Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_hol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_hol (X normal algebraic surface) by algebraic vector bundles.     

Bubble tree compactification of moduli spaces of vector bundles on surfaces

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c ₁ = 0, c ₁ = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.     

Finding ein components in the moduli spaces of stable rank 2 bundles on the projective 3-space

Some method is proposed for finding Ein components in moduli spaces of stable rank 2 vector bundles with first Chern class c₁ = 0 on the projective 3-space. We formulate and illustrate a conjecture on the growth rate of the number of Ein components in dependence on the numbers of the second Chern class. We present a method for calculating the spectra of the above bundles, a recurrent formula, and an explicit formula for computing the number of the spectra of these bundles.     

Some results on special stable vector bundles of rank 3 on algebraic curves

The authors discuss the existence and classification of stable vector bundles of rank 3, with 2 3 or 4 linearly independent holomorphic sections. The sets of all such bundles are denoted by ω3^2,d and w3 respectively. Our argument leads to sufficient and necessary conditions for the existence of both kinds of bundles. The conclusion is very interesting because of its contradiction to the conjectured dimension formula of stable bundles. Finally, we give a preliminary classification of ω3^2,4 and a complete discussion on the structure of ω3^3,2/3g＋2.     

On the Brill-Noether theory for K3 surfaces

Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c ₁(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces.     

Picard groups of the moduli spaces of semistable sheaves I

We compute the Picard group of the moduli spaceU′ of semistable vector bundles of rankn and degreed on an irreducible nodal curveY and show thatU′ is locally factorial. We determine the canonical line bundles ofU′ andU _L ′, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification ofU′.     

Garrett's pullback formula for vector valued Siegel modular forms

Let be the Siegel Eisenstein series of degree n and weight k. Garrett showed a formula of on Hp×Hq, where Hn is the Siegel upper half space of degree n. This formula was useful for investigating the Fourier coefficients of the Klingen Eisenstein series and the algebraicity of the space of Siegel modular forms and of special values of the standard L-functions. We would like to generalize this formula in the case of vector valued Siegel modular forms. In this paper, using a differential operator D by Ibukiyama which sends a scalar valued Siegel modular form to the tensor product of two vector valued Siegel modular forms, under a certain condition, we give a formula of and investigate the Fourier coefficients of the Klingen Eisenstein series.     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.     

Automorphism group of exceptional symmetric domainsRV

The exceptional symmetric Siegel domainR _v(16) in ℂ¹⁶ is defined. The exceptional classical domain ℛ_v(16) = t(R_v(16)) is computed, where t is the Bergman mapping of the Siegel domain R_v(16). And holomorphical automorphism group Aut (R_v(16)) of the exceptional symmetric Siegel domainR _v(16) is presented     

Residual intersection theory with reducible schemes

We develop a formula (Theorem 5.2) which allows to compute top Chern classes of vector bundles on the vanishing locus V(s) of a section of this bundle. This formula particularly applies in the case when V(s) is the union of local complete intersections giving the individual contribution of each component and their mutual intersections. We conclude with applications to the enumeration of rational curves in complete intersections in projective space.     

Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

We study moduli of vector bundles on a two-dimensional neighbourhood Z _k of an irreducible curve ? ? ?¹ with ?² = ?k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z _k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.     

Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

A new definition of canonical conformal differential operators P _k (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham–Jenne–Mason–Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the P _k operators and hence on Einstein manifolds the GJMS operators factor into a product of second-order Laplacian type operators. In even dimension n the GJMS operators are defined only for 1 ≤ k ≤ n/2 and so, on conformally Einstein manifolds, the P _k give an extension of this family of operators to operators of all even orders. For n even and k > n/2 the operators P _k are each given by a natural formula in terms of an Einstein metric but they are not natural conformally invariant operators in the usual sense. They are shown to be nevertheless canonical objects on conformally Einstein structures. There are generalisations of these results to operators between weighted tractor bundles. It is shown that on Einstein manifolds the Branson Q-curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature. As part of development, conformally invariant tractor equations equivalent to the conformal Killing equation are presented.     

Pullbacks of Siegel Eisenstein series and weighted averages of critical L-values

In this paper we obtain a weighted average formula for special values of L-functions attached to normalized elliptic modular forms of weight k and full level. These results are obtained by studying the pullback of a Siegel Eisenstein series and working out an explicit spectral decomposition.     

A Note on Residue Formulas for the Euler Class of Sphere Fibrations

This paper presents a definition of residue formulas for the Euler class of cohomology-oriented sphere fibrations ξ. If the base of ξ is a topological manifold, a Hopf index theorem can be obtained and, for the smooth category, a generalization of a residue formula is derived for real vector bundles given in [2].     

Poincaré polynomial of the moduli spaces of parabolic bundles

In this paper we use Weil conjectures (Deligne’s theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of HarderNarasimhan filtration gives us a recursive formula for the Poincaré polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincaré polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.     

On Framed Instanton Bundles and Their Deformations

We consider a compact twistor space P and assume that there is a surface S ⊂ P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space ([20], [18]). Similar to [6] and [5] we examine the restriction of an instanton bundle V equipped with a fixed trivialization along F to a framed vector bundle over (S, F). First we develope inspired by [13] a suitable deformation theory for vector bundles over an analytic space framed by a vector bundle over a subspace of arbitrary codimension. In the second section we describe the restriction as a smooth natural transformation into a fine moduli space. By considering framed U(r)‐instanton bundles as a real structure on framed instanton bundles over P, we show that the bijection between isomorphism classes of framed U(r)‐instanton bundles and isomorphism classes of framed vector bundles over (S, F) due to [5] is actually an isomorphism of moduli spaces.     

Semiample and k-ample vector bundles

Abstract

The main result of this paper is that tensor products of semiample vector bundles over compact complex manifolds are semiample. An easy proof yields the analogous result for direct sums. We also show that tensor products of semiample vector bundles with k-ample vector bundles in the sense of Sommese are k-ample. On the other hand, we show that it is not generally true that tensor products of nef and k-ample vector bundles for positive k are still k-ample. Results of Sommese on k-ampleness are consequently strengthened. As an application of our main theorem we extend to k-ample the vanishing theorem of Ein-Lazarsfeld.     

A mod 2 index theorem for the twisted Signature operator

A mod 2 index theorem for the twisted Signature operator on 4 _q + 1 dimensional manifolds is established. This result generalizes a result of Farber and Turaev, which was proved for the case of orthogonal flat bundles, to arbitrary real vector bundles. It also provides an analytic interpretation of the sign of the Poincare-Reidemeister scalar product defined by Farber and Turaev. Project partially supported by the National Natural Science Foundation of China (Grant No. 19525102), the Fok Ying-Tung Foundation and the Qiu Shi Foundation.     

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).