title: Extremal contractions from 4-dimensional manifolds to 3-folds

abstract:
  Let $g : X \to Y$ be the contraction of an extremal ray of a smooth
projective 4-fold $X$ such that $\dim Y=3$. Then $g$ may have a finite number
of 2-dimensional fibers. We shall classify those fibers. Especially we shall
prove that any two points of such a fiber is joined by a chain of rational
curves of length at most 2 with respect to $-K_X$, and that $|-K_X|$ is
$g\text{-free}$.


title: Traces and Differential Operators over Beilinson Completion Algebras

abstract:
  A Beilinson completion algebra (BCA) A is a complete semilocal algebra over a
perfect field k, whose residue fields are high dimensional local fields. In
addition A is a semi-topological algebra. The completion of the structure sheaf
of an algebraic k-variety along a saturated chain of points is the prototypical
example of a BCA. We single out two kinds of homomorphisms between BCAs:
morphisms and intensifications. The first kind includes residually finite local
homomorphisms, whereas the second kind is a sort of localization. We prove that
every BCA A has a dual module K(A), and these dual modules are contravariant
w.r.t. morphisms and covariant w.r.t. intensifications. For any
semi-topological A-module M we define its dual Dual_{A} M := Hom_{A}^{cont}(M,
K(A)). This duality operation has the remarkable property of respecting
differential operators: given a continuous DO D : M --> N, there is a dual DO
Dual_{A}(D) : Dual_{A} N --> Dual_{A} M. The results above are used (in a
subsequent paper) to construct the Grothendieck residue complex K_{X}^{.} on
any finite type k-scheme X, and to derive many of its properties.


title: Linear bound for abelian automorphisms of varieties of general type

abstract:
  We prove: Let $G$ be a finite abelian group acting faithfully on a complex
smooth project variety $X$ of general type with numerically effective canonical
divisor, of dimension $n$. Then
  $$|G| \le C(n)K_X^n ,$$
  where $C(n)$ depends only on $n$.


title: Algebraic surfaces and Seiberg-Witten invariants

abstract:
  In this revised version, we add some expository material and references and
make some minor corrections.


title: On the blowups of numerical Godeaux surfaces

abstract:
  We give a short proof of the following result: Let $X$ be a complex surface
of general type. If the canonical divisor of the minimal model of $X$ has
selfintersection $= 1$, then $X$ is not diffeomorphic to a rational surface.
Our proof is the natural extension of the argument given in our paper in
Inventiones Mathematicae (1989) for the case when $X$ is minimal. This argument
also gives information about the non--existence of certain smooth embeddings of
$2$--spheres in $X$, if $X$ has geometric genus zero.


title: Non--trivial harmonic spinors on generic algebraic surfaces

abstract:
  We disprove Hitchin's conjecture to the effect that for a generic complex
structure on a simply connected spin complex surface the square root of the
canonical bundle has no more cohomology then is predicted by the Riemann--Roch
theorem. (This paper will appear in Proc. Amer. Math. Soc.)


title: Quasi-parabolic Siegel Formula

abstract:
  The result of Siegel that the Tamagawa number of $SL_r$ over a function field
is 1 has an expression purely in terms of vector bundles on a curve, which is
known as the Siegel formula. We prove an analogous formula for vector bundles
with quasi-parabolic structures. This formula can be used to calculate the
Betti numbers of the moduli of parabolic vector bundles using the Weil
conjucture.


title: Homology of iterated semidirect products of free groups

abstract:
  Let $G$ be a group which admits the structure of an iterated semidirect
product of finitely generated free groups. We construct a finite, free
resolution of the integers over the group ring of $G$. This resolution is used
to define representations of groups which act compatibly on $G$, generalizing
classical constructions of Magnus, Burau, and Gassner. Our construction also
yields algorithms for computing the homology of the Milnor fiber of a
fiber-type hyperplane arrangement, and more generally, the homology of the
complement of such an arrangement with coefficients in an arbitrary local
system.


title:
Reflection groups in hyperbolic spaces and the denominator formula for
  Lorentzian Kac--Moody Lie algebras

abstract:
  This is a continuation of our "Lecture on Kac--Moody Lie algebras of the
arithmetic type" \cite{25}.
  We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear
form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group
$W\subset W(S)$, fundamental polyhedron $\Cal M$ of $W$ and an acceptable
(corresponding to twisting coefficients) set $P({\Cal M})\subset M$ of vectors
orthogonal to faces of $\Cal M$ (simple roots). One can construct the
corresponding Lorentzian Kac--Moody Lie algebra ${\goth g}={\goth
g}^{\prime\prime}(A(S,W,P({\Cal M})))$ which is graded by $M$.
  We show that $\goth g$ has good behavior of imaginary roots, its denominator
formula is defined in a natural domain and has good automorphic properties if
and only if $\goth g$ has so called {\it restricted arithmetic type}. We show
that every finitely generated (i.e. $P({\Cal M})$ is finite) algebra ${\goth
g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1)))$ may be embedded to ${\goth
g}^{\prime\prime}(A(S,W,P({\Cal M})))$ of the restricted arithmetic type. Thus,
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a
natural class to study.
  Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the
best automorphic properties for the denominator function if they have {\it a
lattice Weyl vector $\rho$}. Lorentzian Kac--Moody Lie algebras of the
restricted arithmetic type with generalized lattice Weyl vector $\rho$ are
called {\it elliptic}


title: The canonical class and the $C^\infty$ properties of K\"ahler surfaces

abstract:
  We give a self contained proof using Seiberg Witten invariants that for
K\"ahler surfaces with non negative Kodaira dimension (including those with
$p_g = 0$) the canonical class of the minimal model and the $(-1)$-curves, are
oriented diffeomorphism invariants up to sign. This implies that the Kodaira
dimension is determined by the underlying differentiable manifold (Van de Ven
Conjecture). We use a set up that replaces generic metrics by the construction
of a localised Euler class of an infinite dimensional bundle with a Fredholm
section. This allows us to compute the Seiberg Witten invariants of all
elliptic surfaces with excess intersection theory. We then reprove that the
multiplicities of the elliptic fibration are determined by the underlying
oriented manifold, and that the plurigenera of a surface are oriented
diffeomorphism invariants.


title: An algebraic version of Demailly's asymptotic Morse Inequalities

abstract:
  We give an elementary algebraic proof of some asymptotic estimates (called by
Demailly asymptotic Morse inequalities) for the dimensions of cohomology groups
of the difference of two ample line bundles on a smooth complex projective
variety of any dimension.


title:
The Langlands lemma and the Betti numbers of stacks of $G$--bundles on a
  curve

abstract:
  In this note we show that the Langlands lemma from the theory of Eisenstein
series can be used to invert the recursion relation for the Poincar\'e series
of the open substack of semi-stable $G$-bundles which was established by
Atiyah/Bott and Harder/Narasimhan.


title: Quotient Spaces Modulo Algebraic Groups

abstract:
  The paper proves that if a reductive group scheme acts properly on a scheme
then the geometric quotient exists as an algebraic space. As a consequence we
obtain the existence of the moduli spcace of canonically polarized varieties
over Spec Z.


title: Bloch's conjecture revisited

abstract:
  Let $X$ be a non-singular projective complex surface. We can show that
Bloch's conjecture (i.e., that if $p_g=0$ then the Albanese kernel vanishes) is
equivalent to the following statement: If $p_g(X)=0$ then for any given Zariski
open $U\subset X$ and $\omega\in H^2(U,{\bf C})$ there is a smaller Zariski
open $V\subset U$ such that $$\omega\mid_V =\omega'+\zeta$$ where $\omega'\in
F^2H^2(V,{\bf C})$ and $\zeta$ is integral.


title: Geometric Properties of the Double-Point Divisor

abstract:
  Let $X^n \subset P^N$ be a nonsingular, nondegenerate projective variety of
dimension $n$ and codimension $N-n \ge 2$. Let $|C_X|$ be the linear system
determined by the double-point divisor obtained by generically projecting $X$
to a hypersurface in $P^{n+1}$. We classify those varieties for which $C_X$ is
not ample, or equivalently, does not separate points of $X$. We call such
varieties Roth varieties and prove that they exist for all dimensions $n \ge 2$
and give a description of their properties. For example, in many cases Roth
varieties are Castelnuovo varieties. Positivity results for the double-point
divisor are analogous to positivity results for the ramification divisor which
are studied in adjunction theory.


title: Hitchin systems, higher Gaudin operators and $r$-matrices

abstract:
  We adapt Hitchin's integrable systems to the case of a punctured curve. In
the case of $\CC P^{1}$ and $SL_{n}$-bundles, they are equivalent to systems
studied by Garnier. The corresponding quantum systems were identified by B.
Feigin, E. Frenkel and N. Reshetikhin with Gaudin systems. We give a formula
for the higher Gaudin operators, using results of R. Goodman and N. Wallach on
the center of the enveloping algebras of affine algebras at the critical level.
Finally we construct a dynamical $r$-matrix for Hitchin systems for a punctured
elliptic curve, and $GL_{n}$-bundles, and (for $n=2$) the corresponding quantum
system.

categories: alg-geom hep-th math.AG nlin.SI solv-int


title: Equianalytic and equisingular families of curves on surfaces

abstract:
  We consider flat families of reduced curves on a smooth surface S such that
each member C has the same number of singularities of fixed singularity types
and the corresponding (locally closed) subscheme H of the Hilbert scheme of S.
We are mainly concerned with analytic resp. topological singularity types and
give a sufficient condition for the smoothness of H (at C). Our results for
S=P^2 seem to be quite sharp for families of cuves of small degree d.


title:
The irreducibility of the moduli space of stable vector bundles of rank
  2 on a quintic in $\pp^3$

abstract:
  In this paper I consider a quintic surface in $\pp^3$, general in the sense
of Noether-Lefschetz theory. The vector bundles of rank 2 on this surface which
are $\mu$-stable with respect to the hyperplane section and have $c_1 = K$, the
canonical class of the surface and fixed $c_2$, are parametrized by a moduli
space. This space is known to be irreducible for large $c_2$ (work of K.G.
O'Grady). I give an explicit bound, namely $c_2 \geq 16$.


title:
Moduli spaces of semiquasihomogeneous singularities with fixed principal
  part

abstract:
  We construct coarse moduli spaces of semiquasihomogeneous hypersurface
singularities with respect to right equivalence and contact equivalence. We
have to fix the principal part of the semiquasihomogeneous singularities. For
the moduli spaces with respect to contact equivalence we also fix the Hilbert
function of the Tjurina algebra induced by the weights.


title:
Stable Parabolic Bundles over Elliptic Surfaces and over Orbifold
  Riemann Surfaces

abstract:
  For an elliptic surface $q:Y \to \Sigma$, with prescribed singular fibres,
Stefan Bauer proved directly via algebraic geometry that the stable bundles
over $Y$, whose chern classes are pull backs from $\Sigma$, correspond to the
stable (V-)bundles over $\Sigma$.
  We show, via a short proof in differential geometry, a generalisation to
stable parabolic bundles. This uses extensions of Donaldson's deep result,
giving the existence of Hermitian-Yang-Mills (or anti-self-dual) connections on
stable parabolic bundles. In our cases these connections are flat and hence,
correspond to representations of certain fundamental groups, which in turn are
isomorphic, by Ue's work.
  To generalize Bauer's equivalence of the corresponding moduli spaces of
stable bundles, we combine his arguments with Kronheimer & Mrowka's
construction of the moduli spaces of stable parabolic bundles. Finally, we
consider the pulling back of smooth parabolic bundles via $q$.


