The Oort conjecture on Shimura curves in the Torelli locus of curves

2014

Oort has conjectured that there do not exist Shimura curves contained generically in the Torelli locus of genus-$g$ curves when $g$ is large enough. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized $g$-dimensional abelian varieties isogenous to $g$-fold self-products of elliptic curves for $g>11$. We also prove that there do not exist Shimura curves contained generically in the Torelli locus of hyperelliptic curves of genus $g>7$. As a consequence, we obtain a finiteness result regarding smooth genus-$g$ curves with completely decomposable Jacobians, which is related to a question of Ekedahl and Serre.

On Severi Type Inequalities for Irregular Surfaces

2017

Let X be a minimal surface of general type and maximal Albanese dimension with irregularity q ≥ 2. We show that K2 X ≥ 4χ(OX) + 4(q − 2) if K2 X < 9 2 χ(OX), and also obtain the characterization of the equality. As a consequence, we prove a conjecture of Manetti on the geography of irregular surfaces if K2 X ≥ 36(q−2) or χ(OX) ≥ 8(q−2), and we also prove a conjecture that the surfaces of general type and maximal Albanese dimension with K2 X = 4χ(OX) are exactly the resolution of double covers of abelian surfaces branched over ample divisors with at worst simple singularities.

The Oort conjecture on Shimura curves in the Torelli locus of hyperelliptic curves

2017

Abstract Oort has conjectured that there do not exist Shimura varieties of dimension >0 contained generically in the Torelli locus of genus-g curves when g is sufficiently large. In this paper we prove the analogue of this conjecture for Shimura curves with respect to the hyperelliptic Torelli locus of genus g > 7 .

On the slope of hyperelliptic fibrations with positive relative irregularity

2016

Let $f:\, S \to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $\lambda_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $\lambda_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $\lambda_f<4$.

On the Oort conjecture for Shimura varieties of unitary and orthogonal types

2014

In this paper we study the Oort conjecture on Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety $\mathcal{A}_g$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibred surfaces, we show that a Shimura curve $C$ is not contained generically in the Torelli locus if its canonical Higgs bundles contains a unitary Higgs subbundle of rank at least $(4g+2)/5$. From this we prove that a Shimura subvariety of $\mathbf{SU}(n,1)$-type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus $g$, the dimension $n+1$, the degree $2d$ of CM field of the Hermitian space,…

On Higgs bundles over Shimura varieties of ball quotient type

2016

We prove the generic exclusion of certain Shimura varieties of unitary and orthogonal types from the Torelli locus. The proof relies on a slope inequality on surface fibration due to G. Xiao, and the main result implies that certain Shimura varieties only meet the Torelli locus in dimension zero.

Corrigendum to: Slopes of Non-hyperelliptic Fibrations in Positive Characteristic

2019

On the minimal number of singular fibers with non-compact Jacobians for families of curves over P1

2016

Abstract Let f : X → P 1 be a non-isotrivial family of semi-stable curves of genus g ≥ 1 defined over an algebraically closed field k. Denote by s nc the number of the singular fibers whose Jacobians are non-compact. We prove that s nc ≥ 5 if k = C and g ≥ 5 ; we also prove that s nc ≥ 4 if char ( k ) > 0 and the relative Jacobian of f is non-smooth.

On the gonality and the slope of a fibered surface

2018

Abstract Let f : X → B be a locally non-trivial relatively minimal fibration of curves of genus g ≥ 2 . We obtain a lower bound of the slope λ ( f ) increasing with the gonality of the general fiber of f. In particular, we show that λ ( f ) ≥ 4 provided that f is non-hyperelliptic and g ≥ 16 .

Slopes of Non-hyperelliptic Fibrations in Positive Characteristic

2016

On CM points away from the Torelli locus

2021

5th International Symposium on Focused Ultrasound

2016

Introduction Breast fibroadenomata (FAD) are benign lesions which occur in about 10 % of all women. Diagnosis is made by triple assessment (physical examination, imaging and/or histopathology/cytology). For a definitive diagnosis of FAD, the treatment is conservative unless the patient is symptomatic. For symptomatic patients, the lumps can be surgically excised or removed interventionally by vacuum-assisted mammotomy (VAM). Ablative techniques like high-intensity focused ultrasound (HIFU), cryo-ablation and laser ablation have also been used for the treatment of FAD, providing a minimally invasive treatment without scarring or poor cosmesis. This review summarises current trials using mini…