The moduli space of stable vector bundles on a punctured Riemann surface

  • 53 Pages
  • 2.93 MB
  • English
Riemann surfaces., Vector bundles., Moduli th
Statementby Jonathan Adam Poritz.
LC ClassificationsMicrofilm 94/2534 (Q)
The Physical Object
Paginationv, 53 leaves
ID Numbers
Open LibraryOL1241980M
LC Control Number94628734

Of the moduli space of stable bundles on a Riemann surface Michael Thaddeus Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Mass.

USA The moduli spaces of stable bundles on a Riemann surface have been so exhaustively studied and discussed in recent years that one cannot help wondering what is new to say about them. Narasimhan, M.S. and Seshadri, C.S., Holomorphic vector bundles on a compact Riemann surface, Math. Ann.

() 69–80 MathSciNet CrossRef zbMATH Google Scholar [10] Narasimhan, M.S. and Seshadri, C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann of Math.

82 () – MathSciNet CrossRef zbMATH Google ScholarCited by: 4. Let X be a compact connected Riemann surface of genus g, with g ≥ 3. Fix a holomorphic line bundle L over X and also fix an integer r ≥ 2.

Let M X (r, L) denote the moduli space of stable vector bundles on X of rank r and determinant L, which is a smooth quasiprojective complex variety of dimension (r 2 − 1) (g − 1).Author: Indranil Biswas, Tathagata Sengupta.

Abstract. We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology.

In one case, all except one component of the moduli space has such vector. ble moduli space of Riemann surfaces is a polynomial algebra generated by certain classes κ i of dimension 2i.

For the purpose of calculating rational co-homology, one may replace the stable moduli space of Riemann surfaces by BΓ∞, where Γ∞ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of.

The moduli space M B of all vector bundles on B, whose restriction to any curve of the family is stable, consists of all moduli sections S of families of the corresp onding moduli spaces M C λ. This chapter highlights a compactification of a moduli space of stable vector bundles on a rational surface.

It discusses semi-stable sheaves, semi-stable sheaves on a rational surface, and semi-stability of the universal extension. The chapter presents a convention of stable sheaves and an order among polynomials in Q[x]. This makes the space of holomorphic structures (i.e.

the space of bundles with a fixed topological type) into an affine space. The group of complex automorphisms of the bundle acts on this space, and the quotient is the moduli space of holomorphic bundles.

If you don't restrict to stable bundles, this quotient space fails to be Hausdorff. The moduli space N of stable parabolic vector bundles of rank k on Xis a complex manifold2. The holomorphic tangent space T {E}N at the point {E} ∈ N corresponding to the stable parabolic bundle Eis naturally isomorphic to the space H 0,1(X 0,EndE0) of square integrable harmonic EndE0-valued (0,1)-forms on X0.

Details The moduli space of stable vector bundles on a punctured Riemann surface FB2

The moduli space N carries a natural. classi cation of unstable vector bundles of rank two. Lecture 5. Semistable vector bundles When it comes to the classi cation of vector bundles, the concept of a semistable vector bundle is the central one.

Semistable vector bundles of xed rank and degree possess a so-called moduli space. Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share.

The Hitchin moduli space Fixed data: C, a compact Riemann surface (possibly with punctures D) G = SU(n), G C = SL(n;C) E!C, a complex vector bundle of rank n with Aut(E) = SL(E) Hitchin moduli space, M. Fact #1: Mis a noncompact hyperk ahler manifold with metric g L2)have a CP1-family of K ahler manifolds M = (M;g L2;I ;!).

M =0 is G C-Higgs. moduli space of rank nflat stable vector bundles and the space of n-dimensional representations of a Schottky group modulo conjugation, have the same dimension: n 2 (g−1)+1. On a Riemann surface Xrealized as Ω/Σ for a Schottky group Σ, we can therefore consider.

Differential Geometry of Stable Vector Bundles Brief History Brief History I Grothendieck () showed that for genus 0 the classification is trivial in the sense that every holomorphic vector bundle over P1 is a direct sum of line bundles (a result known in a different language to Hilbert, Plemelj and Birkhoff, and prior.

Moduli of Vector Bundles on Curves with Parabolic Structures moduli space of parabolic semi-stable vector bundles is complete (Theorem ). Then X = H + mod F is a compact Riemann surface, containing Y=HmodF. If o:F-~GL(n, lr) is a representation of F in a complex vector space E, the vector bundle H x E on E has the structure of a F.