title: The nil Hecke ring and singularity of Schubert varieties

abstract:
  We give a criterion for smoothness of a point in any Schubert variety in any
G/B in terms of the nil Hecke ring.


title: On a conjecture of Varchenko

abstract:
  In this note we generalize and prove a recent conjecture of Varchenko
concerning the number of critical points of a (multivalued) meromorphic
function $\phi$ on an algebraic manifold. Under certain conditions, this number
turns out to coincide with the Euler characteristic (up to a sign) of the
complement of the divisor of $\phi$. A few variants of this basic situation are
also discussed. Two independent proofs are given, respectively using Chern
classes and Morse theory
  In its original form, this counting problem arises (in special cases) in
evaluating the asymptotics of correlation functions of Wess--Zumino--Witten
conformal field theory.


title: Akizuki's counterexample

abstract:
  Following Akizuki, I construct a Noetherian local integral domain C_M whose
normalisation (integral closure) is not finite over C_M. My proof follows
closely Akizuki's ingenious calculations.


title: A gem of the modular universe

abstract:
  We introduce one of the most beautiful algebraic varieties known, a quintic
hypersurface in projective five-space, which is invariant under the action of
the Weyl group of $E_6$. This variety is intricately related with many other
moduli problems, some of which are: marked hyperelliptic curves of genus two,
Picard curves of genus four with a $\sqrt{-3}$-level structure, six points on
the projective line, abelian surfaces with (1,3) polarisations, quartic
surfaces invariant under the action of the Heisenberg group in projective
three-space, K3-surfaces which are double covers of the projective plane
branched along six lines, and last but not least, cubic surfaces in projective
three-space. These relationships are developed in some detail, with particular
care on the birational aspects.


title: An orbifold partition of ${\overline{M}_g^n}$

abstract:
  We define a partition of ${\overline{M}_g^n}$ and show that the cohomology of
${\overline{M}_g^n}$ in a given degree admits a filtration whose respective
quotients are isomorphic to the shifted cohomology groups of the parts if $g$
is sufficiently large. This implies that the map $H^k({\overline{M}_g^n}) \ra
H^k(M_g^n)$ is onto and that the Hodge structure of $H^k(M_g^n)$ is pure of
weight $k$ if $g \geq 2k+1$. Our main ingredient is the stability theorem of
Harer and Ivanov.


title:
Th\'eor\`emes De Connexit\'e Pour Les Produits D'Espaces Projectifs et
  Les Grassmanniennes

abstract:
  Let $G$ be the Grassmannian $G(d,n)$, let $X$ and $Y$ be complete irreducible
varieties, and let $X\rightarrow G$ and $Y\rightarrow G$ be morphisms. Hansen
proved that $X \times_G Y$ is connected when $codim f(X) + codim g(Y) < n$. We
show that the conclusion holds under the often weaker hypothesis
$f(X).g(Y).T\ne 0$, where $T$ is the class of $G(d,n-1)$ in $G$.
  We prove similar results when $G$ is a product of projective spaces. In
particular, if $D$ is an irreducible subvariety of $P^n\times P^n$ of dimension
$n$ which dominates both factors, and if $X$ is complete irreducible, with a
morphism $f: X \rightarrow P^n\times P^n$ such that $dim f(X) >n$, $f^{-1}(D)$
is connected. This extends the classical Fulton-Hansen connectedness theorem.
These results illustrate Fulton and Lazarsfeld's idea that connectedness should
be a numerical property.


title:
Moduli of pre-$\cal D$-modules, perverse sheaves and the Riemann-Hilbert
  morphism -I

abstract:
  We construct a moduli scheme for semistable pre-$\D$-modules with prescribed
singularities and numerical data on a smooth projective variety. These
pre-$\D$-modules are to be viewed as regular holonomic $\D$-modules with `level
structure'. We also construct a moduli scheme for perverse sheaves on the
variety with prescribed singularities and other numerical data, and represent
the de Rham functor (which gives the Riemann-Hilbert correspondence) by an
analytic morphism between the two moduli schemes.


title: Roitman's theorem for singular complex projective surfaces

abstract:
  Let $X$ be a complex projective surface with arbitrary singularities. We
construct a generalized Abel--Jacobi map $A_0(X)\to J^2(X)$ and show that it is
an isomorphism on torsion subgroups. Here $A_0(X)$ is the appropriate Chow
group of smooth 0-cycles of degree 0 on $X$, and $J^2(X)$ is the intermediate
Jacobian associated with the mixed Hodge structure on $H^3(X)$. Our result
generalizes a theorem of Roitman for smooth surfaces: if $X$ is smooth then the
torsion in the usual Chow group $A_0(X)$ is isomorphic to the torsion in the
usual Albanese variety $J^2(X)\cong Alb(X)$ by the classical Abel-Jacobi map.


title: The Mumford relations and the moduli of rank three stable bundles

abstract:
  We find a complete set of relations for the rational cohomology ring of the
moduli space of rank three stable bundles over a Riemann surface of genus g and
also show that the Pontryagin ring vanishes in degree 12g-8 and greater. The
results are obtained by introducing some 'dual' Mumford relations and
generalising Kirwan's proofs of the Mumford and Newstead conjectures in the
rank two case. (In this revised version of the paper the vanishing degree of
the Pontryagin ring of the moduli space has been improved from `in and above
degree 12g-4' to `in and above degree 12g-8'. This degree is now known to be
sharp.)


title: Degrees of generators of ideals defining subschemes of projective space

abstract:
  For an arithmetically Cohen--Macaulay subscheme of projective space, there is
a well-known bound for the highest degree of a minimal generator for the
defining ideal of the subscheme, in terms of the Hilbert function. We prove a
natural extension of this bound for arbitrary locally Cohen--Macaulay
subschemes. We then specialize to curves in $\pthree$, and show that the curves
whose defining ideals have generators of maximal degree satisfy an interesting
cohomological property. The even liaison classes possessing such curves are
characterized, and we show that within an even liaison class, all curves with
the property satisfy a strong Lazarsfeld--Rao structure theorem. This allows us
to give relatively complete conditions for when a liaison class contains curves
whose ideals have maximal degree generators, and where within the liaison class
they occur.


title: Local Homology and Cohomology on Schemes

abstract:
  We prove a sheaf-theoretic derived-category generalization of Greenlees-May
duality (a far-reaching generalization of Grothendieck's local duality
theorem): for a quasi-compact separated scheme X and a "proregular" subscheme
Z---for example, any separated noetherian scheme and any closed
subscheme---there is a sort of sheafified adjointness between local cohomology
supported in Z and left-derived completion along Z. In particular, the i-th
left-derived completion functor is the "local homology" sheaf
$Ext^i(\R\Gamma_Z\O_X, -)$. Sheafified generalizations of a number of duality
theorems scattered about the literature result, e.g., the Peskine-Szpiro
duality sequence (generalizing local duality), the Warwick Duality theorem of
Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck
Duality, we also get a generalization of a Formal Duality theorem of
Hartshorne, and of a related local-global duality theorem. In a sequel we will
develop the latter results further, to study Grothendieck duality and residues
on formal schemes.


title: Hyperbolic Planes

abstract:
  In this paper we consider a special class of arithmetic quotients of bounded
symmetric domains which can roughly be described as higher- dimensional
analogues of the Hilbert modular varities. The algebraic groups are defined as
the unitary groups over two-dimensional right vector spaces over a division
algebra with involution. If $d$ denotes the degree of the division algebra,
then $d=1$ is essentially just case giving rise to Hilbert modular varieties.
We determine the class number (number of cusps) of the arithmetic quotients,
and find inter- esting modular subvarities whos existence derives from the
algebraic structure of the division algebras. Also the moduli interpretation,
given by Shimuras theory, is described.


title: On number of hypersurfaces containing projective curves

abstract:
  Generalizing a classical lemma of Castelnuovo, we characterize rational
normal curves (resp. linearly normal elliptic curves) as curves $C\subset
\PP^n$ such that the number of linearly independent hypersurfaces $Z\supset C$
of given degree~$m$ is maximal (resp. next to maximal).


title: Severi Degrees in Cogenus 3

abstract:
  In this short note, a new computation of the degree of the locus of 3-nodal
plane curves in the linear system of degree d plane curves is given. The answer
is expressed as a tautological class on a blow-up of the Hilbert scheme of 3
points in the plane. The class is evaluated by the Bott residue formula.


title:
Intersections of Q-Divisors on Kontsevich's Moduli Space
  $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry

abstract:
  The theory of Q-Cartier divisors on the space of n-pointed, genus 0, stable
maps to projective space is considered. Generators and Picard numbers are
computed. A recursive algorithm computing all top intersection products of
Q-Divisors is established. As a corollary, an algorithm computing all
characteristic numbers of rational curves in P^r is proven (including simple
tangency conditions). Computations of these characteristic numbers are carried
out in many examples. The degree of the 1-cuspidal rational locus in the linear
system of degree d plane curves is explicitly evaluated.


title: Intersection-theoretical computations on \Mgbar

abstract:
  We determine necessary conditions for ample divisors in arbitrary genus as
well as for very ample divisors in genus 2 and 3. We also compute the
intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter
number is relevant for counting curves of higher genus on manifolds, cf. the
recent work of Bershadsky et al.


title:
Siegel automorphic form corrections of some Lorentzian Kac--Moody Lie
  algebras

abstract:
  We find automorphic form corrections which are generalized Lorentzian
Kac--Moody superalgebras without odd real simple roots (see R. Borcherds
\cite{Bo1} -- \cite{Bo7}, V. Kac \cite{Ka1} -- \cite{Ka3}, R. Moody \cite{Mo}
and \S~6 of this paper) for two elliptic Lorentzian Kac--Moody algebras of the
rank $3$ with a lattice Weyl vector, and calculate multiplicities of their
simple and arbitrary imaginary roots (see an appropriate general setting in
\cite{N5}). These Kac--Moody algebras are defined by hyperbolic (i.e. with
exactly one negative square) symmetrized generalized Cartan matrices $$G_1\ =\
\left(\matrix\hphantom{-} 2&-2&-2\\-2 &\hphantom{-}2&-2\\-2&-2& \hphantom{-} 2
\endmatrix\right)\hskip30pt \text{and} \hskip 30pt G_2\ =\
\left(\matrix\hphantom{-}4 & -4 & -12& -4 \\-4 &\hphantom{-}4 &-4&-12


title: Some Donaldson invariants of CP^2

abstract:
  We compute the Donaldson numbers $q_{17}(CP^2)=2540$ and
$q_{21}(CP^2)=233208$.


