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At least 271 records · Page 15Linked to original sources
arXiv1996

On framed instanton bundles and their deformations

We consider a compact twistor space P and assume that there is a surface S in P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space. Similar to Donaldson and Buchdahl we examine the restriction of an instanton bundle V equipped with a fixed trivialisation along F to a framed vector bundle over (S,F). First we develope inspired by Huybrechts and Lehn 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 Buchdahl is actually an isomorphism of moduli spaces.

alg-geom↗
arXiv1996

Segre Numbers and Hypersurface Singularities

We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the Rees-Böger theorem, and the formula for the multiplicity of the product of two ideals. These results are applied to the study of various equisingularity conditions, such as Verdier's condition W, and conditions $A_f$ and $W_f$.

alg-geom↗
arXiv1996

The Second Cohomology with Symplectic Coefficients of the Moduli Space of Smooth Projective Curves

Each finite dimensional irreducible rational representation V of the symplectic group Sp_{2g} determines a generically defined local system \V over the moduli space M_g of genus g smooth projective curves. We study H^2(M_g;\V) and the mixed Hodge structure on it. Specifically, we prove that if g>5, then the natural map IH^2(MS_g;\V)-->H^2(M_g;\V) is an isomorphism where MS_g is the Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H^2(M_g;\V) is pure of weight 2+r if \V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H^2(M_g;\V) for 2<g<6. Results of this article can be applied in the study of relations in the Torelli group T_g.

alg-geom↗
arXiv1996

Birational automorphisms of algebraic varieties with a pencil of cubic surfaces

It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In particular, this variety is non-rational; moreover, it can not be fibered into rational curves. The proof is obtained by means of the method of maximal singularities.

alg-geom↗
arXiv1996

Giambelli-type formula for subbundles of the tangent bundle

Let us consider a generic n-dimensional subbundle V of the tangent bundle TM on some given manifold M. Given V one can define different degeneracy loci S_r(CV), r=(r_1<= r_2<= r_3<=...<=r_k) on M consisting of all points x in M for which the dimension of the subspace V^j(x) in TM(x) spanned by all length <= j commutators of vector fields tangent to V at x is less than or equal to r_j. We calculate 'explicitly' the cohomology classes dual to S_r(V) using determinantal formulas due to W.Fulton and the expression for the Chern classes of the associated bundle of free Lie algebras in terms of the Chern classes of V itself.

alg-geom↗
arXiv1996

A view on contractions of higher dimensional varieties

In this paper we discuss some recent results about extremal contractions of complex algebraic varieties. These are proper surjective maps, $ϕ: X\longrightarrow Z$, of normal varieties with connected fibers such that $X$ has mild singularities (in particular $X$ is $Q$-Gorenstein) and the anticanonical divisor of $X$ is $ϕ$ ample. It will appear on the Proceedings of AMS-SRI on Algebraic Geometry, Santa Cruz 1995.

alg-geom↗
arXiv1997

Irregular canonical double surfaces

We study minimal surfaces X of general type with $K^2_X=6p_g-14$ and $q(X)>0$ such that $K_X$ is ample, the image of the canonical map is a canonically embedded surface of general type and the canonical map is not birational. The main result states that if X satisfies the above assumptions and $q(X)\ge 2$, then, apart from a finite number of exceptions, X belongs to an infinite series of examples due to Beauville. The exceptions are described in detail and some new examples are given.

alg-geom↗
arXiv1996

The semi-classical approximation for modular operads

The semi-classical approximation is an explicit formula of mathematical physics for the sum of Feynman diagrams with a single circuit.In this paper, we study the same problem in the setting of modular operads (see dg-ga/9408003); instead of being a number, the interaction at a vertex of valence n is an S_n-module. As an application, we calculate the S_n-equivariant Hodge polynomials of the moduli spaces \Mbar_{1,n}.

alg-geom↗
arXiv1996

The elliptic Gromov-Witten invariants of CP^3

We present two explicit recursions which determine the elliptic Gromov-Witten invariants of CP^3 in terms of the rational ones, and give a table up to degree 5. Unlike the rational Gromov-Witten invariants, the coefficients are negative and fractional. In a further paper, we will prove that N^1_ab + (2n-1)N^0_ab/12 is the number of elliptic space curves through a generic lines and b generic points.

alg-geom↗
arXiv1998

Strange duality, mirror symmetry, and the Leech lattice

We give a survey on old and new results concerning Arnold's strange duality. We show that most of the features of this duality continue to hold for the extension of it discovered by C. T. C. Wall and the author. The results include relations to mirror symmetry and the Leech lattice.

alg-geom↗
arXiv1996

Weil classes on abelian varieties

Consider a complex abelian variety X on which a field F acts. Generalizing a construction of A. Weil, one associates to this a subspace W_F of the cohomology of X, which we call the space of Weil classes w.r.t. F. The purpose of this paper is to answer the following two questions: Q1: under what conditions on F does the space W_F contain, or even consist of, Hodge classes?, Q2: if W_F contains Hodge classes, under what conditions on F are these exceptional? In case X is defined over a number field, we also answer the analogous questions for Tate classes.

alg-geom↗
arXiv1997

Effective base point freeness on a normal surface

This treats the base-point-freeness of the adjoint bundles on normal surfaces with a boundary. This is an extension of the non-relative version of the theorem of Ein-Lazarsfeld-Masek and the theorem of Kawachi-Masek.

alg-geom↗
arXiv1996

Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b_+=1

We prove structure theorems for the Donaldson invariants of 4-manifolds with b_+=1, similar to those of Kronheimer and Mrowka in the case b_+>1: We show that for a 4-manifold with b_+=1 and two different period points F, G on the boundary of the positive cone, the difference of the Donaldson invariants at F and G satisfies the k^{th}-order simple type condition for a number k, explicitely given in terms of F and G. We give a formula for this difference in terms of a set of basic classes, expressed in a universal formula involving modular forms. The leading terms coincide with the formulas of Kronheimer and Mrowka in the simple type case. The basic classes are related to the Seiberg Witten invariants. The proof of these results is based on the conjecture of Kotschick and Morgan on the structure of the wall-crossing terms. We relate the problem to new theta functions for indefinite lattices.

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