title: A lecture on Arithmetic Mirror Symmetry and Calabi-Yau manifolds

abstract:
  We extend our variant of mirror symmetry for K3 surfaces \cite{GN3} and
clarify its relation with mirror symmetry for Calabi-Yau manifolds. We
introduce two classes (for the models A and B) of Calabi-Yau manifolds fibrated
by K3 surfaces with some special Picard lattices. These two classes are related
with automorphic forms on IV type domains which we studied in our papers
\cite{GN1}-\cite{GN6}. Conjecturally these automorphic forms take part in the
quantum intersection pairing for model A, Yukawa coupling for model B and
mirror symmetry between these two classes of Calabi-Yau manifolds. Recently
there were several papers by physicists where it was shown on some examples. We
propose a problem of classification of introduced Calabi-Yau manifolds. Our
papers \cite{GN1}-\cite{GN6} and \cite{N3}-\cite{N14} give a hope that this is
possible. They describe possible Picard or transcendental lattices of general
K3 fibers of the Calabi-Yau manifolds.


title: A correspondence dual to McKay's

abstract:
  We give a dual to the McKay correspondence, involving conjugacy classes of
subgroups of SU(2). We prove a determinantal formula involving both
correspondences. We pose some questions concerning a non-commutative Fourier
transform.


title:
Intersection theory on $\Mbar_{1,4}$ and elliptic Gromov-Witten
  invariants

abstract:
  The WDVV equation is satisfied by the genus 0 correlation functions of any
topological field theory in two dimensions coupled to topological gravity, and
may be used to determine the genus 0 (rational) Gromov-Witten invariants of
many projective varieties (as was done for projective spaces by Kontsevich).
  In this paper, we present an equation of a similar universal nature for genus
1 (elliptic) Gromov-Witten invariants -- however, it is much more complicated
than the WDVV equation, and its geometric significance is unclear to us. (Our
prove is rather indirect.) Nevertheless, we show that this equation suffices to
determine the elliptic Gromov-Witten invariants of projective spaces.
  In a sequel to this paper, we will prove that this equation is the only one
other than the WDVV equation which relates elliptic and rational correlation
functions for two-dimensional topological field theories coupled to topological
gravity. It is unclear if there are any further equations of this type on the
small phase space in higher genus, but we think it unlikely. (The genus 0 and 1
cases are special, since the correlation functions on the small phase space
determine those on the large phase space.)


title: The semi-classical approximation for modular operads

abstract:
  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}.


title: Geometry of cohomology support loci for local systems I

abstract:
  Let X be a Zariski open subset of a compact Kaehler manifold. In this paper,
we study the set $\Sigma^k(X)$ of one dimensional local systems on X with
nonvanishing kth cohomology. We show that under certain conditions (X compact,
X has a smooth compactification with trivial first Betti number, or k=1)
$\Sigma^k(X)$ is a union of translates of sets of the form $f^*H^1(T,C^*)$,
where $f:X \to T$ is a holomorphic map to a complex Lie group which is an
extension of a compact complex torus by a product of C^*'s (these correspond to
semiabelian varieties in the algebraic category). This generalizes earlier work
of Beauville, Green, Lazarsfeld, Simpson and the author in the compact case.
The main novelty lies in the proofs which involve consideration of Higgs fields
with logarithmic poles. While a completely satifactory theory of such objects
is still lacking, we are able to work out what we need in the rank one case by
borrowing ideas from mixed Hodge theory. This will appear in the Journal of
Algebraic Geometry.


title: Intermediate Jacobians of moduli spaces

abstract:
  Let $SU_X(n,L)$ be the moduli space of rank n semistable vector bundles with
fixed determinant L on a smooth projective genus g curve X. Let $SU_X^s(n,L)$
denote the open subset parametrizing stable bundles. We show that if g>3 and n
> 1, then the mixed Hodge structure on $H^3(SU_X^s(n, L))$ is pure of type
${(1,2),(2,1)}$ and it carries a natural polarization such that the associated
polarized intermediate Jacobian is isomorphic J(X). This is new when deg L and
n are not coprime. As a corollary, we obtain a Torelli theorem that says
roughly that $SU_X^s(n,L)$ (or $SU_X(n,L)$) determines X. This complements or
refines earlier results of Balaji, Kouvidakis-Pantev, Mumford-Newstead,
Narasimhan-Ramanan, and Tyurin.


title:
Instantons on $S^{4}$ and $\cpbar $, rank stabilization, and Bott
  periodicity

abstract:
  We study the large rank limit of the moduli spaces of framed bundles on the
projective plane and the blown-up projective plane. These moduli spaces are
identified with various instanton moduli spaces on the 4-sphere and $\cpbar $,
the projective plane with the reverse orientation. We show that in the direct
limit topology, these moduli spaces are homotopic to classifying spaces. For
example, the moduli space of $Sp(\infty)$ or $SO(\infty)$ instantons on $\cpbar
$ has the homotopy type of $BU(k)$ where $k$ is the charge of the instantons.
We use our results along with Taubes' result concerning the $k\to \infty $
limit to obtain a novel proof of the homotopy equivalences in the eight-fold
Bott periodicity spectrum. We give explicit constructions for these moduli
spaces.


title: The elliptic Gromov-Witten invariants of CP^3

abstract:
  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.


title: Strange duality, mirror symmetry, and the Leech lattice

abstract:
  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.


title:
Relative Bogomolov's inequality and the cone of positive divisors on the
  moduli space of stable curves

abstract:
  Let f : X --> Y be a projective morphism of smooth algebraic varieties over
an algebraically closed field of characteristic zero with dim f = 1. Let E be a
vector bundle of rank r on X. In this paper, we would like to show that if X_y
is smooth and E_y is semistable for some point y of Y, then f_* (2r c_2(E) -
(r-1) c_1(E)^2) is weakly positive at y. We apply this result to obtain the
following description of the cone of weakly positive $\QQ$-Cartier divisors on
the moduli space of stable curves. Let M_g (resp. M_g^0) be the moduli space of
stable (resp. smooth) curves of genus g >= 2. Let h be the Hodge class and
d_i's (i = 0,...,[g/2]) the boundary classes. A Q-Cartier divisor x h + y_0 d_0
+ ... + y_[g/2] d_[g/2] is weakly positive over M_g^0 if and only if x >= 0, g
x + (8g + 4) y_0>= 0, and i(g-i) x + (2g+1) y_i>= 0 for all 1 <= i <= [g/2].


title: Scaling Behavior on the Space of Calabi-Yau Manifolds

abstract:
  Recent work is reviewed which suggests that certain universal quantities,
defined for all Calabi-Yau manifolds, exhibit a specific behavior which is not
present for general K\"ahler manifolds. The variables in question, natural from
a mathematical perspective, are of physical importance because they determine
aspects of the low-energy string physics in four dimensions, such as Yukawa
couplings and threshold corrections. It is shown that these quantities,
evaluated on the complete class of Calabi-Yau hypersurfaces in weighted
projective 4-space, exhibit scaling behavior with respect to a new scaling
variable. (To appear in Mirror Symmetry II.)


title: Desingularization of singular hyperkaehler varieties II

abstract:
  This is a second part of alg-geom/9611015. We construct a natural
hyperkaehler desingularization for all singular hyperkaehler varieties. The
desingularization theorem was proven in alg-geom/9611015 under additional
assumption of local homogeneity. Here we show that local homogeneity is
redundant: every singular hyperkaehler variety has locally homogeneous
singularities.


title: Blowing up determinantal cerves and varieties of codim 2

abstract:
  We study the blow-ups X of P3 along a proj. normal curve C. We look for very
ample divisor classes on X of low degree, and we study the ideal of the
embedding of X. Some result is generalized to higher dimensions.


title: Recent Developments in Seiberg-Witten Theory and Complex Geometry

abstract:
  Recent developments in Seiberg-Witten theory and relations with Complex
Geometry.


title: Integrability of the twistor space for a hypercomplex manifold

abstract:
  A hypercomplex manifold is by definition a smooth manifold equipped with two
anticommuting integrable almost complex structures. For example, every
hyperkaehler manifold is canonically hypercomplex (the converse is not true).
For every hypercomplex manifold M, the two almost complex structures define a
smooth action of the algebra of quaternions on the tangent bundle to M. This
allows to associate to every hypercomplex manifold M of dimension 4n a certain
almost complex manifold X of dimension 4n+2, called the twistor space of M.
When M is hyperkaehler, X is well-known to be integrable. We show that for an
arbitrary hypercomplex manifold its twistor space is also integrable.


title: Weil classes on abelian varieties

abstract:
  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.


title: Effective base point freeness on a normal surface

abstract:
  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.


title:
Is a linear space contained in a variety? - On the number of derivatives
  needed to tell

abstract:
  Let $X^n\subset C^{n+a}$ or $X^n\subset P^{n+a}$ be a patch of an analytic
submanifold of an affine or projective space, let $x\in X$ be a general point,
and let L^k be a linear space of dimension k osculating to order m at x. If m
is large enough, one expects L to be contained in X and thus X contains a
linear space of dimension kthrough almost every point. We show that $L\subset
X$ in the following cases: k=1 and m=n+1; k=n-1, $a\geq 2$, and m=2; $n\geq 4$,
k=n-2 and m=4. We prove these results by first deriving the order of osculation
that generically implies containment and then showing that in these cases
containment must occur.
  If X is a patch of a projective variety, we address the question as to
whether X can be a smooth variety. We show that if there is a P^k through each
point and $codim(X)<\frac{k}{n-k}$ then X cannot be a smooth variety.