title: Integral Subschemes of Codimension Two

abstract:
  In this paper we study the problem of describing the integral subschemes
within a fixed even linkage class $\L$ of subschemes in $\Pn$ of codimension
two. In the case that $\L$ is not the class of arithmetically Cohen-Macaulay
subschemes, we associate to any $X \in \L$ two invariants $\theta_X$ and
$\eta_X$. When taken with the height $h_X$, each of these invariants determines
the location of $X$ in $\L$, thought of as a poset under domination. In terms
of these invariants, necessary conditions are given for integral subschemes.
The necessary conditions are almost sufficient in the sense that if a subscheme
$X$ satisfies the necessary conditions and dominates an integral subscheme $Y$,
then $X$ can be deformed with constant cohomology through subschemes in $\L$ to
an integral subscheme. In particular, if an even linkage class has a minimal
element which is integral, then the conditions are both necessary and
sufficient.


title: The quantum cohomology of homogeneous varieties

abstract:
  We established the associativity of the quantum cohomologies of homogeneous
varieties by using degeneration method in algebraic geometry.


title:
The first two Betti numbers of the moduli spaces of vector bundles on
  surfaces

abstract:
  We determined the first two Betti numbers of the moduli of rank two stable
sheaves on an arbitrary algebraic surface


title:
Picard groups of the moduli spaces of vector bundles over algebraic
  surfaces

abstract:
  We determined the Picard group of the moduli of rank two stable sheaves on an
arbitrary algebraic surface up to finite index


title: On irreducible four--manifolds

abstract:
  We show that minimal symplectic 4--manifolds with $b_2^+ >1$ and with
residually finite fundamental groups are irreducible. We also give examples of
irreducible orientable four--manifolds with indefinite intersection forms which
are not almost complex with respect to either orientation.


title: Making enumerative predictions by means of mirror symmetry

abstract:
  Given two Calabi--Yau threefolds which are believed to constitute a mirror
pair, there are very precise predictions about the enumerative geometry of
rational curves on one of the manifolds which can be made by performing
calculations on the other. We review the mechanics of making these predictions,
including a discussion of two conjectures which specify how the elusive
``constants of integration'' in the mirror map should be fixed. Such
predictions can be useful for checking whether or not various conjectural
constructions of mirror manifolds are producing reasonable answers.


title: Relative Geometric Invariant Theory and Universal Moduli Spaces

abstract:
  We expose in detail the principle that the relative geometric invariant
theory of equivariant morphisms is related to the GIT for linearizations near
the boundary of the $G$-effective ample cone. We then apply this principle to
construct and reconstruct various universal moduli spaces. In particular, we
constructed the universal moduli space over $\overline{M_g}$ of Simpson's
$p$-semistable coherent sheaves and a canonical dominating morphism from the
universal Hilbert scheme over $\overline{M_g}$ to a compactified universal
Picard.


title: Orbifold Riemann Surfaces and the Yang-Mills-Higgs Equations

abstract:
  We extend Hitchin's results on "The self-duality equations on a Riemann
surface" (Proc. LMS (3), vol. 55, 1987) to orbifold Riemann surfaces. We prove
existence results for orbifold solutions of the Yang-Mills-Higgs equations and
construct the moduli space of solutions. These moduli spaces provide
interesting examples of non-compact hyper-Kahler manifolds in all dimensions
divisible by 4 and of completely integrable Hamiltonian systems. We also
reinterpret these moduli spaces as spaces of orbifold Higgs bundles and as
representation varieties of Fuchsian groups.


title:
Gauge fixing for logarithmic connections over curves and the
  Riemann-Hilbert-Problem

abstract:
  We explain in detail the correspondence between algebraic connections over
CP^{1}, logarithmic at X = { x_{1},...,x_{n} } \subset CP^{1}, and flat bundles
over CP^{1}-X with integer weighted filtrations near each x_{j}. Included is a
gauge fixing theorem for logarithmic connections. (Thus far, one could work
over any Riemann surface.) We prove a bound on the splitting type of a
semi-stable logarithmic connection over CP^{1}. Using this we extend and
simplify some results on the Riemann-Hilbert-Problem, which asks for a
logarithmic connection on a holomorphically trivial bundle over CP^{1},
extending a given flat bundle over CP^{1}-X. The work is self contained and
elementary, using only basic knowledge of Gauge Theory and the
Birkhoff-Grothendieck-Theorem.


title: A Non-Linear Deformation of the Hitchin Dynamical System

abstract:
  Mukai's space, parametrizing simple sheaves on a K3 surface S whose numerical
invariants are those of a line bundle on a curve C in S, is interpreted as a
deformation of Hitchin's system on C. This is used to show that the nilpotent
cone in Mukai space is Lagrangian. In rank 2, components of this nilpotent cone
are described as affine bundles over symmetric products of the curve. The
underlying vector bundles give the corresponding components of the Hitchin
nilpotent cone.


title: Pieri's rule for flag manifolds and Schubert polynomials

abstract:
  We establish the formula for multiplication by the class of a special
Schubert variety in the integral cohomology ring of the flag manifold. This
formula also describes the multiplication of a Schubert polynomial by either an
elementary symmetric polynomial or a complete homogeneous symmetric polynomial.
Thus, we generalize the classical Pieri's rule for symmetric
polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our
primary technique is an explicit geometric description of certain intersections
of Schubert varieties. This method allows us to compute additional structure
constants for the cohomology ring, which we express in terms of paths in the
Bruhat order on the symmetric group.


title: Quantum cohomology of flag varieties

abstract:
  We describe a construction of Gromov-Witten invariants for flag varieties and
use it to give a presentation for the quantum cohomology ring, by extending the
ideas used by Bertram in the case of Grassmannians. This provides a proof for
the conjecture of Givental and Kim in [GK].


title:
Bogomolov conjecture over function fields for stable curves with only
  irreducible fibers (Version 2.0)

abstract:
  Let K be a function field and C a non-isotrivial curve of genus g >= 2 over
K. In this paper, we will show that if C has a global stable model with only
geometrically irreducible fibers, then Bogomolov conjecture over function
fields holds.


title:
A weighted version of Zariski's hyperplane section theorem and
  fundamental groups of complements of plane curves

abstract:
  We formulate and prove a weighted version of Zariski's hyperplane section
theorem on the topological fundamental groups of the complements of
hypersurfaces in a projective space. As an application, we calculate
fundamental groups of the complements of certain singular projective plane
curves.


title: Sequences of stable bundles over a compact complex surface

abstract:
  When identified with sequences of irreducible Hermitian-Einstein connections,
sequences of stable holomorphic bundles of fixed topological type and bounded
degree on a compact complex surface equipped with a Gauduchon metric are shown
to have strongly convergent subsequences after blowing-up and pulling-back
sufficiently many times.


title: Blowups and gauge fields

abstract:
  The relationship between stable holomorphic vector bundles on a compact
complex surface and the same such objects on a blowup of the surface is
investigated, where "stability" is with respect to a Gauduchon metric on the
surface and naturally derived such metrics on the blowup.
  The main results are: descriptions of holomorphic vector bundles on a blowup;
conditions relating (semi)-stability of these to that of their direct images on
the surface; sheaf-theoretic constructions for "stabilising" unstable bundles
and desingularising moduli of stable bundles; an analysis of the behaviour of
Hermitian-Einstein connections on bundles over blowups as the underlying
Gauduchon metric degenerates; the definition of a topology on equivalence
classes of stable bundles on blowups over a surface and a proof (using results
of a concurrent preprint) that this topology is compact in many cases---i.e., a
new compactification for moduli of stable bundles on the original surface.


title: A note on Zariski pairs

abstract:
  A couple of complex projective plane curves are said to make a Zariski pair
if they have the same degree and the same type of singularities, but their
embeddings in the projective plane are topologically different. In this paper,
we present a method to construct certain type of Zariski pairs, and make
infinite series of examples.


title: Un crit\`ere d'extension d'un foncteur d\'efini sur les sch\'emas lisses

abstract:
  Let $k$ be a field of characteristic zero. By using Hironaka's
desingularisation theorem, we prove an extension criterion for a functor
defined on nonsingular k-schemes and taking values on a category of complexes.
Roughly speaking, the criterion shows that if such a functor satisfies the
standard exact sequence of a blowing-up, then the functor can be extended to
all separated k-schemes of finite type. The result is applied to the
Grothendieck's theory of motives, to the Hodge-De Rham filtered complex of an
analytic space, and to the rational homotopy of k-schemes in algebraic De Rham
theory.


title: Spectral Covers

abstract:
  This is a survey of various results about spectral covers and their
relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S
corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is
the Weyl group of G) together with a sheaf on \widetilde{S} which in simple
cases is a line bundle, and is W-equivariant up to certain twists and shifts.
Various other types of spectral covers, depending on the choice of a
representation or weight of G, arise as associated objects of \widetilde{S}. We
focus on the decomposition of the Picards of these spectral covers into Pryms
(this includes various well-known Prym identities as special cases) and on the
interpretation, in the spirit of Hitchin's abelianization program, of a
distinguished Prym component as parameter space for higgs bundles.


title: Algebraic Barth-Lefschetz theorems

abstract:
  Using results of Hironaka-Matsumura and Faltings, we prove a strong version
of the well known Fulton-Hansen connectivity theorem for weighted projective
spaces. As a consequence we get the following result. If $Y$ is an irreducible
subvariety of the $n$-dimensional projective space (over a field of arbitrary
characteristic), then the diagonal embedding ${\Delta}_Y$ is $G_3$ in $Y\times
Y$. This fact implies a generalized version (with a characteristic-free proof)
of a result of Ogus (in char. zero) and Speiser (in positive characteristic).


title: Les Invariants de Seiberg-Witten et la Conjecture de Van De Ven

abstract:
  We give a short proof for the fact that rationality of complex surfaces is a
property depending only on the differential structure. Our proof uses the new
Seiberg-Witten invariants.


title:
The Coupled Seiberg-Witten Equations, vortices, and Moduli spaces of
  stable pairs

abstract:
  We introduce coupled Seiberg-Witten equations, and we prove, using a
generalized vortex equation, that, for Kaehler surfaces, the moduli space of
solutions of these equations can be identified with a moduli space of
holomorphic stable pairs. In the rank 1 case, one recovers Witten's result
identifying the space of irreducible monopoles with a moduli space of divisors.
As application, we give a short proof of the fact that a rational surface
cannot be diffeomorphic to a minimal surface of general type.


title: Seiberg-Witten Invariants and the Van De Ven Conjecture

abstract:
  The purpose of this note is to give a short, selfcontained proof of the
following result: A complex surface which is diffeomeorphic to a rational
surface is rational.


title: Seiberg-Witten Invariants and Rationality of Complex Surfaces

abstract:
  The purpose of this paper is: 1) to explain the Seiberg-Witten invariants, 2)
to show that - on a K\"ahler surface - the solutions of the monopole equations
can be interpreted as algebraic objects, namely effective divisors, 3) to give
- as an application - a short selfcontained proof for the fact that rationality
of complex surfaces is a ${\cal C}^{\infty}$-property.


title: Stratified Picard--Lefschetz theory

abstract:
  The monodromy action in the homology of level sets of Morse functions on
stratified singular analytic varieties is studied. The local variation
operators in both the standard and the intersection homology groups defined by
the loops around the critical values of such functions are reduced to similar
operators in the homology groups of the transversal slices of the corresponding
strata.


title: Iitaka-Severi's Conjecture for Complex Threefolds

abstract:
  We prove the following generalization of Severi's Theorem: Let $X$ be a fixed
complex variety. Then there exist, up to birational equivalence, only finitely
many complex varieties $Y$ of general type of dimension at most three which
admit a dominant rational map $f$ from $X$ to Y$.


