family of stable vector bundles, we study the degeneration of the corresponding HYM connections and these can be characterized from the algebro-geometric perspective. In particular, this proves another version of the singular Donaldson–Uhlenbeck–Yau theorem for the normal projective varieties in the central fiber. As an application, we apply the results to the degeneration of stable bundles through the deformation to projective cones, and we explain how our results are related to the Mehta–Ramanathan restriction theorem.
... The celebrated Donaldson-Uhlenbeck-Yau theorem states that E admits a Hermite-Einstein metric if and only if E is slope stable [Don87;UY86]. We will consider the following version: ...
... Proposition 4.1 ( [Don87]) shows that actually S ⊂ H 1,pmax , where p max := 2n 2n−1 , and it seems natural to interpret H 1,p as an analogue of the space E 1 of metrics (or potentials) of finite energy in the study of cscK metrics (although it is not obvious to us that the constant p max is optimal, c.f. Theorem A). Our next result can be viewed as an analogue of [BBJ21, Theorem 2.16]. ...
... Theorem C follows from a formula for M(h t , h 0 ) in terms of the eigenvalues of log(h 0 h −1 t ), due to Donaldson [Don87]. Using this, we show that M(h t ) can only be bounded from above under very restrictive circumstances: essentially, the geodesic ray (h t ) must have come from a construction similar to Theorem A. The weakly holomorphic W 1,2 -projection theorem of Uhlenbeck-Yau [UY86; UY89] can then be used to produce the desired filtration; applying (1.1) shows that at least one of the subsheafs in the filtration has slope larger than µ(E). ...
We give new proofs of two implications in the Donaldson--Uhlenbeck--Yau theorem. Our proofs are based on geodesic rays of Hermitian metrics, inspired by recent work on the Yau--Tian--Donaldson conjecture.
... The first term on the right hand side vanishes by (14) and the second vanishes because T ij is symmetric and trace-free and the Lie derivative L v g is the symmetrisation v i;j + v j;i . ...
... So we have a quadratic differential τ with T = Reτ . The equation (14) goes over to the condition that τ be a holomorphic quadratic differential; this is the Hopf differential defined by a harmonic map from a Riemann surface. In a local complex coordinate z = x + iy ...
... The circle of ideas around this correspondence between the existence of solutions to the Hermitian-Yang-Mills equations and the algebro-geometric notion of stability has been extremely fruitful and influential in developments in complex differential geometry over the past four decades and we only mention a few aspects of this. In the case of a complex projective manifold, with integral Kähler class, the author gave alternative proofs in [13], [14] exploiting a variational point of view. Soon after, Simpson gave another proof which combined the variational point of view with the Uhlenbeck-Yau techniques [47]. ...
This is an expository article discussing some of the work of Uhlenbeck, focusing mainly on work concerning harmonic maps and Yang-Mills fields.
... Indeed, while this paper is aimed at a mathematical audience, we hope that physicallyinclined readers may also appreciate the presentation here of the rigorous construction of gauge-theoretic moduli spaces and the study of their properties with only mild discomfort. We particularly direct such readers' attention to (i) Propositions 4.23 and 5.37 and the map RG C of equation (5.28) as elegant manifestations of the renormalization group, (ii) the formal gauge transformations of Definition 2.39 that feature in the main results stated in §3, (iii) the Donaldson-type functional [Don85,Don87] which features prominently in the proof of the DUY-type theorem below, and (iv) the Ehresmann connection on the family of non-singular D 2 asymptotically locally flat (ALF) manifolds constructed in §6. ...
... The first results are easy to verify, and (6.5) is a straightforward consequence of them if one employs the final expression in (6.1). Multiplying (6.5) on the right by H and then taking the trace, and using the positive-definiteness of H and the Cauchy-Schwarz inequality, gives Following Donaldson [Don85,Don87], we will prove our DUY theorem by extremizing a functional of h. 58 We begin by describing two constructions involving Hermitian matrices. ...
... However, the Euler-Lagrange equation of this latter functional is not µ R (g(B)) = 0 in the present context, thanks to boundary terms due to integration by parts. At stationary points g, these boundary terms enforce (h, µ R (g(B))) L 2 = 0 for mosth ∈ ig 2 , and so stationary points do not satisfy µ R (g(B)) = 0. We therefore work with the Nahm data analog of the functional from [Don85,Don87] because, as we will see, such boundary terms are not present. Finally, a third candidate functional is the dimensional reduction of the Yang-Mills functional on I × T 3 , or equivalently µ R (g(B)) 2 L 2 (since µ C (g(B)) = µ C (B) and the norm-squared of the anti-self-dual part of the curvature is determined by that of the self-dual part because the second Chern class is fixed). ...
