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Comparison of the algebraic and the symplectic Gromov-Witten invariantsJan 01 1998We show that the algebraic and the symplectic GW-inivariants of smooth projective varieties are equivalent.

Toric Degenerations of Fano Varieties and Constructing Mirror ManifoldsDec 30 1997For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete intersections ... More

On the classification of hyperbolic root systems of the rank three. Part IIDec 30 1997Mar 23 1998Here we prove classification results announced in Part I (alg-geom/9711032). We classify maximal hyperbolic root systems of the rank 3 having restricted arithmetic type and a generalized lattice Weyl vector $\rho$ with $\rho^2\ge 0$ (i.e. of elliptic ... More

A morphism of intersection homology induced by an algebraic mapDec 30 1997Let $f:X-->Y$ be a map of algebraic varieties. Barthel, Brasselet, Fieseler, Gabber and Kaup have shown that there exists a homomorphism of intersection homology groups $f^*:IH^*(Y)-->IH^*(X)$ compatible with the induced homomorphism on cohomology. The ... More

Descent for Shimura VarietiesDec 30 1997Dec 07 1998We verify that the descent maps provided by Langlands's Conjugacy Conjecture do satisfy the continuity condition necessary for them to be effective. Thus Langlands's conjecture does imply the existence of canonical models. This replaces an earlier version ... More

The L^2 dbar method, weak Lefschetz theorems, and the topology of Kahler manifoldsDec 29 1997A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the dbar-method, avoids moving arguments and gives much stronger results. In particular, it is proved that if X and Y are connected smooth projective varieties ... More

Reconstruction of a variety from the derived category and groups of autoequivalencesDec 24 1997We consider smooth algebraic varieties with ample either canonical or anticanonical sheaf. We prove that such a variety is uniquely determined by its derived category of coherent sheaves. We also calculate the group of exact autoequivalences for these ... More

Zero-pole interpolation for matrix meromorphic functions on a compact Riemann surface, and a matrix Fay trisecant identityDec 23 1997Feb 08 1999This paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact Riemann surface having a prescribed Weil divisor (i.e., having prescribed zeros and poles with directional as well as multiplicity information ... More

The arithmetic-geometric mean and isogenies for curves of higher genusDec 21 1997Computation of Gauss's arithmetic-geometric mean involves iteration of a simple step, whose algebro-geometric interpretation is the construction of an elliptic curve isogenous to a given one, specifically one whose period is double the original period. ... More

On hypergeometric functions connected with quantum cohomology of flag spacesDec 20 1997Givental's recursion relations for the flag varieties $G/B$ are established.

Projectivity of moment map quotientsDec 19 1997Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the corresponding ... More

Degenerations of (1,3) abelian surfaces and Kummer surfacesDec 19 1997Jun 25 1999We continue our study of the geometry of Nieto's quintic threefold, looking at degenerate surfaces that correspond to certain loci and showing how they arise from a toroidal compactification of a suitable moduli space.

Heisenberg-invariant Kummer surfacesDec 19 1997Jun 25 1999We study the geometry of Nieto's quintic threefold (Barth & Nieto, J. Alg. Geom. 3, 1994) and the Kummer and abelian surfaces that correspond to special loci.

Poisson structures and birational morphisms associated with bundles on elliptic curvesDec 19 1997In this paper we define a Poisson structure on some moduli spaces related to principal G-bundles on elliptic curves, the simplest example being the moduli space of stable pairs: a vector bundle and its global section. We also study birational morphisms ... More

Symplectic cutting of Kaehler manifoldsDec 19 1997We obtain estimates on the character of the cohomology of an $S^1$-equivariant holomorphic vector bundle over a Kaehler manifold $M$ in terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction of $M$. In particular, we prove ... More

A canonical lift of Chern-Mather classesDec 19 1997There are several ways to generalize characteristic classes for singular algebraic varieties. The simplest ones to describe are Chern-Mather classes obtained by Nash blow up. They serve as an ingredient to construct Chern-MacPherson-Schwartz classes. ... More

Analogue of Weil representation for abelian schemesDec 19 1997In this paper we construct a projective action of certain arithmetic group on the derived category of coherent sheaves on an abelian scheme $A$, which is analogous to Weil representation of the symplectic group. More precisely, the arithmetic group in ... More

Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomologyDec 18 1997Jan 06 1998If $X$ is a smooth projective variety moving in a family, we define a secondary Kodaira-Spencer class for nonabelian Dolbeault cohomology $Hom(X_{Dol}, T)$ of $X$ with coefficients in the complexified 2-sphere $T=S^2\otimes \cc$ (which is a 3-stack on ... More

On the cone of curves of an abelian varietyDec 17 1997Let $X$ be a smooth projective variety over the complex numbers. One knows by the Cone Theorem that the closed cone of curves of $X$ is rational polyhedral whenever $c_1(X)$ is ample. For varieties $X$ such that $c_1(X)$ is not ample, however, it is in ... More

Deformations of canonical singularitiesDec 17 1997Dec 22 1997We prove that small deformations of canonical singularities are canonical.

Derived categories of coherent sheaves on abelian varieties and equivalences between themDec 16 1997Mar 06 2009We study derived categories of coherent sheaves on abelian varieties. We give a criterion for the equivalence of the derived categories on two abelian varieties. We describe the autoequivalence group for the derived category of coherent sheaves of an ... More

Homotopy types of complements of 2-arrangements in R^4Dec 16 1997Aug 14 1998We study the homotopy types of complements of arrangements of n transverse planes in R^4, obtaining a complete classification for n <= 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement ... More

Invariance of PlurigeneraDec 15 1997The following conjecture on the deformation invariance of plurigenera is proved. For a smooth projective holomorphic family of compact complex manifolds over the open unit 1-disk such that all the fibers are of general type, every plurigenus of the fiber ... More

The universal regular quotient of the Chow group of 0-cycles on a singular projective varietyDec 15 1997We show the existence of a regular universal quotient as a smooth commutative algebraic group of the Chow group of 0-cycles on a projective reduced variety, and give over the field of complex numbers an analytic description of it. This generalizes the ... More

Subadjunction of log canonical divisors IIDec 15 1997We extend a subadjunction formula of log canonical divisors as in [K3] to the case when the codimension of the minimal center is arbitrary by using the positivity of the Hodge bundles.

Eigenvalues of products of unitary matrices and quantum Schubert calculusDec 14 1997We describe the inequalities on the possible eigenvalues of products of unitary matrices in terms of quantum Schubert calculus. Related problems are the existence of flat connections on the punctured two-sphere with prescribed holonomies, and the decomposition ... More

Hyperholomorphic sheaves and new examples of hyperkaehler manifoldsDec 11 1997Dec 09 2012Given a compact hyperkaehler manifold $M$ and a holomorphic bundle B over $M$, we consider a Hermitian connection $\nabla$ on B which is compatible with all complex structures on $M$ induced by the hyperkaehler structure. Such a connection is unique, ... More

Mirror Principle IDec 11 1997We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute ... More

Structures du cube et fibres d'intersectionDec 10 1997We define the notion of a hypercube structure on a functor between two strictly commutative Picard categories which generalizes the notion of a cube structure on a $G_m$-torsor over an abelian scheme. We use this notion to define the intersection bundle ... More

The quantum Euler class and the quantum cohomology of the GrassmanniansDec 09 1997The Poincare duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical ``characteristic element;'' in the classical case ... More

The Character of the Infinite Wedge RepresentationDec 09 1997Dec 16 1997We study the character of the infinite wedge projective representation of the algebra of differential operators on the circle. We prove quasi-modularity of this character and also compute certain generating functions for traces of differential operators ... More

Transmutations between Singular and Subsingular Vectors of the N=2 Superconformal AlgebrasDec 08 1997Dec 13 1999We present subsingular vectors of the N=2 superconformal algebras other than the ones which become singular in chiral Verma modules, reported recently by Gato-Rivera and Rosado. We show that two large classes of singular vectors of the Topological algebra ... More

PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degreesDec 08 1997Feb 09 2001In this article we complete the proof---for a broad class of four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of ... More

Integrability in 3+1 Dimensions: Relaxing a Tetrahedron RelationDec 08 1997I propose a scheme of constructing classical integrable models in 3+1 discrete dimensions, based on a relaxed version of the problem of factorizing a matrix into the product of four matrices of a special form.

