Restriction theorems for Higgs principal bundles

We prove analogues of Grauert–Mülich and Flenner?s restriction theorems for semistable principal Higgs bundle over any smooth complex projective variety.     

Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G$G$ of rank 2 over a field. We exhibit exceptional collections of the expected length for types _A2$A_2$ and _B2=_C2$B_2=C_2$ and prove that no such collection exists for type _G2$G_2$. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G$G$-varieties for split linear algebraic groups G$G$ of rank at most 2.     

Restriction theorems for principal bundles

Let E be a semistable (or stable) principal bundle over a smooth complex projective variety X, and let DX be a complete intersection. We study the (semi)stability of the restriction E| _D . Some of the results known for vector bundles, such as Grauert–Mülich, Flenner and Mehta–Ramanathan theorems, are generalized to principal bundles. Mathematics Subject Classification (2000): 14F05, 32L05The authors are members of VBAC (Vector Bundles on Algebraic Curves), which is partially supported by EAGER (EC FP5 Contract no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101). T.G. was supported by a postdoctoral fellowship of Ministerio de Educación y Cultura (Spain), and wants to thank the Tata Institute of Fundamental Research, where this work was done while he was a postdoctoral student.     

Stability and unobstructedness of syzygy bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle E_d1,…,dn on P^N defined as the kernel of a general epimorphism

     

Looking out for stable syzygy bundles

We study (slope-)stability properties of syzygy bundles on a projective space PN given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.     

Semistability vs. nefness for (Higgs) vector bundles

Generalizing a result of Miyaoka, we prove that the semistability of a vector bundle E on a smooth projective curve over a field of characteristic zero is equivalent to the nefness of any of certain divisorial classes θs, λs in the Grassmannians Grs(E) of locally-free quotients of E and in the projective bundles PQs, respectively (here 0<s<rkE and Qs is the universal quotient bundle on Grs(E)). The result is extended to Higgs bundles. In that case a necessary and sufficient condition for semistability is that all classes λs are nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the classes λs is equivalent to the semistability of the bundle E together with the vanishing of the characteristic class .     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

On homogeneous vector bundles

Let G be a simply connected linear algebraic group, defined over the field of complex numbers, whose Lie algebra is simple. Let P be a proper parabolic subgroup of G. Let E be a holomorphic vector bundle over G/P such that E admits a homogeneous structure. Assume that E is not stable. Then E admits a homogeneous structure with the following property: There is a nonzero subbundle F?E left invariant by the action of G such that degree(F)/rank(F)?degree(E)/rank(E).     

Numerically effective line bundles associated to a stable bundle over a curve

Let G be a complex semisimple group and χ a character of a parabolic subgroup P⊂G such that the associated line bundle on G/P is ample. For a general stable G-bundle E_G over a compact Riemann surface of genus at least two, the line bundle over E_G/P defined by χ has the property that the restriction of  to any closed subvariety of E_G/P of smaller dimension is ample, although is not ample.     

A relation between the parabolic Chern characters of the de Rham bundles

In this paper, we consider the weight i de Rham–Gauss–Manin bundles on a smooth variety arising from a smooth projective morphism for . We associate to each weight i de Rham bundle, a certain parabolic bundle on S and consider their parabolic Chern characters in the rational Chow groups, for a good compactification S of U. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.     

Torus fixed points of moduli spaces of stable bundles of rank three

By a result of Klyachko the Euler characteristic of moduli spaces of stable bundles of rank two on the projective plane is determined. Using similar methods we extend this result to bundles of rank three. The fixed point components correspond to moduli spaces of the subspace quiver. Moreover, the stability condition is given by a certain system of linear inequalities so that the generating function of the Euler characteristic can be determined explicitly.     

A stability condition for line bundles on reducible varieties

Over a family of varieties with singular special fiber, the relative Picard functor (i.e. the moduli space of line bundles) may fail to be compact. We propose a stability condition for line bundles on reducible varieties that is aimed at compactifying it. This stability condition generalizes the notion of ‘balanced multidegree’ used by Caporaso in compactifying the relative Picard functor over families of curves. Unlike the latter, it is defined ‘asymptotically’; an important theme of this paper is that although line bundles on higher-dimensional varieties are more complicated than those on curves, their behavior in terms of stability asymptotically approaches that of line bundles on curves.Using this definition of stability, we prove that over a one-parameter family of varieties having smooth total space, any line bundle on the generic fiber can be extended to a unique semistable line bundle on the (possibly reducible) special fiber, provided the special fiber is not too complicated in a combinatorial sense.     

Vector bundles on ruled Fano 3-folds

Let be a ruled Fano 3-fold. The goal of this paper is to compute the dimension, prove the irreducibility and smoothness and describe the structure of the moduli space M _L (2;c ₁,c ₂) of L-stable, rank 2 vector bundles E on X with certain Chern classes and for a suitable polarization L closely related to c ₂. More precisely, we will cover the study of some moduli spaces M _L (2;c ₁,c ₂) such that the generic point is given as a non-trivial extension of line bundles. This work nicely reflects the general philosophy that moduli spaces inherits a lot of geometrical properties of the underlying variety. Received: 16 February 1999 / Revised version: 2 July 1999     

Algebraic loop groups and moduli spaces of bundles

We study algebraic loop groups and affine Grassmannians in positive characteristic. The main results are normality of Schubert-varieties, the construction of line-bundles on the affine Grassmannian, and the proof that they induce line-bundles on the moduli-stack of torsors. Received March 15, 2001 / final version received April 11, 2002?Published online November 19, 2002