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Journal articles on the topic 'Moduli space of vector bundles'

Author: Grafiati

Published: 14 March 2023

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1

Aprodu, Marian, and Vasile Brînzănescu. "Moduli spaces of vector bundles over ruled surfaces." Nagoya Mathematical Journal 154 (1999): 111–22. http://dx.doi.org/10.1017/s0027763000025332.

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AbstractWe study moduli spaces M(c1, c2, d, r) of isomorphism classes of algebraic 2-vector bundles with fixed numerical invariants c1, c2, d, r over a ruled surface. These moduli spaces are independent of any ample line bundle on the surface. The main result gives necessary and sufficient conditions for the non-emptiness of the space M(c1, c2, d, r) and we apply this result to the moduli spaces ML(c1, c2) of stable bundles, where L is an ample line bundle on the ruled surface.

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2

Beck, N. "Moduli of decorated swamps on a smooth projective curve." International Journal of Mathematics 26, no. 10 (September 2015): 1550086. http://dx.doi.org/10.1142/s0129167x1550086x.

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In order to unify the construction of the moduli space of vector bundles with different types of global decorations, such as Higgs bundles, framed vector bundles and conic bundles, A. H. W. Schmitt introduced the concept of a swamp. In this work, we consider vector bundles with both a global and a local decoration over a fixed point of the base. This generalizes the notion of parabolic vector bundles, vector bundles with a level structure and parabolic Higgs bundles. We introduce a notion of stability and construct the coarse moduli space for these objects as the GIT-quotient of a parameter space. In the case of parabolic vector bundles and vector bundles with a level structure our stability concept reproduces the known ones. Thus, our work unifies the construction of their moduli spaces.

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3

Basu, Suratno, and Sourav Das. "A Torelli type theorem for nodal curves." International Journal of Mathematics 32, no. 07 (April 23, 2021): 2150041. http://dx.doi.org/10.1142/s0129167x21500415.

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The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal-crossing singularities and it provides flat degeneration of the moduli of vector bundles over a smooth projective curve. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than [Formula: see text]) and fixed degree such that rank and degree are co-prime.

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4

Almeida, C., M. Jardim, A. S. Tikhomirov, and S. A. Tikhomirov. "New moduli components of rank 2 bundles on projective space." Sbornik: Mathematics 212, no. 11 (November 1, 2021): 1503–52. http://dx.doi.org/10.1070/sm9490.

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Abstract We present a new family of monads whose cohomology is a stable rank 2 vector bundle on . We also study the irreducibility and smoothness together with a geometrical description of some of these families. These facts are used to construct a new infinite series of rational moduli components of stable rank 2 vector bundles with trivial determinant and growing second Chern class. We also prove that the moduli space of stable rank 2 vector bundles with trivial determinant and second Chern class equal to 5 has exactly three irreducible rational components. Bibliography: 40 titles.

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5

PORITZ, JONATHAN A. "PARABOLIC VECTOR BUNDLES AND HERMITIAN-YANG-MILLS CONNECTIONS OVER A RIEMANN SURFACE." International Journal of Mathematics 04, no. 03 (June 1993): 467–501. http://dx.doi.org/10.1142/s0129167x9300025x.

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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.

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6

CASTRAVET, ANA-MARIA. "RATIONAL FAMILIES OF VECTOR BUNDLES ON CURVES." International Journal of Mathematics 15, no. 01 (February 2004): 13–45. http://dx.doi.org/10.1142/s0129167x0400220x.

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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.

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7

GAVIOLI, FRANCESCA. "THETA FUNCTIONS ON THE MODULI SPACE OF PARABOLIC BUNDLES." International Journal of Mathematics 15, no. 03 (May 2004): 259–87. http://dx.doi.org/10.1142/s0129167x04002272.