Quantum Hyperplane Section Theorem For Homogeneous SpacesDec 05 1997Jan 02 1998We formulated a mirror-free approach to the mirror conjecture, namely, quantum hyperplane section conjecture, and proved it in the case of nonnegative complete intersections in homogeneous manifolds. For the proof we followed the scheme of Givental's ... More

Toric varieties and minimal complexesDec 05 1997We translate the equivariant decomposition theorem (in the case of a proper morphism of toric varieties) in to the language of combinatorially defined ``shifted minimal complexes''.

Special Kahler ManifoldsDec 04 1997Aug 24 1998We give an intrinsic definition of the special geometry which arises in global N=2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kahler manifold is so ... More

Homological infiniteness of Torelli groupsDec 03 1997We prove that rational homology of the Torelli group of genus g is infinite dimensional, provided g>6. This means that rational homology of the Torelli space of genus g>6 is infinite dimensional. The Torelli groups with marked points are also considered. ... More

Real Algebraic Threefolds II: Minimal Model ProgramDec 02 1997This is the second of a series of papers studying real algebraic threefolds using the minimal model program. The main result is the following. Let $X$ be a smooth projective real algebraic 3-fold. Assume that the set of real points is an orientable 3-manifold ... More

Real Algebraic Threefolds I: Terminal SingularitiesDec 02 1997This is the first of a series of papers studying real algebraic threefolds using the minimal model program. The main results are outlined in Part II. The present part I. contains the necessary preliminary work concerning terminal singularities. First ... More

Real Algebraic SurfacesDec 02 1997These are the notes for my lectures at the Trento summer school held September 1997. The aim of the lectures is to provide an introduction to real algebraic surfaces using the minimal model program. This leads to a fairly complete understanding of real ... More

Tamagawa numbers of polarized algebraic varietiesDec 01 1997Dec 17 1997Let ${\cal L} = (L, \| \cdot \|_v)$ be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety $V$ defined over a number field. Denote by $N(V,{\cal L},B)$ the number of rational points in $V$ having ${\cal L}$-height $\leq B$. ... More

Factorizable sheaves and quantum groupsDec 01 1997Apr 15 1998The category $\cal{C}$ (studied by Andersen-Jantzen-Soergel) of representations of a Lusztig's small quantum group at a root of unity, together with its modular structure, is defined geometrically, using configuration spaces.

Five Lectures on Soliton EquationsNov 30 1997This is a self-contained review of a new approach to soliton equations of KdV type developed by the author together with B. Feigin and B. Enriquez.

Coupled Contact Systems and Rigidity of Maximal Dimensional Variations of Hodge StructureNov 30 1997In this article we prove that locally Griffiths' horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system. As a consequence of this description we obtain strong local rigidity properties ... More

On the Euler Characteristic of Generalized Kummer VarietiesNov 28 1997We apply the Yau-Zaslow-Beauville method to compute the Euler characteristic of the generalized Kummer varieties attached to a complex abelian surface (a calculation also done by Goettsche and Soergel by different methods). It is related to the number ... More

Minimal discrepancies of hypersurface singularitiesNov 26 1997We give an upper bound for the minimal discrepancies of hypersurface singularities. As an application, we show that Shokurov's conjecture is true for log-terminal threefolds.

Uniform Zariski's Theorem On Fundamental GroupsNov 25 1997The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space of dimension at least 3 and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental ... More

On the classification of hyperbolic root systems of the rank three. Part INov 25 1997It was recently understood that from the point of view of automorphic Lorentzian Kac-Moody algebras and some aspects of Mirror Symmetry, interesting hyperbolic root systems should have restricted arithmetic type and a generalized lattice Weyl vector. ... More

The enumerative geometry of K3 surfaces and modular formsNov 24 1997We prove the conjectures of Yau-Zaslow and Gottsche concerning the number curves on K3 surfaces. Specifically, let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric ... More

Quantum cohomology of the moduli space of stable bundles over a Riemann surfaceNov 24 1997We determine the quantum cohomology of the moduli space of odd degree rank two stable vector bundles over a Riemann surface $\Sigma$ of any genus. This work together with dg-ga/9710029 prove that this quantum cohomology is isomorphic to the instanton ... More

Classification of Exceptional Complements: Elliptic Curve CaseNov 24 1997We classify the log del Pezzo surface (S,B) of rank 1 with no 1-,2-,3-,4-, or 6-complements with the additional condition that B has one irreducible component C which is an elliptic curve, and that C has the coefficient b in B with (1/n)floor((n+1)b)=1 ... More

On moduli spaces of 4- or 5-instanton bundlesNov 21 1997We study the scheme of multi-jumping lines of an $n$-instanton bundle mainly for $n\leq 5$. We apply it to prove the irreducibility and smoothness of the moduli space of 5-instanton. Some particular situations with higher $c_2$ are also studied.

