We study period integrals of CY hypersurfaces in a partial flag variety. We construct a regular holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can be described explicitly. The results are also generalized to CY complete intersections. The construction of these new systems of differential equations has lead us to the notion of a tautological system.

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...