In this paper we investigate the moduli space of parabolic Higgs bundles over a punctured Riemann surface with varying weights at the punctures.

Let $\cMx$ be the moduli space of stable vector. Modulus Space Vector Bundle Riemann Surface Line Bundle Central Extension These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. Let M be the moduli space of rank 3 parabolic vector bundles over a Riemann surface with several punctures.

By the Mehta-Seshadri correspon-dence, this is the space of rank 3 unitary representations of the fundamental group of the punctured surface with specified conjugacy classes of the images. (Recall that the relationship between vector bundles and repre-sentations of the fundamental groups was first investigated by A.

Weil [39].) Finally, Seshadri gave the GIT construction of the moduli space of stable vector bundles on a Riemann surface together with its compactification by S-equivalence classes of semistable vector bundles [36].

Description The moduli space of stable vector bundles on a punctured Riemann surface EPUB

VECTOR BUNDLES ON RIEMANN SURFACES A4. Same picture, but assume now that the "special fibre" Co acquires a node 8 (fig. 2a and 2b); we assume that the points Pi (0) stay away from 8. Let Co be the normalization of Co, i.e.

the Riemann surface obtained by separating the two branches at 8 to get two distinct points 8' and 8".There is an isomorphism. We study a certain moduli space of irreducible Hermitian-Yang-Mills connections on a unitary vector bundle over a punctured Riemann surface.

The connections used have non-trivial holonomy around the punctures lying in fixed conjugacy classes of U (n) and differ from each other by elements of a weighted Sobolev space; these connections give rise to parabolic bundles in the sense of Mehta and.

The homotopy types of gauge groups of nonorientable surfaces and applications to moduli spaces Theriault, Stephen, Illinois Journal of Mathematics, ; Rationality of moduli spaces of vector bundles on rational surfaces Costa, Laura and Miro-Ŕoig, Rosa M., Nagoya Mathematical Journal, ; Moduli spaces of vector bundles over ruled surfaces Aprodu, Marian and Brînzănescu, Vasile, Nagoya.

[zag] D. Zagier, "On the cohomology of moduli spaces of rank two vector bundles over curves," in The Moduli Space of Curves, Boston, MA: Birkhäuser,pp. Show bibtex @incollection {zag, MRKEY = {}. Stable and unitary vector bundles on a compact Riemann surface By M.

Narasimhan and C. Seshadri 1. Introduction D. Mumford has defined the notion of a stable vector bundle on a compact Riemann surface X and proved that the set of equivalence classes of stable bundles (of fixed rank and degree) has a natural structure of a non-singular.

The moduli space of stable vector bundles over of rank and degree was first given by Mumford andGieseker gave a different construction which generalized to higher dimensions. Simpson invented a more natural and general method using Grothendieck's Quot scheme which also extends to singular curves and higher dimensions (see).

is simply the Picard variety we. Many moduli spaces associated with Riemann surfaces have such Kahler structures: the Jacobi variety, Teichmüller space, moduli spaces of stable vector bundles and even the first real cohomology group have such structures.

In all of these examples the topology of the associated spaces depends, remarkably, only on the topology of the Riemann. Teichmüller space. moduli space of Riemann surfaces. super Riemann surface. stable vector bundle. elliptic curve.

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worldsheet. beta-gamma system. References. Historical references include. Hermann Weyl, Die Idee der Riemannschen Fläche, (_The concept of a Riemann surface_) (on the book, by Peter Schreiber, web) Lecture notes include. Stable vector bundles over curves.

A slope of a holomorphic vector bundle W over a nonsingular algebraic curve (or over a Riemann surface) is a rational number μ(W) = deg(W)/rank(W).A bundle W is stable if and only if bundle is stable if it is "more. Moduli Space of Parabolic Higgs Bundles Pengfei Huang Universit e C´ ote d'Azur, Nice, France History I Metha and Seshadri introduced parabolic structure to vector bundles over Riemann surface,   20 Decemberto VENUE Ramanujan Lecture Hall, ICTS, Bangalore The theory of holomorphic vector bundles on a compact Riemann surface is a vast "non-abelian" generalisation of the.

In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic is thus a special case of a moduli ing on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different.We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface $\Sigma$.The book grew out of lecture courses.

The presentation style is therefore similar to a lecture. Graduate students of theoretical and mathematical physics will appreciate this book as textbook. Students of mathematics who are looking for a short introduction to the various aspects of modern geometry and their interplay will also find it useful.