We propose an infinite-dimensional generalization of Kronheimer's construction of families of hyperkahler manifolds resolving flat orbifold quotients of $\mathbb{R}^4$. As in [Kro89], these manifolds are constructed as hyperkahler quotients of affine spaces. This leads to a study of \emph{singular equivariant instantons} in various dimensions. In this paper, we study singular equivariant Nahm data to produce the family of $D_2$ asymptotically locally flat (ALF) manifolds as a deformation of the flat orbifold $(\mathbb{R}^3 \times S^1)/Z_2$. We furthermore introduce a notion of stability for Nahm data and prove a Donaldson-Uhlenbeck-Yau type theorem to relate real and complex formulations. We use these results to construct a canonical Ehresmann connection on the family of non-singular $D_2$ ALF manifolds. In the complex formulation, we exhibit explicit relationships between these $D_2$ ALF manifolds and corresponding $A_1$ ALE manifolds. We conjecture analogous constructions and results for general orbifold quotients of $\mathbb{R}^{4-r} \times T^r$ with $2 \le r \le 4$. The case $r = 4$ produces K3 manifolds as hyperkahler quotients.
... • (cf. section 2.5, 2.6) The Hermitian Yang-Mills equation [27][26] [79] was a major inspiration to Thomas and Yau. The deformed Hermitian Yang-Mills equation [18] [22][23] [16] has been advocated as a mirror version of special Lagrangians. ...
... • Remark 2.11. An alternative approach by Donaldson [27] in the projective manifold case, is also based on the flow method, but uses a dimensional induction in which the stability condition appears indirectly through alternative algebrogeometric characterisations. The approach of Uhlenbeck and Yau [79] uses the continuity method instead, where the stability condition appears in a way similar to the above. ...
... The subgroup U (E) of unitary gauge transformations is analogous to the maximal compact subgroup. The space GL(E, C) U (E) can be formally identified with the space of Hermitian metrics on E. The HYM equation arises naturally from considerations of the moment map, and the Donaldson-Uhlenbeck-Yau theorem can be formally motivated from this picture [27]. ...
The main theme of this paper is the Thomas-Yau conjecture, primarily in the setting of exact, (quantitatively) almost calibrated, unobstructed Lagrangian branes inside Calabi-Yau Stein manifolds. In our interpretation, the conjecture is that Thomas-Yau semistability is equivalent to the existence of special Lagrangian representatives. We clarify how holomorphic curves enter this conjectural picture, through the construction of bordism currents between Lagrangians, and in the definition of the Solomon functional. Under some extra hypothesis, we shall prove Floer theoretic obstructions to the existence of special Lagrangians, using the technique of integration over moduli spaces. In the converse direction, we set up a variational framework with the goal of finding special Lagrangians under the Thomas-Yau semistability asumption, and we shall make sufficient progress to pinpoint the outstanding technical difficulties, both in Floer theory and in geometric measure theory.
... This proposition allows one to obtain a slightly different expression for the functional M K , which can be found for example in [15,16,10,4]. Indeed, let us write M (t) for the value M (t) = M K (Ke ts ). ...
... We now follow [4] to write the second term on the right-hand side with a local expression involving frames. Let us fix a smooth unitary (with respect to K) frame for E for which the matrix of s with respect to this frame is diagonal with eigenvalues λ 1 , . . . ...
... where the summand is interpreted as 1 2 |(∂s) β α | 2 if α = β. Following [16] and [4,Lemma 24] it is then possible to obtain the following estimate, which we won't really need, but which we collect for completeness. ...
For a holomorphic vector bundle over a compact K\"ahler orbifold, the slope stability of the bundle is shown to be equivalent to the existence of a Hermitian-Einstein metric or to the properness of a certain functional introduced by Donaldson.
... 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. ...
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. ...
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. ...
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]. ...
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. ...
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 G$_2$ 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 $\alpha'$ 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 case of algebraic surfaces was proven by Donaldson [Don85], and the full correspondence for compact Kähler manifolds using a continuity method by Uhlenbeck and Yau the following year [UY86]. Donaldson afterwards gave a new proof of the theorem for all projective manifolds using a different technique to Uhlenbeck and Yau, with an inductive argument along the lines of the case of algebraic surfaces [Don87]. ...