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

abstract:
  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.


title: A smooth surface in P4 not of general type has degree at most 66

abstract:
  We continue the work of Braun and Floystad, and Cook bounding the degree of
smooth surfaces in P4 not of general type using generic initial ideal theory.


title: A smooth surface in P4 not of general type has degree at most 46

abstract:
  This is the continuation of papers by Braun and Floystad, Cook, Braun and
Cook. We use Generic Initial Ideal Theory in conjunction with Liaison Theory to
further restrict the possible generic initial ideals of hyperplane sections of
smooth surfaces not of general type in P4.


title: Hodge numbers attached to a polynomial map

abstract:
  Given a polynomial map $f:\Bbb C^{n+1}\to\Bbb C$, one can attach to it a
geometrical variation of mixed Hodge structures (MHS) which gives rise to a
limit MHS. The equivariant Hodge numbers of this MHS are analytical invariants
of the polynomial map and reflect its asymptotic behaviour. In this paper we
compute them for a class of generic polynomials, in terms of Hodge numbers
attached to isolated hypersurface singularities and Hodge numbers of cyclic
coverings of projective space branched along a hypersurface.


title: Simultaneous minimal model of homogeneous toric deformation

abstract:
  For a flat family of Du Val singularities, we can take a simultaneous
resolution after finite base change. It is an interesting problem to consider
this analogy for a flat family of higher dimensional canonical singularities.
In this note, we consider an existence of simultaneous terminalization for K.
Altmann's homogeneous toric deformation whose central fibre is an affine
Gorenstein toric singularity. We obtain examples that there are no simultaneous
terminalization even after finite base change and give a sufficient condition
for an existence of simultaneous terminalization. Some examples of
4-dimensional flop are obtained as an application.


title:
Quotients par des groupes non r\'eductifs et vari\'et\'es de modules de
  complexes

abstract:
  I give a way to construct moduli spaces of complexes, as quotients of open
subsets of the space of all complexes by the product of the groups of
automorphisms of the terms.


title: On the cone of divisors of Calabi-Yau fiber spaces

abstract:
  We prove some version of Morrison's conjecture on the cone of divisors for
Calabi-Yau fiber spaces with non-trivial base pace whose total space is
3-dimensional.


title: On Hitchin's Connection

abstract:
  The aim of this paper is to give an explicit expression for Hitchin's
connection in the case of rank 2 bundles with trivial determinant over curves
of genus 2. We recall the definition of this connection (which arose in Quantum
Field Theory) and characterize it for families of genus two curves. We then
consider a particular family (over a configuration space), the corresponding
vector bundles are (almost) trivial. The Heat equations which give the
connection these bundles are related to the Lie algebra so(6). We compute some
local monodromy representations. As a byproduct we obtain, for any
representation space V of so(2g+2), a flat connection on the trivial bundle
with fiber V over a configuration space and thus a monodromy representation of
a pure braid group.


title:
Adelic Formulas for Gamma- and Beta-functions in Algebraic Numbers
  Fields

abstract:
  On the basis of analysis on the adele ring of any algebraic numbers field
(Tate's formula) a regularization for divergent adelic products of gamma- and
beta-functions for all completions of this field are proposed, and
corresponding regularized adelic formulas are obtained.


title: On the singularities of the theta divisors on Jacobians

abstract:
  We study the intersection cohomology of the theta divisors on Jacobians of
nonhyperelliptic curves.


title: Subvarieties of SU_C(2) and 2\theta-divisors in the Jacobian

abstract:
  We explore some of the interplay between Brill-Noether subvarieties of the
moduli space SU_C(2,K) of rank 2 bundles with canonical determinant on a smooth
projective curve and 2\theta divisors, via the inclusion of the moduli space
into |2\theta|, singular along the Kummer variety. In particular we show that
the moduli space contains all the trisecants of the Kummer and deduce that
there are quadrisecant lines only if the curve is hyperelliptic; we show that
for generic curves of genus <6, though no higher, bundles with >2 sections are
cut out by \Gamma_00; and that for genus 4 this locus is precisely the
Donagi-Izadi nodal cubic threefold associated to the curve.


title: K3 surfaces with interesting groups of automorphisms

abstract:
  By the fundamental result of I.I. Piatetsky-Shapiro and I.R. Shafarevich
(1971), the automorphism group Aut(X) of a K3 surface X over C and its action
on the Picard lattice S_X are prescribed by the Picard lattice S_X. We use this
result and our method (1980) to show finiteness of the set of Picard lattices
S_X of rank $\ge 3$ such that the automorphism group Aut(X) of the K3 surface X
has a non-trivial invariant sublattice S_0 in S_X where the group Aut(X) acts
as a finite group. For hyperbolic and parabolic lattices S_0 it has been proved
by the author before (1980, 1995). Thus we extend this results to negative
sublattices S_0.
  We give several examples of Picard lattices S_X with parabolic and negative
S_0. We also formulate the corresponding finiteness result for reflective
hyperbolic lattices of hyperbolic type over purely real algebraic number
fields.
  These results are important for the theory of Lorentzian Kac--Moody algebras
and Mirror Symmetry.


title: A Prym construction for the cohomology of a cubic hypersurface

abstract:
  Mumford defined a natural isomorphism between the intermediate jacobian of a
conic-bundle over $P^2$ and the Prym variety of a naturally defined \'etale
double cover of the discrminant curve of the conic-bundle. Clemens and
Griffiths used this isomorphism to give a proof of the irrationality of a
smooth cubic threefold and Beauville later generalized the isomorphism to
intermediate jacobians of odd-dimensional quadric-bundles over $P^2$. We
further generalize the isomorphism to the primitive cohomology of a smooth
cubic hypersurface in $P^n$.


title: Families of K3 surfaces

abstract:
  We use automorphic forms to prove that a compact family of Kaehler K3
surfaces with constant Picard number is isotrivial.


title: Geometry of cohomology support loci II: integrability of Hitchin's map

abstract:
  In very rough terms, the main theorem is that the set, which consists of
semistable vector bundles with trivial rational Chern classes and nontrivial
kth cohomology on a smooth complex projective variety, is a degeneration of a
union of abelian varieties. More precisely, we consider the subset of the
moduli space of Higgs bundles satisfying the analogous cohomological condition.
We show that this set is Zariski closed and that if Sigma is the normalization
of an irreducible component containing a stable point, then a connected
component of a general fiber of the restriction of the Hitchin map to Sigma is
an abelian variety. This should be interpreted as a nonabelian version of a
theorem of Green and Lazarsfeld. The Hitchin map in this setting is in fact
Simpson's generalization of it. The key point is to show that the general
fibers of this map are lagrangian (where the target of the map is taken to be
the image). This hinges on the fact that this property is essentially
hereditary for hyperkaehler submanifolds. We establish various other properties
of these sets including a codimension estimate which may be viewed as a generic
vanishing theorem.


title: A remark on an algebraic Riemann-Roch formula for flat bundles

abstract:
  We propose an algebraic Riemann-Roch formula for moving flat bundles on
contant families in characteristic zero with values in the ring of algebraic
differential characters. This formula lifts Grothendieck-Riemann-Roch formula
in the Chow groups and Bismut-Lott formula in the odd dimensional Betti
cohomology with $\C/\Z$ coefficients over the field of complex numbers.


title: A mirror theorem for toric complete intersections

abstract:
  We prove a generalized mirror conjecture for non-negative complete
intersections in symplectic toric manifolds. Namely, we express solutions of
the PDE system describing quantum cohomology of such a manifold in terms of
suitable hypergeometric functions. Revision 03.03.97: we correct an error in
Introduction.


title: Heights and Geometric Invariant Theory

abstract:
  Let $K$ be a number field, $\OK$ be its ring of integers. We introduce the
notion of compactified representation of $GL_N(\OK)$ and, we see how to
associate to a hermitian vector bundle $\E$ over $\Spec(\OK)$ and a
compactified representation $\T$, a hermitian tensor bundle $\E_T$. We can
prove then that there exists a lower bound for the heights of points
$x\in\P(\E_T)$ with $SL_N(K)$--semistable generic fibre in terms of the degree
of $\E$ and some universal constants depending only on the compactified
representation. We give then three applications: a universal lower bound for
general flag varieties, an application to the adjoint representation of
$SL_N(K)$ and a construction of a height on the moduli space of semistable
vector bundles over algebraic curves.


title:
Exemples de composantes irr\'eductibles et non r\'eduites du sch\'ema de
  Hilbert des courbes lisses et connexes de ${\bf P}^3$ (II)

abstract:
  Soit $H_{d,g}$ le sch\'ema de Hilbert des courbes lisses et connexes de
degr\'e $d$ et genre $g$ de l'espace projectif ${\bf P}^3$ sur un corps $k$
alg\'ebriquement clos de caract\'eristique nulle. Le but principal de cet
article est d'exhiber des composantes irr\'eductibles et non r\'eduites de
$H_{d,g}$ dont l'\'el\'ement g\'en\'eral correspond \`a une courbe trac\'ee sur
une surface quartique \`a droite double.


title: Counting rational curves on K3 surfaces

abstract:
  The aim of these notes is to explain the remarkable formula found by Yau and
Zaslow to express the number of rational curves on a K3 surface. Projective K3
surfaces fall into countably many families F(g) (g>0); a surface in F(g) admits
a g-dimensional linear system of curves of genus g. Such a system contains a
positive number, say n(g), of rational (highly singular) curves. The formula is
\sum n(g) q^g = q/D((q), where D(q) = q \prod (1-q^n)^{24} is the well-known
modular form of weight 12.


title: Residues and Resultants

abstract:
  Resultants, Jacobians and residues are basic invariants of multivariate
polynomial systems. We examine their interrelations in the context of toric
geometry. The global residue in the torus, studied by Khovanskii, is the sum
over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$
variables. Cox introduced the related notion of the toric residue relative to
$n+1$ divisors on an $n$-dimensional toric variety. We establish denominator
formulas in terms of sparse resultants for both the toric residue and the
global residue in the torus. A byproduct is a determinantal formula for
resultants based on Jacobians.


title: Principal bundles on elliptic fibrations

abstract:
  A central role in recent investigations of the duality of F-theory and
heterotic strings is played by the moduli of principal bundles, with various
structure groups G, over an elliptically fibered Calabi-Yau manifold on which
the heterotic theory is compactified. In this note we propose a simple
algebro-geometric technique for studying the moduli spaces of principal
G-bundles on an arbitrary variety X which is elliptically fibered over a base
S: The moduli space itself is naturally fibered over a weighted projective base
parametrizing spectral covers $\tilde{S}$ of S, and the fibers are identified
as translates of distinguished Pryms of these covers. In nice situations, the
generic Prym fiber is isogenous to the product of a finite group and an abelian
subvariety of $Pic(\tilde{S})$.


title: Dual varieties and the duality of the second fundamental form

abstract:
  First, we consider a compact real-analytic irreducible subvariety $M$ in a
sphere and its dual variety $M^\vee$. We explain that two matrices of the
second fundamental forms for both varieties $M$ and $M^\vee$ can be regarded as
the inverse matrices of each other. Also generalization in hyperbolic space is
explained.


title:
Semistable reduction of abelian varieties over extensions of small
  degree

abstract:
  We obtain necessary and sufficient conditions for abelian varieties to
acquire semistable reduction over fields of low degree. Our criteria are
expressed in terms of torsion points of small order defined over unramified
extensions.


title: Moishezon Manifolds

abstract:
  Let X be a compact Moishezon manifold which becomes projective after blowing
up a smooth subvariety $Y \subset X$. We assume also that there exists a proper
map $\rho :X \to X'$ onto a projective variety X' with $\rho(Y)$ a point, such
that $Pic(X/X') = \Z$ and $K_X$ is $\rho$-big. We prove some inequalities
between the dimensions of Y and X and we construct examples which shows the
optimality of the inequalities. Then we discuss some differential geometry
properties of these examples which lead to a conjecture.


title: On exponential sums

abstract:
  Let f be a polinomial with coefficients in a finite field F. Let $\Psi : F
\to C^{\ast}$ be a non-trivial additive character. In this paper we give bounds
for the exponential sums $\sum_{x\in F^n} \Psi (Tr_{F/F_p} (f(x)))$ in some
cases where the highest degree form of f defines a singular projective
hypersurface X (e.g. when X is an arrangement of lines in P^2). The bound
involves the Milnor numbers of the singularities of X. The proof goes via the
classical cohomological interpretation of this exponential sums through
Grothendieck's trace formula.


