Brill-Noether theory on Hirzebruch surfaces

In this paper, we investigate higher rank Brill-Noether problems for stable vector bundles on Hirzebruch surfaces. Using suitable non-splitting extensions, we deal with the non-emptiness. Results concerning the emptiness follow as a consequence of a generalization of Clifford’s theorem for line bundles on curves to vector bundles on surfaces.     

The d-very ampleness of adjoint line bundles on quasi-elliptic surfaces

In this paper, we give a numerical criterion of Reider-type for the d-very ampleness of the adjoint line bundles on quasi-elliptic surfaces, and meanwhile we give a new proof of the vanishing theorem on quasi-elliptic surfaces emailed from Langer and show that it is the optimal version.     

Exceptional sequences on rational {\mathbb{C}^{*}} -surfaces

Inspired by Bondal’s conjecture, we study the behavior of exceptional sequences of line bundles on rational ${\mathbb{C}^{*}}$ -surfaces under homogeneous degenerations. In particular, we provide a sufficient criterion for such a sequence to remain exceptional under a given degeneration. We apply our results to show that, for toric surfaces of Picard rank 3 or 4, all full exceptional sequences of line bundles may be constructed via augmentation. We also discuss how our techniques may be used to construct noncommutative deformations of derived categories.     

Embeddings of general blowups of abelian surfaces

Blowups of algebraic surfaces polarized by tensor powers of ample line bundles were studied by Coppens [8]. In this note we complete the picture in the case of abelian surfaces studying blowups of surfaces polarized by primitive line bundles. Received: 16 November 1999 / Revised version: 13 June 2000     

Projective normality of abelian surfaces¶given by primitive line bundles

In this paper, we study projective normality of abelian surfaces, with embeddings given by ample line bundles of type (1,d). We show that if d≥ 7, the generic abelian surface is projectively normal. Received: 12 June 1998     

Okounkov bodies and Seshadri constants

Okounkov bodies, which are closed convex sets defined for big line bundles, have rich information on the line bundles. On the other hand, Seshadri constants are invariants which measure the positivity of line bundles. In this paper, we prove that Okounkov bodies give lower bounds of Seshadri constants.     

A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z, except for three exceptional cases. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety. Oblatum 17-VI-1999 & 10-IX-2001?Published online: 19 November 2001     

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.     

$\Omega$-admissible theory II. Deligne pairings over moduli spaces of punctured Riemann surfaces

In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics. This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for -admissible metrized line bundles depend on . In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric on the moduli space is algebraic. Received February 14, 2000 / Accepted August 18, 2000 / Published online February 5, 2001     

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.     

Seshadi constants via functions on Newton–Okounkov bodies

The aim of this note is to establish a somewhat surprising connection between functions on Newton–Okounkov bodies and Seshadri constants of line bundles on algebraic surfaces.     

A Note on General Blow-Ups of Elliptic Quasi-Bundles

In this paper we give some numerical conditions for a line bundle on a general blow-up of elliptic quasi bundles to give an embedding of order k.     

一类第一 Betti数为奇数的曲面上的向量丛

本文得到一些有关一类第一Betti数为奇数的曲面上的全纯向量丛的结果,以及例外Hopf曲面上的集合IS2(X,0)的描述．     

On syzygies of projective bundles over abelian varieties

Syzygies or $N_p$-property of an ample line bundles on abelian varieties are well known. In this paper, we study defining equations and syzygies among them of projective bundles over abelian varieties. We prove an analogue of Pareschi's theorem (or Lazarsfeld's conjecture) on abelian varieties, extended to projective bundles over an abelian variety.     

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.     

Some results about the projective normality of abelian varieties

We reduce the problem of the projective normality of polarized abelian varieties to check the rank of very explicit matrices. This allows us to prove some results on normal generation of primitive line bundles on abelian threefolds and fourfolds. We also give two situations where the projective normality always fails. Finally we make some conjecture. Received: 1 September 2004; revised: 10 March 2005     

Derived categories of Keum's fake projective planes

We conjecture that derived categories of coherent sheaves on fake projective n  -spaces have a semi-orthogonal decomposition into a collection of n+1$n+1$ exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group (such as Keum's surface). Then by passing to equivariant categories we construct new examples of phantom categories with both Hochschild homology and Grothendieck group vanishing.     

Hom stacks

We study Hom 2-functors parameterizing 1-morphisms of algebraic stacks, and prove that they are representable by algebraic stacks under certain conditions, using Artin's criterion. As an application we study Picard 2-functors which parameterize line bundles on algebraic stacks. An erratum to this article is available at .