A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over and let L be a line bundle on X. In this paper we shall construct a moduli space parametrizing -cohomology L-twisted Higgs pairs, i.e., pairs where E is a vector bundle on X and . If we take , the canonical line bundle on X, the variety is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section , one can define, in a natural way, a Poisson structure on . We also analyze the relations between this Poisson structure on and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves. Received November 4, 1997; in final form May 28, 1998     

Generalized parabolic sheaves on an integral projective curve

We extend the notion of a parabolic vector bundle on a smooth curve to define generalized parabolic sheaves (GPS) on any integral projective curve X. We construct the moduli spacesM(X) of GPS of certain type onX. IfX is obtained by blowing up finitely many nodes inY then we show that there is a surjective birational morphism from M(X) to M (Y). In particular, we get partial desingularisations of the moduli of torsion-free sheaves on a nodal curveY.     

Donagi–Morrison’s examples on 2-elementary K3 surfaces

Let X be an algebraic K3 surface, and let L be a base point free and big line bundle on X. If X admits a map of degree 2 to the projective plane branched over a smooth sextic and L is the pullback of the line bundle O_\mathbbP²(3),{\mathcal{O}_{\mathbb{P}^{2}}(3),} then the gonality of the smooth curves of the complete linear system |L| is not constant. The polarized K3 surface (X, L) consisting of the K3 surface X and the line bundle L is called Donagi–Morrison’s example. In this paper, we give a necessary and sufficient condition for the polarized K3 surface (X, L) consisting of a 2-elementary K3 surface X and an ample line bundle L to be Donagi–Morrison’s example.     

Semistability of Logarithmic Cotangent Bundle on Some Projective Manifolds

In this article, we investigate the semistability of logarithmic de Rham sheaves on a smooth projective variety (X, D), under suitable conditions. This is related to existence of Kähler–Einstein metric on the open variety. We investigate this problem when the Picard number is one. Fix a normal crossing divisor D on X and consider the logarithmic de Rham sheaf Ω _X (log D) on X. We prove semistability of this sheaf, when the log canonical sheaf K _X  + D is ample or trivial, or when ?K _X  ? D is ample, i.e., when X is a log Fano n-fold of dimension n ≤ 6. We also extend the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.     

Geodesics On The Symplectomorphism Group

Let M be a compact manifold with a symplectic form ω and consider the group D_w{\mathcal{D}_\omega} consisting of diffeomorphisms that preserve ω. We introduce a Riemannian metric on M which is compatible with ω and use it to define an L ²-inner product on vector fields on M. Extending by right invariance we get a weak Riemannian metric on D_w{\mathcal{D}_\omega} . We show that this metric has geodesics which come from integral curves of a smooth vector field on the tangent bundle of D_w{\mathcal{D}_\omega} . Then, estimating the growth of such geodesics, we show that they extend globally.     

Vector bundles with a fixed determinant on an irreducible nodal curve

LetM be the moduli space of generalized parabolic bundles (GPBs) of rankr and degree dona smooth curveX. LetM _−L be the closure of its subset consisting of GPBs with fixed determinant− L. We define a moduli functor for whichM _−L is the coarse moduli scheme. Using the correspondence between GPBs onX and torsion-free sheaves on a nodal curveY of whichX is a desingularization, we show thatM _−L can be regarded as the compactified moduli scheme of vector bundles onY with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves onY. The relation to Seshadri-Nagaraj conjecture is studied.     

Foliations by curves on threefolds

We study the conormal sheaves and singular schemes of one-dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ-stable whenever the tangent bundle $TX$ is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on ${\mathbb{P}}^3$ and on a smooth quadric hypersurface $Q_3\subset {\mathbb{P}}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q₃.     

Splitting of vector bundles on algebraic curves and elementary transformations

Let Ebe a rank nvector bundle on a smooth projective curve X. It is known that Emay be obtained from a splitted bundle +1≤i≤ L_i;, rank(L_i) = 1, by a finite number of elementary transformations. Here we give upper bounds for their minimal number. If n= 2 this is related to the order of stability of E.     