title:
On projective plane curves whose complements have finite non-abelian
  fundamental groups

abstract:
  We construct new examples of singular projective plane curves whose
complements have finite and non-abelian fundamental groups, by generalizing the
classical three cuspidal quartic curve discovered by Zariski.


title: Rational Blowdowns of Smooth 4-Manifolds

abstract:
  In this paper we introduce a surgical procedure, called a rational blowdown,
for a smooth 4-manifold X and determine how this procedure affects both the
Donaldson and Seiberg-Witten invariants of X.


title: Mapping threefolds onto three-quadrics

abstract:
  We prove that the degree of a nonconstant morphism from a smooth projective
3-fold $X$ with N\'{e}ron-Severi group ${\bf Z}$ to a smooth 3-dimensional
quadric is bounded in terms of numerical invariants of $X$. In the special case
where $X$ is a 3-dimensional cubic we show that there are no such morphisms.
The main tool in the proof is Miyaoka's bound on the number of double points of
a surface.


title: Lang's conjectures, Conjecture H, and uniformity

abstract:
  The purpose of this note is to wish a happy birthday to Professor Lucia
Caporaso.*
  We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together
with Lang's conjecture implies the uniformity of rational points on varieties
of general type, as predicted in [CHarM]; a few applications in arithmetic and
geometry are stated.
  Let X be a variety of general type defined over a number field K. It was
conjectured by S. Lang that the set of rational points X(K) is not Zariski
dense in X. In the paper [CHarM] of L. Caporaso, J. Harris and B. Mazur it is
shown that the above conjecture of Lang implies the existence of a uniform
bound on the number of K-rational points of all curves of fixed genus g over K.
  The paper [CHarM] has immediately created a chasm among arithmetic geometers.
This chasm, which often runs right in the middle of the personalities involved,
divides between loyal believers of Lang's conjecture, who marvel in this
powerful implication, and the disbelievers, who try (so far in vain) to use
this implication to derive counterexamples to the conjecture. In this paper we
will attempt to deepen this chasm, using the techniques of [CHarM] and
continuing [aleph], by proving more implications, some of which very strong, of
various conjectures of Lang. Along the way we will often use a conjecture
donned by Caporaso et. al. Conjecture H (see again [CHarM], sec. 6) about
Higher dimensional varieties, which is regarded very plausible among experts of
higher dimensional algebraic geometry. In particular, we will show


title: Rational surfaces in P^4 containing a plane curve

abstract:
  The families of smooth rational surfaces in $\PP^4$ have been classified in
degree $\le 10$. All known rational surfaces in $\PP^4$ can be represented as
blow-ups of the plane $\PP^2$. The fine classification of these surfaces
consists of giving explicit open and closed conditions which determine the
configurations of points corresponding to all surfaces in a given family. Using
a restriction argument originally due independently to Alexander and Bauer we
achieve the fine classification in two cases, namely non-special rational
surfaces of degree 9 and special rational surfaces of degree 8. The first case
completes the fine classification of all non-special rational surfaces. In the
second case we obtain a description of the moduli space as the quotient of a
rational variety by the symmetric group $S_5$. We also discuss in how far this
method can be used to study other rational surfaces in $\PP^4$.


title:
A Geometric Invariant Theory Compactification of M_{g,n} Via the
  Fulton-MacPherson Configuration Space

abstract:
  A compactification over $\overline{M}_g$ of $M_{g,n}$ is obtained by
considering the relative Fulton-MacPherson configuration space of the universal
curve. The resulting compactification differs from the Deligne-Mumford space
$\overline{M}_{g,n}$. In case $n=2$, the compactification constructed here and
the Deligne-Mumford compactification are essentially the distinct minimal
resolutions of the fiber product over $\overline{M}_g$ of the universal curve
with itself.


title: A Note On Elliptic Plane Curves With Fixed j-Invariant

abstract:
  Let N_d be the number of degree d, nodal, rational plane curves through 3d-1
points in the complex projective plane. The number of degree d>=3, nodal,
elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is
found to be {d-1 \choose 2}*N_d.


title:
Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the
  Universal Commensurability Teichm\"uller Space

abstract:
  There exists on each Teichm\"uller space $T_g$ (comprising compact Riemann
surfaces of genus $g$), a natural sequence of determinant (of cohomology) line
bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''.
There is a remarkable connection, (Belavin-Knizhnik), between the Mumford
isomorphisms and the existence of the Polyakov string measure on the
Teichm\"uller space. This suggests the question of finding a genus-independent
formulation of these bundles and their isomorphisms. In this paper we combine a
Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q$ bundles
to construct such an universal version. Our universal objects exist over the
universal space, $T_\infty$, which is the direct limit of the $T_g$ as the
genus varies over the tower of all unbranched coverings of any base surface.
The bundles and the connecting isomorphisms are equivariant with respect to the
natural action of the universal commensurability modular group.


title: Topological invariants of a fan associated to a toric variety

abstract:
  Associated to a toric variety $X$ of dimension $r$ over a field $k$ is a fan
$\Delta$ on $\Bbb R^r$. The fan $\Delta$ is a finite set of cones which are in
one-to-one correspondence with the orbits of the torus action on $X$. The fan
$\Delta$ inherits the Zariski topology from $X$. In this article some
cohomological invariants of $X$ are studied in terms of whether or not they
depend only on $\Delta$ and not $k$. Secondly some numerical invariants of $X$
are studied in terms of whether or not they are topological invariants of the
fan $\Delta$. That is, whether or not they depend only on the finite
topological space defined on $\Delta$. The invariants with which we are mostly
concerned are the class group of Weil divisors, the Picard group, the Brauer
group and the dimensions of the torsion free part of the \'etale cohomology
groups with coefficients in the sheaf of units. The notion of an open
neighborhood of a fan is introduced and examples are given for which the above
invariants are sufficiently fine to give nontrivial stratifications of an open
neighborhood of a fan all of whose maximal cones are nonsimplicial.


title: On the local geometry of moduli spaces of vector bundles

abstract:
  We give a canonical description of the formal moduli space of a vector bundle
on a variety; as an application, we prove the closedness of certain
differential forms on moduli corresponding to the trace form on the
endomorphism algebra of the bundle.


title: Determinant of Period Integral

abstract:
  Period integrals arise from the comparison between Betti cohomologies and de
Rham cohomologies. In this paper, we give a formula for the determinant of
period integrals in terms of the canonical pairing between Picard group of rank
1 realizations and relative Chow group. It is compatible with the similar
formula for Galois modules proved by the first author, and as a consequence,
under the assumption of Gersten resolution, the determinant of the cohomology
of a determinantally motivic object is motivic.


title:
Flips from semi-stable 4-folds whose degenerate fibers are unions of
  Cartier divisors which are terminal factorial 3-folds

abstract:
  We shall investigate flipping contractions from a semi-stable 4-fold $X$
whose degenerate fiber is a union of Cartier divisors which are terminal
factorial 3-folds. Especially we shall prove that $X$ is smooth along the
flipping locus, and that the flip exists for such contractions.


title: Quaternionic Monopoles

abstract:
  We present the simplest non-abelian version of Seiberg-Witten theory:
Quaternionic monopoles. These monopoles are associated with
Spin^h(4)-structures on 4-manifolds and form finite-dimensional moduli spaces.
On a Kahler surface the quaternionic monopole equations decouple and lead to
the projective vortex equation for holomorphic pairs. This vortex equation
comes from a moment map and gives rise to a new complex-geometric stability
concept. The moduli spaces of quaternionic monopoles on Kahler surfaces have
two closed subspaces, both naturally isomorphic with moduli spaces of
canonically stable holomorphic pairs. These components intersect along
Donaldsons instanton space and can be compactified with Seiberg-Witten moduli
spaces. This should provide a link between the two corresponding theories.
Notes: To appear in CMP The revised version contains more details concerning
the Uhlenbeck compactfication of the moduli space of quaternionic monopoles,
and possible applications are discussed. Attention ! Due to an ununderstandable
mistake, the duke server had replaced all the symbols "=" by "=3D" in the
tex-file of the revised version we sent on February, the 2-nd. The command
"\def{\ad}" had also been damaged !


title:
How to tell you're hearing a Calabi-Yau: Universal variations of Hodge
  structure and local Schottky relations for Calabi-Yau manifolds

abstract: A revised version with a number of corrections and refinements.


title: Minimal sections of conic bundles

abstract:
  Let the threefold X be a general smooth conic bundle over the projective
plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this
paper we prove the existence of two natural families C(+) and C(-) of curves on
X, such that the Abel-Jacobi map F sends one of these families onto a copy of
the theta divisor (Theta), and the other -- onto the jacobian J(X). The general
curve C of any of these two families is a section of the conic bundle
projection, and our approach relates such C to a maximal subbundle of a rank 2
vector bundle E(C) on C, or -- to a minimal section of the ruled surface
P(E(C)). The families C(+) and C(-) correspond to the two possible types of
versal deformations of ruled surfaces over curves of fixed genus g(C). As an
application, we find parameterizations of J(X) and (Theta) for certain classes
of Fano threefolds, and study the sets Sing(Theta) of the singularities of
(Theta).


title:
The fiber of the Griffiths map for the non-hyperelliptic Fano threefolds
  of genus 6

abstract:
  Among smooth non-rational Fano 3-folds, the non-hyperelliptic Fano 3-fold
X(10) of genus 6 has the unique property to admit a non-trivial orbit of
birationally isomorphic 3-folds, inside its moduli space. Here we prove that
these orbits are, in fact, the same as the fibers of the Griffiths period map
for X(10). This leads upto the main result of the paper: The general fiber of
the period map for X(10) is a union of two irreducible families of 3-folds:
F(1) + F(2), each F(i) -- isomorphic to the Fano surface of conics of any of
its elements. As an application, we give a negative answer to a Tjurin's
conjecture: The general X(10) is birational to a quartic double solid.


title: Symmetric subgroups of rational groups of hermitian type

abstract:
  A rational group of hermitian type is an algebraic group over the rational
numbers whose symmetric space is a hermitian symmetric space. We assume such a
group $G$ to be given, which we assume is isotropic. Then, for any rational
parabolic $P$ in the group $G$, we find a reductive rational subgroup $N$
closely related with $P$ by a relation we call incidence. This has implications
to the geometry of arithmetic quotients of the symmetric space by arithmetic
subgroups of $G$, in the sense that $N$ defines a subvariety on such an
arithmetic quotient which has special behaviour at the cusp corresponding to
the parabolic with which $N$ is incident.


title: Semistable 3-fold flips

abstract:
  We piece together ingredients, which are well known and documented in the
literature, into a new proof of the existence of semistable 3-fold flips


title: Del Pezzo surfaces over Dedekind schemes

abstract:
  Let S be a Dedekind scheme with fraction field K. We study the following
problem: given a Del Pezzo surface X, defined over K, construct a distinguished
integral model of X, defined over all of S. We provide a satisfactory answer if
S is a smooth complex curve, and a conjectural answer if X is a cubic Del Pezzo
surface over (nearly) arbitrary S.