Mathematical Instantons In Characteristic TwoNov 20 1997On P3, we show that mathematical instantons in characteristic two are unobstructed. We produce upper bounds for the dimension of the moduli space of stable rank two bundles on P3 in characteristic two. In cases where there is a phenomenon of good reduction ... More

Symmetric matrices and quantum codesNov 20 1997Dec 22 1997This paper has been withdrawn since a Gilbert-Varshamov bound for general quantum codes has already appeared in Ekert and Macchiavello, Prys. Rev. Lett. 77, p. 2585, and a Gilbert-Varshamov bound for stabilizer codes connected with orthogonal geometry, ... More

Cycles on Siegel 3-folds and derivatives of Eisenstein seriesNov 20 1997We consider the Siegel modular variety of genus 2 and a p-integral model of it for a good prime p>2, which parametrizes principally polarized abelian varieties of dimension two with a level structure. We consider cycles on this model which are characterized ... More

Complements on surfacesNov 20 1997Mar 28 2001The main result of the paper is a boundedness for $n$-complements on algebraic surfaces. In addition, applications of this theorem to a classification of log Del Pezzo surfaces and of birational contractions for 3-folds are formulated.

Cellular Resolutions of Monomial ModulesNov 19 1997We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals.

Equations of the moduli of pointed curves in the infinite GrassmannianNov 19 1997Feb 24 1999The main result of this paper is the explicit computation of the equations defining the moduli space of triples $(C,p,z)$ (where $C$ is an integral and complete algebraic curve, $p$ a smooth rational point and $z$ a formal trivialization around $p$) in ... More

A geometric approach to the fundamental lemma for unitary groupsNov 19 1997We consider from a geometric point of view the conjectural fundamental lemma of Langlands and Shelstad for unitary groups over a local field of positive characteristic. We introduce projective algebraic varieties over the finite residue field $k$ and ... More

Semistable reduction in characteristic 0 for families of surfaces and three-foldsNov 17 1997Let F:X->B be a morphism of varieties in characteristic zero. The problem of semistable reduction of F was stated as a problem in the combinatorics of polyhedral complexes by Abramovich and Karu (alg-geom/9707012). In this paper we solve the combinatorial ... More

Stringy Hodge numbers and Virasoro algebraNov 16 1997Let $X$ be an arbitrary smooth $n$-dimensional projective variety. It was discovered by Libgober and Wood that the product of the Chern classes $c_1(X)c_{n-1}(X)$ depends only on the Hodge numbers of $X$. This result has been used by Eguchi, Jinzenji ... More

Chern Classes of Bundles over Rational SurfacesNov 15 1997Jul 27 1998Consider the blow up $\pi: \widetilde{X} \to X$ of a rational surface $X$ at a point. Let $\widetilde{V}$ be a holomorphic bundle over $\widetilde{X}$ whose restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$ and define $V ... More

On the theta divisor of SU(2,1)Nov 13 1997Let SU(2,1) be the moduli space of stable rank two vector bundles having fixed determinant of odd degree over a compact Riemann surface C. In this paper it is shown that the Theta divisor of SU(2,1) is very ample for every C. The proof is related to the ... More

Boundary Manifolds of Line ArrangementsNov 12 1997In this paper we describe the complement of real line arrangements in the complex plane in terms of the boundary three-manifold of the line arrangement. We show that the boundary manifold of any line arrangement is a graph manifold with Seifert fibered ... More

Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fieldsNov 12 1997Dec 31 1997We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local - global approach, we compute completely the ... More

Topological Triviality of $μ$-constant Deformations of Type f(x) + tg(x)Nov 11 1997We show that every $\mu$-constant family of isolated hypersurface singularities of type f(x) + tg(x), where t is a parameter, is topologically trivial. In the proof we construct explicitely a vector field trivializing the family. The proof uses only the ... More