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In this paper we extend the result on base point freeness of the powers of the determinant bundle on the moduli space of vector bundles on a curve. We describe the parabolic analogues of parabolic theta functions, then we determine a uniform bound depending only on the rank of the parabolic bundles. In order to get this bound, we construct a parabolic analogue of Grothendieck's scheme of quotients, which parametrizes quotient bundles of a parabolic bundle, of fixed parabolic Hilbert polynomial. We prove an estimate for its dimension, which extends the result of Popa and Roth on the dimension of the Quot scheme. As an application of the theorem on base point freeness, we characterize parabolic semistability on the algebraic stack of quasi-parabolic bundles as the base locus of the linear system of the parabolic determinant bundle.

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8

Bhosle, Usha N., and Sanjay Kumar Singh. "Fourier–Mukai Transform on a Compactified Jacobian." International Mathematics Research Notices 2020, no. 13 (June 19, 2018): 3991–4015. http://dx.doi.org/10.1093/imrn/rny136.

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Abstract We use Fourier–Mukai transform to compute the cohomology of the Picard bundles on the compactified Jacobian of an integral nodal curve $Y$. We prove that the transform gives an injective morphism from the moduli space of vector bundles of rank $r \ge 2$ and degree $d$ ($d$ sufficiently large) on $Y$ to the moduli space of vector bundles of a fixed rank and fixed Chern classes on the compactified Jacobian of $Y$. We show that this morphism induces a morphism from the moduli space of vector bundles of rank $r \ge 2$ and a fixed determinant of degree $d$ on $Y$ to the moduli space of vector bundles of a fixed rank with a fixed determinant and fixed Chern classes on the compactified Jacobian of $Y$.

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9

Dey, Arijit, Sampa Dey, and Anirban Mukhopadhyay. "Statistics of moduli space of vector bundles." Bulletin des Sciences Mathématiques 151 (March 2019): 13–33. http://dx.doi.org/10.1016/j.bulsci.2018.12.003.

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10

BISWAS, INDRANIL, and TOMÁS L. GÓMEZ. "HECKE CORRESPONDENCE FOR SYMPLECTIC BUNDLES WITH APPLICATION TO THE PICARD BUNDLES." International Journal of Mathematics 17, no. 01 (January 2006): 45–63. http://dx.doi.org/10.1142/s0129167x06003357.

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We construct a Hecke correspondence for a moduli space of symplectic vector bundles over a curve. As an application we prove the following. Let X be a complex smooth projective curve of genus g(X) > 2 and L a line bundle over X. Let [Formula: see text] be the moduli space parametrizing stable pairs of the form (E,φ), where E is a vector bundle of rank 2n over X and φ : E ⊗ E → L a skew-symmetric nondegenerate bilinear form on the fibers of E. If deg (E) ≥ 4n(g(X)-1), then there is a projectivized Picard bundle on [Formula: see text], which is a projective bundle whose fiber over any point [Formula: see text] is ℙ(H0(X,E)). We prove that this projective bundle is stable.

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11

Pal, Sarbeswar. "Moduli of Rank 2 Stable Bundles and Hecke Curves." Canadian Mathematical Bulletin 59, no. 4 (December 1, 2016): 865–77. http://dx.doi.org/10.4153/cmb-2016-058-9.

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AbstractLet X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this short note we will show that the moduli space of rank 2 stable vector bundles with determinant isomorphic to Lx , where Lx denotes the line bundle corresponding to a point x ∊ X, is isomorphic to a certain variety of lines in the moduli space of S-equivalence classes of semistable bundles of rank 2 with trivial determinant.

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12

Choe, Insong, and G. H. Hitching. "A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve." International Journal of Mathematics 25, no. 05 (May 2014): 1450047. http://dx.doi.org/10.1142/s0129167x14500475.

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A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.

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13

Greb, Daniel, Julius Ross, and Matei Toma. "Moduli of vector bundles on higher-dimensional base manifolds — Construction and variation." International Journal of Mathematics 27, no. 07 (June 2016): 1650054. http://dx.doi.org/10.1142/s0129167x16500543.