title: Hodge groups of abelian varieties with purely multiplicative reduction

abstract:
  The main result of the paper is that if $A$ is an abelian variety over a
subfield $F$ of ${\bold C}$, and $A$ has purely multiplicative reduction at a
discrete valuation of $F$, then the Hodge group of $A$ is semisimple. Further,
we give necessary and sufficient conditions for the Hodge group to be
semisimple. We obtain bounds on certain torsion subgroups for abelian varieties
which do not have purely multiplicative reduction at a given discrete
valuation, and therefore obtain bounds on torsion for abelian varieties,
defined over number fields, whose Hodge groups are not semisimple.


title: Uniformity of stably integral points on elliptic curves

abstract:
  A common practice in arithmetic geometry is that of generalizing rational
points on projective varieties to integral points on quasi-projective
varieties.
  Following this practice, we demonstrate an analogue of a result of L.
Caporaso, J. Harris and B. Mazur, showing that the Lang - Vojta conjecture
implies a uniform bound on the number of stably integral points on an elliptic
curve over a number field, as well as the uniform boundedness conjecture
(Merel's theorem).


title: Composite Differentiable Functions

abstract:
  We introduce a new point of view towards Glaeser's theorem on composite
$C^\infty$ functions [Ann. of Math. 1963], with respect to which we can
formulate a ``$C^k$ composite function property" that is satisfied by all
semiproper real analytic mappings. As a consequence, we see that a closed
subanalytic set $X$ satisfies the $C^\infty$ composite function property if and
only if the ring $C^\infty (X)$ of $C^\infty$ functions on $X$ is the
intersection of all finite differentiability classes.


title: The index of $grad f(x,y)$

abstract:
  Let $f(x,y)$ be a real polynomial of degree $d$ with isolated critical
points, and let $i$ be the index of $grad f$ around a large circle containing
the critical points. An elementary argument shows that $|i| \leq d-1$. In this
paper we show that $ i \leq max \{1, d-3 \}$. We also show that if all the
level sets of $f$ are compact, then $i = 1$, and otherwise $ |i| \leq \dr -1 $
where $\dr$ is the sum of the multiplicities of the real linear factors in the
homogeneous term of highest degree in $f$. The technique of proof involves
computing $i$ from information at infinity. The index $i$ is broken up into a
sum of components $i_{p,c}$ corresponding to points $p$ in the real line at
infinity and limiting values $c \in \realinf$ of the polynomial. The numbers
$i_{p,c}$ are computed in three ways: geometrically, from a resolution of
$f(x,y)$, and from a Morsification of $f(x,y)$. The $i_{p,c}$ also provide a
lower bound for the number of vanishing cycles of $f(x,y)$ at the point $p$ and
value $c$.


title: A note on the genus of certain curves over finite fields

abstract:
  We prove the following result which was conjectured by Stichtenoth and Xing:
let $g$ be the genus of a projective, irreducible non-singular curve over the
finite field $\Bbb F_{q^2}$ and whose number of $\Bbb F_{q^2}$-rational points
attains the Hasse-Weil bound; then either $4g\le (q-1)^2$ or $2g=(q-1)q$.


title: On Alternative Supermatrix Reduction

abstract:
  We consider a nonstandard odd reduction of supermatrices (as compared with
the standard even one) which arises in connection with possible extension of
manifold structure group reductions. The study was initiated by consideration
of the generalized noninvertible superconformal-like transformations. The
features of even- and odd-reduced supermatrices are investigated on a par. They
can be unified into some kind of "sandwich" semigroups. Also we define a
special module over even- and odd-reduced supermatrix sets, and the generalized
Cayley-Hamilton theorem is proved for them. It is shown that the odd-reduced
supermatrices represent semigroup bands and Rees matrix semigroups over a unit
group.


title: On exotic algebraic structures on affine spaces

abstract:
  By an exotic algebraic structure on the affine space ${\bf C}^n$ we mean a
smooth affine algebraic variety which is diffeomorphic to ${\bf R}^{2n}$ but
not isomorphic to ${\bf C}^n$. This is a survey of the recent developement on
the subject, which emphasizes its analytic aspects and points out some open
problems.


title: Open problems on open algebraic varieties

abstract:
  This report records a large number of open problems in Affine Algebraic
Geometry that were proposed by participants in a Conference on Open Algebraic
Varieties at the Centre de Recherches en Mathematiques in Montreal at December
1994.


title: On localization and Riemann-Roch numbers for symplectic quotients

abstract:
  Suppose $(M,\omega)$ is a compact symplectic manifold acted on by a compact
Lie group $K$ in a Hamiltonian fashion, with moment map $\mu: M \to \Lie(K)^*$
and Marsden-Weinstein reduction $M_{red} = \mu^{-1}(0)/K$. In this paper, we
assume that $M$ has a $K$-invariant K\"ahler structure. In an earlier paper, we
proved a formula (the residue formula) for $\eta_0 e^{\omega_0}[M_{red}]$ for
any $\eta_0 \in H^*(M_{red})$, where $\omega_0$ is the induced symplectic form
on $M_{red}$. Here we apply the residue formula in the special case $\eta_0 =
Td(M_{red})$; when $K$ acts freely on $\mu^{-1}(0)$ this yields a formula for
the Riemann-Roch number $RR (L_{red})$ of a holomorphic line bundle $L_{red}$
on $M_{red}$ that descends from a holomorphic line bundle $L$ on $M$ for which
$c_1(L) = \omega$. Using the holomorphic Lefschetz formula we similarly obtain
a formula for the $K$-invariant Riemann-Roch number $RR^K(L) $ of $L$. In the
case when the maximal torus $T$ of $K$ has dimension one (except in a few
special circumstances), we show the two formulas are the same. Thus in this
special case the residue formula is equivalent to the result of Guillemin and
Sternberg that $RR(L_{red}) = RR^K(L)$. (The residue formula was proved under
the assumption that 0 is a regular value of $\mu$, and was given in terms of
the restrictions of classes in the equivariant cohomology $H^*_T(M) $ of $M$ to
the


title: Hermitian-Einstein Inequalities and Harder-Narasimhan Filtration

abstract:
  Unstable holomorphic bundles can be described algebraically by
Harder-Narasimhan filtrations. In this note we show how such filtrations
correspond to the existence of special metrics defined by Hermitian-Einstein
inequalities.


title: Curves that change genus can have arbitrarily many rational points

abstract:
  A singular curve over a non-perfect field K may not have a smooth model over
K. Those are said to "change genus". If K is a global field of positive
characteristic and C/K a curve that change genus, then C(K) is known to be
finite. The purpose of this note is to give examples of curves with fixed
relative genus, defined over K for which #C(K) is arbitrarily large. The
motivation for considering this problem comes from the work of Caporaso et al.
[CHM], where they show that a conjecture of Lang implies that, for a number
field K, #C(K) can be bounded in terms of g and K only for all curves C/K of
genus g > 1.


title: Rang maximal pour $T_P^n$

abstract:
  In this paper, I compute the last non-trivial term of the minimal free
resolution of the homogeneous ideal of $s$ points of $P^n$ in sufficiently
general position, for any $s$, showing that this term is the one conjectured by
the Minimal Resolution Conjecture of Anna Lorenzini. I use a geometrical
method, the "vectorial m\'ethode d'Horace" developped by Andr\'e Hirschowitz
and Carlos Simpson to get a proof of the MRC for $s$ large enough.


title: Stability of the Poincar\'e bundle

abstract:
  Let $C$ be a nonsingular projective curve of genus $g\ge2$ defined over the
complex numbers, and let $M_{\xi}$ denote the moduli space of stable bundles of
rank $n$ and determinant $\xi$ on $C$, where $\xi$ is a line bundle of degree
$d$ on $C$ and $n$ and $d$ are coprime. It is shown that the universal bundle
$\cu_{\xi}$ on $C\times M_{\xi}$ is stable with respect to any polarisation on
$C\times M_{\xi}$. It is shown further that the connected component of the
moduli space of $\cu_{\xi}$ containing $\cu_{\xi}$ is isomorphic to the
Jacobian of $C$.


title: Semiorthogonal decomposition for algebraic varieties

abstract:
  A criterion for a functor between derived categories of coherent sheaves to
be full and faithful is given. A semiorthogonal decomposition for the derived
category of coherent sheaves on the intersection of two even dimensional
quadrics is obtained. The behaviour of derived categories with respect to
birational transformations is investigated. A theorem about reconstruction of a
variety from the derived category of coherent sheaves is proved.


title: Fundamental group of locally symmetric varieties

abstract:
  Take a bounded symmetric domain $D$ and an arithmetic subgroup $\Gamma$ of
${\rm Aut}(D)$. Take the quotient $D/\Gamma$, compactify and resolve the
singularities. We study the fundamental group of the compact complex manifolds
that result from this procedure, and in particular the case of Siegel modular
threefolds.


title: Higher cohomology triples and holomorphic extensions

abstract:
  We introduce equations for special metrics, and notions of stability for some
new types of augmented holomorphic bundles. These new examples include
holomorphic extensions, and in this case we prove a Hitchin-Kobayashi
correspondence between a certain deformation of the Hermitian-Einstein
equations and our definition of stability for an extension.


title: On the separate algebraicity along the families of algebraic curves

abstract:
  A new generalization of the classical separate algebraicity theorem is
suggested and proved.


title: Cycles alg\'ebriques sur les surfaces K3 r\'eelles

abstract:
  For a real algebraic K3 surface $X(R)$, we give all possible values of the
dimension $h^1_{alg}(X(R)$ of the group $\H^1_{alg}(X(R),Z/2)$ of algebraic
cycles of $X(R)$. In particular, we prove that if $X(R)$ is not an M-surface,
$X(R)$ can always be deformed to some $X'(R)$ with
$h^1_{alg}(X'(R))=\dim\H^1(X(R),Z/2)$. Furthermore, we obtain that in certain
moduli space of real algebraic K3 surfaces, the collection of real isomorphism
classes of K3 surfaces $X(R)$ such that $h^1_{alg}(X(R))$is greater or equal
than $k$ is a countable union of subspaces of dimension $20-k$.


title: Minimal Siegel modular threefolds

abstract:
  In this paper we study the maximal extension $\Gamma_t^*$ of the subgroup
$\Gamma_t$ of $\operatorname{Sp}_4 (\bq)$ which is conjugate to the paramodular
group. The index of this extension is $2^{\nu(t)}$ where $\nu(t)$ is the number
of prime divisors of $t$. The group $\Gamma_t^*$ defines the minimal modular
threefold ${\Cal A}_t^*$ which is a finite quotient of the moduli space ${\Cal
A}_t$ of $(1,t)$-polarized abelian surfaces. A certain degree 2 quotient of
${\Cal A}_t$ is a moduli space of lattice polarized $K3$ surfaces. The space
${\Cal A}_t^*$ can be interpreted as the space of Kummer surfaces associated to
$(1,t)$-polarized abelian surfaces. Using the action of $\Gamma_t^*$ on the
space of Jacobi forms we show that many spaces between ${\Cal A}_t$ and ${\Cal
A}_t^*$ posess a non-trivial 3-form, i.e. the Kodaira dimension of these spaces
is non-negative. Finally we determine the divisorial part of the ramification
locus of the finite map ${\Cal A}_t\rightarrow {\Cal A}_t^*$ which is a union
of Humbert surfaces. We interprete the corresponding Humbert surfaces as
Hilbert modular surfaces.