Seshadri-exceptional foliations

 The maximal Seshadri number μ(L) of an ample line bundle L on a smooth projective variety X measures the local positivity of the line bundle L at a general point of X. By refining the method of Ein-Küchle-Lazarsfeld, lower bounds on μ(L) are obtained in terms of L ⁿ , n=dim(X), for a class of varieties. The main idea is to show that if a certain lower bound is violated, there exists a non-trivial foliation on the variety whose leaves are covered by special curves. In a number of examples, one can show that such foliations must be trivial and obtain lower bounds for μ(L). The examples include the hyperplane line bundle on a smooth surface in ℙ³ and ample line bundles on smooth threefolds of Picard number 1. Received: 29 June 2001 / Published online: 16 October 2002 RID="⋆" ID="⋆" Supported by Grant No. 98-0701-01-5-L from the KOSEF. RID="⋆⋆" ID="⋆⋆" Supported by Grant No. KRF-2001-041-D00025 from the KRF.     

Estimates of sections of determinant line bundles on Moduli spaces of pure sheaves on algebraic surfaces

Let X be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on X, i.e. ${M_X^H(u)}$ with u?=?(0, L, χ(u)?=?0) and L an effective line bundle on X, together with a series of determinant line bundles associated to ${r[\mathcal{O}_X]-n[\mathcal{O}_{pt}]}$ in the Grothendieck group of X. Let g _L denote the arithmetic genus of curves in the linear system |L|. For g _L ?≤?2, we give a upper bound of the dimensions of sections of these line bundles by restricting them to a generic projective line in |L|. Our result gives, together with G?ttsche’s computation, a first step of a check for the strange duality for some cases for X a rational surface.     

On the Existence of a Compact Generator on the Derived Category of a Noetherian Formal Scheme

In this paper, we prove that for a noetherian formal scheme \mathfrak X\mathfrak X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(\mathfrak X)\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X) is generated by a single compact object. In an Appendix we prove that the category of compact objects in D_qct(\mathfrak X)\boldsymbol{\mathsf{D}}_\mathsf{qct}(\mathfrak X) is skeletally small.     

Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

Log canonical thresholds of divisors on Grassmannians

Let L be the Plücker line bundle on the Grassmannian. Given D ∈ |kL|, we show that the log canonical threshold of D is at least . The main ingredients of the proof are Kapranov's result on the derived category of coherent sheaves on the Grassmannian, Nadel's vanishing theorem for multiplier ideal sheaves, and Demailly's vanishing theorem for vector bundles.     

Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives

This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve and n is its multiplicity, then there is a filtration C₁ = C ? C₂ ? … ? C_n = Y such that C is the reduced curve associated to Y, and for every P ∈ C, if z ∈ O_Y,P is an equation of C then (zⁱ) is the ideal of C_i in O_Y,P. A coherent sheaf on Y is called torsion free if it does not have any non zero subsheaf with finite support. We prove that torsion free sheaves are reflexive. We study then the quasi locally free sheaves, i.e., sheaves which are locally isomorphic to direct sums of the O_Ci.We define an invariant for these sheaves, the complete type, and prove the irreducibility of the set of sheaves of given complete type. We study the generic quasi locally free sheaves, with applications to the moduli spaces of stable sheaves on Y (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Principal bundles whose restriction to curves are trivial

Let k be an algebraically closed field and X a smooth projective variety defined over k. Let E_G be a principal G–bundle over X, where G is an algebraic group defined over k, with the property that for every smooth curve C in X the restriction of E_G to C is the trivial G–bundle. We prove that the principal G–bundle E_G over X is trivial. We also give examples of nontrivial principal bundle over a quasi-projective variety Y whose restriction to every smooth curve in Y is trivial.     

Best simultaneous approximation in L∞(μ, X)

Let Y be a reflexive subspace of the Banach space X, let (Ω, Σ, μ) be a finite measure space, and let L_∞(μ, X) be the Banach space of all essentially bounded μ ‐Bochner integrable functions on Ω with values in X, endowed with its usual norm. Let us suppose that Σ₀ is a sub‐σ ‐algebra of Σ, and let μ₀ be the restriction of μ to Σ₀. Given a natural number n, let N be a monotonous norm in ?ⁿ . We prove that L_∞(μ, Y) is N ‐simultaneously proximinal in L_∞(μ,X), and that if X is reflexive then L_∞(μ₀, X) is N ‐simultaneously proximinal in L_∞(μ, X) in the sense of Fathi, Hussein, and Khalil [3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)