Moduli Space Of Vector Bundles Research Articles

Vector bundles on blown-up Hopf surfaces

We show that certain moduli spaces of vector bundles over blown-up primary Hopf surfaces admit no compact components. These are the moduli spaces used by Andrei Teleman in his work on the classification of class VII surfaces.

Moduli space of parabolic vector bundles on a curve

Let X be an irreducible smooth complex projective curve. Faltings gave a cohomological criterion for vector bundles on X to be semistable. Using this criterion he gave a construction of the moduli spaces of vector bundles on X (Faltings in J Alg Geom 2:507–568, 1993). An analogous cohomological criterion for semistable parabolic vector bundles on X is now known. Our aim here is to give a construction of the moduli spaces of parabolic vector bundles using this criterion.

The Brauer group of moduli spaces of vector bundles over a real curve

LetXXbe a geometrically connected smooth projective curve of genusgX≥2g_X \geq 2overR\mathbb {R}. LetM(r,ξ)M(r, \xi )be the coarse moduli space of geometrically stable vector bundlesEEoverXXof rankrrand determinantξ\xi, whereξ\xiis a real point of the Picard varietyPic_d(X)\underline {\mathrm {Pic}}^d( X). IfgX=r=2g_X = r = 2, then letddbe odd. We compute the Brauer group ofM(r,ξ)M(r,\xi ).

Brill-Noether theory for moduli spaces of sheaves on algebraic varieties

Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be an ample line bundle on $X$. Let $M_{X,H}(r;c_1, ..., c_{s})$ be the moduli space of $H$-stable vector bundles $E$ on $X$ of rank $r$ and Chern classes $c_i(E)=c_i$ for $i=1, ..., s:=min\{r,n\}$. We define the Brill-Noether filtration on $M_{X,H}(r;c_1, ..., c_{s})$ as $W_{H}^{k}(r;c_1,..., c_{s})= \{E \in M_{X,H}(r;c_1, ..., c_{s}) | h^0(X,E) \geq k \}$ and we realize $W_{H}^{k}(r;c_1,..., c_{s})$ as the $k$th determinantal variety of a morphism of vector bundles on $M_{X,H}(r;c_1, ..., c_{s})$, provided $H^i(E)=0$ for $i \geq 2$ and $E \in M_{X,H}(r;c_1, ..., c_{s})$. We also compute the expected dimension of $W_{H}^{k}(r;c_1,..., c_{s})$. Very surprisingly we will see that the Brill-Noether stratification allow us to compare moduli spaces of vector bundles on Hirzebruch surfaces stables with respect to different polarizations. We will also study the Brill-Noether loci of the moduli space of instanton bundles and we will see that they have the expected dimension.

Degeneration of the strange duality map for symplectic bundles

Global sections of the line bundles on a moduli space of vector bundles (or, more generally, principal G-bundles) are called generalized theta functions. The dimension of the vector spaces of generalized theta functions is given by the celebrated Verlinde formula. If you compute several Verlinde numbers, you will find that some of them coincide unexpectedly. Behind the coincidence, there is often geometric meaning. Let SU(r) be the moduli space of rank r bundles with trivial determinant on a smooth projective curve C of genus g, and let L be the ample generator of PicSU(r). Let U(n) be the moduli space of rank n bundles of slope g − 1 on C, and let U(n) ⊃ Θ be the locus of E ∈ U(n) such that H(C,E) 6= 0. Then by the Verlinde formula, you see that the dimensions of the vector spaces H0(SU(r),L⊗n) and H0(U(n),O(Θ)⊗r) are equal. There is a strange duality map (cf. [B1]) H0(SU(r),L⊗n)∗ → H0(U(n),O(Θ)⊗r),

Maps between moduli spaces of vector bundles and the base locus of the theta divisor

We consider maps between different spaces of vector bundles on curves obtained by taking wedge powers, elementary transformations or kernels of evaluation maps and studying their respective fibers. We apply the results to construct large dimensional sets in the base locus of the generalized theta divisor.