First quantum correction for the moduli space of stable bundles over a Riemann surfaceNov 11 1997We compute some Gromov-Witten invariants of the moduli space of odd degree rank two stable vector bundles over a Riemann surface of any genus. Next we find the first correction term for the quantum product of this moduli space and hence get the two leading ... More

A conjectural generating function for numbers of curves on surfacesNov 11 1997I give a conjectural generating function for the numbers of $\delta$-nodal curves in a linear system of dimension $\delta$ on an algebraic surface. It reproduces the results of Vainsencher for the case $\delta\le 6$ and Kleiman-Piene for the case $\delta\le ... More

Hypergeometric functions on reductive groupsNov 10 1997Apr 06 1998The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced by an arbitrary ... More

Constructing curves over finite fields with many points by solving linear equationsNov 10 1997In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of class field theory. ... More

Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' SpacesNov 08 1997Sep 30 2008For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating ... More

Stringy Hodge numbers of varieties with Gorenstein canonical singularitiesNov 06 1997Mar 16 1998We introduce the notion of stringy E-function for an arbitrary normal irreducible algebraic variety X with at worst log-terminal singularities. We prove some basic properties of stringy E-functions and compute them explicitly for arbitrary Q-Gorenstein ... More

Construction de familles minimales de courbes gauchesNov 06 1997Let $A$ be a local noetherian ring and $N$ be a locally sheaf on the projective space $P^3_A$ : one proves easily that there exists a family $C$ of (smooth connected) curves contained in $P^3_A$, flat over $A$, and an integer $h$ such that the ideal sheaf ... More

Sheaves on Toric Varieties for PhysicsNov 05 1997May 12 1998In this paper we give an inherently toric description of a special class of sheaves (known as equivariant sheaves) over toric varieties, due in part to A. A. Klyachko. We apply this technology to heterotic compactifications, in particular to the (0,2) ... More

Beyond the Manin obstructionNov 05 1997We construct a (smooth, projective) surface over the field of rational numbers, which is a counterexample to the Hasse principle not accounted for by the Manin obstruction. The construction relies on the classical 4-descent on elliptic curves. By combining ... More

Some fundamental problems on real-analytic setsNov 04 1997Jun 18 2010The category of real-analytic sets and real-analytic maps is the most important category in application. However, in spite of efforts by F. Bruhat, H. Cartan, H. Whitney et al., the basic theory of real-analytic category does not yet seem to be well-developed. ... More

Approximation by smooth curves near the tangent coneNov 03 1997Sep 15 2003We show that through a point of an affine variety there always exists a smooth plane curve inside the ambient affine space, which has the multiplicity of intersection with the variety at least 3. This result has an application to the study of affine schemes. ... More

Pure Hodge structure on the $L_2$-cohomology of varieties with isolated singularitiesNov 03 1997Let $V$ be a complex projective variety with isolated singularities. Let the smooth part be given the metric induced by a projective imbedding. Then we develop the $L_2$ harmonic theory and construct a pure Hodge structure on the $L_2$-cohomology of $V$. ... More

Resonant Hypergeometric Systems and Mirror SymmetryNov 03 1997The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they give solutions for certain resonant systems of GKZ hypergeometric differential equations. For this some complex parameters in the Gamma-series are replaced by nilpotent elements ... More

Universal R-matrix for null-plane quantized Poincar{é} algebraNov 03 1997Dec 01 1997The universal ${\cal R}$--matrix for a quantized Poincar{\'e} algebra ${\cal P}(3+1)$ introduced by Ballesteros et al is evaluated. The solution is obtained as a specific case of a formulated multidimensional generalization to the non-standard (Jordanian) ... More

On higher order embeddings of Fano threefolds by the anticanonical linear systemNov 02 1997Nov 05 1997The map given by the anticanonical bundle of a Fano manifold is investigated with respect to a number of natural notions of higher order embeddings of projective manifolds. This is of importance in the understanding of higher order embeddings of the special ... More

On the variety of rational space curvesOct 30 1997We consider the locus of irreducible nonsingular rational curves of degree d Pn, n>2, meeting a generic collection of linear subspaces. When this locus is 0 (resp 1)- dimensional, we compute (recursively) its degree (resp. geometric genus). The method ... More