title: Modular forms and Donaldson invariants for 4-manifolds with $b_+=1$

abstract:
  We study the Donaldson invariants of simply connected $4$-manifolds with
$b_+=1$, and in particular the change of the invariants under wall-crossing. We
assume the conjecture of Kotschick and Morgan about the shape of the
wall-crossing terms (which Oszva\'th and Morgan are now able to prove), and are
determine a generating function for the wall-crossing terms in terms of modular
forms. As an application we determine all the Donaldson invariants of the
projective plane in terms of modular forms. The main tool are the blowup
formulas, which are used to obtain recursive relations.


title:
Wall-crossing formulas, Bott residue formula and the Donaldson
  invariants of rational surfaces

abstract:
  We study the Donaldson invariants of rational surfaces and their dependence
on the chambers in the ample cone. We build on a previous joint paper in which
we have expressed the change of the Donaldson invariants on an algebraic
surface $S$ under crossing a so-called good wall in terms of certain
intersection numbers on Hilbert schemes of points on $S$. In the current paper
we assume that $S$ is rational. Thish enables us to apply the Bott residue
theorem to evaluate these intersection numbers with the help of a computer.
Combining this with the blowup formulas we obtain an algorithm for computing
the Donaldson invariants of all rational surfaces in almost all chambers. In
particular we compute all the $SU(2)$- and $SO(3)$-Donaldson invariants of the
projective plane of degree at most 50.


title:
Instantons and affine algebras I: The Hilbert scheme and vertex
  operators

abstract:
  This is the first in a series of papers which describe the action of an
affine Lie algebra with central charge $n$ on the moduli space of
$U(n)$-instantons on a four manifold $X$. This generalises work of Nakajima,
who considered the case when $X$ is an ALE space. In particular, this describes
the combinatorial complexity of the moduli space as being precisely that of
representation theory, and thus will lead to a description of the Betti numbers
of moduli space as dimensions of weight spaces. This Lie algebra acts on the
space of conformal blocks (i\.e\., the cohomology of a determinant line bundle
on the moduli space) generalising the ``insertion'' and ``deletion'' operations
of conformal field theory, and indeed on any cohomology theory. In the
particular case of $U(1)$-instantons, which is essentially the subject of this
present paper, the construction produces the basic representation after
Frenkel-Kac. Then the well known quadratic nature of $ch_2$,
  $$ch_2 = \frac{1}{2} c_1\cdot c_1 - c_2 $$
  becomes precisely the formula for the eigenvalue of the degree operator,
i\.e\. the well known quadratic behaviour of affine Lie algebras.


title: A Vector Bundle of Rank 2 on P1 x P3

abstract:
  The paper studies a rank 2 vector bundle on P1 x P3. Similarly to the
Horrocks - Mumford bundle on P4 this vector bundle encodes a lot of geometric
information. It is defined via the Serre construction by an abelian surface in
P1 x P3. The bundle is stable with respect to O(1,1), its jumping lines are
determined. Morover it is shown that it is closely related to a double structue
on P1 x P1 embedded in P1 x P3 as well as to a certain pencil of quadrics of
degree 4 in P3.


title: Deforming Calabi-Yau Threefolds

abstract:
  This paper first generalises the Bogomolov-Tian-Todorov unobstructedness
theorem to the case of Calabi-Yau threefolds with canonical singularities. The
deformation space of such a Calabi-Yau threefold is no longer smooth, but the
general principle is that the obstructions to deforming such a threefold are
precisely the obstructions to deforming the singularities of the threefold.
Secondly, these results are applied to smoothing singular Calabi-Yau threefolds
with crepant resolutions. Any such Calabi-Yau threefold with isolated complete
intersection singularities which are not ordinary double points is smoothable.
A Calabi-Yau threefold with non-complete intersection isolated singularities is
proved to be smoothable under much stronger hypotheses.


title: Stacks of Stable Maps and Gromov-Witten Invariants

abstract:
  We correct some errors in the earlier version of this paper. Most
importantly, the definition of isogeny of marked stable graphs has changed.


title: Residues in Toric Varieties

abstract:
  We study residues on a complete toric variety X, which are defined in terms
of the homogeneous coordinate ring of X. We first prove a global transformation
law for toric residues. When the fan of the toric variety has a simplicial cone
of maximal dimension, we can produce an element with toric residue equal to 1.
We also show that in certain situations, the toric residue is an isomorphism on
an appropriate graded piece of the quotient ring. When X is simplicial, we
prove that the toric residue is a sum of local residues. In the case of equal
degrees, we also show how to represent X as a quotient (Y-{0})/C* such that the
toric residue becomes the local residue at 0 in Y.


title: Automorphisms of crepant resolutions for quotient spaces

abstract:
  A formula for calculating the Lefschetz number of an automorphism acting on a
crepant resolution for a quotient of a Kahler manifold derived from an
equivariant version of McKay correspondence. The latter is proven in some
cases. As an application the Lefschetz numbers of of involutions acting on
Calabi-Yau threefolds and their mirrors are compared.


title: On triple coverings of irrational curves

abstract:
  Given a triple covering $X$ of genus $g$ of a general (in the sense of
Brill-Noether) curve $C$ of genus $h$, we show the existence of base-point-free
pencils of degree $d$ which are not composed with the triple covering for any
$d\ge g-[{3h+1\over 2}]-1$ by utilizing some enumerative methods and
computations. We also discuss about the sharpness of our main result and the
so-called Castelnuovo-Severi bound byexhibiting some examples.


title:
General n-canonical divisors on two-dimensional smoothable
  semi-log-terminal singularities

abstract:
  In this paper we calculate genaral n-canonical divisors on smoothable
semi-log-terminal singularities in dimension 2, in other words, the full
sheaves associated to the double dual of the nth tensor power of the dualizing
sheaves of these singularities. And as its application we give the inequality
which bound the Gorenstein index by the local self intersection number of the
n-canonical divisor of these singularities.


title:
The line bundles on the stack of parabolic $G$-bundles over curves and
  their sections

abstract:
  Let $X$ be a smooth, complete and connected curve and $G$ be a simple and
simply connected algebraic group over $\comp$. We calculate the Picard group of
the moduli stack of quasi-parabolic $G$-bundles and identify the spaces of
sections of its members to the conformal blocs of Tsuchiya, Ueno and Yamada. We
describe the canonical sheaf on these stacks and show that they admit a unique
square root, which we will construct explicitly. Finally we show how the
results on the stacks apply to the coarse moduli spaces and recover (and
extend) the Drezet-Narasimhan theorem. We show moreover that the coarse moduli
spaces of semi-stable $SO_r$-bundles are not locally factorial for $r\geq 7$.


title: A note on scrolls of smallest embedded codimension

abstract:
  Let $M$ be a submanifold of ${\Bbb P}^N$ of dimension $n>2$. Suppose that
$(M,{\Cal O}_M(1))\cong{\Bbb P}({\Cal E}),{\Cal O}(1))$ for some vector bundle
${\Cal E}$ on a surface $S$. Then $N\ge 2n-1$ by Barth-Lefschetz Theorem. We
are interested in the case $N=2n-1$. In 1994 Ionescu and Toma gave a
classification of the cases where $S$ is not of general type. Here we propose a
conjecture concerning this remaining case, which is verified for $n\le 1100$ by
a computer programm.


title: On a class of rational cuspidal plane curves

abstract:
  We obtain new examples and the complete list of the rational cuspidal plane
curves $C$ with at least three cusps, one of which has multiplicity ${\rm
deg}\,C - 2$. It occurs that these curves are projectively rigid. We also
discuss the general problem of projective rigidity of rational cuspidal plane
curves.


title: On the number of singular points of plane curves

abstract:
  This is an extended, renovated and updated report on a joint work which the
second named author presented at the Conference on Algebraic Geometry held at
Saitama University, 15-17 of March, 1995. The main result is an inequality for
the numerical type of singularities of a plane curve, which involves the degree
of the curve, the multiplicities and the Milnor numbers of its singular points.
It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality
due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the
authors independently in the particular case of an irreducible cuspidal curve
at 1992. Our proof is based on the localization, the local Zariski--Fujita
decomposition and uses a graph discriminant calculus. The key point is a local
analog of the BMY-inequality for a plane curve germ. As a corollary, a
boundedness criterium for a family of plane curves has been obtained. Another
application of our methods is the following fact: a rigid rational cuspidal
plane curve cannot have more than 9 cusps.


title: A Noether-Lefschetz theorem for vector bundles

abstract:
  In this note we use the monodromy argument to prove a Noether-Lefschetz
theorem for vector bundles.


title: On the diffeomorphism groups of elliptic surfaces

abstract:
  In this paper we determine for relatively minimal elliptic surfaces with
positive Euler number the image of the natural representation of the group of
orientation preserving self-diffeomorphisms on $\Hbar$, the second homology
group reduced modulo torsion. To this end we construct as many embedded spheres
of square -2 such that an isometry not induced from any combination of
reflections at such spheres or from 'complex conjugation' can be shown not to
be induced from some diffeomorphism at all. This is done with the help of
Seiberg-Witten-invariants.


title:
The Gauss map and the dual variety of real-analytic submanifolds in a
  sphere or in a hyperbolic space

abstract:
  We study the Gauss map and the dual variety of a real-analytic immersion of a
connected compact real-analytic manifold into a sphere or into a hyperbolic
space. The dual variety is defined to be the set of all normal directions of
the immersion. First, we show that the image of the Gauss map characterizes the
manifold. Also we show that the dual variety characterizes the manifold.
Besides, duality of the second fundamental form and some results on
degeneration are obtained. In Algebraic Geometry E-prints eight files are
tar-compressed and uuencoded.


title: Remarks on Seshadri constants

abstract:
  Given a smooth complex projective variety $X$ and an ample line bundle $L$ on
$X$. Fix a point $x\in X$.
  We consider the question, are there conditions which guarantee the maxima of
the Seshadri constant of $L$ at $x$, i.e $\eps(L,x)=\root n \of {L^n}$?
  We give a partial answer for surfaces and find examples where the answer to
our question is negative. If $(X,\Theta)$ is a general principal polarized
abelian surface, then $\eps(\Theta,x)={4/3}<\sqrt{2}=\sqrt{\Theta^2}$ for all
$x\in X$.


title: Koszul duality and Galois cohomology

abstract:
  It it shown that the Bloch-Kato conjecture on the norm residue homomorphism
$K^M(F)/l \to H^*(G_F,Z/l)$ follows from its (partially known) low-degree part
under the assumption that the Milnor K-theory algebra $K^M(F)/l$ modulo $l$ is
Koszul. This conclusion is a case of a general result on the cohomology of
nilpotent (co-)algebras and Koszulity.


title: A Projective Surface of Degree Eight with 168 Nodes

abstract:
  The estimate for the maximal number of ordinary double points of a projective
surface of degree eight is improved to $168\leq\mu(8)\leq 174$ by constructing
a projective surface of degree eight with 168 nodes.


title: Heisenberg algebra and Hilbert schemes of points on projective surfaces

abstract:
  I have just replaced the first line by %&amslplain in order to be compiled by
AMS-LaTeX.