title: Commutator coverings of Siegel threefolds

abstract:
  We investigate the existence and non-existence of modular forms of low weight
with a character with respect to the paramodular group $\Gamma_t$ and discuss
the resulting geometric consequences. Using an advanced version of Maa\ss\
lifting one can construct many examples of such modular forms and in particular
examples of weight 3 cusp forms. Consequently we find many abelian coverings of
low degree of the moduli space ${\Cal A}_t$ of (1,t)-polarized abelian surfaces
which are not unirational. We also determine the commutator subgroups of the
paramodular group $\Gamma_t$ and its degree 2 extension $\Gamma^+_t$. This has
applications for the Picard group of the moduli stack ${\underline{\Cal A}}_t$.
Finally we prove non-existence theorems for low weight modular forms. As one of
our main results we obtain the theorem that the maximal abelian cover ${\Cal
A}_t^{com}$ of ${\Cal A}_t$ has geometric genus 0 if and only if t=1, 2, 4 or
5. We also prove that ${\Cal A}_t^{com}$ has geometric genus 1 for t=3 and 7.


title: The Gromov Invariants of Ruan-Tian and Taubes

abstract:
  Taubes has recently defined Gromov invariants for symplectic four-manifolds
and related them to the Seiberg-Witten invariants. Independently, Ruan and Tian
defined symplectic invariants based on ideas of Witten. In this note, we show
that Taubes' Gromov invariants are equal to certain combinations of Ruan-Tian
invariants. This link allows us to generalize Taubes' invariants. For each
closed symplectic four-manifold, we define a sequence of symplectic invariants
$Gr_{\delta}$, $\delta=0,1,2,...$. The first of these, $Gr_0$, generates
Taubes' invariants, which count embedded J-holomorphic curves. The new
invariants $Gr_{\delta}$ count immersed curves with $\delta$ double points. In
particular, these results give an independent proof that Taubes' invariants are
well-defined. They also show that some of the Ruan-Tian symplectic invariants
agree with the Seiberg-Witten invariants.


title: The Chow Ring of the Moduli Space of Abelian Threefolds

abstract:
  In this paper we determine the structure of the Chow ring of the
Delaunay-Voronoi compactification $\tilde{\cal A}_3$ of the moduli space of
principally polarized abelian threefolds. This compactification was introduced
by Namikawa and studied by Tsushima. We use equivariant classes on level
coverings of $\tilde{\cal A}_3$. We also compare this ring with the Chow ring
of the moduli space of stable genus 3 curves as determined by Faber.


title: Global Intersection Cohomology of Quasimaps' Spaces

abstract:
  Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of
complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple
$\alpha\in\BN[I]$ of positive integers one can consider the space $\CQ_\alpha$
of algebraic maps of degree $\alpha$ from $C$ to $\CB$. This space admits some
remarkable compactifications $\CQ^D_\alpha$ (Quasimaps), $\CQ^L_\alpha$
(Quasiflags), $\CQ^K_\alpha$ (Stable Maps) of $\CQ_\alpha$ constructed by
Drinfeld, Laumon and Kontsevich respectively. It has been proved that the
natural map $\pi: \CQ^L_\alpha\to \CQ^D_\alpha$ is a small resolution of
singularities. The aim of the present note is to study the cohomology
$H^\bullet(\CQ^L_\alpha,\BQ)$ of Laumon's spaces or, equivalently, the
Intersection Cohomology $H^\bullet(\CQ^L_\alpha,IC)$ of Drinfeld's Quasimaps'
spaces. We calculate the generating function $P_G(t)$ (``Poincar\'e
polynomial'') of the direct sum
$\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^D_\alpha,IC)$ and construct a natural
action of the Lie algebra ${\frak{sl}}_n$ on this direct sum by some
middle-dimensional correspondences between Quasiflags' spaces. We conjecture
that this module is isomorphic to distributions on nilpotent cone supported at
nilpotent subalgebra.


title:
On extremal contractions from threefolds to surfaces: the case of one
  non-Gorenstein point and non-singular base surface

abstract:
  In this paper we study a local structure of extrmal contractions $f\colon
X\to S$ from threffolds $X$ with only terminal singularities onto a surface
$S$. If the surface $S$ is non-singular and $X$ has a unique non-Gorenstein
point on a fiber we prove that either the linear system $|-K_X|$, $|-2K_X|$ or
$|-3K_X|$ contains a "good" divisor.


title: Universal Schubert polynomials

abstract:
  We introduce polynomials that represent general degeneracy loci for maps of
vector bundles. These polynomials specialize to the known classical and quantum
forms of single and double Schubert polynomials. This is the final version of
the paper, to appear in Duke Math. J. The results are strengthened because
Lakshmibai and Magyar have proved a conjecture from the previous version: in
the universal case, these loci are Cohen-Macaulay.


title:
Numerical Criteria for vey Ampleness of Divisors on Projective Bundles
  over an elliptic curve

abstract:
  In Butler, J.Differential Geom. 39 (1):1--34,1994, the author gives a
sufficient condition for a line bundle associated with a divisor D to be
normally generated on $X=P(E)$ where E is a vector bundle over a smooth curve
C. A line bundle which is ample and normally generated is automatically very
ample. Therefore the condition found in Butler's work, together with Miyaoka's
well known ampleness criterion, give a sufficient condition for the very
ampleness of D on X. This work is devoted to the study of numerical criteria
for very ampleness of divisors D which do not satisfy the above criterion, in
the case of C elliptic. Numerical conditions for the very ampleness of D are
proved,improving existing results. In some cases a complete numerical
characterization is found.


title: Semisimple Frobenius (super)manifolds and quantum cohomology of $P^r$

abstract:
  We introduce and study a superversion of Dubrovin's notion of semisimple
Frobenius manifolds. We establish a correspondence between semisimple Frobenius
(super)manifolds and special solutions to the (supersymmetric) Schlesinger
equations. Finally, we calculate the Schlesinger initial conditions for
solutions describing quantum cohomology of projective spaces.


title: Degenerations of Planar Linear Systems

abstract:
  Fixing $n$ general points $p_i$ in the plane, what is the dimension of the
space of plane curves of degree $d$ having multiplicity $m_i$ at $p_i$ for each
$i$? In this article we propose an approach to attack this problem, and
demonstrate it by successfully computing this dimension for all $n$ and for
$m_i$ constant, at most 3. This application, while previously known (see
\cite{hirschowitz1}), demonstrates the utility of our approach, which is based
on an analysis of the corresponding linear system on a degeneration of the
plane itself, leading to a simple recursion for these dimensions. We also
obtain results in the ``quasi-homogeneous'' case when all the multiplicities
are equal except one; this is the natural family to consider in the recursion.


title: McKay correspondence

abstract:
  This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS
workshops in Dec 1996, on work in progress that has not yet reached any really
worthwhile conclusion, but contains lots of fun calculations. History of Vafa's
formula, how the McKay correspondence for finite subgroups of SL(n,C) relates
to mirror symmetry. The main aim is to give numerical examples of how the 2
McKay correspondences
  (1) representations of G <--> cohomology of resolution
  (2) conjugacy classes of G <--> homology must work, and to restate my 1992
Conjecture as a tautology, like cohomology or K-theory of projective space.
Another aim is to give an introduction to Nakamura's results on the Hilbert
scheme of G-clusters, following his preprints and his many helpful
explanations. This is partly based on joint work with Y. Ito, and has benefited
from encouragement and invaluable suggestions of S. Mukai.


title:
Braid Group, Algebraic Surfaces and Fundamental Groups of complements of
  Branch Curves

abstract:
  This paper will appear in the Santa Cruz proceedings. An overview of the
braid group techniques in the theory of algebraic surfaces, from Zariski to the
latest results, is presented. An outline of the Van Kampen algorithm for
computing fundamental groups of complements of curves and the modification of
Moishezon-Teicher regarding branch curves of generic projections are given. The
paper also contains a description of a quotient of the braid group, namely
$\tilde B_n$ which plays an important role in the description of fundamental
groups of complements of branch curves.It turns out that all such groups are
``almost solvable'' $\tildeB_n$-groups. Finally, possible applications to the
study of moduli spaces of surfaces of general type are described and new
examples of positive signature spin surfaces whose fundamental groups can be
computed using the above algorithm (Galois cover of Hirzebruch surfaces) are
presented.


title:
On $\delta_m$ constant locus of versal deformations of nondegenerate
  hypersurface simple K3 singularities

abstract:
  Hypersurface simple K3 singularities defined by nondegenerate
quasi-homogeneous polynomials are classified into ninety five classes in term
of the weight of the polynomial by T. Yonemura. We consider versal deformations
of them. It has been conjectured that the stratum $\mu$ =const of the versal
deformation of any nondegenerate hypersurface simple K3 singularity is
equivalent to the $\delta_m$ constant locus. It holds true for the case
deformations are also nondegenerate by K. Watanabe. On the other hand, it
follows from Reid that the $\delta_m$ constant locus includes the $\mu$
constant locus generally. We show the conjecture holds true in general for
No.10-14, 46-51 and 83 in the table of Yonemura.


title: Five Definitions of Critical Point at Infinity

abstract:
  This survey paper discusses five equivalent ways of defining a ``critical
point at infinity'' for a complex polynomial of two variables.


title:
Schubert polynomials, the Bruhat order, and the geometry of flag
  manifolds

abstract:
  We illuminate the relation between the Bruhat order on the symmetric group
and structure constants (Littlewood-Richardson coefficients) for the cohomology
of the flag manifold in terms of its basis of Schubert classes. Equivalently,
the structure constants for the ring of polynomials in variables $x_1,x_2,...$
in terms of its basis of Schubert polynomials. We use combinatorial, algebraic,
and geometric methods, notably a study of intersections of Schubert varieties
and maps between flag manifolds. We establish a number of new identities among
these structure constants. This leads to formulas for some of these constants
and new results on the enumeration of chains in the Bruhat order. A new graded
partial order on the symmetric group which contains Young's lattice arises from
these investigations. We also derive formulas for certain specializations of
Schubert polynomials.


title: Complex projective surfaces and infinite groups

abstract:
  The paper contains a general construction which produces new examples of non
simply-connected smooth projective surfaces. We analyze the resulting surfaces
and their fundamental groups. Many of these fundamental groups are expected to
be non-residually finite. Using the construction we also suggest a series of
potential counterexamples to the Shafarevich conjecture which claims that the
universal covering of smooth projective variety is holomorphically convex. The
examples are only potential since they depend on group theoretic questions,
which we formulate, but we do not know how to answer. At the end we formulate
an arithmetic version of the Shafarevich conjecture.


