Holomorphic symmetric differentials and a birational characterization of abelian varieties

A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.     

A bound on the Castelnuovo-Mumford regularity for curves

Recall that a projective curve in with ideal sheaf is said to be n-regular if for every integer and that in this case, it is cut out scheme-theoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus satisfies a smaller regularity bound than the optimal one . For example, if then a curve is -regular unless it is embedded by a complete linear system of degree . Received: 29 May 2000 / Published online: 24 September 2001     

A combinatorial classification of 2-regular simple modules for Nakayama algebras

Enomoto showed for finite dimensional algebras that the classification of exact structures on the category of finitely generated projective modules can be reduced to the classification of 2-regular simple modules. In this article, we give a combinatorial classification of 2-regular simple modules for Nakayama algebras and we use this classification to answer several natural questions such as when there is a unique exact structure on the category of finitely generated projective modules for Nakayama algebras. We also classify 1-regular simple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of global dimension at most two. It turns out that most classes are enumerated by well-known combinatorial sequences, such as Fibonacci, Riordan and Narayana numbers. We first obtain interpretations in terms of the Auslander-Reiten quiver of the algebra using homological algebra, and then apply suitable bijections to relate these to combinatorial statistics on Dyck paths.     

Generic projections, the equations defining projective varieties and Castelnuovo regularity

For a reduced, irreducible projective variety X of degree d and codimension e in the Castelnuovo-Mumford regularity is defined as the least k such that X is k-regular, i.e., for , where is the sheaf of ideals of X. There is a long standing conjecture about k-regularity (see [5]): . Here we show that for any smooth fivefold and for any smooth sixfold by extending methods used in [10]. Furthermore, we give a bound for the regularity of a reduced, connected and equidimensional locally Cohen-Macaulay curve or surface in terms of degree d, codimension e and an arithmetic genus (see Theorem 4.1). Received November 12, 1998; in final form May 4, 1999     

Local rigidity of quasi-regular varieties

For a G-variety X with an open orbit, we define its boundary ∂ X as the complement of the open orbit. The action sheaf S _X is the subsheaf of the tangent sheaf made of vector fields tangent to ∂ X. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups H ⁱ (X, S _X ) for i > 0, extending results of Bien and Brion (Compos. Math. 104:1–26, 1996). We apply these results to study the local rigidity of the smooth projective varieties with Picard number one classified in Pasquier (Math. Ann., in press).     

Exterior products of forms and the cohomology ring of a complex space

In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology.     

Some Examples of Tilt-Stable Objects on Threefolds

We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarization ω gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.     

Higher order dual varieties of projective surfaces

We investigate higher order dual varieties of projective manifolds whose osculatory behavior is the best possible. In particular, for a k-jet ample surface we prove the nondegeneratedness of the k-th dual variety and for 2-regular surfaces we investigate the degree of the second dual variety.     

Polyanalytic forms on compact Riemann surfaces

A sheaf of differentials on a compact Riemann surface supplied with a projective structure is said to be n-analytic if, in local projective coordinates, sections of the sheaf satisfy the differential equation { } For the projective structure induced by a covering mapping from the disk, an explicit characterization of the space of cross sections and of the space of first cohomologies of an n-analytic sheaf is given in terms of known spaces of sections of certain holomorphic sheaves. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 15–25. Translated by S. V. Kislyakov.     

Künneth decompositions for quotient varieties

In this paper we discuss Künneth decompositions for finite quotients of several classes of smooth projective varieties. The main result is the existence of an explicit (and readily computable) Chow-Künneth decomposition in the sense of Murre with several pleasant properties for finite quotients of abelian varieties. This applies in particular to symmetric products of abelian varieties and also to certain smooth quotients in positive characteristics which are known to be not abelian varieties, examples of which were considered by Enriques and Igusa. We also consider briefly a strong Künneth decomposition for finite quotients of projective smooth linear varieties.     

Viehwegʼs hyperbolicity conjecture is true over compact bases

We prove Viehweg?s hyperbolicity conjecture over compact bases and over bases with non-uniruled compactification. The most general case of the conjecture states that the base space of a maximal variation family of smooth projective manifolds with semi-ample canonical sheaf is of log-general type.     

ON ABELIAN AUTOMORPHISM GROUPS OF SURFACES OF GENERAL TYPE

1.IntroductionItiswell-knownthatforacomplexcurveofgenusg22,itstotalautomorphismgroup(resp.abelianautomorphismgroup)isoforderS84(g-l)=42degKC(resp.S4g 4),whereKCisthecanonicaldivisorofC(cL[3,4]).Itisanintriguingproblemtogeneralisetheseboundstohigherdimensions.Severalauthorshavestudiedthisproblem(see[5,6]fordetails).Recently,Xiaohasgeneralisedtheseresultstosurfacesofgeneraltype,ingoodanalogywiththecaseofcurves.HehasprovedthatforacomplexminimalsmoothprojectivesurfaceSofgeneraltype,itstota1au…     

Identifying quadric bundle structures on complex projective varieties

In this paper we characterize smooth complex projective varieties that admit a quadric bundle structure on some dense open subset in terms of the geometry of certain families of rational curves.      

Holomorphic vector fields with totally degenerate zeroes

A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of C^*-actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component,then the manifold is stably birational to that component of the zero set.When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.     

Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.

     

Bogomolov inequality for Higgs parabolic bundles

We show that over some smooth projective varieties every semistable Higgs logarithmic vector bundle is semistable in the ordinary sense, hence satisfies Bogomolov inequality. More generaly, we prove that semistable Higgs parabolic vector bundles of rank two over smooth projective varieties of dimension ≥ 2 satisfy the “parabolic” 'Bogomolov inequality Received: 1 March 1999 / Revised version: 11 June 1999     

A theorem on the adjoint system for vector bundles

In this paper, we study the classification theory of uniruled varieties by means of the adjoint system for vector bundles on the varieties. We prove that ifE is an ample vector bundle on a smooth projective varietyX with rank(E)=dimX-2, thenK _X +C ₁ (E) is numerically effective except in a few cases. In all of the exceptional cases,X is a uniruled variety. As consequences, we generalized a result of Fujita [Fu3] and Ionescu [Io] and improve upon a theorem of Wiśniewski [Wi1].