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 \centerline{\large \bf Schedule: Algebraic Vector Bundles}

 %\centerline{\large \bf Title to be Continued Here if Too Long}

 \bigskip

 \centerline{Thursday, June 13, 2002}
 \centerline{Organizers: V. Ancona, N.M. Kumar, G. Ottaviani, C. Peterson, A.P. Rao}
 %\centerline{e-mail {\tt trm@mathematik.uni-kl.de}}

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 %\centerline{\it Institution Name, Country}
 %\centerline{e-mail {\tt Author 2}}
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 \bigskip
9:00 - 9:30 Jean Vall\`es \ \ {\it Jumping lines of Logarithmic
bundles on $\bf P^2$}

9:40 - 10:10 Vikram Mehta \ \ {\it Harder-Narasimhan Filtration of
Principal Bundles}

{\bf Break}

10:45 - 11:15 Edoardo Ballico \ \ {\it Holomorphic vector bundles}

11:20 - 11:50 G\"unther Trautmann \ \ {\it The Moduli Problem for
Instanton Bundles}

11:55 - 12:25 Luca Chiantini \ \ {\it Vector bundles and curves on
hypersurfaces of $\bf P^4$}

12:30 - 13:00 Joseph Le Potier \ \ {\it Strange duality for
surfaces}

{\bf Break}

15:00 - 15:30 Hideyasu Sumihiro \ \ {\it A splitting theorem of
rank two vector bundles}

15:40 - 16:10 Wolfram Decker \ \ {TBA}

{\bf Break}

16:45 - 17:15 Laurent Gruson \ \ {\it Deformation of instantons}

17:20 - 17:50 TBA

\pagebreak

 \baselineskip=15pt

 \centerline{\large \bf Jumping lines of Logarithmic bundles on $\bf P^2$}

 %\centerline{\large \bf Title to be Continued Here if Too Long}

 \bigskip

 \centerline{Jean Vall\`es}
 \centerline{\it Universit\'e de Versailles, France}
 \centerline{e-mail {\tt valles@math.uvsq.fr}}

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In the paper {\it Arrangement of hyperplanes and vector bundles on
$\bf P^n$} the authors I.Dolgachev and M.Kapranov prove that the
scheme of jumping lines of logarithmic
 bundles is uniquely defined by their finite set
of superjumping lines. They show, for an appropriate first Chern
class, that this scheme is  a (so-called) mono\"{\i}dal complex
(Thm 5.2). In this talk I would like to show that on $\bf P^2$ the
study of this mono\"{\i}dal complex is related to the study of
osculating hyperplane sections of  $\bf P^2$ blown up along a
finite subscheme.

\pagebreak

 \baselineskip=15pt

 \centerline{\large \bf On the Harder-Narasimhan Filtration of Principal Bundles}

 %\centerline{\large \bf Title to be Continued Here if Too Long}

 \bigskip

 \centerline{Vikram Mehta}
 \centerline{\it Tata Institute for Fundamental Research, India}
 \centerline{e-mail {\tt vikram@math.tifr.res.in}}

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The stability and semistability of vector bundles has been studied
for a long time. If a vector bundle is not semistable, then there
is a canonical filtration, the Harder- Narasimhan filtration,
which can be defined and which has certain rationality properties.
In this talk we define and prove the rationality of the
Harder-Narasimhan filtration for nonsemistable principal
G-bundles, where G is any semisimple group. This is joint work
with S.Subramanian

\pagebreak \baselineskip=15pt \centerline{\large \bf Holomorphic
vector bundles}
\bigskip
\centerline{E. Ballico} \centerline{\it Dept. of Mathematics,
University of Trento, 38050 Povo (TN), Italy} \centerline{e-mail
{\tt ballico@science.unitn.it}}
\bigskip
\par
Here I will present my recent work on holomorphic vector bundles
on non-compact complex spaces (mainly on open subsets of Stein
spaces and on quasi-affine algebraic varieties) and on
infinite-dimensional complex manifolds.
\bigskip

\pagebreak
 \baselineskip=15pt

 \centerline{\large \bf The Moduli Problem for Instanton Bundles}

 %\centerline{\large \bf Title to be Continued Here if Too Long}

 \bigskip

 \centerline{G\"unther Trautmann}
 \centerline{\it University of Kaiserslautern, Germany}
 \centerline{e-mail {\tt trm@mathematik.uni-kl.de}}

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 \bigskip
 Instanton bundles over the complex projective 3-space had been introduced in
 connection with Yang-Mills fields on the 4-sphere in the late 1970's. An
 n-instanton is defined to be a stable algebraic vector bundle of rank 2 over
 the projective 3-space with Chern classes $c_1=0$ and $c_2=n > 0$. It is an
 open problem since the time when such bundles appeared, whether the moduli
 space $MI(n)$ of n-instantons is nonsingular or irreducible. Presently the
 answer is yes for $n\leq 5$. Together with a survey on instanton bundles a
 proof for this will be indicated stressing the recent efforts for the
 case $n=5$.