title:
Basepoint freeness for big line bundles in positive characteristic, with
  applications to M_g and to 3-fold MMP

abstract:
  I give a necessary and sufficient condition for a nef and big line bundle in
positive characteristic to be semi-ample, and then give two applications: I
show that the relative dualizing sheaf of the universal curve is semi-ample, in
positive characteristic, and give a simple example which shows that this, and
the semi-ampleness criterion, fail in characteristic zero. My second
application is to Mori's program for minimal models of 3-folds. I prove a
version of the Basepoint Free Theorem (for big line bundles on 3-folds of
positive characteristic) and a simple Cone Theorem, for 3-folds of positive
Kodaira dimension.


title: Geometry of Moduli Spaces of Flat Bundles on Punctured Surfaces

abstract:
  We consider the moduli spaces of flat $SL(n, C)$-bundles on Riemann surfaces
with one puncture when we fix the conjugacy class ${\cal C}$ of the monodromy
transformation around the puncture. We show that under a certain condition on
the class ${\cal C}$ (namely the product of $k<n$ eigenvalues is not equal to
$1$) that we call property P the moduli space in question is smooth and its
natural closure is a normal algebraic variety with rational singularities. The
set of conjugacy classes having property P constitutes a Zariski open subset of
$SL(n, C)$ and it is also possible to define property P for the groups $SO(n,
C)}$ and $Sp(2n, C)$ to prove similar results. There are a few other
applications of our techniques, one of which is that if $G$ is a classical
reductive algebraic group and $A_1, A_2, ..., A_p\in G$, $p>1$ and
$A_1A_2\cdots A_pA_1^{-1}A_2^{-1}\cdots A_p^{-1}$ belongs to a class which
satisfies property P then the $p$ -tuple $(A_1, ..., A_p)$ algebraically
generates the whole group $G$.


title:
Applications of Braid Group Techniques to algebraic Surfaces, New
  examples

abstract:
  Every smooth minimal complex algebraic surface of general type, $X$, may be
mapped into a moduli space, $\MM_{c_1^2(X), c_2(X)}$, of minimal surfaces of
general type, all of which have the same Chern numbers. Using the braid group
and braid monodromy,we construct infinitely many new examples of pairs of
minimal surfaces of general type which have the same Chern numbers and
non-isomorphic fundamental groups. Unlike previous examples, our results
include $X$ for which $|\pi_1(X)|$ is arbitrarily large. Moreover, the surfaces
are of positive signature. This supports our goal of using the braid group and
fundamental groupsto decompose $\MM_{c_1^2(X),c_2(X)}$ into connected
components.


title: Examples of smooth non-general type surfaces in P^4

abstract:
  The aim of the paper is to provide a series of new examples of smooth
surfaces in P^4, not of general type, in degrees varying from 12 up to 14, and
to describe their geometry. By using mainly syzygies and liaison techniques, we
construct the following families of surfaces:
  - minimal proper elliptic surfaces of degree 12 and sectional genus 13,
  - two types of non-minimal proper elliptic surfaces of degree 12 and
sectional genus 14,
  - non-minimal K3 surfaces of degree 13 and sectional genus 16, and -
non-minimal K3 surfaces of degree 14 and sectional genus 19.


title:
The degree of the generators of the canonical ring of surfaces with
  p_g=0

abstract:
  It is shown that the canonical ring of a minimal surface of general type with
$p_g=0, K^2\geq 2$ is generated by its elements of degree lesser or equal to 5,
provided $|2K|$ has no fixed components, and that this bound can be lowered to
4 under the assumption that $|2K|$ is base-point free.


title: The deformation theory of local complete intersections

abstract:
  This is an expository paper on the subject of the title. It assumes basic
scheme theory, commutative and homological algebra.


title: A Remark on the Geometry of Elliptic Scrolls and Bielliptic Surfaces

abstract:
  The union of two quintic elliptic scrolls in P^4 intersecting transversally
along an elliptic normal quintic curve is a singular surface Z which behaves
numerically like a bielliptic surface. In the appendix to the paper [W. Decker
et al.: Syzygies of abelian and bielliptic surfaces in P^4, alg-geom/9606013]
where the equations of this singular surface were computed, we proved that Z
defines a smooth point in the appropriate Hilbert scheme and that Z cannot be
smoothed in P^4. Here we consider the analogous situation in higher dimensional
projective spaces P^{n-1}, where, to our surprise, the answer depends on the
dimension n-1. If n is odd the union of two scrolls cannot be smoothed, whereas
it can be smoothed if n is even. We construct an explicit smoothing.


title: Perverse Sheaves on Loop Grassmannians and Langlands Duality

abstract:
  We outline a proof of a geometric version of the Satake isomorphism. Given a
connected, complex algebraic reductive group G we show that the tensor category
of representations of the dual group $\check G$ is naturally equivalent to a
certain category of perverse sheaves on the loop Grassmannian of G. The above
result has been announced by Ginsburg in \cite{G} and some of the arguments are
borrowed from \cite{G}. However, we use a more "natural" commutativity
constraint for the convolution product, due to Drinfeld. Secondly, we give a
direct geometric proof that the global cohomology functor is exact and
decompose this cohomology functor into a direct sum of weights. We avoid the
use of the decomposition theorem of \cite{BBD} which makes our techniques
applicable to perverse sheaves with coefficients over arbitrary commutative
rings. This note includes sketches of (some of the) proofs. The details, as
well as the generalization to representations over arbitrary fields and
commutative rings will appear elsewhere.


title: Moduli of flat bundles on open Kaehler manifolds

abstract:
  We consider the moduli space MN of flat unitary connections on an open
Kaehler manifold U (complement of a divisor with normal crossings) with
restrictions on their monodromy transformations. Using intersection and L2
cohomologies with degenerating coefficients we construct a natural symplectic
form F on MN. When U is quasi-projective we prove that F is actually a Kaehler
form.


title: Moduli of regular holonomic D-modules with normal crossing singularities

abstract:
  This paper solves the global moduli problem for regular holonomic D-modules
with normal crossing singularities on a nonsingular complex projective variety.
This is done by introducing a level structure (which gives rise to
``pre-D-modules''), and then introducing a notion of (semi-)stability and
applying Geometric Invariant Theory to construct a coarse moduli scheme for
semistable pre-D-modules. A moduli is constructed also for the corresponding
perverse sheaves, and the Riemann-Hilbert correspondence is represented by an
analytic morphism between these moduli spaces.


title: Contact Cohomology of the Projective Plane

abstract:
  We construct an associative ring which is a deformation of the quantum
cohomology ring of the projective plane. Just as the quantum cohomology encodes
the incidence characteristic numbers of rational plane curves, the contact
cohomology encodes the tangency characteristic numbers.


title: Derivation algebras of toric varieties

abstract:
  Normal affine algebraic varieties in characteristic 0 are uniquely determined
(up to isomorphism) by the Lie algebra of derivations of their coordinate ring.
This is not true without the hypothesis of normality. But, we show that (in
general, non-normal) toric varieties defined by simplicial affine semigroups
are uniquely determined by their Lie algebra if they are supposed to be
Cohen-Macaulay of dimension at least 2 or Gorenstein of dimension 1. Moreover,
every automorphism of the Lie algebra is induced from a unique automorphism of
the variety and every derivation of the Lie algebra is inner, i.e., the first
cohomology of the Lie algebra with coefficients in the adjoint representation
vanishes.


title: Associativity relations in quantum cohomology

abstract:
  We describe interdependencies among the quantum cohomology associativity
relations. We strengthen the first reconstruction theorem of Kontsevich and
Manin by identifying a subcollection of the associativity relations which
implies the full system of WDVV equations. This provides a tool for identifying
non-geometric solutions to WDVV.


title: Hypercomplex Varieties

abstract:
  We give a number of equivalent definitions of hypercomplex varieties and
construct a twistor space for a hypercomplex variety. We prove that our
definition of a hypercomplex variety (used, e. g., in alg-geom/9612013) is
equivalent to a definition proposed by Deligne and Simpson, who used twistor
spaces. This gives a way to define hypercomplex spaces (to allow nilpotents in
the structure sheaf). We give a self-contained proof of desingularization
theorem for hypercomplex varieties: a normalization of a hypercomplex variety
is smooth and hypercomplex.


title: Chern Classes of Fibered Products of Surfaces

abstract:
  In this paper we introduce a formula to compute Chern classes of fibered
products of algebraic surfaces. For f, a generic projection to CP^2, of an
algebraic surface X, we define X_k (for k smaller than degf) to be the k
products of X over f minus the big diagonal. For k=degf, X_k is called the
Galois cover of f w.r.t. full symmetric group. Let S be the branch curve of f.
We give a formula for c_1^2 and c_2 of X_k, in terms of degf, degS, and the
number of cusps, nodes and branch points of S. We apply the formula in 2
examples and add a conjecture concerning the spin structure of fibered products
of Veronese surfaces.


title: Mixed Segre Numbers and Integral Closure of Ideals

abstract:
  We introduce mixed Segre numbers of ideals which generalize the notion of
mixed multiplicities of ideals of finite colength and show how many results on
mixed multiplicities can be extended to results on mixed Segre numbers. In
particular, we give a necessary and sufficient condition in terms of these
numbers for two ideals to have the same integral closure. Also, our theory
yields a new proof of a generalization of Rees' theorem that links the integral
closure of an ideal to its multiplicity. Finally, we give a quick application
of our results to Whitney equisingularity.


title: A New look at the vortex equations and dimensional reduction

abstract:
  In order to use the technique of dimensional reduction, it is usually
necessary for there to be a symmetry coming from a group action. In this paper
we consider a situation in which there is no such symmetry, but in which a type
of dimensional reduction is nevertheless possible. We obtain a relation between
the Coupled Vortex equations on a closed Kahler manifold, $X$, and the
Hermitian-Einstein equations on certain $P^1$-bundles over $X$. Our results
thus generalize the dimensional reduction results of Garcia-Prada, which apply
when the Hermitian-Einstein equations are on $X\times P^1$.


title: Standard Monomial Theory for Bott-Samelson Varieties of GL(n)

abstract:
  We construct an explicit basis for the coordinate ring of the Bott-Samelson
variety Z_i associated to G = GL(n) and an arbitrary sequence of simple
reflections i. Our basis is parametrized by certain standard tableaux and
generalizes the Standard Monomial basis for Schubert varieties. Our standard
tableaux have a natural crystal graph structure.


title: Determinant bundles for abelian schemes

abstract:
  To a symmetric, relatively ample line bundle on an abelian scheme one can
associate a linear combination of the determinant bundle and the relative
canonical bundle, which is a torsion element in the Picard group of the base.
We improve the bound on the order of this element found by Faltings and Chai.
In particular, we obtain an optimal bound when the degree of the line bundle d
is odd and the set of residue characteristics of the base does not intersect
the set of primes p dividing d, such that $p\equiv -1\mod(4)$ and p<2g, where g
is the relative dimension of the abelian scheme. Also, we show that in some
cases these torsion elements generate the entire torsion subgroup in the Picard
group of the corresponding moduli stack.