title: Descent, Motives and K-theory

abstract:
  To an arbitrary variety over a field of characteristic zero, we associate a
complex of Chow motives, which is, up to homotopy, unique and bounded. We
deduce that any variety has a natural Euler characteristic in the Grothendieck
group of Chow motives. We show that the cohomology with integer coefficients of
any singular variety over the complex numbers has a natural weight filtration.
We define algebraic K-theory with "compact supports".


title:
All strictly exceptional collections in $D^b_{coh}(P^m)$ consist of
  vector bundles

abstract:
  It is proved that any strictly exceptional collection generating the derived
category of coherent sheaves on a smooth projective variety X with \rk K_0(X) =
\dim X + 1 constists of locally free sheaves up to a common shift.


title: Correlation for Surfaces of General Type

abstract:
  The main geometric result of this paper is that given any family of surfaces
of general type f:X-->B, for sufficiently large n the fiber product X^n_B
dominates a variety of general type. This result is especially interesting when
it is combined with Lang's Conjecture. This states that for a variety V of
general type over a number field K, the K rational points V(K) are not Zariski
dense in V. Assuming Lang's Conjecture, we prove the existence of a uniform
bound on the degree of the Zariski closure of the K-rational points of a
surface of general type.


title: Degenaration of Calabi-Yau Manifold with W-P Metric

abstract:
  In this paper we focus on determining for which degenerations the central
fibre is at finite distance with respect to Weil-Petersson metric. We obtain a
simple condition on the limiting mixed Hodge structure. Then we combine the
result with the canonical mixed Hodge structure of the central fibre and obtain
a simple cohomological condition for the central fibre to be at finite
distance. As a corollary, we prove that a central fibre with simple nodes is at
finite distance. This issue has been raised in the Physics literature including
the recent development of so-called "mass-less black holes".


title:
Spectral curves, algebraically completely integrable Hamiltonian
  systems, and moduli of bundles

abstract:
  This is the expanded text of a series of CIME lectures. We present an
algebro-geometric approach to integrable systems, starting with those which can
be described in terms of spectral curves. The prototype is Hitchin's system on
the cotangent bundle of the moduli space of stable bundles on a curve. A
variant involving meromorphic Higgs bundles specializes to many familiar
systems of mathematics and mechanics, such as the geodesic flow on an ellipsoid
and the elliptic solitons. We then describe some systems in which the spectral
curve is replaced by various higher dimensional analogues: a spectral cover of
an arbitrary variety, a Lagrangian subvariety in an algebraically symplectic
manifold, or a Calabi-Yau manifold. One peculiar feature of the CY system is
that it is integrable analytically, but not algebraically: the Liouville tori
(on which the system is linearized) are the intermediate Jacobians of a family
of Calabi-Yau manifolds. Most of the results concerning these three types of
non-curve-based systems are quite recent. Some of them, as well as the
compatibility between spectral systems and the KP hierarchy, are new, while
other parts of the story are scattered through several recent preprints. As
best we could, we tried to maintain the survey style of this article, starting
with some basic notions in the field and building gradually to the recent
developments.


title:
Localization in equivariant intersection theory and the Bott residue
  formula

abstract:
  The purpose of this paper is to prove the localization theorem for torus
actions in equivariant intersection theory. Using the theorem we give another
proof of the Bott residue formula for Chern numbers of bundles on smooth
complete varieties. In addition, our techniques allow us to obtain residue
formulas for bundles on a certain class of singular schemes which admit torus
actions. This class is rather special, but it includes some interesting
examples such as complete intersections and Schubert varieties.


title:
Modular subvarieties of arithmetic quotients of bounded symmetric
  domains

abstract:
  Arithmetic quotients are quotients of bounded symmetric domains by arithmetic
groups, and modular subvarieties of arithmetic quotients are themselves
arithmetic quotients of lower dimension which live on arithmetic quotients, by
an embedding induced from an inclusion of groups of hermitian type. We show the
existence of such modular subvarieties, drawing on earlier work of the author.
If $\Gamma$ is a fixed arithmetic subgroup, maximal in some sense, then we
introduce the notion of ``$\Gamma$-integral symmetric'' subgroups, which in
turn defines a notion of ``integral modular subvarieties'', and we show that
there are finitely many such on an (isotropic, i.e, non-compact) arithmetic
variety.


title: Heights for line bundles on arithmetic surfaces

abstract:
  For line bundles on arithmetic varieties we construct height functions using
arithmetic intersection theory. In the case of an arithmetic surface,
generically of genus g, for line bundles of degree g equivalence is shown to
the height on the Jacobian defined by the Theta divisor.


title: Grothendieck topologies and deformation theory II

abstract:
  This paper is a continuation of ``Operads, Grothendieck topologies and
deformation theory'' (alg-geom/9502010). We show how to develop a cohomology
theory that would control deformations of a sheaf of associative algebras over
a scheme by means of introducing an appropriate site and considering sheaves on
it.


title:
Canonical desingularization in characteristic zero by blowing up the
  maximum strata of a local invariant

abstract:
  This article contains an elementary constructive proof of resolution of
singularities in characteristic zero. Our proof applies in particular to
schemes of finite type and to analytic spaces (so we recover the great theorems
of Hironaka). We introduce a discrete local invariant $\inv_X (a)$ whose
maximum locus determines a smooth centre of blowing up, leading to
desingularization. To define $\inv_X$, we need only to work with a category of
local-ringed spaces $X=(|X|,{\cal O}_X)$ satisfying certain natural conditions.
If $a\in |X|$, then $\inv_X(a)$ depends only on $\widehat{\cal O}_{X,a}$. More
generally, $\inv_X(a)$ is defined inductively after any sequence of blowings-up
whose centres have only normal crossings with respect to the exceptional
divisors and lie in the constant loci of $\inv_X(\cdot)$. The paper is
self-contained and includes detailed examples. One of our goals is that the
reader understand the desingularization theorem, rather than simply ``know'' it
is true.


title:
Topological Classifying Spaces of Lie Algebras and the Natural
  Completion of Contractions

abstract:
  The space K^n of all n-dimensional { Lie} algebras has a natural
non-Hausdorff topology k^n, which has characteristic limits, called
transitions, A -> B, between distinct Lie algebras A and B. The entity of these
transitions are the natural transitive completion of the well known
Inonu-Wigner contractions and their partial generalizations by Saletan.
Algebras containing a common ideal of codimension 1 can be characterized by
homothetically normalized Jordan normal forms of one generator of their adjoint
representation. For such algebras, transitions A -> B can be described by limit
transitions between corresponding normal forms. The topology k^n is presented
in detail for n < 5. Regarding the orientation of the algebras as vector spaces
has a non-trivial effect for the corresponding topological space K^n_or: There
exist both, selfdual points and pairs of dual points w.r.t. orientation
reflection.


title: Geometry of plane curves via Tschirnhausen resolution tower

abstract:
  We show the existence of toric resolution tower for an irreducible curve
singularity which is explicitly described by Tschirnhausen polynomials. We
deduce for a smooth affine plane curve from its topology restrictions for its
singularity at infinity. For instance, we discribe the singularities at
infinity (up to equisingular deformation) for curves of genus 0,1 and 2. From
this follows an extension of the Abhyankar-Moh-Suzuki theorem to genus 1 and 2.


title: Zagier's conjecture on $L(E,2)$

abstract:
  In this paper we introduce an elliptic analog of the Bloch-Suslin complex and
prove that it (essentially) computes the weight two parts of the groups
$K_2(E)$ and $K_1(E)$ for an elliptic curve $E$ over an arbitrary field $k$.
Combining this with the results of Bloch and Beilinson we proved Zagier's
conjecture on $L(E,2)$ for modular elliptic curves over $\Bbb Q$.


title: Frobenius morphisms over Z/p^2 and Bott vanishing

abstract:
  Let $X$ be a smooth projective algebraic variety over $Z/p$, which has a flat
lift to a scheme $X'$ over $Z/p^2$. If the absolute Frobenius morphism $F$ on
$X$ lifts to a morphism on $X'$, then an old trick by Mazur shows that
push-down of the de Rham complex under $F$ decomposes. We show that the
quasi-isomorphism in question is split. This is then applied to toric varieties
(where a glueing argument gives lifting of Frobenius to $Z/p^2$) and we derive
natural characteristic $p$ proofs of Bott vanishing and degeneration of the
Danilov spectral sequence. For flag varieties we obtain generalizations of a
result of Paranjape and Srinivas about non-lifting of Frobenius to the Witt
vectors.


title: Actions of groups of birationally extendible automorphisms

abstract:
  We study the actions of a Lie group $G$ by birationally extendible
automorphisms on a domain $D\subset C^n$. For a large class of such domains
defined by polynomial inequalities, all automorphisms are of this type. In the
cases 1) $G$ has finitely many components or 2) the degree of the automorphisms
is bounded, we prove that the action of $G$ is projectively linearizable, i.e.
there exist a linear representation of $G$ on some $ C^{N+1}$ and a holomorphic
$G$-equivariant embedding $i: D\to P^N$, which is a restriction of a rational
mapping. As a corollary we obtain as many rational invariant functions as the
dimension of generic orbits allows. A hard copy is available from
Dmitri.Zaitsev@rz.ruhr-uni-bochum.de


title: On the quantum cohomology of the plane, old and new

abstract:
  We describe a method for counting maps of curves of given genus (and variable
moduli) to $\Bbb P^2$, essentially by splitting the $\Bbb P^2$ in two; then
specialising to the case of genus 0 we show that the method of quantum
cohomology may be viewed as the 'mirror' of the former method where one splits
the $\Bbb P^1$ rather than the $\Bbb P^2$, and we indicate a proof of the
associativity of quantum multiplication based on this idea.


title: Quotients by Groupoids

abstract:
  We show that if a flat group scheme acts properly, with finite stabilizers,
on an algebraic space, then a quotient exists as a separated algebraic space.
More generally we show any flat groupid for which the family of stabilizers is
finite has a uniform geometric, uniform categorical quotient in the category of
algebraic spaces. Our argument is elementary and essentially self contained.


title: Anticanonical Rational Surfaces

abstract:
  A determination of the fixed components, base points and irregularity is made
for arbitrary numerically effective divisors on any smooth projective rational
surface having an effective anticanonical divisor. All of the results are
proven over an algebraically closed field of arbitrary characteristic.
Applications, treated in separate papers, include questions involving: points
in good position, birational models of rational surfaces in projective space,
and resolutions for ideals of fat point subschemes of $P^2$.


title: Free Resolutions of Fat Point Ideals on $P^2$

abstract:
  By defining a fat point subscheme of $P^2$ to be a 0-dimensional subscheme
defined by a sheaf of integrally closed ideals one extends the notion of fat
point subschemes to allow infinitely near points. With this notion of fat
points, this preprint determines the number of generators in each degree in a
minimal homogeneous set of generators for the homogeneous ideal defining any
fat point subscheme supported at points of any plane conic, smooth or not.
Special cases for points on a cubic are also studied. All work is over an
algebraically closed field of arbitrary characteristic.