\pagebreak

\baselineskip=15pt \centerline{\large \bf Vector bundles and
curves on hypersurfaces of $\bf P^4$ }
\bigskip

\centerline{Luca Chiantini} \centerline{\it Universit\'a di Siena,
Italy} \centerline{e-mail {\tt chiantini@unisi.it}}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
%   Text of the abstract
Let $X$ be a smooth hypersurface of degree $d$ in $\bf P^4$. The
study of vector bundles on $X$ is relevant for our knowledge of
subvarieties of $X$ and their families. We focus mainly on rank
$2$ vector bundles $E$, which then correspond to subcanonical
curves on $X$. When $E$ is stable, indeed, its sections are almost
in one-to-one correspondence with curves of this type and ${\bf
P}H^0 (E)$ can be viewed as a surrogate of a linear system of
curves in codimension $2$.\par Horrock's celebrated splitting
criterion says that bundles on $X=\bf P^3$ without intermediate
cohomology, called arithmetically Cohen-Macaulay (ACM) bundles,
split in a sum of line bundles. The criterion is known to fail as
soon as $d=$deg$(X)\geq 2$, but the exceptions must have Chern
classes limited in a narrow numerical range. We review the bounds
and the consequent classification of non-splitting ACM bundles on
hypersurfaces of degree $d\leq 4$. For $d=5$, the Calabi-Yau case,
a complete classification in yet unavailable but, as a particular
case of a conjecture of Tyurin, we prove that all stable ACM
bundles are (infinitesimally) rigid in their Moduli space.
Extensions of such results to general hypersurface of higher
degree are discussed.\par Finally we introduce a similar theory
for the wider class of reflexive sheaves on $X$ (which are
associated to general curves) and discuss the study of these
objects in connection with the theory of curves on some threefolds
of general type.


\bigskip
\pagebreak

\baselineskip=15pt

\centerline{\large \bf  Strange duality for   surfaces}



\bigskip

\centerline{Joseph Le Potier} \centerline{\it Institut de
Math\'ematiques de Jussieu, France } \centerline{e-mail: {\tt
jlp@math.jussieu.fr}}


\bigskip
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\par
%
Let $X$ be a complex algebraic smooth projective surface; assume
for simplicity that $X$ is simply connected. Denote by $K(X)$
the topological  Grothendieck algebra of $X$; on $K(X)$ we have a
natural quadratic form $c\mapsto \chi(c^2)$. Let $M_{c}$ be the
moduli space of semi-stable sheaves on $X$ with fixed Grothendieck
class in $K(X)$,  respect to a generic  polarization of $X.$ If
$c^*\in K(X)$ is  orthogonal to $c$ we can associate to $c^*$ a
line bundle ${\cal D}_{c^*}$ on $M_{c}$ called determinant line
bundle which generalizes the usual Donaldson determinant line
bundle.  With some other conditions  we can define a natural
bilinear form on $H^0(M_{c},{\cal D}_{c^*}) \times
H^{0}(M_{c^*},{\cal D}_{c}).$  We explain some examples where this
bilinear form gives a duality.


\bigskip

\pagebreak
\baselineskip=15pt

\centerline{\large \bf A splitting theorem of rank two vector
bundles on projective spaces}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Hideyasu Sumihiro} \centerline{\it Hiroshima
University, Japan} \centerline{e-mail {\tt
sumihiro@math.sci.hiroshima-u.ac.jp}}

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\par
  As for splitting problems for rank two vector bundles on complex projective
$n-$ space ${\bf P}^{n}$,R.Hartshorne posed the following famous
conjectures which are equivalent to each other.\par
  $S_{n}$:{\it Every rank two vector bundle on ${\bf P}^{n}\ ( n \ge 7 )$
splits into line bundles.}\par $C_{n}$:{\it Every smooth closed
subvariety of codimension 2 in ${\bf P}^{n}\ ( n \ge 7 )$ is a
complete intersection.}\par Later, H.Grauert and M.Schneider tried
to solve the following important problem. However there was a gap
in their proof unfortunately.\par $GS$:{\it Every unstable rank
two vector bundle on ${\bf P}^{4}$ is a direct sum of line
bundles.}\par Though many mathematicians have attempted to solve
the conjectures $S_{n}$, $C_{n}$ and $GS$, one does not have
obtained any complete answers yet.\par We shall introduce
determinantal varieties $X$ associated to any very ample rank two
vector bundle $E$ on projective $n-$space ${\bf P}^{n}$ defined
over an algebraically closed field of arbitrary characteristic
which are smooth closed subvarieties of dimension $m$ if $n = 2m$
in ${\bf P}^{n}$ (resp. $m+1$ if $n = 2m+1$) and investigate the
following algebro-geometric properties of those determinantal
varieties $X$ : 1) Topology of $X$, 2) Divisors on $X$, 3)
Comparison Theorems of Cohomologies, 4) Normal and Tangent bundles
of $X$, 5) Hilbert Schemes of ${\bf P}^{n}$.\par As a by-product,
we can establish some splitting theorems for rank two vector
bundles on  ${\bf P}^{n}\ ( n \ge 4 )$. The following splitting
theorem is obtained by studying some geometric structures of the
Hilbert scheme at determinantal varieties.\par {\bf {Theorem} {\it
Let $E$ be a rank two vector bundle on ${\bf P}^{n}\ ( n \ge 4 )$
defined over an algebraically closed field of arbitrary
characterisic, $P$ a 4- or 5- dimensional linear subspace of ${\bf
P}^{n}$ and let ${\overline {E} = E \vert P}$ be the restriction
of $E$ to $P$, Then $E$ is a direct sum of line bundles if and
only if $H^{1}(P, End({\overline {E})) = 0.}$

\bigskip
\pagebreak \baselineskip=15pt

\centerline{\large \bf TBA}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Wolfram Decker} \centerline{\it Universit\"at des
Saarlandes, Germany} \centerline{e-mail {\tt
decker@math.uni-sb.de}}

\pagebreak \baselineskip=15pt

\centerline{\large \bf Deformation of Instantons}

%\centerline{\large \bf Title to be Continued Here if Too Long}

\bigskip

\centerline{Laurent Gruson} \centerline{\it Universit\'e de
Versailles, France}
 \centerline{e-mail {\tt gruson@math.uvsq.fr}}
 \bigskip
\rm I will discuss deformation of instantons into torsion free
modules with dimension one singular locus. This locus acquires a
theta-characteristic with additional restrictions.
\end{document}
