Smooth Projective Curve Research Articles

A functorial approach to the stability of vector bundles

On a normal projective variety the locus of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable bundles that remain μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable on all Galois covers prime to the characteristic is open in the moduli space of Gieseker semistable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable bundles. As an application we obtain a very different behaviour of the étale fundamental group in positive vs. characteristic 0.

Counting maximal isotropic subbundles of orthogonal bundles over a curve

Let C be a smooth projective curve and V an orthogonal bundle over C. Let IQe(V) be the isotropic Quot scheme parameterizing degree e isotropic subsheaves of maximal rank in V. We give a closed formula for intersection numbers on components of IQe(V) whose generic element is saturated. As a special case, for g≥2, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in [20], and of the symplectic case studied in [8].

The Dirac–Higgs Complex and Categorification of (BBB)-Branes

Abstract Let ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$. Following Gaiotto’s suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.

The canonical representation of the Drinfeld curve

AbstractIf is a smooth projective curve over an algebraically closed field and is a group of automorphisms of , the canonical representation of is given by the induced ‐linear action of on the vector space of holomorphic differentials on . Computing it is still an open problem in general when the cover is wildly ramified. In this paper, we fix a prime power , we consider the Drinfeld curve, that is, the curve given by the equation over together with its standard action by , and decompose as a direct sum of indecomposable representations of , thus solving the aforementioned problem in this case.

Slope inequality for an arbitrary divisor

Let [Formula: see text] be a surjective morphism with connected fibers from a smooth complex projective surface [Formula: see text] to a smooth complex projective curve [Formula: see text] with general fiber [Formula: see text]. In this paper, we develop a more general version of the slope inequality for data [Formula: see text], where [Formula: see text] is an arbitrary relatively effective divisor on [Formula: see text] and [Formula: see text] is a locally free sub-sheaf of [Formula: see text]. We see how the speciality of [Formula: see text], restricted to the general fiber, plays a role in the results. Moreover, we compute some natural examples and provide applications.

Extension groups of tautological bundles on punctual Quot schemes of curves

We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot–Chow morphism of tensor and wedge products of duals of tautological bundles.

Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components

Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components

The spectrum of Grothendieck monoid: Classifying Serre subcategories and reconstruction theorem

The Grothendieck monoid of an exact category is a monoid version of the Grothendieck group. We use it to classify Serre subcategories of an exact category and to reconstruct the topology of a noetherian scheme. We first construct bijections between (i) the set of Serre subcategories of an exact category, (ii) the set of faces of its Grothendieck monoid, and (iii) the monoid spectrum of its Grothendieck monoid. By using (ii), we classify Serre subcategories of exact categories related to a finite dimensional algebra and a smooth projective curve. For this, we determine the Grothendieck monoid of the category of coherent sheaves on a smooth projective curve. By using (iii), we introduce a topology on the set of Serre subcategories. As a consequence, we recover the topology of a noetherian scheme from the Grothendieck monoid.

Automorphisms of moduli spaces of principal bundles over a smooth curve

For any almost-simple group [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero, we describe the automorphism group of the moduli space of semistable [Formula: see text]-bundles over a connected smooth projective curve [Formula: see text] of genus at least [Formula: see text]. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers.

On actions of Frobenius morphisms for moduli stacks of principal bundles over algebraic curves

We study the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the l-adic cohomology of the moduli stack in terms of Chern classes.

Torsors on moduli spaces of principal G-bundles on curves

Let [Formula: see text] be a semisimple complex algebraic group with a simple Lie algebra [Formula: see text], and let [Formula: see text] denote the moduli stack of topologically trivial stable [Formula: see text]-bundles on a smooth projective curve [Formula: see text]. Fix a theta characteristic [Formula: see text] on [Formula: see text] which is even in case [Formula: see text] is odd. We show that there is a nonempty Zariski open substack [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text], for all [Formula: see text]. It is shown that any such [Formula: see text] has a canonical connection. It is also shown that the tangent bundle [Formula: see text] has a natural splitting, where [Formula: see text] is the restriction of [Formula: see text] to the semi-stable locus. We also produce an isomorphism between two naturally occurring [Formula: see text]-torsors on the moduli space of regularly stable [Formula: see text].

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

Let L=(L,[⋅,⋅],δ) be an algebraic Lie algebroid over a smooth projective curve X of genus g≥2 such that L is a line bundle whose degree is less than 2−2g. Let r and d be coprime numbers. We prove that the motivic class of the moduli space of L-connections of rank r and degree d over X does not depend on the Lie algebroid structure [⋅,⋅] and δ of L and neither on the line bundle L itself, but only on the degree of L (and of course on r, d and X). In particular it is equal to the motivic class of the moduli space of KX(D)-twisted Higgs bundles of rank r and degree d, for D any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding E-polynomials. Some applications of these results are then deduced.

Periodicity of Hitchin’s Uniformizing Higgs Bundles

Abstract Let $C$ be a smooth projective curve over ${{\mathbb{C}}}$. We link the periodicity of Hitchin’s uniformizing Higgs bundle of $C$ with the underlying arithmetic geometry of the curve. Some new relations are discovered. We also speculate on the whole class of periodic Higgs bundles.

