On the existence of Hermitian-Yang-Mills connections in stable vector bundles.

*(English)*Zbl 0615.58045Solutions of the self-dual Yang-Mills equations on 4-manifolds have become a major tool in differential topology, following the existence theorems of Taubes and the analysis of moduli spaces of Donaldson. When the 4-manifold is a Kähler manifold, the anti-self duality equations become the statement that the curvature form is orthogonal to the Kähler form, or in the nomenclature of this paper, the connection is Hermitian-Yang-Mills.

It was suggested by the reviewer [Nonlinear problems in geometry, Proc. Sixth Int. Symp., Sendai/Japan (1979; Zbl 0433.53002)] and independently S. Kobayashi [Proc. Jap. Acad., Ser. A 58, 158-162 (1982; Zbl 0538.32021)] that stability of a holomorphic vector bundle should be a necessary and sufficient condition for the existence of such a connection. Necessity was proved by M. Lübke [Manuscr. Math. 42, 245-257 (1983; Zbl 0558.53037)]. Sufficiency is proved in this paper, motivating evidence having been always present in the theorem of M. S. Narasimhan and C. S. Seshadri, who had long ago proved the theorem for curves [Ann. Math., II. Ser. 82, 540-567 (1965; Zbl 0178.048)], the more recent fact that the solutions of the self-duality equations on \(S^ 4\) pull back to be Hermitian-Yang-Mills on the twistor space \({\mathbb{C}}P^ 3\), and the proof of the theorem for algebraic surfaces by S. K. Donaldson [Proc. Lond. Math. Soc., III. Ser. 50, 1-26 (1985; Zbl 0529.53018)]. This particular problem began in fact as Donaldson’s thesis topic, but was soon overtaken by his spectacular results on 4-manifolds. His interest in the algebraic surface situation sprang from the possibility of using the theorem to identify spaces of stable bundles, which can sometimes be calculated, with moduli spaces of anti-self-dual connections.

The paper under review proves the theorem in full generality. The authors first of all prove an existence theorem for the perturbed equation which says that the Kähler form component of the curvature is \(\in\) log h, h being the self-adjoint Hermitian transformation defining the required metric, relative to a fixed metric, and then analysing the situation as \(\in \to 0\). Given stability of the original bundle, the solutions persist to \(\in =0\). The most difficult part of the theorem concerns this limiting process where if convergence is violated, one needs to produce a filtration of the vector bundle by torsion free sheaves. To obtain such regularity from the analysis the authors use the solution to the plateau problem, and results of Sacks and the first author on minimal surfaces. As a by-product of their methods they produce an independently interesting theorem on weakly holomorphic maps.

Corollaries of the theorem include inequalities between Chern classes of Bogomolov type, and more importantly explicit information about the case of equality. In particular, a stable holomorphic bundle on a compact complex manifold \(X^ n\) with Kähler class w is flat if and only if \(C_ 1^ 2w^{n-2}=C_ 2w^{n-2}=0\).

It was suggested by the reviewer [Nonlinear problems in geometry, Proc. Sixth Int. Symp., Sendai/Japan (1979; Zbl 0433.53002)] and independently S. Kobayashi [Proc. Jap. Acad., Ser. A 58, 158-162 (1982; Zbl 0538.32021)] that stability of a holomorphic vector bundle should be a necessary and sufficient condition for the existence of such a connection. Necessity was proved by M. Lübke [Manuscr. Math. 42, 245-257 (1983; Zbl 0558.53037)]. Sufficiency is proved in this paper, motivating evidence having been always present in the theorem of M. S. Narasimhan and C. S. Seshadri, who had long ago proved the theorem for curves [Ann. Math., II. Ser. 82, 540-567 (1965; Zbl 0178.048)], the more recent fact that the solutions of the self-duality equations on \(S^ 4\) pull back to be Hermitian-Yang-Mills on the twistor space \({\mathbb{C}}P^ 3\), and the proof of the theorem for algebraic surfaces by S. K. Donaldson [Proc. Lond. Math. Soc., III. Ser. 50, 1-26 (1985; Zbl 0529.53018)]. This particular problem began in fact as Donaldson’s thesis topic, but was soon overtaken by his spectacular results on 4-manifolds. His interest in the algebraic surface situation sprang from the possibility of using the theorem to identify spaces of stable bundles, which can sometimes be calculated, with moduli spaces of anti-self-dual connections.

The paper under review proves the theorem in full generality. The authors first of all prove an existence theorem for the perturbed equation which says that the Kähler form component of the curvature is \(\in\) log h, h being the self-adjoint Hermitian transformation defining the required metric, relative to a fixed metric, and then analysing the situation as \(\in \to 0\). Given stability of the original bundle, the solutions persist to \(\in =0\). The most difficult part of the theorem concerns this limiting process where if convergence is violated, one needs to produce a filtration of the vector bundle by torsion free sheaves. To obtain such regularity from the analysis the authors use the solution to the plateau problem, and results of Sacks and the first author on minimal surfaces. As a by-product of their methods they produce an independently interesting theorem on weakly holomorphic maps.

Corollaries of the theorem include inequalities between Chern classes of Bogomolov type, and more importantly explicit information about the case of equality. In particular, a stable holomorphic bundle on a compact complex manifold \(X^ n\) with Kähler class w is flat if and only if \(C_ 1^ 2w^{n-2}=C_ 2w^{n-2}=0\).

Reviewer: N.Hitchin

##### MSC:

58J90 | Applications of PDEs on manifolds |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

32L05 | Holomorphic bundles and generalizations |

14L24 | Geometric invariant theory |

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\textit{K. Uhlenbeck} and \textit{S. T. Yau}, Commun. Pure Appl. Math. 39, S257--S293 (1986; Zbl 0615.58045)

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