Singularities of moduli spaces of vector bundles over curves in characteristic 0 and p

Singularities of moduli spaces of vector bundles over curves in characteristic 0 and p

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Atiyah and Bott used equivariant Morse theory applied to the Yang–Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SLn. This article attempts to survey and extend our understanding of this link between Yang–Mills theory and Tamagawa numbers, and to explain how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth projective curve over ℂ can be adapted to the setting of 𝔸1-homotopy theory to study the motivic cohomology of these moduli spaces over an algebraically closed field.

TWO REMARKS ON SUBVARIETIES OF MODULI SPACES

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.

On the rational homotopy type of a moduli space of vector bundles over a curve

We study the rational homotopy of the moduli space N-X that parametrizes the isomorphism classes of all stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g, with g >= 2. The symplectic group Aut(H-1(X, Z)) congruent to Sp(2g, Z) has a natural action on the rational homotopy groups pi(n)(N-X)circle times(Z)Q. We prove that this action extends to an action of Sp(2g, C) on pi(n)(N-X)circle times C-Z. We also show that pi(n)(N-X)circle times C-Z is a non-trivial representation of Sp(2g, C) congruent to Aut (H-1(X, C)) for all n >= 2g - 1. In particular, N-X is a rationally hyperbolic space. In the special case where g = 2, for each n is an element of N, we compute the leading Sp(2g, C) representation occurring in pi(n)(N-X)circle times C-Z.

HITCHIN SYSTEMS ON SINGULAR CURVES II: GLUING SUBSCHEMES

In this paper we continue our studies of Hitchin systems on singular curves (started in [1]). We consider a rather general class of curves which can be obtained from the projective line by gluing two subschemes together (i.e. their affine part is: Spec {f ∈ ℂ[z] : f(A(∊)) = f(B(∊)); ∊N = 0}, where A(∊), B(∊) are arbitrary polynomials). The most simple examples are the generalized cusp curves which are projectivizations of Spec {f ∈ ℂ[z] : f′(0) = f″(0) = ⋯ fN-1(0) = 0}. We describe the geometry of such curves; in particular we calculate their genus (for some curves the calculation appears to be related with the iteration of polynomials A(∊), B(∊) defining the subschemes). We obtain the explicit description of moduli space of vector bundles, the dualizing sheaf, Higgs field and other ingredients of the Hitchin integrable systems; these results may deserve the independent interest. We prove the integrability of Hitchin systems on such curves. To do this we develop r-matrix formalism for the functions on the truncated loop group GLn(ℂ[z]), zN = 0. We also show how to obtain the Hitchin integrable systems on such curves as hamiltonian reduction from the more simple system on some finite-dimensional space.

Raynaud’s vector bundles and base points of the generalized Theta divisor

We study base points of the generalized Θ-divisor on the moduli space of vector bundles on a smooth algebraic curve X of genus g defined over an algebraically closed field. To do so, we use the derived categories Db(Pic0(X)) and Db(Jac(X)) and the equivalence between them given by the Fourier–Mukai transform \({\rm FM}_\mathcal {P}\) coming from the Poincaré bundle. The vector bundles P m on the curve X defined by Raynaud play a central role in this description. Indeed, we show that E is a base point of the generalized Θ-divisor, if and only if there exists a non-trivial homomorphism Prk(E)g+1 → E.

The generalized Verschiebung map for curves of genus 2

Let C be a smooth curve, and M r (C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We examine the generalized Verschiebung map $$V_{r}: M_{r}(C^{(p)}) \dashrightarrow M_{r}(C)$$ induced by pulling back under Frobenius. Our main result is a computation of the degree of V 2 for a general C of genus 2, in characteristic p > 2. We also give several general background results on the Verschiebung in an appendix.