Infinitesimal Deformations of Some Quot Schemes

Abstract Let $E$ be a vector bundle on a smooth complex projective curve $C$ of genus at least two. Let $\mathcal{Q}(E,d)$ be the Quot scheme parameterizing the torsion quotients of $E$ of degree $d$. We compute the cohomologies of the tangent bundle $T_{\mathcal{Q}(E,d)}$. In particular, the space of infinitesimal deformations of $\mathcal{Q}(E,d)$ is computed. Kempf and Fantechi computed the space of infinitesimal deformations of $\mathcal{Q}({\mathcal O}_{C},d)\,=\, C^{(d)}$ [ 11, 19].

Tamely Ramified Geometric Langlands Correspondence in Positive Characteristic

Abstract We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for $GL_{n}(k)$, where $k$ is an algebraically closed field of characteristic $p> n$. Let $X$ be a smooth projective curve over $k$ with marked points, and fix a parabolic subgroup of $GL_{n}(k)$ at each marked point. We denote by $\operatorname{Bun}_{n,P}$ the moduli stack of (quasi-)parabolic vector bundles on $X$, and by $\mathcal{L}oc_{n,P}$ the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}({\mathcal{L}oc_{n,P}^{0}}))$ of quasi-coherent sheaves on an open substack $\mathcal{L}oc_{n,P}^{0}\subset \mathcal{L}oc_{n,P}$, and the bounded derived category $D^{b}(\mathcal{D}^{0}_{{\operatorname{Bun}}_{n,P}}\operatorname{-mod})$ of $\mathcal{D}^{0}_{{\operatorname{Bun}}_{n,P}}$-modules, where $\mathcal{D}^{0}_{\operatorname{Bun}_{n,P}}$ is a localization of $\mathcal{D}_{\operatorname{Bun}_{n,P}}$ the sheaf of crystalline differential operators on $\operatorname{Bun}_{n,P}$. Thus, we extend the work of Bezrukavnikov–Braverman [ 8] to the tamely ramified case. We also prove a correspondence between flat connections on $X$ with regular singularities and meromorphic Higgs bundles on the Frobenius twist $X^{(1)}$ of $X$ with first-order poles.

The virtual intersection theory of isotropic Quot schemes

Isotropic Quot schemes parameterize rank r isotropic subsheaves of a vector bundle equipped with symplectic or symmetric quadratic form. We define a virtual fundamental class for isotropic Quot schemes over smooth projective curves. Using torus localization, we prescribe a way to calculate top intersection numbers of tautological classes, and obtain explicit formulas when r=2. These include and generalize the Vafa-Intriligator formula. In this setting, we compare the Quot scheme invariants with the invariants obtained via the stable map compactification.

Genuinely ramified maps and monodromy

For any genuinely ramified morphism f:Y⟶X between irreducible smooth projective curves we prove that (Y×XY)∖Δ‾ is connected, where Δ⊂Y×XY is the diagonal. Using this result the following are proved:(1)If f is further Morse then the Galois closure is the symmetric group Sd, where d=degree(f).(2)The Galois group of the general projection, to a line, of any smooth curve X⊂Pn of degree d, which is not contained in a hyperplane and contains a non-flex point, is Sd.

The Whittaker Functional Is a Shifted Microstalk

AbstractFor a smooth projective curve X and reductive group G, the Whittaker functional on nilpotent sheaves on $$Bun _G(X)$$ B u n G ( X ) is expected to correspond to global sections of coherent sheaves on the spectral side of Betti geometric Langlands. We prove that the Whittaker functional calculates the shifted microstalk of nilpotent sheaves at the point in the Hitchin moduli where the Kostant section intersects the global nilpotent cone. In particular, the shifted Whittaker functional is exact for the perverse t-structure and commutes with Verdier duality. Our proof is topological and depends on the intrinsic local hyperbolic symmetry of $$Bun _G(X)$$ B u n G ( X ) . It is an application of a general result relating vanishing cycles to the composition of restriction to an attracting locus followed by vanishing cycles.

On the standard conjecture for a fourfold with $1$-parameter fibration by Abelian varieties

It is proved that the Grothendieck standard conjecture $B(X)$ of Lefschetz type holds for a smooth complex projective 4-dimensional variety $X$ provided that there exists a morphism of $X$ onto a smooth projective curve whose generic scheme fibre is an Abelian variety with bad semi-stable reduction at some place of the curve.

Geometry of the logarithmic Hodge moduli space

AbstractWe show the smoothness over the affine line of the Hodge moduli space of logarithmic ‐connections of coprime rank and degree on a smooth projective curve with geometrically integral fibers over an arbitrary Noetherian base. When the base is a field, we also prove that the Hodge moduli space is geometrically integral. Along the way, we prove the same results for the corresponding moduli spaces of logarithmic Higgs bundles and of logarithmic connections. We use smoothness to derive specialization isomorphisms on the étale cohomology rings of these moduli spaces; this includes the special case when the base is of mixed characteristic. In the special case where the base is a separably closed field of positive characteristic, we show that these isomorphisms are filtered isomorphisms for the perverse filtrations associated with the corresponding Hitchin‐type morphisms.