... on the space of integrable unitary connections A(h) on a Hermitian vector bundle (E, h) → (X, ω). Donaldson constructed a determinant line bundle L → A(h) for which the Atiyah-Bott symplectic form is the curvature of a Hermitian metric [Don87]. Donaldsons construction uses a local version of the index theorem and ideas due to Bismut-Freed, and indeed going back to Quillen, to identify the correct determinant bundle [Qui85,BF86a,BF86b]. ...
In this thesis we study the principle that extremal objects in differential geometry correspond to stable objects in algebraic geometry. In our introduction we survey the most famous instances of this principle with a view towards the results and background needed in the later chapters. In Part I we discuss the notion of a $Z$-critical metric recently introduced in joint work with Ruadha\'i Dervan and Lars Martin Sektnan. We prove a correspondence for existence with an analogue of Bridgeland stability in the large volume limit, and study important properties of the subsolution condition away from this limit, including identifying the analogues of the Donaldson and Yang-Mills functionals for the equation. In Part II we study the recent theory of optimal symplectic connections on K\"ahler fibrations in the isotrivial case. We prove a correspondence with the existence of Hermite-Einstein metrics on holomorphic principal bundles.
... A corner stone in gauge theory is the Hitchin-Kobayashi correspondence ( [17,20,30,12]). This celebrated generalisation of the Narasimhan and Seshadri theorem asserts that a holomorphic vector bundle over a Kähler manifold carries an Hermite-Einstein metric if and only if it is polystable in the sense of Mumford and Takemoto ([22,29]). The interplay between the differential geometric side, hermitian Yang-Mills connections (HYM for short) that originated from physics, and the algebro-geometric side, the stability notion motivated by moduli constructions, has had many applications and became a very fertile source of inspiration. ...
... 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 : 17,20,30,12]). 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. ...
We investigate hermitian Yang--Mills connections for pullback vector bundles on blow-ups of K\"ahler manifolds along submanifolds. Under some mild asumptions on the graded object of a simple and semi-stable vector bundle, we provide a necessary and sufficent numerical criterion for the pullback bundle to admit a sequence of hermitian Yang--Mills connections for polarisations that make the exceptional divisor sufficiently small, and show that those connections converge to the pulled back hermitian Yang-Mills connection of the graded object.
... 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. [15,44]. Thus, this allows one to translate the differential geometric problem of constructing Hermitian Yang-Mills connections to an algebraic geometry problem. ...
... When P 6 = CP 3 (see (2) above) we know that all holomorphic line bundles are given by O CP 3 (k) for k ∈ Z: these are determined by integer multiples of the Fubini-Study form which themselves correspond to the curvature forms of the Hermitian Yang-Mills connections. The Donaldson-Uhlenbeck-Yau correspondence asserts that these are all the abelian Hermitian Yang-Mills connections [15,44]. 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. ...
We consider a dimensional reduction of the Hermitian Yang-Mills condition on $S^1$-invariant K\"ahler Einstein $6$-manifolds. This allows us to reformulate the Hermitian Yang-Mills equations in terms of data on the quotient K\"ahler $4$-manifold. In particular, we apply this construction to the canonical bundle of $\mathbb{C}\mathbb{P}^2$ endowed with the Calabi ansatz metric to find explicit abelian $SU(3)$ instantons and we show that these are determined by the spectrum of $\mathbb{C}\mathbb{P}^2$. As a by-product of our investigation we find a coordinate expression for its holomorphic volume form which leads us to construct a special Lagrangian foliation of $\mathcal{O}_{\mathbb{C}\mathbb{P}^2}(-3)$.
... In Section 3.3, we study the extension of Donaldson's program to the study of Hermite-Einstein metrics on a stable holomorphic vector bundle E over (X, ω), whose existence and uniqueness has been established by Donaldson in [10] and Uhlenbeck-Yau in [38]. This fundamental result implies in particular that a holomorphic vector bundle E is Mumford stable if and only if E is simple, meaning that it cannot be decomposed as a direct sum of holomorphic vector bundles of smaller dimension, and admits a Hermitian metric h E satisfying the following weak Hermite-Einstein equation ...
... Recall from [10] and [38] (1.21). Then there is p 0 ∈ N such that for any p p 0 , there exists a unique ν-balanced product q p ∈ Prod(H p ) up to multiplicative constant. ...