title:
Geometric Realization of Whittaker Functions and the Langlands
  Conjecture

abstract:
  We prove the equivalence of two conjectural constructions of unramified
cuspidal automorphic functions on the adelic group GL_n(A) associated to an
irreducible l-adic local system of rank n on an algebraic curve X over a finite
field. The existence of such a function is predicted by the Langlands
conjecture.
  The first construction, which was proposed by Shalika and Piatetski-Shapiro
following Weil and Jacquet-Langlands (n=2), is based on considering the
Whittaker function. The second construction, which was proposed recently by
Laumon following Drinfeld (n=2) and Deligne (n=1), is geometric: the
automorphic function is obtained via Grothendieck's ``faisceaux-fonctions''
correspondence from a complex of sheaves on an algebraic stack.
  Our proof of their equivalence is based on a local result about the spherical
Hecke algebra, which we prove for an arbitrary reductive group. We also discuss
a geometric interpretation of this result.


title:
On Betti numbers and Chern classes of varieties with trivial odd
  cohomology groups

abstract:
  It was noticed in a very recent preprint of T. Eguchi, K. Hori, and Ch.-Sh.
Xiong (hep-th/9703086) that a curious identity between Betti numbers and Chern
classes holds for many examples of Fano varieties. The goal of this paper is to
prove that for varieties with trivial odd cohomology groups this identity is
equivalent to having zero Hodge numbers $h^{p,q}$ for $p\neq q$.


title: A note on P-resolutions of cyclic quotient singularities

abstract:
  P-resolutions of a cyclic quotient singularity are known to be in one-to-one
correspondence with the components of the base space of its semi-universal
deformation. Stevens and Christophersen have shown that P-resolutions are
parametrized by so-called chains representing zero or, equivalently, certain
subdivisions of polygons. I give here a purely combinatorial proof of the
correspondence between subdivisions of polygons and P-resolutions.


title:
On the quotient of the braid group by commutators of transversal
  half-twists and its group actions

abstract:
  The group described in this paper appeared while studying fundamental groups
of complements of branch curves. It turned out that a certain quotient of the
braid group acts on those fundamental groups and studying this action is
essential for understanding the structure of the fundamental groups. We
describe here the quotient of the Artin braid group by commutators of
transversal half-twists and we investigates its group actions. We denote the
quotient by B_n~ and refer to the groups which admit an action of B_n~ , as
B_n~-groups.
  We distinguish special elements in B_n~-groups which we call prime elements
and we give a criterion for an element to be prime. This criterion will be
applied to the study of the structure of fundamental groups of complements of
branch curves. The group B_n~ itself turns out to be an extension of a solvable
group by a symmetric group.


title: Heisenberg invariant quartics and SU_C(2) for a curve of genus four

abstract:
  If C is a curve of genus 4 without vanishing theta-nulls then there exists a
unique (irreducible) Heisenberg-invariant quartic Q_C in |2\Theta| = P^{15}
such that Sing Q_C contains the image of SU_C(2), the moduli space of rank 2
vector bundles with trivial determinant. Moreover, in each eigen-P^7 of the
Heisenberg action on |2\Theta|, Q_C restricts to the classical Coble quartic of
the corresponding Prym-Kummer variety. We compare Q_C with the hypersurface G_3
in |2\Theta| of divisors containing a translate of C in J(C), and show that in
the eigen-P^7s G_3 recovers Beauville--Debarre's quadrisecant planes of the
Prym-Kummers (this works for any genus). Using the Recillas construction this
enables us to deduce, contrary to the analogous result for genus 3, that Q_C
and G_3 are distinct.


title: Three-block exceptional collections over Del Pezzo surfaces

abstract:
  We study complete exceptional collections of coherent sheaves over Del Pezzo
surfaces, which consist of three blocks such that inside each block all Ext
groups between the sheaves are zero. We show that the ranks of all sheaves in
such a block are the same and the three ranks corresponding to a complete
3-block exceptional collection satisfy a Markov-type Diophantine equation that
is quadratic in each variable. For each Del Pezzo surface, there is a finite
number of these equations; the complete list is given. The 3-string braid group
acts by mutations on the set of complete 3-block exceptional collections. We
describe this action. In particular, any orbit contains a 3-block collection
with the sum of ranks that is minimal for the solutions of the corresponding
Markov-type equation, and the orbits can be obtained from each other via
tensoring by an invertible sheaf and with the action of the Weyl group. This
allows us to compute the number of orbits up to twisting.


title: Interpolation on Jets

abstract:
  We show that for any finite generic union of pairs $(x_i,L_i)_i$ where
$x_i\in L_i$ is a point of the line $L_i$ in projective $n$-space, the divisors
$m_ix_i$ on the $L_i$ have maximal rank with respect to homogeneous $d$ forms
for all $d\geq 0$ and all $m_i\geq 0$ modulo the expected numerical
restrictions.


title: Generic Hypersurface Singularities

abstract:
  We give a new differential proof of our result on the maximal rank of generic
unions of points of multiplicity two in projective space in degrees greater
than five. This simplifies somewhat our proof of the Waring conjecture.


title: Alexander Invariants of Complex Hyperplane Arrangements

abstract:
  Let A be an arrangement of complex hyperplanes. The fundamental group of the
complement of A is determined by a braid monodromy homomorphism from a finitely
generated free group to the pure braid group. Using the Gassner representation
of the pure braid group, we find an explicit presentation for the Alexander
invariant of A. From this presentation, we obtain combinatorial lower bounds
for the ranks of the Chen groups of A. We also provide a combinatorial
criterion for when these lower bounds are attained.


title: On the topology of holomorphic bundles

abstract:
  In this work we study the topology of holomorphic rank two bundles over
complex surfaces. We consider bundles that are constructed by glueing and show
that under certain conditions the topology of the bundle does not depend on the
glueing. We present a simple classification of bundles on blown-up surfaces.


title:
Projections of Grassmannians of lines and characterization of Veronese
  varieties

abstract:
  We characterize the double Veronese embedding of P^n as the only variety
that, under certain general conditions, can be isomorphically projected from
the Grassmannian of lines in P^{2n+1} to the Grassmannian of lines in P^{n+1}.


title: Rankin-Cohen Type Differential Operators for Siegel Modular Forms

abstract:
  Let H_n be the Siegel upper half space and let F and G be automorphic forms
on H_n of weights k and l, respectively. We give explicit examples of
differential operators D acting on functions on H_n x H_n such that the
restriction of D(F(Z_1) G(Z_2)) to Z = Z_1 = Z_2 is again an automorphic form
of weight k+l+v on H_n. Since the elliptic case, i.e. n=1, has already been
studied some time ago by R. Rankin and H. Cohen we call such differential
operators Rankin-Cohen type operators. We also discuss a generalisation of
Rankin-Cohen type operators to vector valued differential operators.


title: Testing S-duality conjecture and exceptional bundles

abstract:
  In this paper we deal with algebro-geometrical problems connected with
testing S-duality conjecture for super-symmetric Yang-Mills quantum field
theories in four dimensions. We describe all field configurations such that
beta function coefficient for gauge groups of rank 1 and 2 is zero or negative.
We also give special series of such field configurations for gauge groups of an
arbitrary rank. Realization of one of the series discovers a connection with
exceptional bundles. That points again at relation between S-duality and string
duality.


title: The Ideal Generation Problem for Fat Points

abstract:
  This paper is concerned with determining the number of generators in each
degree for minimal sets of homogeneous generators for saturated ideals defining
fat point subschemes $Z=m_1p_1+ ... +m_rp_r$ for general sets of points $p_i$
of $P^2$. For thin points (i.e., m_i=1 for all i), a solution is known, in
terms of a maximal rank property. Although this property in general fails for
fat points, we show it holds in an appropriate asymptotic sense. In the uniform
(i.e., $m_1= ... =m_r$) case, we determine all failures of this maximal rank
property for $r\le 9$, and we develop evidence for the conjecture that no other
failures occur for $r > 9$.


title:
Very Ampleness and Higher Syzygies for Algebraic Surfaces and Calabi-Yau
  Threefolds

abstract:
  This work consists of two parts. In the first part we develop new techniques
to compute Koszul cohomology groups for several classes of varieties. As
applications we prove results on projective normality and syzygies for
algebraic surfaces. From more general results we obtain in particular the
following:
  a) Mukai's conjecture (and stronger variants of it) regarding projective
normality and normal presentation for surfaces with Kodaira dimension 0, and
uniform bounds for higher syzygies associated to adjoint linear series,
  b) effective bounds along the lines of Mukai's conjecture regarding
projective normality and normal presentation for surfaces of positive Kodaira
dimension, and,
  c) results on projective normality for pluricanonical models of surfaces of
general type (recovering and strengthening results by Ciliberto) and
generalizations of them to higher syzygies.
  In the second part we prove results on very ampleness, projective normality
and higher syzygies for both singular and smooth Calabi-Yau threefolds. The
general results that we prove are analogues of the results of St. Donat for K3
surfaces. From these general results we obtain bounds very close to Fujita's
conjecture regarding very ampleness of powers of an ample line bundle A (for
instance, if A^3 is greater than 1, our bound is one short of Fujita's). We
generalize our results on very ampleness and projective normality to higher
syzygies.


title: An Assympotic Vanishing Theorem for Generic Unions of Multiple Points

abstract:
  In this revised form, the proof of the principal lemma has been simplified
and the main theorem has been extended to all characteristics for those
varieties which are smooth in codimension one.
  This principal theorem essentially says the following : given an ample line
bundle O(1) on a projective variety X and a fixed upper bound M on the
multiplicities, there exists a lower bound D such that any generic union of
multiple points of multiplicity at most M imposes independent conditions on the
sections of O(d) for d>D. Here a multiple point is the closed subscheme defined
by a power m of the ideal of a smooth point in X and m is its multiplicity.


title: On Punctual Quot Schemes for Algebraic Surfaces

abstract:
  The punctual Quot scheme parametrizes all length d quotients of a (locally)
trivial rank r sheaf which are supported at a fixed point. The author shows
that this scheme is irreducible and (rd-1)-dimensional. The same result was
proved independently by Ellingsrud and Lehn.


title:
Tempered currents and the cohomology of the remote fiber of a real
  polynomial map

abstract:
  Let $p:R^n\to R$ be a polynomial map. Consider the complex
$S'\Omega^*(\RR^n)$ of tempered currents on $R^n$ with the twisted differential
$d_p=d-dp$ where $d$ is the usual exterior differential and $dp$ stands for the
exterior multiplication by $dp$. Let $t\in R$ and let $F_t=p^{-1}(t)$. In this
paper we prove that the reduced cohomology $\tilda H^k(F_t;C)$ of $F_t$ is
isomorphic to $H^{k+1}(S'\Omega^*(\RR^n),d_p)$ in the case when $p$ is
homogeneous and $t$ is any positive real number. We conjecture that this
isomorphism holds for any polynomial $p$, for $t$ large enough (we call the
$F_t$ for $t >> 0$ the remote fiber of $p$) and we prove this conjecture for
polynomials that satisfy certain technical condition. The result is analogous
to that of A. Dimca and M. Saito, who give a similar (algebraic) way to compute
the reduced cohomology of the generic fiber of a complex polynomial.


