Search results for "14D07"
showing 4 items of 4 documents
Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups
2019
Let $k $ be the algebraic closure of a finite field of odd characteristic $p$ and $X$ a smooth projective scheme over the Witt ring $W(k)$ which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow and prove that the category of periodic Higgs-de Rham flows over $X/W(k)$ is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the \'{e}tale fundamental group $\pi_1(X_K)$ of the generic fiber of $X$, after Fontaine-Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber $X_k$ of $X$ of rank $\leq p$ initiates a semistable Higgs-de Rham …
Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds
2002
We prove that the moduli spaces A_3(D) of polarized abelian threefolds with polarizations of types D=(1,1,2), (1,2,2), (1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space H_{3,A}(Y) which parameterizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^{-1}.
The monodromy groups of Dolgachev's CY moduli spaces are Zariski dense
2014
Let $\mathcal{M}_{n,2n+2}$ be the coarse moduli space of CY manifolds arising from a crepant resolution of double covers of $\mathbb{P}^n$ branched along $2n+2$ hyperplanes in general position. We show that the monodromy group of a good family for $\mathcal{M}_{n,2n+2}$ is Zariski dense in the corresponding symplectic or orthogonal group if $n\geq 3$. In particular, the period map does not give a uniformization of any partial compactification of the coarse moduli space as a Shimura variety whenever $n\geq 3$. This disproves a conjecture of Dolgachev. As a consequence, the fundamental group of the coarse moduli space of $m$ ordered points in $\mathbb{P}^n$ is shown to be large once it is not…
Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces
2003
Let M(d,n) be the moduli stack of hypersurfaces of degree d > n in the complex projective n-space, and let M(d,n;1) be the sub-stack, parameterizing hypersurfaces obtained as a d fold cyclic covering of the projective n-1 space, ramified over a hypersurface of degree d. Iterating this construction, one obtains M(d,n;r). We show that M(d,n;1) is rigid in M(d,n), although the Griffiths-Yukawa coupling degenerates for d<2n. On the other hand, for all d>n the sub-stack M(d,n;2) deforms. We calculate the exact length of the Griffiths-Yukawa coupling over M(d,n;r), and we construct a 4-dimensional family of quintic hypersurfaces, and a dense set of points in the base, where the fibres ha…