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We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson–Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker–Maruyama moduli spaces with respect to two different chosen polarizations are related via Thaddeus-flips through other “multi-Gieseker”-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson–Uhlenbeck moduli space.

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14

Kim, Hoil. "Moduli spaces of stable vector bundles on Enriques surfaces." Nagoya Mathematical Journal 150 (June 1998): 85–94. http://dx.doi.org/10.1017/s002776300002506x.

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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.

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15

JOSHI, KIRTI. "TWO REMARKS ON SUBVARIETIES OF MODULI SPACES." International Journal of Mathematics 19, no. 02 (February 2008): 237–43. http://dx.doi.org/10.1142/s0129167x08004649.

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We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties.

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16

Fu, Lie, Victoria Hoskins, and Simon Pepin Lehalleur. "Motives of moduli spaces of rank $ 3 $ vector bundles and Higgs bundles on a curve." Electronic Research Archive 30, no. 1 (2021): 66–89. http://dx.doi.org/10.3934/era.2022004.

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<abstract><p>We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank $ 3 $ and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Białynicki-Birula decompositions associated with a scaling action, together with the variation of stability and wall-crossing for moduli spaces of rank $ 2 $ pairs, which occur in the fixed locus of this action.</p></abstract>

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17

LIU, MIN. "SMALL RATIONAL CURVES ON THE MODULI SPACE OF STABLE BUNDLES." International Journal of Mathematics 23, no. 08 (July 10, 2012): 1250085. http://dx.doi.org/10.1142/s0129167x12500851.

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For a smooth projective curve C with genus g ≥ 2 and a degree 1 line bundle [Formula: see text] on C, let [Formula: see text] be the moduli space of stable vector bundles of rank r over C with the fixed determinant [Formula: see text]. In this paper, we study the small rational curves on M and estimate the codimension of the locus of the small rational curves. In particular, we determine all small rational curves when r = 3.

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18

BISWAS, INDRANIL, and MAINAK PODDAR. "CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II." International Journal of Mathematics 21, no. 04 (April 2010): 497–522. http://dx.doi.org/10.1142/s0129167x10006094.

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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.

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19

Ben-Zvi, David, and Thomas Nevins. "Perverse bundles and Calogero–Moser spaces." Compositio Mathematica 144, no. 6 (November 2008): 1403–28. http://dx.doi.org/10.1112/s0010437x0800359x.

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AbstractWe present a simple description of moduli spaces of torsion-free 𝒟-modules (𝒟-bundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with Calogero–Moser quiver varieties. Namely, we show that the moduli of 𝒟-bundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes T*X[n] in the rank-one case). The proof is based on the description of the derived category of 𝒟-modules on X by a noncommutative version of the Beilinson transform on P1.

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20

Fassarella, Thiago, and Frank Loray. "Flat parabolic vector bundles on elliptic curves." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 761 (April 1, 2020): 81–122. http://dx.doi.org/10.1515/crelle-2018-0006.

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21

Biswas, Indranil. "Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space." Annales de l’institut Fourier 47, no. 3 (1997): 885–914. http://dx.doi.org/10.5802/aif.1584.

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22

Baird, Thomas. "Moduli Spaces of Vector Bundles over a Real Curve: ℤ/2-Betti Numbers." Canadian Journal of Mathematics 66, no. 5 (October 1, 2014): 961–92. http://dx.doi.org/10.4153/cjm-2013-049-1.

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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.

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23

Bhosle, Usha N., and Indranil Biswas. "Moduli space of parabolic vector bundles on a curve." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 53, no. 2 (May 11, 2011): 437–49. http://dx.doi.org/10.1007/s13366-011-0053-7.

