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...

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base $Y^{\circ }$, and suppose the family is non-isotrivial. If $Y$ is a smooth compactification of $Y^{\circ }$, such that $D:=Y\setminus Y^{\circ }$ is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along $D$. Viehweg and Zuo have shown that for some $m\>0$, the $m^{\mathrm{th}}$ symmetric power of this sheaf admits many sections. More precisely, the $m^{\mathrm{th}}$ symmetric power contains an invertible...

The pre-Tango structure is an ample invertible sheaf of locally exact differentials on a variety of positive characteristic. It is well known that pre-Tango structures on curves often induce pathological uniruled surfaces. We show that almost all pre-Tango structures on varieties induce higher-dimensional pathological uniruled varieties, and that each of these uniruled varieties also has a pre-Tango structure. For this purpose, we first consider the p-closed rational vector field induced...

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.

Given a finite $p$-group $G$ acting on a smooth projective curve $X$ over an algebraically closed field $k$ of characteristic $p$, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of $G$ acting on the space $V$ of global holomorphic quadratic differentials on $X$. We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when $G$ is cyclic or when the action of $G$ on $X$ is weakly...

In a previous paper, the author introduced an integral structure in quantum cohomology defined by the $K$-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation of toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism of...