We describe explicitly the moduli spaces ${ℳ}_g^{\mathrm{pst}}(S,E)$ of polystable holomorphic structures $ℰ$ with $detℰ\cong 𝒦$ on a rank two vector bundle $E$ with $c_1\left(E\right)=c_1\left(K\right)$ and $c_2\left(E\right)=0$ for all minimal class VII surfaces $S$ with $b_2\left(S\right)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, ${ℳ}_g^{\mathrm{pst}}(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely...

Let $F$ be a field and $Gr\left(i,F^n\right)$ be the Grassmannian of $i$-dimensional linear subspaces of $F^n$. A map $f:Gr\left(i,F^n\right)\to Gr\left(j,F^n\right)$ is called nesting if $l\subset f\left(l\right)$ for every $l\in Gr\left(i,F^n\right)$. Glover, Homer and Stong showed that there are no continuous nesting maps $Gr\left(i,{\mathbb{C}}^n\right)\to Gr\left(j,{\mathbb{C}}^n\right)$ except for a few obvious ones. We prove a similar result for algebraic nesting maps $Gr\left(i,F^n\right)\to Gr\left(j,F^n\right)$, where $F$ is an algebraically closed field of arbitrary characteristic. For $i=1$ this yields a description of the algebraic sub-bundles of the tangent bundle to the projective space $P_F^n$.

To any finite covering $f:Y\to X$ of degree $d$ between smooth complex projective manifolds, one associates a vector bundle $E_f$ of rank $d-1$ on $X$ whose total space contains $Y$. It is known that $E_f$ is ample when $X$ is a projective space ([Lazarsfeld 1980]), a Grassmannian ([Manivel 1997]), or a Lagrangian Grassmannian ([Kim Maniel 1999]). We show an analogous result when $X$ is a simple abelian variety and $f$ does not factor through any nontrivial isogeny $X^{\text{'}}\to X$. This result is obtained by showing that $E_f$ is $M$-regular in the...

In this short survey, we would like to overview the recent development of the study on Deligne-Malgrange lattices and resolution of turning points for algebraic meromorphic flat bundles. We also explain their relation with wild harmonic bundles. The author hopes that it would be helpful for access to his work on wild harmonic bundles.