title:
Generators for Symbolic Powers of Ideals Defining General Points of
  $P^2$

abstract:
  Given distinct points $p_1,\cdots,p_r$ of the projective plane $P^2$ and a
positive integer $m$, the homogeneous ideal defining the fat point subscheme
$Z=m(p_1+\cdots+p_r)$ is the symbolic power $I^{(m)}$ of the homogeneous ideal
$I$ defining the smooth union of the $r$ points $p_1,\ldots,p_r$. If
$p_1,\ldots,p_r$ are sufficiently general, it is known that the maximal rank
conjecture holds for $I$; i.e., for every $d$ the multiplication map
$I_1\otimes I_d\to I_{(d+1)}$ on homogeneous components has maximal rank
(meaning the map is either injective or surjective). One easily sees this fails
for symbolic powers of ideals defining general points; this preprint relates
the failure to the occurrence of (in Nagata's terminology) uniform abnormal
curves, and, for $r<10$, takes complete account of the failure, thereby
completely determining the modules in a minimal free resolution of $I^{(m)}$
when $r<10$. It is also conjectured that maximal rank holds if $r>9$. Assuming
this and a previous conjecture of the author, one can completely determine the
modules in a minimal free resolution of $I^{(m)}$ for any $r>0$ general points
and any $m>0$.
  The author's www site, http://www.math.unl.edu/~bharbour, makes available, in
addition to plainTeX textfile and dvi versions of this preprint, a Macintosh
(stuffed and bin hexed) executable and a C source textfile program which output
the (conjectural for $r>9$) modules in a minimal free resolution of $I^{(m)}$
for any $r>0$ general plane points and any $m>0$. Web visitors can also run a
version of


title:
The Canonical Class of $\overline{M}_{0,n}(P^r,d)$ And Enumerative
  Geometry

abstract:
  D. Abramovich found an error in the singularity analysis in the first posting
of this paper affecting one formula (Thanks Dan).The error has been corrected.
Only the arithmetic genus formula has changed. The geometric genus and all the
canonical class formulas are the same.


title:
Gorenstein algebras, symmetric matrices, self-linked ideals, and
  symbolic powers

abstract:
  Inspired by recent work in the theory of central projections onto
hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those
with a Hilbert--Burch matrix that has a maximal symmetric subblock. We also
prove that every Gorenstein perfect algebra of grade 1 can be presented, as a
module, by a symmetric matrix. Both results are derived from the same
elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor,
a symmetric syzygy matrix. In addition, we establish a correspondence, roughly
speaking, between Gorenstein perfect algebras of grade 1 that are birational
onto their image, on the one hand, and self-linked perfect ideals of grade 2
that have one of the self-linking elements contained in the second symbolic
power, on the other hand. Finally, we provide another characterization of these
ideals in terms of their symbolic Rees algebras, and we prove a criterion for
these algebras to be normal.


title: The Toda hierarchy and the KdV hierarchy

abstract:
  The Toda hierarchy of size $N$ is well known to be analogous to the KdV
hierarchy at $N$ goes to infinity. This paper shows that given $f$ a periodic
function, there is a canonical way of defining the initial data for the Toda
lattice equations so that the evolution of this data under the Toda lattice
hierarchy looks asymptotically like the evolution of $f$ under the KdV
hierarchy. Further, the conserved quantities of $f$ and those of the Toda
hierarchy match.


title: Obstruction bundles, semiregularity, and Seiberg-Witten invariants

abstract:
  We compare the deformation theory and the analytic structure of the
Seiberg-Witten moduli spaces of a K\"ahler surface to the corresponding
components of the Hilbert scheme, and show that they are isomorphic. Next we
show how to compute the invariant in case the moduli space is smooth but not of
the expected dimension, and apply this study to elliptic surfaces. Finally we
discuss ruled surfaces, both products and more general ruled surfaces. For
product ruled surfaces we relate the infinitesimal structure of the moduli
spaces to Brill-Noether theory and compute the invariant in special cases. For
more general ruled surfaces, we relate the geometry of the Hilbert scheme to
properties of stable bundles and give more general computations.


title: Bogomolov conjecture for curves of genus 2 over function fields

abstract:
  In this note, we will show that Bogomolov conjecture holds for a
non-isotrivial curve of genus 2 over a function field.


title: Mirror duality and string-theoretic Hodge numbers

abstract:
  We prove in full generality the mirror duality conjecture for
string-theoretic Hodge numbers of Calabi-Yau complete intersections in
Gorenstein toric Fano varieties. The proof is based on properties of
intersection cohomology


title:
The weight-two Hodge structure of moduli spaces of sheaves on a K3
  surface

abstract:
  We prove that the weight-two Hodge structure of moduli spaces of torsion-free
sheaves on a K3 surface is as described by Mukai (the rank is arbitrary but we
assume the first Chern class is primitive). We prove the moduli space is an
irreducible symplectic variety (by Mukai's work it was known to be symplectic).
By work of Beauville, this implies that its $H^2$ has a canonical integral
non-degenrate quadratic form; Mukai's recepee for $H^2$ includes a description
of Beauville's quadratic form. As an application we compute higher-rank
Donaldson polynomials of $K3$ surfaces.


title: Finiteness Theorem for Sp(4,Z)

abstract:
  We consider Siegel upper half space of rank two ${\cal H}^2$ and different
subgroups $H\subseteq {\bf Sp(4,Z)}$ of finite index. The purpose of this paper
is to prove that the field of rational functions of ${\cal H}^2/H$ has general
type for all but the finite number of $H$.


title: Instantons and affine Lie algebras

abstract:
  Various constructions of the affine Lie algebra action on the homology group
of moduli spaces of instantons on 4-manifolds are discussed. The analogy
between the local-global principle and the role of mass is also explained. The
detailed proofs are given in separated papers \cite{Na-algebra,Na-Hilbert}.


title: On Fujita's freeness conjecture for 3-folds and 4-folds

abstract:
  The result on 3-fold is now optimal by the result of Ein-Lazarsfeld and
Helmke.


title: On the Brill-Noether Problem for Vector Bundles

abstract:
  On an arbitrary compact Riemann surface, necessary and sufficient conditions
are found for the existence of semistable vector bundles with slope between
zero and one and a prescribed number of linearly independent holomorphic
sections. Existence is achieved by minimizing the Yang-Mills-Higgs functional.


title: Some Remarks on Beilinson Adeles

abstract:
  This paper contains two remarks on Beilinson's adeles with values in the De
Rham complex of a scheme. The first is an interpretation, in terms of adeles,
of the decomposition of the De Rham complex on a scheme defined modulo $p^{2}$
(the result of Deligne-Illusie). The second remark is about the possible
relation between adeles and Hodge decomposition. We work out a counter example.


title: Smooth Formal Embeddings and the Residue Complex

abstract:
  Let \pi : X -> S be a finite type morphism of noetherian schemes. A smooth
formal embedding of X (over S) is a bijective closed immersion X -> \frak{X},
where \frak{X} is a noetherian formal scheme, formally smooth over S. An
example of such an embedding is the formal completion \frak{X} = Y_{/X} where X
\subset Y is an algebraic embedding. Smooth formal embeddings can be used to
calculate algebraic De Rham (co)homology. Our main application is an explicit
construction of the Grothendieck residue complex when S is a regular scheme. By
definition the residue complex is the Cousin complex of \pi^{!} \cal{O}_{S}. We
start with Huang's theory of pseudofunctors on modules with 0-dimensional
support, which provides a graded sheaf \cal{K}^{.}_{X/S}. We then use smooth
formal embeddings to obtain the coboundary operator on \cal{K}^{.}_{X / S}. We
exhibit a canonical isomorphism between the complex (\cal{K}^{.}_{X/S}, \delta)
and the residue complex of Grothendieck. When \pi is equidimensional of
dimension n and generically smooth we show that H^{-n} \cal{K}^{.}_{X/S} is
canonically isomorphic to the sheaf of regular differentials of Kunz-Waldi.
Another issue we discuss is Grothendieck Duality on a noetherian formal scheme
\frak{X}. Our results on duality are used in the construction of
\cal{K}^{.}_{X/S}.


title: K3 surfaces, Lorentzian Kac--Moody algebras and Mirror Symmetry

abstract:
  We consider the variant of Mirror Symmetry Conjecture for K3 surfaces which
relates "geometry" of curves of a general member of a family of K3 with
"algebraic functions" on the moduli of the mirror family. Lorentzian Kac--Moody
algebras are involved in this construction. We give several examples when this
conjecture is valid.


title: Nilpotent groups and universal coverings of smooth projective varieties

abstract:
  In this paper we prove that the universal cover of a smooth projective
variety with nilpotent fundamental group is holomorphically convex.


title: Quantum cohomology of projective bundles over $\Pee^n$

abstract:
  Results in the preliminary version have been strengthed. In addition,
Batyrev's conjectural formula for quantum cohomology of projective bundles
associated to direct sum of line bundles over $\Pee^n$ is partially verified.


title: Parabolic Higgs bundles and Teichm\"uller spaces for punctured surfaces

abstract:
  In this paper we study the relation between parabolic Higgs bundles and
irreducible representations of the fundamental group of punctured Riemann
surfaces established by Simpson. We generalize a result of Hitchin, identifying
those parabolic Higgs bundles that correspond to Fuchsian representations. We
also study the Higgs bundles that give representations whose image is
contained, after conjugation, in SL($k,\Bbb R$). We compute the real dimension
of one of the components of this space of representations, which in the absence
of punctures is the generalized Teichm\"uller space introduced by Hitchin, and
which in the case of $k=2$ is the Teichm\"uller space of the Riemann surface.


title: Lectures on Seiberg-Witten Invariants

abstract:
  These are yet another lecture notes on Seiberg-Witten invariants, where no
claim of originality is made, they contain a discussion of some related results
from the recent literature.


title: Second N=1 Superanalog of Complex Structure

abstract:
  We found another N=1 odd superanalog of complex structure (the even one is
widely used in the theory of super Riemann surfaces). New N=1
superconformal-like transformations are similar to anti-holomorphic ones of
nonsupersymmetric complex function theory. They are dual to the ordinary
superconformal transformations subject to the Berezinian addition formula
presented, noninvertible, highly degenerated and twist parity of the tangent
space in the standard basis. They also lead to the ''mixed cocycle condition''
which can be used in building noninvertible objects analogous to super Riemann
surfaces. A new parametrization for the superconformal group is presented which
allows us to extend it to a semigroup and to unify the description of old and
new transformations.


title: Manin's conjecture for toric varieties

abstract:
  We prove an asymptotic formula conjectured by Manin for the number of
$K$-rational points of bounded height with respect to the anticanonical line
bundle for arbitrary smooth projective toric varieties over a number field $K$.


title: Rational curves of degree at most 9 on a general quintic threefold

abstract:
  We prove the following form of the Clemens conjecture in low degree. Let
$d\le9$, and let $F$ be a general quintic threefold in $\IP^4$. Then (1)~the
Hilbert scheme of rational, smooth and irreducible curves of degree $d$ on $F$
is finite, nonempty, and reduced; moreover, each curve is embedded in $F$ with
normal bundle $\O(-1)\oplus\O(-1)$, and in $\IP^4$ with maximal rank. (2)~On
$F$, there are no rational, singular, reduced and irreducible curves of degree
$d$, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher).
(3)~On $F$, there are no connected, reduced and reducible curves of degree $d$
with rational components.


title: Infinite Grassmannians and Moduli Spaces of G-bundles

abstract:
  This lecture note is devoted to prove that the space of vacua is isomorphic
with the space of generalized theta functions (joint work with M.S. Narasimhan,
and A. Ramanathan)