scholarly journals Moduli Spaces of Vector Bundles over a Real Curve: ℤ/2-Betti Numbers

2014 ◽  
Vol 66 (5) ◽  
pp. 961-992 ◽  
Author(s):  
Thomas Baird
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Lagrangian Submanifolds ◽  
Betti Numbers ◽  
Stable Bundles ◽  
Complex Curve ◽  
Yang Mills ◽  
Real Curve

AbstractModuli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi–stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute ℤ/2–Betti numbers of these spaces.

scholarly journals Moduli spaces of stable vector bundles on Enriques surfaces

1998 ◽  
Vol 150 ◽  
pp. 85-94 ◽  
Author(s):  
Hoil Kim
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
K3 Surface ◽  
Enriques Surface ◽  
Stable Bundles ◽  
Stable Vector Bundles ◽  
Enriques Surfaces

Abstract.We show that the image of the moduli space of stable bundles on an Enriques surface by the pull back map is a Lagrangian subvariety in the moduli space of stable bundles, which is a symplectic variety, on the covering K3 surface. We also describe singularities and some other features of it.

scholarly journals CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

2010 ◽  
Vol 21 (04) ◽  
pp. 497-522 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAINAK PODDAR
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Prime Number ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Line Bundles ◽  
Holomorphic Line Bundle ◽  
Stable Vector Bundles

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

scholarly journals RATIONAL FAMILIES OF VECTOR BUNDLES ON CURVES

2004 ◽  
Vol 15 (01) ◽  
pp. 13-45 ◽  
Author(s):  
ANA-MARIA CASTRAVET
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Rational Curves ◽  
Complex Curve ◽  
Stable Vector Bundles ◽  
Irreducible Components ◽  
Rationally Connected ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

Let C be a smooth projective complex curve of genus g≥2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k≥1, we find all the irreducible components of the space of rational curves on M, of degree k. In particular, we find the maximal rationally connected fibrations of these components. We prove that there is a one-to-one correspondence between moduli spaces of rational curves on M and moduli spaces of rank 2 vector bundles on ℙ1×C.

PARABOLIC VECTOR BUNDLES AND HERMITIAN-YANG-MILLS CONNECTIONS OVER A RIEMANN SURFACE

1993 ◽  
Vol 04 (03) ◽  
pp. 467-501 ◽  
Author(s):  
JONATHAN A. PORITZ
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Sobolev Space ◽  
Moduli Space ◽  
Vector Bundles ◽  
Weighted Sobolev Space ◽  
Conjugacy Classes ◽  
Stable Bundle ◽  
Yang Mills ◽  
Parabolic Bundles

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 Seshadri. We show in fact that the moduli space of stable parabolic bundles can be identified with our moduli space of HYM connections, by proving that every stable bundle admits a unique unitary gauge orbit of Hermitian-Yang-Mills connections.

scholarly journals Moduli spaces of the stable vector bundles over abelian surfaces

1980 ◽  
Vol 77 ◽  
pp. 47-60 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Ample Line Bundle ◽  
Abelian Surfaces ◽  
Singular Variety ◽  
Stable Bundles ◽  
Stable Vector Bundles ◽  
Closed Subvariety

Let X be a projective non-singular variety and H an ample line bundle on X. The moduli space of H-stable vector bundles exists by Maruyama [4]. If X is a curve defined over C, the structure of the moduli space (or its compactification) M(X, d, r) of stable vector bundles of degree d and rank r on X is studied in detail. It is known that the variety M(X, d, r) is irreducible. Let L be a line bundle of degree d and let M(X, L, r) denote the closed subvariety of M(X, d, r) consisting of all the stable bundles E with det E = L.

TORELLI THEOREM FOR MODULI SPACES OF SL(r,ℂ)-CONNECTIONS ON A COMPACT RIEMANN SURFACE

2009 ◽  
Vol 11 (01) ◽  
pp. 1-26
Author(s):  
INDRANIL BISWAS ◽  
VICENTE MUÑOZ
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Complex Structure ◽  
Real Numbers ◽  
Stable Vector Bundles ◽  
Complex Variety ◽  
Projective Curves ◽  
Torelli Theorem

Let X be any compact connected Riemann surface of genus g, with g ≥ 3. For any r ≥ 2, let [Formula: see text] denote the moduli space of holomorphic SL (r,ℂ)-connections over X. It is known that the biholomorphism class of the complex variety [Formula: see text] is independent of the complex structure of X. If g = 3, then we assume that r ≥ 3. We prove that the isomorphism class of the variety [Formula: see text] determines the Riemann surface X uniquely up to an isomorphism. A similar result is proved for the moduli space of holomorphic GL (r,ℂ)-connections on X. We also show that the Torelli theorem remains valid for the moduli spaces of connections, as well as those of stable vector bundles, on geometrically irreducible smooth projective curves defined over the field of real numbers.

HOMOTOPY DECOMPOSITIONS OF GAUGE GROUPS OVER RIEMANN SURFACES AND APPLICATIONS TO MODULI SPACES

2011 ◽  
Vol 22 (12) ◽  
pp. 1711-1719 ◽  
Author(s):  
STEPHEN D. THERIAULT
Keyword(s):  
Riemann Surface ◽  
Gauge Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Vector Bundles ◽  
Homotopy Groups ◽  
Gauge Groups ◽  
Stable Vector Bundles

For a prime p, the gauge group of a principal U(p)-bundle over a compact, orientable Riemann surface is decomposed up to homotopy as a product of spaces, each of which is commonly known. This is used to deduce explicit computations of the homotopy groups of the moduli space of stable vector bundles through a range, answering a question of Daskalopoulos and Uhlenbeck.

scholarly journals THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hyperelliptic Curve ◽  
Geometric Description ◽  
Birational Model ◽  
Rank 2 ◽  
Semistable Vector Bundles ◽  
Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

scholarly journals COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

2012 ◽  
Vol 23 (04) ◽  
pp. 1250037 ◽  
Author(s):  
MICHELE BOLOGNESI ◽  
SONIA BRIVIO
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Low Rank ◽  
Coherent Systems ◽  
Projective Varieties ◽  
Complex Curve ◽  
Smooth Complex ◽  
Moduli Theory ◽  
Modular Varieties ◽  
Small Genus

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

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