title:
Birational Models of the Moduli Spaces of Stable Vector Bundles over
  Curves

abstract:
  We give a method to construct stable vector bundles whose rank divides the
degree over curves of genus bigger than one. The method complements the one
given by Newstead. Finally, we make some systematic remarks and observations in
connection with rationality of moduli spaces of stable vector bundles.


title: Empty real Enriques surfaces and Enriques-Einstein-Hitchin 4-manifolds

abstract:
  We prove that the moduli space of empty real Enriques surfaces (and, thus,
the moduli space of compact orientable 4-dimensional Einstein manifolds whose
universal covering is a K3-surface and \pi_1(E) = Z/2 x Z/2) is connected. The
proof is based on a systematic study of real elliptic pencils and gives
explicit models of all empty real Enriques surfaces.


title: Bend, break and count

abstract:
  We give a 'recursive' formula (in terms of reducible limits) for counting
rational curves on a variety moving in any sufficiently large and well-behaved
family. Our approach is completely elementary and makes no use of moduli spaces
for maps or the like.


title: The global nilpotent variety is Lagrangian

abstract:
  The purpose of this note is to present a short elementary proof of a theorem
due to Faltings and Laumon, saying that the global nilpotent cone is a
Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on
a complex compact curve. This result plays a crucial role in the Geometric
Langlands program, due to Beilinson-Drinfeld, since it insures that the
D-modules on the moduli space of G-bundles whose characteristic variety is
contained in the global nilpotent cone are automatically holonomic, hence, e.g.
have finite length.

categories: alg-geom dg-ga math.AG math.DG math.QA q-alg


title:
A closed model structure for $n$-categories, internal $Hom$, $n$-stacks
  and generalized Seifert-Van Kampen

abstract:
  We define a closed model category containing the $n$-nerves defined by
Tamsamani, and admitting internal $Hom$. This allows us to construct the
$n+1$-category $nCAT$ by taking the internal $Hom$ for fibrant objects. We
prove a generalized Seifert-Van Kampen theorem for Tamsamani's Poincar\'e
$n$-groupoid of a topological space. We give a still-speculative discussion of
$n$-stacks, and similarly of comparison with other possible definitions of
$n$-category.


title:
All Abelian Quotient C.I.-Singularities Admit Projective Crepant
  Resolutions in All Dimensions

abstract:
  For Gorenstein quotient spaces $C^d/G$, a direct generalization of the
classical McKay correspondence in dimensions $d\geq 4$ would primarily demand
the existence of projective, crepant desingularizations. Since this turned out
to be not always possible, Reid asked about special classes of such quotient
spaces which would satisfy the above property. We prove that the underlying
spaces of all Gorenstein abelian quotient singularities, which are embeddable
as complete intersections of hypersurfaces in an affine space, have
torus-equivariant projective crepant resolutions in all dimensions. We use
techniques from toric and discrete geometry.


title:
Divisors on elliptic Calabi-Yau 4-folds and the superpotential in
  F-theory, I

abstract:
  Each smooth elliptic Calabi-Yau 4-fold determines both a three-dimensional
physical theory (a compactification of ``M-theory'') and a four-dimensional
physical theory (using the ``F-theory'' construction). A key issue in both
theories is the calculation of the ``superpotential''of the theory. We propose
a systematic approach to identify these divisors, and derive some criteria to
determine whether a given divisor indeed contributes. We then apply our
techniques in explicit examples, in particular, when the base B of the elliptic
fibration is a toric variety or a Fano 3-fold. When B is Fano, we show how
divisors contributing to the superpotential are always "exceptional" (in some
sense) for the Calabi-Yau 4-fold X. This naturally leads to certain transitions
of X, that is birational transformations to a singular model (where the image
of D no longer contributes) as well as certain smoothings of the singular
model. If a smoothing exists, then the Hodge numbers change. We speculate that
divisors contributing to the superpotential are always "exceptional" (in some
sense) for X, also in M-theory. In fact we show that this is a consequence of
the (log)-minimal model algorithm in dimension 4, which is still conjectural in
its generality, but it has been worked out in various cases, among which toric
varieties.


title: Rozansky-Witten invariants via Atiyah classes

abstract:
  Recently, L.Rozansky and E.Witten (hep-th/9612216) associated to any
hyperKaehler manifold X a system of "weights" (numbers, one for each trivalent
graph) and used them to construct invariants of topological 3-manifolds. We
give a very simple cohomological definition of these weights in terms of the
Atiyah class of X (the obstruction to existence of a holomorphic connection).
We show that the analogy between the tensor of curvature of a hyperKaehler
metric and the tensor of structure constants of a Lie algebra observed by
Rozansky and Witten, holds in fact for any complex manifold, if we work on the
level of cohomology and for any Kaehler manifold, if we work on the level of
Dolbeault forms. In particular, for any sheaf A of commutative algebras on any
complex manifold the shifted cohomology of the tangent sheaf tensored with A is
a graded Lie algebra. We also show that a certain system of Gilkey-type
complexes of "natural tensors" on Kaehler manifolds is, up to suspension
(accounting for various changes of signs), identified with the PROP (in the
sense of Adams and MacLane) describing Lie algebras "up to higher homotopies".
As an outcome of our considerations, we give a formula for the Rozansky-Witten
classes using any Kaehler metric on a holomorphic symplectic manifold.


title: Plane curves of minimal degree with prescribed singularities

abstract:
  We prove that there exists a>0 such that for any integer d>2 and any
topological types S_1,...,S_n of plane curve singularities, satisfying
$\mu(S_1)+...+\mu(S_n) \leq ad^2$, there exists a reduced irreducible plane
curve of degree d with exactly n singular points of types S_1,...,S_n,
respectively. This estimate is optimal with respect to the exponent of d. In
particular, we prove that for any topological type S there exists an
irreducible polynomial of degree $d \leq 14\sqrt{\mu(S)}$ having a singular
point of type S.


title: On Degenerations of Fibre Spaces of Curves of Genus >= 2

abstract:
  In this note, we show that for surfaces admitting suitable fibrations, any
given degeneration X / Delta is bimeromorphic to a fiber space over Delta and
we apply this result to the study of the degenerate fiber.


title: On fundamental groups of elliptically connected surfaces

abstract:
  A compact complex manifold $X$ is called elliptically connected if any pair
of points in $X$ can be connected by a chain of elliptic or rational curves. We
prove that the fundamental group of an elliptically connected compact complex
surface is almost abelian. This confirms a conjecture which states that the
fundamental group of an elliptically connected K\"ahler manifold must be almost
abelian.


title:
Connected components in the intersection of two open opposite Schubert
  cells in real complete flag manifold

abstract:
  In this paper we reduce the problem of counting the number of connected
components in the intersection of two opposite open Schubert cells in the
variety of real complete flags to a purely combinatorial question of counting
the number of orbits of a certain intriguing group action in the space of upper
triangular matrices with {0,1}-valued entries. The crucial step of our
reduction uses the parametrization of the space of real unipotent totally
positive upper triangular matrices introduced by Lusztig and Berenstein, Fomin,
Zelevinski.


title: Intersection Homology of Toric Varieties and a Conjecture of Kalai

abstract:
  We prove that the intersection homology Poincare' polynomial P(X) of an
affine toric variety X is bounded below by the product P(Y)P(X/Y), where Y is
the closure of any orbit in X and X/Y is a slice transverse to the orbit. This
proves a combinatorial conjecture of Kalai for rational polytopes.


title: Non-Abelian Brill-Noether theory and Fano 3-folds

abstract:
  A Brill-Noether locus is a subscheme of the moduli of bundles E over a curve
C defined by requiring E to have a given number of sections, or homomorphisms
from another bundle. There are a number of different types, that can be treated
by determinantal methods, with symmetry or skewsymmetry arising from Serre
duality. They have many beautiful applications to curves, K3 surfaces and Fano
3-folds.


title: On the irreducibility of the punctual Quotient Scheme of a Surface

abstract:
  We prove that the Quot-scheme of finite quotients of a vector bundle which
are of a given length and supported in one point, is irreducible and of the
expected dimension.


title: An algorithm for computing the integral closure

abstract:
  In this article we give an algorithm for computing the integral closure of a
reduced Noetherian ring R, in case this integral closure is finitely generated
over R.


title: Bertini and his two fundamental theorems

abstract:
  After reviewing Bertini's life story, a fascinating drama, we make a critical
examination of the old statements and proofs of Bertini's two fundamental
theorems, the theorem on variable singular points and the theorem on reducible
linear systems. We explain the content of the statements in a way that is
accessible to a nonspecialist, and we develop versions of the old proofs that
are complete and rigorous by current standards. In particular, we prove a new
extension of Bertini's first theorem, which treats variable $r$-fold points for
any $r$.


title: Towards a Halphen theory of linear series on curves

abstract:
  A linear series on a curve C in $P^3$ is "primary" when it does not contain
the series cut by planes. We provide a lower bound for the degree of these
series, in terms of deg(C), g(C) and of the number $s = min{i: h^0(I_C(i))\neq
0}$; as a corollary, we obtain bounds for the gonality of C. Examples show that
our bound is sharp. Extensions to the case of general linear series and to the
case of curves in higher projective spaces are considered.


title: Second order theta divisors on Pryms

abstract:
  Van Geemen and van der Geer, Donagi, Beauville and Debarre proposed
characterizations of the locus of jacobians which use the linear system of
$2\Theta$-divisors. We give new evidence for these conjectures in the case of
Prym varieties.


title: Le probleme de Brill-Noether pour des fibres stables de petites pentes

abstract:
  We deal with the Brill-Noether problem for stable vector bundles of slope
between one and two.


title: A simply connected numerical Godeaux surface with ample canonical class

abstract:
  We prove that a recent construction of a numerical Godeaux surface due to P.
Craighero and R. Gattazzo is simply connected, and show how to realize their
construction as a double plane. By proving that the surface contains no
(-2)-curves, we obtain that this is the first example of a simply connected
surface with vanishing geometric genus and ample canonical class.


title: Mirror symmetry on K3 surfaces via Fourier-Mukai transform

abstract:
  We use a relative Fourier-Mukai transform on elliptic K3 surfaces $X$ to
describe mirror symmetry. The action of this Fourier-Mukai transform on the
cohomology ring of $X$ reproduces relative T-duality and provides an
infinitesimal isometry of the moduli space of algebraic structures on $X$
which, in view of the triviality of the quantum cohomology of K3 surfaces, can
be interpreted as mirror symmetry.