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24

ALBIN, PIERRE, and FRÉDÉRIC ROCHON. "SOME INDEX FORMULAE ON THE MODULI SPACE OF STABLE PARABOLIC VECTOR BUNDLES." Journal of the Australian Mathematical Society 94, no. 1 (February 2013): 1–37. http://dx.doi.org/10.1017/s144678871200047x.

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AbstractWe study natural families of $\bar {\partial } $-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.

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Bradlow, S. B., O. García-Prada, V. Muñoz, and P. E. Newstead. "Coherent Systems and Brill–Noether Theory." International Journal of Mathematics 14, no. 07 (September 2003): 683–733. http://dx.doi.org/10.1142/s0129167x03002009.

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Let X be a curve of genus g. A coherent system on X consists of a pair (E,V), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyze the cases k=1,2,3 and n=2 explicitly. For small values of α, the moduli spaces of coherent systems are related to the Brill–Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections. The study of coherent systems is applied to find the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill–Noether loci with k≤3.

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THERIAULT, STEPHEN D. "HOMOTOPY DECOMPOSITIONS OF GAUGE GROUPS OVER RIEMANN SURFACES AND APPLICATIONS TO MODULI SPACES." International Journal of Mathematics 22, no. 12 (December 2011): 1711–19. http://dx.doi.org/10.1142/s0129167x11007690.

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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.

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27

MUÑOZ, VICENTE. "HODGE STRUCTURES OF THE MODULI SPACES OF PAIRS." International Journal of Mathematics 21, no. 11 (November 2010): 1505–29. http://dx.doi.org/10.1142/s0129167x10006604.

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Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let [Formula: see text] be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of [Formula: see text] are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

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28

Ein, Lawrence. "Generalized null correlation bundles." Nagoya Mathematical Journal 111 (September 1988): 13–24. http://dx.doi.org/10.1017/s0027763000000970.

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It is well known that the moduli space of stable rank 2 vector bundles on ℙ2 of the fixed topological type is an irreducible smooth variety ([1], and [8]). There are also many known results on the classification of stable rank 2 vector bundles on ℙ3 with “small” Chern classes.

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29

HITCHING, GEORGE H. "RANK FOUR SYMPLECTIC BUNDLES WITHOUT THETA DIVISORS OVER A CURVE OF GENUS TWO." International Journal of Mathematics 19, no. 04 (April 2008): 387–420. http://dx.doi.org/10.1142/s0129167x08004716.

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The moduli space [Formula: see text] of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from [Formula: see text] to the space of even 4Θ divisors on the Jacobian variety of the curve.

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Reede, Fabian, and Ziyu Zhang. "Stability of some vector bundles on Hilbert schemes of points on K3 surfaces." Mathematische Zeitschrift 301, no. 1 (December 3, 2021): 315–41. http://dx.doi.org/10.1007/s00209-021-02920-6.

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AbstractLet X be a projective K3 surfaces. In two examples where there exists a fine moduli space M of stable vector bundles on X, isomorphic to a Hilbert scheme of points, we prove that the universal family $${\mathcal {E}}$$ E on $$X\times M$$ X × M can be understood as a complete flat family of stable vector bundles on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.

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31

Biswas, Indranil, and L. Brambila-Paz. "A note on the Picard bundle over a moduli space of vector bundles." Mathematische Nachrichten 279, no. 3 (February 2006): 235–41. http://dx.doi.org/10.1002/mana.200310358.

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32

Li, Jun. "Kodaira dimension of moduli space of vector bundles on surfaces." Inventiones Mathematicae 115, no. 1 (December 1994): 1–40. http://dx.doi.org/10.1007/bf01231752.

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Naruki, Isao. "On the moduli space $M(0, 4)$ of vector bundles." Journal of Mathematics of Kyoto University 27, no. 4 (1987): 723–30. http://dx.doi.org/10.1215/kjm/1250520608.

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Giansiracusa, Noah. "Fibonacci, Golden Ratio, and Vector Bundles." Mathematics 9, no. 4 (February 21, 2021): 426. http://dx.doi.org/10.3390/math9040426.