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 towards balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson's iterations towards balanced metrics on K\"ahler manifolds with constant scalar curvature.
... Uhlenbeck and Yau proved that E is polystable with respect to ω if and only if E admits a Hermitian metric that satisfies the Hermite-Einstein equation defined using ω [UY]. This result was proved earlier by Donaldson under the extra assumptions that X is a complex projective manifold and ω represents a rational cohomology class [Do2]. As a consequence of these theorems of Uhlenbeck-Yau and Donaldson, a holomorphic vector bundle on X admits a flat unitary connection if and only if E is polystable with c 1 (E) = 0 = c 2 (E). ...
... Proposition 2.1] and [Do2]. ...
Given a compact Kähler manifold X, there is an equivalence of categories between the completely reducible flat vector bundles on X and the polystable Higgs bundles \((E, \theta )\) on X with \(c_1(E)= 0= c_2(E)\) (Simpson in J Am Math Soc 1(4):867–918, 1988; Corlette in J Differ Geom 28:361–382, 1988; Uhlenbeck and Yau in Commun Pure Appl Math 39:257–293, 1986; Donaldson in Duke Math J 54(1):231–247, 1987). We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat vector bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that any stable basic Higgs bundle over a compact Sasakian manifold admits a basic Hermitian metric that satisfies the Yang–Mills–Higgs equation.
... 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, . . . ...
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).
... [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 ...
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]). ...
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 $\mathbb{C}^*$ 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. ...
We exhibit examples of slope-stable and modular vector bundles on a hyperk\"ahler manifold of K3$^{[2]}$-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.
... Eventually, Donaldson [16] and Uhlenbeck-Yau [73] established the equivalence on any dimensional complex projective manifolds. Note that Uhlenbeck-Yau proved it for any compact Kähler manifolds, more generally. ...
We prove the Kobayashi-Hitchin correspondence between good wild harmonic bundles and polystable good filtered λ-flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way to construct Frobenius manifolds.
... Lemma 6.6.3, [20]) (which is an adaptation of Donaldson's proof in [6]), we must have {H j } ∞ j=1 uniformly bounded in L p 2 −norm. This is a contradiction to the assumption that s j L p 2 → ∞. ...
We consider a variant of the Seiberg-Witten equations for multiple-spinors. The moduli space of solutions to our generalized Seiberg-Witten equations in the setting of K\"ahler surfaces has a direct relation with ASD connections of holomorphic vector bundle. Also in K\"ahler setting, we construct a numerical invariant from the equations that detects a notion of $\phi-$stability of $SU(n)-$holomorphic vector bundles where $\phi$ is some prescribed non-trivial holomorphic section.
... The Donaldson-Uhlenbeck-Yau theorem for twisted Higgs bundles ( [2,8,23,39]) guarantees the existence of Higgs-Hermitian-Einstein metrics for the polystable case. It was originally proved by Narasimhan-Seshadri ( [34]), Donaldson ([15,16]) and Uhlenbeck-Yau ( [42]) for holomorphic bundles. There are also many interesting and important generalized Donaldson-Uhlenbeck-Yau theorems (see [4,5,6,9,25,31,32,33,37,44,45] and references therein). ...
In this paper, we consider the Yang-Mills-Higgs flow for twisted Higgs pairs over K\"ahler 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--eshadri filtration of the initial twisted Higgs bundle.
... On the other hand, we have the Hitchin-Kobayashi correspondence which states that the stability of holomorphic vector bundles is equivalent to the existence of irreducible Hermitian-Einstein metrics. It was proved by Narasimhan-Seshadri ( [22]) for Riemann surfaces, Donaldson ([7,8]) for algebraic manifolds and Uhlenback-Yau ( [27]) for higher dimension Kähler manifolds. Hitchin ([15]) and Simpson ([24]) proved the Hitchin-Kobayashi correspondence for Higgs bundles. ...
In this paper, we study the non-Hermitian Yang-Mills (NHYM for short) bundles over compact K\"ahler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in \cite{KV}.
... This result was first proved by Narasimhan and Seshadri ( [34]) for compact Riemannian surface by using the methods of algebraic geometry. Subsequently, Donaldson ([15,16,17]) gave a new proof of the Narasimhan-Seshadri theorem, and generalized to algebraic surfaces and algebraic manifolds by using Hermitian-Yang-Mills flow. Uhlenbeck and Yau ([40,41]) proved it for general compact Kähler manifold by using continuity method. ...