title: Generalized Fourier-Mukai Transforms

abstract:
  The paper sets out a generalized framework for Fourier-Mukai transforms and
illustrates their use via vector bundle transforms. A Fourier-Mukai transform
is, roughly, an isomorphism of derived categories of (sheaves) on smooth
varieties X and Y. We show that these can only exist if the first Chern class
of the varieties vanishes and, in the case of vector bundle transforms, will
exist if and only if there is a bi-universal bundle on XxY which is "strongly
simple" in a suitable sense. Some applications are given to abelian varieties
extending the work of Mukai.


title: Fourier-Mukai transforms for elliptic surfaces

abstract:
  We compute a large number of moduli spaces of stable bundles on a general
algebraic elliptic surface using a new class of Fourier-Mukai type transforms.


title: The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

abstract:
  Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of
complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple
$\alpha\in N[I]$ of positive integers one can consider the space $Q_\alpha$ of
algebraic maps of degree $\alpha$ from $C$ to $B$. This space admits some
remarkable compactifications $Q^D_\alpha$ (Quasimaps), $Q^L_\alpha$
(Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it
was proved that the natural map $\pi: Q^L_\alpha\to Q^D_\alpha$ is a small
resolution of singularities. The aim of the present note is to study the
singular support of the Goresky-MacPherson sheaf $IC_\alpha$ on the Quasimaps'
space $Q^D_\alpha$. Namely, we prove that this singular support $SS(IC_\alpha)$
is irreducible. The proof is based on the factorization property of Quasimaps'
space and on the detailed analysis of Laumon's resolution $\pi: Q^L_\alpha\to
Q^D_\alpha$.


title: Trianalytic subvarieties of the Hilbert scheme of points on a K3 surface

abstract:
  Let X be a hyperkaehler manifold. Trianalytic subvarieties of X are
subvarieties which are complex analytic with respect to all complex structures
induced by the hyperkaehler structure. Given a K3 surface M, the Hilbert scheme
classifying zero-dimensional subschemes of M admits a hyperkaehler structure.
We show that for M generic, there are no trianalytic subvarieties of the
Hilbert scheme. This implies that a generic deformation of the Hilbert scheme
of K3 has no complex subvarieties.


title: A base point free theorem for log canonical surfaces

abstract:
  We prove a base point free theorem for nef and log big divisors on log
canonical surfaces.


title: Mixed twistor structures

abstract:
  The purpose of this paper is to introduce the notion of mixed twistor
structure, a generalization of the notion of mixed Hodge structure. The utility
of this notion is to make possible a theory of weights for various things
surrounding arbitrary representations of the fundamental group of a smooth
projective variety. We give some examples of generalizations of classical
results for variations of mixed Hodge structure, to the twistor setting. This
supports a ``meta-theorem'' (which we state but don't prove) that one can
everywhere replace the word ``Hodge'' by the word ``twistor''.
  We show that the jet spaces of hyperk\"ahler or more generally hypercomplex
manifolds have natural mixed twistor structures which determine the
hypercomplex structure in a formal neighborhood of a point.


title: Some notes on the moduli of stable sheaves on elliptic surfaces

abstract:
  Under some assumptions, we compute the Picard group of moduli of stable
sheaves on Abelian surfaces. Considering relative moduli spaces, it is
sufficient to consider the moduli of stable sheaves on the product of elliptic
curves. By using elementary transformatios (which was used by Friedman to treat
rank 2 moduli spaces on elliptic surfaces), we treat this case. We also show
that some moduli spaces on P^2 are rational.


title: On the rationality of $SU(r,d)$

abstract:
  Let SU(r,d) be the moduli space of rank r, degree d vector bundles over a
smooth projective curve of genus $g\ge 2$. If (r,d)=1 and d divides r+1, then
SU is rational. Furthermore, if $0<\delta < r$ and all prime divisors of
$\delta$ divide r, and if d divides $r-\delta$, then SU is rational. The proof
is a variation on a result of Newstead and modifications due to Ballico and
then Boden and Yokogawa.


title: Birational maps of moduli of Brill-Noether pairs

abstract:
  Let $C$ be a smooth projective irreducible curve of genus $g$. And let
$G_{\alpha}(n,d,l)$ be the moduli space of $\alpha$ stable pairs of a vector
bundle of $\rank n, \deg d$ and a subspace of $H^0(C,E)$ of $\dim = l $. We
find an explicit birational map from $G_{\alpha} (n, d, n+1)$ to $G_{\alpha}
(1, d, n+1)$ for $C$ general, $1/\alpha \gg 0$ and $g \ge n^2-1$. Because of
this and other examples, we conjecture $G_{\alpha} (a, d, a+z)$ maps
birationally to $G_{\alpha} (z, d, a+z)$ for $1/ \alpha \gg 0$ and $C$ general
with $g>2$.


title: Seiberg-Witten integrable systems

abstract:
  This is a survey of the work of Seiberg and Witten on 4-dimensional N=2
supersymmetric Yang-Mills theory and of some of its recent extensions, written
for mathematicians. The point of view is that of algebraic geometry and
integrable systems. An introductory chapter tries to describe some of the
relevant physics for a reader with no physics background. This is followed by a
review of the relevant properties of integrable systems. The remaining chapters
describe the specific integrable systems used, and include a detailed study of
the applications to quantum field theory.


title: On the Pontrjagin-Viro form

abstract:
  A new invariant, the Pontrjagin-Viro form, of algebraic surfaces is
introduced and studied. It is related to various Rokhlin-Guillou-Marin forms
and generalizes Mikhalkin's complex separation. The form is calculated for all
real Enriques surfaces for which it is well defined.


title: Existence of good divisors on Mukai varieties

abstract:
  A Mukai variety is a Fano n-fold of index n-2. In this paper we study the
fundamental divisor of a Mukai variety with at worst log terminal
singularities. The main result is a complete classification of log terminal
Mukai varieties which have not good divisors, examples of "bad" varieties are
given. In such a way we also give a shorter proof of Mukai Conjecture, solved
in our previous paper alg-geom/9611024.


title:
Identities of Structure Constants for Schubert polynomials and Orders on
  S_n

abstract:
  We use geometry to prove a number of new identities among the
Littlewood-Richardson coefficients for Schubert polynomials (Schubert classes
in a flag manifold). For many of these identities, there is a companion result
about the Bruhat order which should imply the identity, were it known how to
express these coefficients in terms of the Bruhat order. This analysis leads
the determination of many of these constants. We conclude with an outline of
geometric proofs for these identities.


title: Riemann-Roch theorems via deformation quantization

abstract:
  We give a proof of a conjecture of P. Schapira and J.-P. Schneiders on the
characteristic classes of D-modules.


title: Boundedness for codimension two submanifolds of quadrics

abstract:
  Arrondo, Sols and De Cataldo proved that there are only finitely many
families of codimension two subvarieties not of general type in the smooth
quadric of dimension $n+2$ for $n\ge 2 $, $n\neq 4$. In this paper we drop the
assumption $n\neq 4$ from the previous result (obviously the assumption $n\ge
2$ cannot be removed).


title: A geometric construction of Getzler's relation

abstract:
  A geometric construction of Getzler's cohomological relation in the moduli
space of 4 pointed elliptic curves is given by a push-forward of a natural
rational equivalence in a space of admissible covers. In particular, Getzler's
relation is shown to be a rational equivalence. The recursion for the elliptic
Gromov-Witten invariants of P^2 predicted by Eguchi, Hori, and Xiong from the
Virasoro conjecture is proven via Getzler's equation and the WDVV-equations.


title: Stable extensions by line bundles

abstract:
  Let C be an algebraic curve of genus g. Consider extensions E of a vector
bundle F'' of rank n'' by a vector bundle F' of rank n'. The following
statement was conjectured by Lange: If 0<n'deg F''-n''degF'\le n'n''(g-1), then
there exist extensions like this with E stable. We prove this result for the
generic curve when F' is a line bundle. Our method uses a degeneration argument
to a reducible curve.


title: Constructing new ample divisors out of old ones

abstract:
  We prove a gluing theorem which allows to construct an ample divisor on a
rational surface from two given ample divisors on simpler surfaces. This
theorem combined with the Cremona action on the ample cone gives rise to an
algorithm for constructing new ample divisors. We then propose a conjecture
relating continued fractions approximations and Seshadri-like constants of line
bundles over rational surfaces. By applying our algorithm recursively we verify
our conjecture in many cases and obtain new asymptotic estimates on these
constants. Finally, we explain the intuition behind the gluing theorem in terms
of symplectic geometry and propose generalizations.


title: On Lange's Conjecture

abstract:
  Let C be an algebraic curve of genus g. Let E be a vector bundle of rank n
and degree d. Consider among all subbundles F' of E of rank n' those of maximal
degree d'. Then s_n'(E)= n'd-nd'\le n'(n-n')g. If E is stable s_n'(E)>0 while
if E is generic s_n'(E)\ge n'(n-n')(g-1) . The following statement was
conjectured by Lange: If 0<s\le n'(n-n')(g-1), then there exist stable vector
bundles with s_n'(E)=s. We prove this result for the generic curve. We also
clarify what happens in the interval n'(n-n')(g-1)<s\le n'(n-n')g Our method
uses a degeneration argument to a reducible curve. A similar result has been
obtained by L.Bambrila-Paz and H.Lange using a different method.


title: Curves in Grassmannians

abstract:
  This paper considers curves in Grassmannians which are themselves immersed in
projective space by the Plucker map. It is shown that for a generic vector
bundle of high enough degree, the image curve lies in a proper linear
subvariety of this projective space and satisfies good conditions on syzygies
as a curve in this subspace. For very small degree and generic vector bundle,
the curve is non- degenerate.


title:
Values of zeta functions at negative integers, Dedekind sums and toric
  geometry

abstract:
  This is an expanded version. We study relations among special values of zeta
functions, invariants of toric varieties, and generalized Dedekind sums. In
particular, we use invariants arising in the Todd class of a toric variety to
give a new explicit formula for the values of the zeta function of a real
quadratic field at nonpositive integers. We also express these invariants in
terms of the generalized Dedekind sums studied previously by several authors.
The paper includes conceptual proofs of the above mentioned relations and
explicit computations of the various zeta values and Dedekind sums involved.


title: On surfaces in P^6 with no trisecant lines

abstract:
  We construct examples of smooth surfaces S in P^6 with no trisecant lines.
This list includes examples of surfaces not cut out by quadrics. We prove that
unless S has a finite number of disjoint $(-1)$-lines, and each one meets some
other line L on the surface with L^2\leq -2, and S is not an inner projection
from P^7 the surface belongs to the list of examples.


