%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The abstract must be written in LaTeX2e %% %% Make sure that your abstract can be compiled before %% %% you send it to the organizer. %% %% Do not use personal macros in your TeX file, and, %% %% for the sake of making your abstract understandable %% %% to a large audience and to avoid difficulties with %% %% the handling of your file, please limit the use of %% %% formulas as much as possible. %% %% We suggest that you do not include references in %% %% your abstract. In case that you must have references, %% %% then do not use \cite commands. %% %% %% %% For labels, try to write : \label{yourname01}, %% %% \label{yourname02}... %% %% Your abstract should be no more than one page long. %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{article} \usepackage{amsthm,amsmath,amssymb,latexsym} \pagestyle{empty} \textwidth=12.8cm \textheight=21.7cm \hoffset=-0.3in \voffset=-0.6in \parskip=6pt \lineskip=18pt \begin{document} \baselineskip=15pt \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}} %%% in case of several authors: %\smallskip %\centerline{Author 2} %\centerline{\it Institution Name, Country} %\centerline{e-mail {\tt Author 2}} %%% \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}} %%% in case of several authors: %\smallskip %\centerline{Author 2} %\centerline{\it Institution Name, Country} %\centerline{e-mail {\tt Author 2}} %%% \bigskip 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}} %%% in case of several authors: %\smallskip %\centerline{Author 2} %\centerline{\it Institution Name, Country} %\centerline{e-mail {\tt Author 2}} %%% \bigskip 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}} %%% in case of several authors: %\smallskip %\centerline{Author 2} %\centerline{\it Institution Name, Country} %\centerline{e-mail {\tt Author 2}} %%% \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% \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}} %%% in case of several %authors: %\smallskip %\centerline{Author 2} %\centerline{\it Institution Name,Country} %\centerline{e-mail {\tt Author2}} %%% \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% \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}