Let $X$ be a smooth projective surface, $K$ the canonical divisor, $H$ a very ample divisor and $M_H(c_1,c_2)$ the moduli space of rank-two vector bundles, $H$-stable with Chern classes $c_1$ and $c_2$. We prove that, if there exists $c_1^{\text{'}}$ such that $c_1$ is numerically equivalent to $2c_1^{\text{'}}$ and if $c_2-\frac{1}{4}{c}_1^2$ is even, greater or equal to $H^2+HK+4$, then there is no Poincaré bundle on $M_H(c_1,c_2)\times X$. Conversely, if there exists $c_1^{\text{'}}$ such that the number $c_1^{\text{'}}·c_1$ is odd or if $\frac{1}{2}{c}_1^2-\frac{1}{2}{c}_1·K-c_2$ is odd, then there exists a Poincaré bundle on $M_H(c_1,c_2)\times X$.

Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi :P\to \Sigma$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N\left(\xi \right)$ of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from $N\left(\xi \right)$ onto a certain representation space ${\mathrm{Rep}}_{\xi }(\Gamma ,G)$, in fact a diffeomorphism, with reference to suitable smooth structures $C^{\infty }\left(N\right(\xi \left)\right)$ and $C^{\infty }\left({\mathrm{Rep}}_{\xi }(\Gamma ,G)\right)$, where $\Gamma$ denotes the universal central extension of...

Let $\pi :Z\to X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group $G$. For any dominant weight $\lambda$ consider the curve $Y=Z/\mathrm{Stab}\left(\lambda \right)$. The Kanev correspondence defines an abelian subvariety $P_{\lambda }$ of the Jacobian of $Y$. We compute the type of the polarization of the restriction of the canonical principal polarization of $\mathrm{Jac}\left(Y\right)$ to $P_{\lambda }$ in some cases. In particular, in the case of the group $E_8$ we obtain families of Prym-Tyurin varieties. The main idea is the use of...

A “relative” $K$-theory group for holomorphic or algebraic vector bundles on a compact or quasiprojective complex manifold is constructed, and Chern-Simons type characteristic classes are defined on this group in the spirit of Nadel. In the projective case, their coincidence with the Abel-Jacobi image of the Chern classes of the bundles is proved. Some applications to families of holomorphic bundles are given and two Riemann-Roch type theorems are proved for these classes.