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There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.

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Giorgadze, Grigori, and Gega Gulagashvili. "On the splitting type of holomorphic vector bundles induced from regular systems of differential equation." Georgian Mathematical Journal 29, no. 1 (November 3, 2021): 25–35. http://dx.doi.org/10.1515/gmj-2021-2113.

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Abstract We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

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36

Biswas, Indranil, and Arijit Dey. "Brauer group of a moduli space of parabolic vector bundles over a curve." Journal of K-theory 8, no. 3 (February 18, 2011): 437–49. http://dx.doi.org/10.1017/is011001009jkt138.

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AbstractLet be a moduli space of stable parabolic vector bundles of rank n ≥ 2 and fixed determinant of degree d over a compact connected Riemann surface X of genus g(X) ≥ g(X) = 2, then we assume that n > 2. Let m denote the greatest common divisor of d, n and the dimensions of all the successive quotients of the quasi–parabolic filtrations. We prove that the Brauer group Br is isomorphic to the cyclic group ℤ/mℤ. We also show that Br is generated by the Brauer class of the Brauer–Severi variety over obtained by restricting the universal projective bundle over X × .

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37

VASSELLI, EZIO. "GAUGE-EQUIVARIANT HILBERT BIMODULES AND CROSSED PRODUCTS BY ENDOMORPHISMS." International Journal of Mathematics 20, no. 11 (November 2009): 1363–96. http://dx.doi.org/10.1142/s0129167x09005807.

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C*-endomorphisms arising from superselection structures with nontrivial center define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz–Pimsner algebra of a vector bundle having the above-mentioned rank, first Chern class and can be used to construct a duality for abstract (nonsymmetric) tensor categories versus group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e. the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogs of gauge-equivariant vector bundles in the sense of Nistor–Troitsky.

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38

Roy, Sumit. "Spectral data for parabolic projective symplectic/orthogonal Higgs bundles." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 121702. http://dx.doi.org/10.1063/5.0119058.

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Hitchin [Duke Math. J. 54(1), 91–114 (1987)] introduced a proper morphism from the moduli space of stable G-Higgs bundles [[Formula: see text] and [Formula: see text]] over a curve to a vector space of invariant polynomials, and he described the generic fibers of that morphism. In this paper, we first describe the generic Hitchin fibers for the moduli space of stable parabolic projective symplectic/orthogonal Higgs bundles without fixing the determinant. We also describe the generic fibers when the determinant is trivial.

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39

Serman, Olivier. "Moduli spaces of orthogonal and symplectic bundles over an algebraic curve." Compositio Mathematica 144, no. 3 (May 2008): 721–33. http://dx.doi.org/10.1112/s0010437x07003247.

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AbstractWe prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism $\mathcal {M}_{\mathbf {O}_r} \longrightarrow \mathcal {M}_{\mathbf {GL}_r}$ (respectively $\mathcal M_{\mathbf {Sp}_{2r}}\longrightarrow \mathcal M_{\mathbf {GL}_{2r}}$) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

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40

Brambila-Paz, L., and Herbert Lange. "A stratification of the moduli space of vector bundles on curves." Journal für die reine und angewandte Mathematik (Crelles Journal) 1998, no. 494 (January 15, 1998): 173–87. http://dx.doi.org/10.1515/crll.1998.005.

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41

Laszlo, Yves. "Local structure of the moduli space of vector bundles over curves." Commentarii Mathematici Helvetici 71, no. 1 (December 1996): 373–401. http://dx.doi.org/10.1007/bf02566426.

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42

Bigas, Montserrat Teixidor I. "Moduli Spaces of Vector Bundles on Reducible Curves." American Journal of Mathematics 117, no. 1 (February 1995): 125. http://dx.doi.org/10.2307/2375038.