Infinitesimal deformations of the tangent bundle of a moduli space of vector bundles over a curve

Fix a line bundle $\xi$ on a connected smooth complex projective curve $X$ of genus at least three. Let $\mathcal{N}$ denote the moduli space of all stable vector bundles over $X$ of rank $n$ and determinant $\xi$. We assume that $n\geq 3$ and coprime to $\operatorname{degree}(\xi)$; If $\operatorname{genus}(X)\leq 4$, then we also assume that $n \geq 4$. We prove that $H^i(\mathcal{N}, \End(T\mathcal{N}\mkern2mu)) = H^i(X, \mathcal{O}_X)$ for $i= 0,1$.

Vector bundles and theta functions on curves of genus 2 and 3

Let SU C (r) be the moduli space of vector bundles of rank r and trivial determinant on a curve C . A general E in SU C (r) defines a divisor Θ E in the linear system | r Θ|, where Θ is the canonical theta divisor in Pic g -1 ( C ). This defines a rational map Θ: SU C (r) → | r Θ|, which turns out to be the map associated to the determinant bundle on SU C (r) (the positive generator of Pic ( SU C (r) ). In genus 2 we prove that this map is generically finite and dominant. The same method, together with some classical work of Morin, shows that in rank 3 and genus 3 the theta map is a finite morphism - in other words, every vector bundle in SU C (3) admits a theta divisor.

Chow group of 1-cycles on the moduli space of vector bundles of rank 2 over a curve

Let X be a non-singular complex projective curve of genus ≥3. Choose a point x ∈ X. Let Mx be the moduli space of stable bundles of rank 2 with determinant Open image in new window We prove that the Chow group CHQ1(Mx) of 1-cycles on Mx with rational coefficients is isomorphic to CHQ0(X). By studying the rational curves on Mx, it is not difficult to see that there exits a natural homomorphism CH0(J)→CH1(Mx) where J denotes the Jacobian of X. The crucial point is to show that this homomorphism induces a homomorphism CH0(X)→CH1(Mx), namely, to go from the infinite dimensional object CH0(J) to the finite dimensional object CH0(X). This is proved by relating the degeneration of Hecke curves on Mx to the second term I*2 of Bloch's filtration on CH0(J).

Moduli of Framed Parabolic Sheaves

This paper gives a construction of the moduli space of framed parabolic sheaves on a Riemann surface. This space serves as a universal, master, space for the well known moduli space of parabolic bundles, as well as moduli spaces of vector bundles, which can all be obtained from this space by torus quotients. The construction is given for the structure group SL(N, C), and indeed is adapted to this case. At the end of the paper, an approach is suggested for dealing with the case of arbitrary reductive groups, involving the associated loop group.

A Gieseker type degeneration of moduli stacks of vector bundles on curves

We construct a new degeneration of the moduli stack of vector bundles over a smooth curve when the curve degenerates to a singular curve which is irreducible with one double point. We prove that the total space of the degeneration is smooth and its special fibre is a divisor with normal crossings. Furthermore; we give a precise description of how the normalization of the special fibre of the degeneration is related to the moduli space of vector bundles over the desingularized curve.

Hitchin System on Singular Curves

We study the Hitchin system on singular curves. We consider curves obtainable from the projective line by matching at several points or by inserting cusp singularities. It appears that on such singular curves, all basic ingredients of Hitchin integrable systems (moduli space of vector bundles, dualizing sheaf, Higgs field, etc.) can be explicitly described, which can be interesting in itself. Our main result is explicit formulas for the Hitchin Hamiltonians. We also show how to obtain the Hitchin integrable system on such curves by Hamiltonian reduction from a much simpler system on a finite-dimensional space. We pay special attention to a degenerate curve of genus two for which we find an analogue of the Narasimhan–Ramanan parameterization of the moduli space of SL(2) bundles as well as the explicit expressions for the symplectic structure and Hitchin-system Hamiltonians in these coordinates. We demonstrate the efficiency of our approach by rederiving the rational and trigonometric Calogero–Moser systems, which are obtained from Hitchin systems on curves with a marked point and with the respective cusp and node.

THETA FUNCTIONS ON THE MODULI SPACE OF PARABOLIC BUNDLES

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.