Before, the hermitian structure on a holomorphic vector bundle was used as. A connection r in e is said to be compatible with the holomorphic structure in e if p0,1rss for all sections s of e. B outlines fundamental results about kahlereinstein and hermiteeinstein metrics. Assume that we have a holomorphic vector bundle es over each member ms of the family so that these bundles together form a holomorphic bundle over the whole family. A hermitian metric on a complex vector space is a positivedefinite hermitian form on. Let ex be a holomorphic vector bundle over a complex manifold x. Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections by raoul bott and s.
The main purpose of this book is to lay a foundation for the theory of einstein hermitian vector bundles. We shall not give a detailed introduction here in this preface since the table of contents is fairly selfexplanatory and, furthermore, each chapter is headed by a brief introduction. Singular hermitian metrics on holomorphic vector bundles. Hermitianeinstein metrics on holomorphic vector bundles. Pdf document information annals of mathematics fine hall washington road princeton university princeton, nj 08544, usa phone. A hermitian metric on a complex vector bundle e over a smooth manifold m is a smoothly varying positivedefinite hermitian form on each fiber. Vector bundles, connections, and curvature tony perkins. Any two hermitian metrics on can be transferred into each other by an automorphism of. Curvature of vector bundles associated to holomorphic. Mbe a hermitian vector bundle with hermitian metric h. The existence of hermitianeinstein metrics on holomorphic vector bundles over gauduchon surfaces has been used by liyauzheng 16, 17, and also teleman 23, based on ideas in 16, to provide a. In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space x for example x could be a topological space, a manifold, or an algebraic variety. Let e be a smooth complex vector bundle on a smooth manifold m.
Kobayashi 16, a hermitian metric h in a holomorphic vector bundle e on x is called a ghermitianeinstein metric if it satis. Homogeneous hermitian holomorphic vector bundles and the. Let e be a holomorphic vector bundle over a compact complex manifold. An hermitian structure h in e can be expressed, in terms of a local holomorphic frame field s l9, s r of e, by a. Hermitian yang mills metrics, are called hermitian einstein metrics in this reference. We consider a complex nmanifold x and a holomorphic vector bundle e over x whose fiber dimension equals the dimension of x and wish to study the zerosets. In this paper, we prove the longtime existence of the hermitian einstein flow on a holomorphic vector bundle over a compact hermitian nonkahler manifold, and solve the dirichlet problem for. Introduction in 5,6,7 i introduced the concept of einsteinhermitian vector bundle. The space endowed with a hermitian metric is called a unitary or complexeuclidean or hermitian vector space, and the hermitian metric on it is called a hermitian scalar product.
With respect to a local frame, a hermitian structure is given by a hermitian matrixvalued function h hij, with hij hsi,sjih e. Differential geometry of holomorphic vector bundles on a curve. Hermitian einstein metrics on vector bundles and stability. Pdf homogeneous hermitian holomorphic vector bundles and. The main purpose of this book is to lay a foundation for the theory of einsteinhermitian vector bundles. Homogeneous hermitian holomorphic vector bundles and the cowendouglas class over bounded symmetric domains adam koranyi and gadadhar misra abstract. In we have already seen that there is a decomposition.
On holomorphic sections of certain hermitian vector bundles. Hermitian vector bundles and the equidistribution of the. Every indecomposable holomorphic homogeneous hermitian vector bundle e can be written as a tensor product l. Holomorphic vector bundles on almost complex manifolds. The metric connections of hermitian vector bundles behave well with respect to bundle operations, as we see in the next two lemmas. Hermitian vector bundles and the equidistribution of. Approximations and examples of singular hermitian metrics on. We write h for an hermtian metric on e,which of course induces a metric also denoted by h on e. M, we can produce a locally constant holomorphic vector bundle of the same rank ec. A connection r in e is said to be compatible with the holomorphic structure in e if p0,1rs. E the sheaf of germs of local holomorphic sections of e. Stability and hermitianeinstein metrics for vector. Let f be a finsler metric on a holomorphic vector bundle.
The complex vector bundles t1,0 x and tx,j are identi. Then there is a unique connection in e compatible with both the hermitian and holomorphic structures. A holomorphic vector bundle has holomorphic transition functions, but a vector bundle cannot have antiholomorphic. We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following berndtsson and paun. Moreover, the hermitian extension gc of g to tcx restricted to t l,0 x is 1 2 g.
If the curvature form fh of ah satis es p 1 fh id 1. Lambdastructure on grothendieck groups of hermitian. We show in this article that if a holomorphic vector bundle has a nonnegative hermitian metric in the sense of bott and chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of chern forms are strongly nonnegative. Let e be a holomorphic vector bundle of rank r over a complex manifold m. Introduction hermitian vector bundles and dirac operators. We define what it means for such a metric to be positively and negatively curved in the sense of griffiths and investigate the assumptions needed in order to locally define the curvature.
We suppose that the underlying complex manifold m is. In this paper, we prove the longtime existence of the hermitianeinstein flow on a holomorphic vector bundle over a compact hermitian non. Then f is itself a hermitian bundle with metric connection d. For a holomorphic vector bundle e over a complex manifold, we denote by. Asingular hermitian metric h on e is a measurable map from the.
Chern connection on e let e be a holomorphic vector bundle over a complex manifold. This particularly implies that all the chern numbers of such a holomorphic vector bundle are nonnegative. In order to discuss moduli of holomorphic vector bundles, it is essential to. Hermitian vector bundle which is the restriction to g and d of a g c homo gene ous vector bundle over g c k c p. Suppose s0 2 s and hs0 is a hermitianeinstein metric of es0 with respect to the k. M there is a unique hermitianeinstein metric h on e such that h. We shall not give a detailed introduction here in this preface since the table of contents is fairly selfexplanatory and, furthermore, each chapter is. Poisson equation and hermitianeinstein metrics on vector. Given any hermitian metric h on the holomorphic vector bundle e there exists one and only one complex metric connection ah. X is the complex bundle underlying the holomorphic tangent bundle and hence is the same as the holomorphic tangent bundle. Nonnegative hermitian vector bundles and chern numbers. The metric connections of hermitian vector bundles behave well with respect to bundle operations, as we see. Hermitian vector bundles and value distribution for schubert cyclesi by michael j. Chern used the theory of characteristic differential forms of a holomorphic hermitian vector bundle to study the distribution of zeroes of a holomorphic section.
Let e be a holomorphic subbundle of a holomorphic vector bundle e over a manifold with hermitian metric. A hermitian bundle e e, h on x is a geometric vector bundle e on x, together with a conjugation invariant hermitian metric h on ec, the holomorphic vector bundle associated to e on xc. It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic lie algebra on nite dimensional inner product spaces. The comparatively easier converse, asserting that every hermitianeinstein vector bundle is semistable and splits into a direct sum of stable subbundles, had previously been proved by.
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