X which is locally trivial, that is, for any x 2 x there exists an open neighborhood u of x and an isomorphism of. A lot of the foundational results in this section can be found in bv, appendix. Recall that the canonical bundle may be viewed as a representative of the rst chern class. Koszul complexes yield ampleexamples of2periodic complexes. The koszul complex kos v is a di erential graded algebra whose underlying algebra is kos v v v sym v. An algebraic vector bundle of rank r on x is a linear. We apply it to prove some acyclicity results of the koszul complex of a module and. Our geometric setting consists of two vector subbundles f1,f2 of a trivial vector bundle x. A bundle over a manifold is trivial if it is simply the cartesian product of.
In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle. The case of quaternionic vector bundles can be handled in the same way. In particular, we say that a line bundle o x1 is koszul ample if it is 2n koszul. X be a vector bundle of rank r over an algebraic variety x. To any complex vector bundle eover x is associated an integer, the degree dege c 1ex. Introduction lagrangian physics derives equations of motion from a variational principle of least action. V over a smooth and proper complex algebraic variety x. A note on cayleybacharach property for vector bundles shengli tan and eckart viehweg. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. The okagrauert principle states that for any stein manifold x x the holomorphic and the topological classification of complex vector bundles on x x. The complex form of bott periodicity asserts simply that kse nis zfor neven. M is a complex vector bundle together with the structure of a complex manifold on e, such that for any x2mthere exists u3x in mand trivialization. E is not only a module over the ring of all realvalued functions c. Let us write it down a little more generally, namely with coe.
A complex vector bundle with complex 1dimensional fibers is a complex line bundle. We apply it to prove some acyclicity results of the koszul complex of a module and to compute the cohomology of the sheaves of relative and absolute differential pforms of a projective bundle. Both are constructed via a suitable truncation of the koszul complex over an x for the regular sequence t 1. For a vector bundle e on the ndimensional projective space pn over k, giving the hilbert polynomial of e is equivalent to giving the chern classes of e. Thanks for contributing an answer to mathematics stack exchange. Ramanan no part of this book may be reproduced in any form by print, micro. Positselskis book on koszul algebras uses n koszul in a di. Let fbe a vector bundle on p1 that is globally generated. The following koszul malgrange criterion 6, see also 2, 2.
For v2v, we write vfor the corresponding element in v 1 v, and v for the corresponding element in sym1 v. Application of the geometric technique of calculating. Pdf the koszul complex of a moment map researchgate. In particular we generalize botts formula for the projective space to a projective bundle over a scheme of characteristic zero. Then we can apply the geometric technique of calculating syzygies to deal with properties of smand rm. Kempf was that the direct image via a map q of a koszul complex of a resolution. In particular, we say that a line bundle o x1 is koszulample if it is 2nkoszul. It is a koszul complex on the vector bundle y on x x v. An introduction to the topology of the moduli space of stable. For example, the standard line bundle o1 on pn is koszulample. A simple example of a non trivial vector bundle is the in. Holomorphic koszulbrylinski homology international.
Steven sam joint work with jerzy weyman these are notes for a talk given at the commutative algebra seminar at university of michigan on january 19, 2012. The koszul complex of a moment map article pdf available in journal of symplectic geometry 1 may 2012 with 47 reads how we measure reads. Chapter 7 vector bundles louisiana state university. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The appearance of the koszul complex in the description of the bott element makes contact with. Rk,the base space of the bundle being understood by the context.
An important example of this sort occurs when m is the module of cycles in the koszul complex on variables in a polynomial ring. However, since his not commutative, one has to be careful when writing down formulas. We apply it to prove some acyclicity results of the koszul complex of a module and to. The riemannroch theorem for complex spaces 151 algebra of all toeplitz operators on u a, i. But there are many other interesting vector bundles. Equivariant algebraic ktheory northeastern university. Koszul complexes, birkhoffnormal form and the magnetic dirac operator. With respect to a local frame, a hermitian structure is given by a hermitian matrixvalued function h hij, with hij hsi,sjih e. U ck that is a biholomorphic map of complex manifolds. Pdf we construct relative and global euler sequences of a module.
For example, the standard line bundle o1 on pn is koszul ample. The complex version of koxg, called kxe, is constructed in the same way as koxg but using vector bundles whose. A complex vector bundle is a vector bundle with respect complex vector spaces. Things become simpler if one passes from real vector spaces to complex vector spaces. In this case we prove that the answers to questions ac are positive if the characteristic of the.
M but also a module over the ring of all complexvalued functions c. Moreover, if one takes the tth graded piece of this complex one obtains the cech complex for ft with respect to the same open cover. Chern classes a connection and curvature in a complex. First, let us be a bit more precise about what we mean by koszul duality. The serre construction, and vector bundles on k3 surfaces. So, from the result of maruyama, it follows that the set of isomorphism classes of stable rank r vector bundles on pn with chern classes c 1. This algebra is essentially commutativel and is a fredholm representation of the algebra cbu. The most important example is the tangent bundle, a real vector bundle. Stable rank three vector bundles on 4 with chern classes c. 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. Linear koszul duality and fourier transform for convolution. This explicit description is necessary to prove the following fact see theorem 8.
If the dimension of the vector space is mthen the bundle is often called an mplane bundle. Such a trivialization is called a holomorphic trivialization. Euler sequence and koszul complex of a module springerlink. Positselskis book on koszul algebras uses nkoszul in a di. The present paper deals with an extension of this concept. Koszul homology and classical invariant theory speaker. Then we obtain a free resolution of c am called the koszul complex. Topological vector bundles over x are classi ed up to isomorphism by their rank and degree. In this note we prove the vanishing of twisted koszul cohomology groups k p. Modeled on classical treatments of the koszul complex cf. But avoid asking for help, clarification, or responding to other answers. The homological properties of koszul complex include the information. The powers of lhave very interesting geometric properties.
This integer actually gives a complete topological classi cation of complex vector bundles on x. Introduction a fundamental object associated to complex manifolds x is its canonical line bundle topt x. The bundle eand its exterior powers ieare the main object. Koszul algebras and koszul duality university of ottawa.
Let lbe an ample line bundle on a complex abelian variety x. On degeneracy loci of morphisms between vector bundles core. We say that ais a koszul algebra iff kis koszul as an amodule. Suppose e m is an ndimensional complex vector bundle. Local cohomology of l module um lsa um college of lsa. Two applications are considered, the curvature of the vertical line bundle of the universal curve, and the positivity of the pushdown of the square of the line bundle. Pdf koszul complexes, birkhoffnormal form and the magnetic. In the early 1950s, jeanlouis koszul in 10 introduced a complex today, we call it the koszul complex for the theory of lie algebra cohomology.