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43

Kilaru, Sambaiah. "Rational curves on moduli spaces of vector bundles." Proceedings of the Indian Academy of Sciences - Section A 108, no. 3 (October 1998): 217–26. http://dx.doi.org/10.1007/bf02844479.

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44

Costa, Laura, and Rosa M. Miro-Ŕoig. "Rationality of moduli spaces of vector bundles on rational surfaces." Nagoya Mathematical Journal 165 (March 2002): 43–69. http://dx.doi.org/10.1017/s0027763000008138.

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Let X be a smooth rational surface. In this paper, we prove the rationality of the moduli space MX,L(2; c1; c2) of rank two L-stable vector bundles E on X with det (E) = c1 ∈ Pic(X) and c2(E) = c2 ≫ 0.

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45

BOLOGNESI, MICHELE, and SONIA BRIVIO. "COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$." International Journal of Mathematics 23, no. 04 (April 2012): 1250037. http://dx.doi.org/10.1142/s0129167x12500371.

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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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46

Dhillon, Ajneet. "On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn." Canadian Journal of Mathematics 58, no. 5 (October 1, 2006): 1000–1025. http://dx.doi.org/10.4153/cjm-2006-038-8.

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AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.

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Chung, Kiryong, and Sanghyeon Lee. "Stable maps of genus zero in the space of stable vector bundles on a curve." International Journal of Mathematics 28, no. 11 (October 2017): 1750078. http://dx.doi.org/10.1142/s0129167x17500781.

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Let [Formula: see text] be a smooth projective curve with genus [Formula: see text]. Let [Formula: see text] be the moduli space of stable rank two vector bundles on [Formula: see text] with a fixed determinant [Formula: see text] for [Formula: see text]. In this paper, as a generalization of Kiem and Castravet’s works, we study the stable maps in [Formula: see text] with genus [Formula: see text] and degree [Formula: see text]. Let [Formula: see text] be a natural closed subvariety of [Formula: see text] which parametrizes stable vector bundles with a fixed subbundle [Formula: see text] for a line bundle [Formula: see text] on [Formula: see text]. We describe the stable map space [Formula: see text]. It turns out that the space [Formula: see text] consists of two irreducible components. One of them parameterizes smooth rational cubic curves and the other parameterizes the union of line and smooth conics.

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CASANELLAS, MARTA, ROBIN HARTSHORNE, FLORIAN GEISS, and FRANK-OLAF SCHREYER. "STABLE ULRICH BUNDLES." International Journal of Mathematics 23, no. 08 (July 10, 2012): 1250083. http://dx.doi.org/10.1142/s0129167x12500838.

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The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces X ⊂ ℙ3. We give necessary and sufficient conditions on the first Chern class D for the existence of stable Ulrich bundles on X of rank r and c1 = D. When such bundles exist, we prove that the corresponding moduli space of stable bundles is smooth and irreducible of dimension D2 - 2r2 + 1 and consists entirely of stable Ulrich bundles (see Theorem 1.1). We are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in ℙ4, and we show that the restriction map from bundles on the threefold to bundles on the surface is generically étale and dominant.

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Hu, Yi, and Wei-Ping Li. "Birational Models of the Moduli Spaces of Stable Vector Bundles Over Curves." International Journal of Mathematics 08, no. 06 (September 1997): 781–808. http://dx.doi.org/10.1142/s0129167x97000391.

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We give a method to construct stable vector bundles whose rank divides the degree over curves of genus bigger than one. The method complements the one given by Newstead. Finally, we make some systematic remarks and observations in connection with rationality of moduli spaces of stable vector bundles.

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Tikhomirov, Aleksandr S. "Moduli of mathematical instanton vector bundles with odd $ c_2$ on projective space." Izvestiya: Mathematics 76, no. 5 (October 26, 2012): 991–1073. http://dx.doi.org/10.1070/im2012v076n05abeh002613.

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