In this paper, we study the semi-stable twisted holomorphic vector bundles over compact Gauduchon manifolds. By using Uhlenbeck--Yau's continuity method, we show that the existence of approximate Hermitian--Einstein structure and the semi-stability of twisted holomorphic vector bundles are equivalent over compact Gauduchon manifolds. As its application, we show that the Bogomolov type inequality is also valid for a semi-stable twisted holomorphic vector bundle.
... Geometric flows are now well-recognized as a powerful tool for geometry and topology. Major successes include the Eells-Sampson theorem on harmonic maps [ES64], Hamilton's Ricci flow [Ham82] and Perelman's proof of the Poincaré conjecture [Per02,Per03b,Per03a], Donaldson's heat flow proof of the Donaldson-Uhlenbeck-Yau theorem on Hermitian-Yang-Mills connections [Don87], the differentiable sphere theorem of Brendle-Schoen [BS09], and many other developments. However, these successes were mostly in the settings of Riemannian and complex geometry, leaving the subject of symplectic geometric flows underdeveloped. ...
Several geometric flows on symplectic manifolds are introduced which are potentially of interest in symplectic geometry and topology. They are motivated by the Type IIA flow and T-duality between flows in symplectic geometry and flows in complex geometry. Examples include the Hitchin gradient flow on symplectic manifolds, and a new flow which is called the dual Ricci flow.
... This correspondence has led to many applications in topology, gauge theory and theoretical physics throughout these years. It was first proved for Riemann surfaces by Narasimhan and Seshadri [31], and then for Kähler manifolds by Donaldson-Uhlenbeck-Yau [12][13][14]37] and for complex manifolds with Gauduchon metrics by Buchdahl-Li-Yau [10,26]. There are many generalizations. ...
In this paper, we investigate canonical metrics on bi-holomorphic bundles with a nontrivial global holomorphic section, and we prove that the I±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{\pm }$$\end{document}-holomorphic pair (E,∂¯+,∂¯-,ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(E,\bar{\partial }_{+},\bar{\partial }_{-},\phi )$$\end{document} is (α,τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,\tau ) $$\end{document}-semi-stable if and only if it admits an approximate (α,τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,\tau )$$\end{document}-Hermitian–Einstein structure over the compact bi-Hermitian manifold.
... The earliest, and still most influential, geometric structure which emerged as a candidate for vacuum configuration of unified string theories was that of a Calabi-Yau manifold, equipped with a holomorphic vector bundle admitting a Hermitian-Einstein metric. This was proposed by Candelas, Horowitz, Strominger, and Witten [5] for the heterotic string, and it could rely on the fundamental existence theorems for such structures proved earlier by Yau [35] and Donaldson-Uhlenbeck-Yau [8,33]. But other geometric structures have emerged since, motivated by the other string theories, for which analogous existence theorems are not yet available. ...
The Type IIB flow is a flow of conformally balanced complex manifolds introduced by Phong, Picard, and Zhang, about whose singularities little is as yet known. We formulate convergence criteria for the Gromov-Cheeger-Hamilton convergence of sequences of Hermitian manifolds, and apply them to precompactness theorems and the existence of singularity models for the Type IIB flow, in analogy with Hamilton's classic compactness theorems and classification of singularities for the Ricci flow.
... In [Bog78], Bogomolov proved the celebrated inequality asserting that the discriminant ∆(E) := 2rc 2 (E) − (r − 1)c 2 1 (E) of any rank r vector bundle E on a projective surface that is slope H-polystable with respect to an ample divisor H is non-negative. One analytic method of proof consists of using the existence of a Hermitian-Einstein metric on E. This was proven by Simon Donaldson for projective algebraic surfaces [Don85] and later for projective algebraic manifolds [Don87], by Karen Uhlenbeck and Shing-Tung Yau [UY86] for compact Kähler manifolds. ...
In this note, we prove the Bogomolov's inequality over a reduced, compact, irreducible, K\"ahler complex space that is smooth in codimension 2. The proof is obtained by a reduction to the smooth case, using Hironaka's resolution of singularities.
... Donaldson's result motivated Kobayashi and Hitchin, indipendently, to conjecture that this result holds for every holomorphic vector bundle on a compact Kähler manifold: it is this correspondence which is usually referred to as Kobayashi-Hitchin correspondence. The rst proof of this correspondence for higher dimensional manifolds was given by Donaldson in [16] for algebraic surfaces, and then in [17] for algebraic manifolds. ...
