We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound. As an application we show that the
Coleman-Oort conjecture holds for Shimura curves associated to partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.
In this short notes, we prove a stronger version of Theorem 0.6 in our previous paper arXiv:1709.01485: Given a smooth log scheme $(mathcal{X} supset mathcal{D})_{W(mathbb{F}_q)}$, each stable twisted $f$-periodic logarithmic Higgs bundle $(E,theta)$
over the closed fiber $(X supset D)_{mathbb{F}_q}$ will correspond to a $mathrm{PGL}_r(mathbb{F}_{p^f})$-crystalline representation of $pi_1((mathcal{X} setminus mathcal{D})_{W(mathbb{F}_q)[frac{1}{p}]})$ such that its restriction to the geometric fundamental group is absolutely irreducible.
This paper contains three new results. {bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of etale fundamental groups introduced
in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {bf 2.}We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. {bf 3.} We investigate the dynamic of self-maps induced by the Higgs-de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on $mathbb{P}^1$ with logarithmic structure on marked points $D:={x_1,,...,x_n}$ for $ngeq 4$ and construct infinitely many geometrically absolutely irreducible $mathrm{PGL_2}(mathbb Z_p^{mathrm{ur}})$-crystalline representations of $pi_1^text{et}(mathbb{P}^1_{{mathbb{Q}}_p^text{ur}}setminus D)$. We find an explicit formula of the self-map for the case ${0,,1,,infty,,lambda}$ and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve $mathcal{C}_lambda$ defined by $ y^2=x(x-1)(x-lambda)$ with the order coprime to $p$.
In this paper we study various aspects of the Ekedahl-Serre problem. We formulate questions of Ekedahl-Serre type and Coleman-Oort type for general weakly special subvarieties in the Siegel moduli space, propose a conjecture relating these two questi
ons, and provide examples supporting these questions. The main new result is an upper bound of genera for curves over number fields whose Jacobians are isogeneous to products of elliptic curves satisfying the Sato-Tate equidistribution, and we also refine previous results showing that certain weakly special subvarieties only meet the open Torelli locus in at most finitely many points.
In this paper we study the Coleman-Oort conjecture for superelliptic curves, i.e., curves defined by affine equations $y^n=F(x)$ with $F$ a separable polynomial. We prove that up to isomorphism there are at most finitely many superelliptic curves of
fixed genus $ggeq 8$ with CM Jacobians. The proof relies on the geometric structures of Shimura subvarieties in Siegel modular varieties and the stability properties of Higgs bundles associated to fibred surfaces.
Let $f:,X to mathbb{P}^1$ be a non-isotrivial semi-stable family of varieties of dimension $m$ over $mathbb{P}^1$ with $s$ singular fibers. Assume that the smooth fibers $F$ are minimal, i.e., their canonical line bundles are semiample. Then $kappa(X
)leq kappa(F)+1$. If $kappa(X)=kappa(F)+1$, then $s>frac{4}m+2$. If $kappa(X)geq 0$, then $sgeqfrac{4}m+2$. In particular, if $m=1$, $s=6$ and $kappa(X)=0$, then the family $f$ is Teichmuller.
Let $X$ be a minimal surface of general type and maximal Albanese dimension with irregularity $qgeq 2$. We show that $K_X^2geq 4chi(mathcal O_X)+4(q-2)$ if $K_X^2<frac92chi(mathcal O_X)$, and also obtain the characterization of the equality. As a con
sequence, we prove a conjecture of Manetti on the geography of irregular surfaces if $K_X^2geq 36(q-2)$ or $chi(mathcal O_X)geq 8(q-2)$, and we also prove a conjecture that surfaces of general type and maximal Albanese dimension with $K_X^2=4chi(mathcal O_X)$ are exactly the resolution of double covers of abelian surfaces branched over ample divisors with at worst simple singularities.
Let $f:,S to B$ be a locally non-trivial relatively minimal fibration of genus $ggeq 2$ with relative irregularity $q_f$. It was conjectured by Barja and Stoppino that the slope $lambda_fgeq frac{4(g-1)}{g-q_f}$. We prove the conjecture when $q_f$ is
small with respect to $g$; we also construct counterexamples when $g$ is odd and $q_f=(g+1)/2$.
In this paper we extend the construction of the canonical polarized variation of Hodge structures over tube domain considered by B. Gross in cite{G} to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of Hodge
structures, which we call characteristic subvarieties. We prove that the characteristic subvariety of the canonical polarized variations of Hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by N. Mok in cite{M}. We verified the generating property of B. Gross for all irreducible bounded symmetric domains, which was predicted in cite{G}.
In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $ggeq 1$ over characteristic $p$ with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic cur
ves of $p-$rank zero in a semi-stable family over characteristic $p$ with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. An extension of the above results to smooth families of Abelian varieties over $k$ with $W_2$-lifting assumption is also included.