On the k-normality of some projective manifoldsOct 29 1997A long standing conjecture, known to us as the Eisenbud Goto conjecture, states that an n-dimensional variety embedded with degree $d$ in the $N$- dimensional projective space is $(d-(N-n)+1)$-regular in the sense of Castelnuovo-Mumford. In this work ... More

PU(2) Monopoles, I: Regularity, Uhlenbeck Compactness, and TransversalityOct 29 1997We prove the existence of perturbations for the PU(2) monopole equations, yielding transversality on the complement of the anti-self-dual or reducible solutions, and the existence of an Uhlenbeck compactification for the moduli space of solutions to these ... More

Frobenius Manifolds and Formality of Lie Algebras of Polyvector FieldsOct 28 1997Jan 07 1998We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariants

Involutions on Moduli Spaces and Refinements of the Verlinde FormulaOct 28 1997Dec 14 1998The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant over a complex curve carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion point, we define ... More

The universal rank-(n-1) bundle on G(1,n) restricted to subvarietiesOct 27 1997We classify those smooth (n-1)-folds in G(1,n) for which the restriction of the rank-(n-1) universal bundle has more than n+1 independent sections. As an aplication, we classify also those (n-1)-folds for which that bundle splits.

Irreducibility of the moduli space of vector bundles on surfaces and Brill-Noether theory on singular curvesOct 27 1997Mar 23 2000We prove the irreducibility of the moduli space of rank 2 semistable torsion free sheaves (with a generic polarization and any value of c_2) on a K3 or a del Pezzo surface. In the case of a K3 surface, we need to prove a result on the connectivity of ... More

Self-duality of the SL_2 Hitchin integrable system at genus twoOct 27 1997Jan 20 1998We revisit the Hitchin integrable system whose phase space is the bundle cotangent to the moduli space $N$ of holomorphic $SL_2$-bundles over a smooth complex curve of genus two. $N$ may be identified with the 3-dimensional projective space of theta functions ... More

On the infinitesimal rigidity of homogeneous varietiesOct 26 1997Oct 29 1997Let $X\subset P^N$ be a variety (respectively a patch of an analytic submanifold) and let $x\in X$ be a general point. We show that if the projective second fundamental form of $X$ at $x$ is isomorphic to the second fundamental form of a point of a Segre ... More

Injective resolutions of BG and derived moduli spaces of local systemsOct 24 1997It was suggested on several occasions by Deligne, Drinfeld and Kontsevich that all the moduli spaces arising in the classical problems of deformation theory should be extended to natural "derived" moduli spaces which are always smooth in an appropriate ... More

Hyperkaehler structures on total spaces of holomorphic cotangent bundlesOct 23 1997Let $M$ be a Kaehler manifold, and consider the total space $T^*M$ of the cotangent bundle to $M$. We show that in the formal neighborhood of the zero section $M \subset T^*M$ the space $T^*M$ admits a canonical hyperkaehler structure, compatible with ... More

Minimal even sets of nodesOct 20 1997We extend some results on even sets of nodes which have been proved for surfaces up to degree 6 to surfaces up to degree 10. In particular, we give a formula for the minimal cardinality of a nonempty even set of nodes.

The Geometry of the Space of Holomorphic Maps from a Riemann Surface into Complex Projective SpaceOct 20 1997In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint preserving) holomorphic maps of a given degree k from a Riemann surface M of genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric products of the surface ... More

Real deformations and complex topology of plane curve singularitiesOct 20 1997Apr 03 2000This is the paper as published. The topology of a complex plane curve singularity with real branches is deduced from any real deformation having delta crossings. An example of the computation of the global geometric monodromy of a polynomial mapping is ... More

Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete Intersections in GrassmanniansOct 19 1997In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. ... More

Cohomology of complete intersections in toric varietiesOct 17 1997Dec 20 2000This paper explicitly describes Hodge structures of complete intersections of ample hypersurfaces in compact simplicial toric varieties.

On a conjecture of LangeOct 16 1997Nov 14 1997Let C be a projective smooth curve of genus g> 1. Let E be a vector bundle of rank r on C. For each integer r'<r, associate to E the invariant s_{r'}(E)=r'deg(E)-rdeg(E') where E'is a subbundle of E of rank r' and maximal degree. For every r', one can ... More

Birational Calabi--Yau n-folds have equal Betti numbersOct 16 1997Mar 16 1998Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are birational, they ... More