title: Hodge and Tate conjectures for hypergeometric sheaves

abstract:
  A constructible sheaf corresponding to Gel'fand Zelevinski hypergeometric
functions on a torus is called hypergeometric sheaf. We consider Hodge and Tate
conjectrue for hypergeomtric sheaves. Hodge conjecture is formulated in terms
of variation of Hodge strucure and Tate conjecture is done for l-adic sheaves
on an open set of torus. We prove Hodge and Tate conjecture up to Hodge and
Tate cycle of Fermat motifes. We use cohomological Mellin transform to get the
main theorem. This is the final revision for preprint.


title: Quantum multiplication of Schur polynomials

abstract:
  We give formulas for the products of classes of Schubert varieties in the
quantum cohomology rings of Grassmannians, in terms of the combinatorics of
partitions and tableaux.


title: Compact Hyperkaehler Manifolds: Basic Results

abstract:
  The paper generalizes some of the well-known results for K3 surfaces to
higher-dimensional irreducible symplectic (or, equivalently, compact
irreducible hyperkaehler) manifolds. In particular, we discuss the projectivity
of such manifolds and their ample resp. Kaehler cones. It is proved that the
period map surjects any non-empty connected component of the moduli space of
marked manifolds onto the corresponding period domain. We also establish an
anlogue of the transitivity of the Weyl-action on the set of chambers of a K3
surface. As a converse of a higher-dimensional version of the `Main Lemma' of
Burns and Rapoport we prove that two birational irreducible symplectic
manifolds are deformation equivalent. This compares nicely with a result of
Batyrev and Kontsevich.


title: Minimal model theorem for toric divisors

abstract:
  Minimal model conjecture for a proper variety $X$ is that if $\kappa(X)\geq
0$, then $X$ has a minimal model with the abundance and if $\kappa =-\infty$,
then $X$ is birationally equivalent to a variety $Y$ which has a fibration $Y
\to Z$ with $-K_Y$ relatively ample. In this paper, we prove this conjecture
for a $\D$-regular divisor on a proper toric variety by means of successive
contractions of extremal rays and flips of ambient toric variety. Furthermore,
for such a divisor $X$ with $\kappa(X)\geq 0$ we construct a projective minimal
model with the abundance in a different way; by means of "puffing up" of the
polytope, which gives an algorithm of a construction of a minimal model.


title:
An application of exceptional bundles to the moduli of stable sheaves on
  a K3 surface

abstract:
  Let M(v) be the moduli of stable sheaves on K3 surfaces X of Mukai vector v.
If v is primitive, than it is expected that M(v) is deformation equivalent to
some Hilbert scheme and weight two hogde structure can be described by
H^*(X,Z). These are known by Mukai, O'Grady and Huybrechts if rank is 1 or 2,
or the first Chern class is primitive. Under some conditions on the dimension
of M(v), we shall show that these assertion are true. For the proof, we shall
use Huybrechts's results on symplectic manifolds.


title: Through the Looking Glass

abstract:
  It is frequently possible to produce new Calabi-Yau threefolds from old ones
by a process of allowing the complex structure to degenerate to a singular one,
and then performing a resolution of singularities. (Some care is needed to
ensure that the Calabi-Yau condition be preserved.) There has been speculation
that all Calabi-Yau threefolds could be linked in this way, and considerable
evidence has been amassed in this direction. We propose here a natural way to
relate this construction to the string-theoretic phenomenon known as ``mirror
symmetry.'' We formulate a conjecture which in principle could predict mirror
partners for all Calabi-Yau threefolds, provided that all were indeed linked by
the degeneration/resolution process. The conjecture produces new mirrors from
old, and so requires some initial mirror manifold construction---such as
Greene-Plesser orbifolding---as a starting point. (Lecture given at the CIRM
conference, Trento, June 1994, and at the Workshop on Complex Geometry and
Mirror Symmetry, Montr\'eal, March 1995.)


title:
Generalization of Arnold-Viro inequalities for real singular algebraic
  curves

abstract:
  The Arnold inequalities characterizing the topology of non-singular plane
real algebraic curves and the generalization of these inequalities for nodal
curves by Viro are extended in this paper for the curves whose singularities
have non-degenerated Milnor form. This involves an invariant of such real
singularities, the real linking form, which is calculated.


title: Subgroups of inertia groups arising from abelian varieties

abstract:
  Given an abelian variety over a field with a discrete valuation, Grothendieck
defined a certain open normal subgroup of the absolute inertia group. This
subgroup encodes information on the extensions over which the abelian variety
acquires semistable reduction. We study this subgroup, and use it to obtain
information on the extensions over which the abelian variety acquires
semistable reduction.


title:
Intersection Numbers and Rank One Cohomological Field Theories in Genus
  One

abstract:
  We obtain a simple, recursive presentation of the tautological (\kappa, \psi,
and \lambda) classes on the moduli space of curves in genus zero and one in
terms of boundary strata (graphs). We derive differential equations for the
generating functions for their intersection numbers which allow us to prove a
simple relationship between the genus zero and genus one potentials. As an
application, we describe the moduli space of normalized, even, rank one
cohomological field theories in genus one in coordinates which are additive
under taking tensor products. Our results simplify and generalize those of
Kaufmann, Manin, and Zagier.


title: Numerical Schubert calculus

abstract:
  We develop numerical homotopy algorithms for solving systems of polynomial
equations arising from the classical Schubert calculus. These homotopies are
optimal in that generically no paths diverge. For problems defined by
hypersurface Schubert conditions we give two algorithms based on extrinsic
deformations of the Grassmannian: one is derived from a Gr\"obner basis for the
Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its
projective coordinate ring. The more general case of special Schubert
conditions is solved by delicate intrinsic deformations, called Pieri
homotopies, which first arose in the study of enumerative geometry over the
real numbers. Computational results are presented and applications to control
theory are discussed.


title:
G\'eom\'etrie d'Arakelov des vari\'et\'es toriques et fibr\'es en
  droites int\'egrables

abstract:
  En nous appuyant sur une construction due \`a Bedford et Taylor, et certains
r\'esultats r\'ecents de Demailly, nous pr\'esentons une extension (partielle)
de la g\'eom\'etrie d'Arakelov aux fibr\'es en droites int\'egrables. (Ces
derniers sont les fibr\'es en droites hermitiens sur une vari\'et\'e
arithm\'etique $X$ pouvant se d\'ecomposer sous la forme $\ov{E} =
\ov{E}_{1}\otimes (\ov{E}_{2})^{-1}$, o\`u $\ov{E}_{1}$ et $\ov{E}_{2}$ sont
des fibr\'es en droites munis \`a l'infini d'une m\'etrique continue
approchable uniform\'ement sur $X(C)$ par des m\'etriques positives
$C^{\infty}$). Nous appliquons notre th\'eorie aux fibr\'es en droites sur une
vari\'et\'e torique munis \`a l'infini de leur m\'etrique canonique. Nous en
d\'eduisons, entre autres choses, la d\'emonstration d'un analogue
arithm\'etique du th\'eor\`eme de Bernstein-Koushnirenko.


title:
Algorithms for computing intersection numbers on moduli spaces of
  curves, with an application to the class of the locus of Jacobians

abstract:
  We describe algorithms for computing the intersection numbers of divisors and
of Chern classes of the Hodge bundle on the moduli spaces of stable pointed
curves. We also discuss the implementations and the results obtained. There are
several applications. We discuss one in particular: the calculation of the
projection in the tautological ring of the moduli space of abelian varieties of
the class of the locus of Jacobians.


title: Drinfeld Moduli Schemes and Infinite Grassmannians

abstract:
  The aim of this paper is to construct an immersion of the Drinfeld moduli
schemes in a finite product of infinite Grassmannians, such that they will be
locally closed subschemes of these Grassmannians which represent a kind of flag
varieties.


title: A Vanishing Result for the Universal Bundle on a Toric Quiver Variety

abstract:
  Let Q be a finite quiver without oriented cycles. Denote by U --> M the fine
moduli space of stable thin sincere representations of Q with respect to the
canonical stability notion. We prove Ext^i(U,U) = 0 for all i >0 and compute
the endomorphism algebra of the universal bundle U. Moreover, we obtain a
necessary and sufficient condition for when this algebra is isomorphic to the
path algebra kQ of the quiver Q. If so, then the bounded derived category of
finitely generated right kQ-modules is embedded into that of coherent sheaves
on M.


title: Prym varieties and the infinite Grassmannian

abstract:
  In this paper we study Prym varieties and their moduli space using the well
known techniques of the infinite Grassmannian. There are three main results of
this paper: a new definition of the BKP hierarchy over an arbitrary base field
(that generalizes the classical one over the complex numbers; a
characterization of Prym varieties in terms of dynamical systems, and explicit
equations for the moduli space of (certain) Prym varieties. For all of these
problems the language of the infinite Grassmannian, in its algebro-geometric
version, allows us to deal with these problems from the same point of view.


title: Painlev\'{e} - Calogero correpondence

abstract:
  It is proved that the Painlev\'{e} VI equation $(PVI_{\al,\be,\ga,\de})$ for
the special values of constants $(\al=\frac{\nu^2}{4},\be=-\frac{\nu^2}{4},
\ga=\frac{\nu^2}{4},\de=\f1{2}-\frac{\nu^2}{4})$ is a reduced hamiltonian
system. Its phase space is the set of flat $SL(2,C)$ connections over elliptic
curves with a marked point and time of the system is given by the elliptic
module. This equation can be derived via reduction procedure from the free
infinite hamiltonian system. The phase space of later is the affine space of
smooth connections and the "times are the Beltrami differentials. This approach
allows to define the associate linear problem, whose isomonodromic deformations
is provided by the Painlev\'{e} equation and the Lax pair. In addition, it
leads to description of solutions by a linear procedure. This scheme can be
generalized to $G$ bundles over Riemann curves with marked points, where $G$ is
a simple complex Lie group. In some special limit such hamiltonian systems
convert into the Hitchin systems. In particular, for $\SL$ bundles over
elliptic curves with a marked point we obtain in this limit the elliptic
Calogero N-body system. Relations to the classical limit of the Knizhnik-
Zamolodchikov-Bernard equations is discussed.


title:
Solvable fundamental groups of algebraic varieties and K\"ahler
  manifolds

abstract:
  This is a long overdue write up of the following: If the fundamental group of
a normal complex algebraic variety (respectively Zariski open subset of a
compact K\"ahler manifold) is a solvable group of matrices over Q (respectively
polycyclic group), then it is virtually nilpotent. This should be interpreted
as saying that a large class of solvable groups will not occur as fundamental
groups of such spaces. The essential strategy is to first "complete" these
groups to algebraic groups (generalizing constructions of Malcev and Mostow),
and then check that the identity component is unipotent. This is done by Galois
theoretic methods in the algebraic case, and is reduced to homological
properties of one dimensional characters in the analytic case.


title: The Deligne-Lusztig curve associated to the Suzuki group

abstract:
  We give a characterization of the Deligne-Lusztig curve associated to the
Suzuki group based on the genus and the number of rational points of the curve.