We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g −polystable if and only if it is g −Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X , then a twisted holomorphic vector bundle on X is g −semistable if and only if it is approximate g −Hermite-Einstein.
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
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.
In this paper, we establish the Kobayashi–Hitchin correspondence, that is, the equivalence of the existence of an Einstein–Hermitian metric and $\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 $\psi $-stable Poisson modules over complex surfaces that are not stable in the ordinary sense.
In this paper, we define the concept of twisted λ-flat bundles over complex manifolds and establish the Hitchin-Kobayashi correspondence for it over Gauduchon manifolds, that is, we prove that there exists a Poisson metric on the twisted λ-flat bundle (λ≠0) if and only if it is polystable. This is a generalization of the work of Donaldson [10], Corlette [6] and Mochizuki [22].
In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold X which admits an Astheno–Kähler metric. We prove that numerically flatness is equivalent to approximate Hermitian flatness and the existence of a filtration by sub-bundles whose quotients are Hermitian flat. This gives an affirmative answer to the question proposed by Demailly, Peternell and Schneider.
The moduli space of Higgs bundles can be defined as a quotient of an infinite-dimensional space. Moreover, by the Kuranishi slice method, it is equipped with the structure of a normal complex space. In this paper, we will use analytic methods to show that the moduli space is quasi-projective. In fact, following Hausel's method, we will use the symplectic cut to construct a normal and projective compactification of the moduli space, and hence prove the quasi-projectivity. The main difference between this paper and Hausel's is that the smoothness of the moduli space is not assumed.
In this paper, we introduce a family of generalized Donaldson functionals on holomorphic vector bundles, whose Euler–Lagrange equations are a vector bundle version of the complex k-Hessian equations. We also study the uniqueness of solutions to these equations.
In this paper, we use the affine Hermitian-Yang-Mills flow to prove a generalized Donaldson-Uhlenbeck-Yau theorem on flat Higgs bundles over a class of non-compact affine Gauduchon manifolds.
The mathematical development of Yang–Mills theory is an extremely fruitful subject. The purpose of this paper is to give non-experts and researchers in interdisciplinary areas a quick overview of some history, key ideas and recent developments in this subject.
In this continuation of \cite{BK} we investigate the non-abelian Hodge correspondence on compact Sasakian manifolds with emphasis on the quasi-regular case. On quasi-regular Sasakian manifolds, we introduce the notions of quasi-regularity and regularity of basic vector bundles. These notions are useful in relating the vector bundles over a quasi-regular Sasakian manifold with the orbibundles over the orbifold defined by the orbits of the Reeb foliation of the Sasakian manifold. We note that the non-abelian Hodge correspondence on quasi-regular Sasakian manifolds gives a canonical correspondence between the semi-simple representations of the orbifold fundamental groups and the Higgs orbibundles on locally cyclic complex orbifolds admitting Hodge metrics. Under the quasi-regularity of Sasakian manifolds and vector bundles, we extend this correspondence to one between the flat bundles and the basic Higgs bundles. We also prove a Sasakian analogue of the characterization of numerically flat bundles given by Demailly, Peternell and Schneider.
In this paper, we study holomorphic pairs over a class of non-compact Gauduchon manifolds. We prove that the stability implies the existence of Hermitian–Yang–Mills metric, and the semi-stability implies the existence of approximate Hermitian–Yang–Mills structure. We generalize the result in [S Bradlow, J Differ Geom, 1991] to the non-compact and non-Kähler case. Our proof is a combination of heat flow method and continuity method.
In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the $L^2$ -bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgs k -functional with Higgs self-interaction, we show that, provided $\dim (M) , for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local $L^2$ -derivative estimates, energy estimates and blow-up analysis.
In this paper, we prove a Chern number inequality for Higgs bundles over some Kähler manifolds. As an application, we get the Bogomolov inequality for semi-stable parabolic Higgs bundles over smooth projective varieties.
Given a compact Kähler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying Kähler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain determinant line bundle equipped with a Quillen metric. It gives rise to a holomorphic line bundle on the classifying GIT space together with a continuous hermitian metric.
C.S. Seshadri passed away in July 2020. The first part of this article contains some reminiscences from the early 1980s, when as a graduate student in the Tata Institute of Fundamental Research (TIFR), Mumbai, I had the good fortune to learn from him. The second part is on the Narasimhan-Seshadri theorem and my encounter with it.
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.