Related papers: A note on Shimura subvarieties in the hyperellipti…

We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to…

We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in P^4, there are only finitely many smooth rational curves of degree 10, and each curve is embedded in F with normal bundle…

In 2010, Brasselet, Sch\"urmann and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky-MacPherson $L$-class $L_*(X)$ and the Hirzebruch homology class $T_{1*}(X)$ for a compact complex…

In this paper we formulate some conjectures about algebraic flows on Shimura varieties. In the first part of the paper we prove the `logarithmic Ax-Lindemann theorem'. We then prove a result concerning the topological closure of the images…

We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.

We prove an analogue of the classical Ax-Lindemann theorem in the context of compact Shimura varieties. Our work is motivated by J. Pila's strategy for proving the Andr\'e-Oort conjecture unconditionally

We propose a conjecture on the $p$-adic nowhere density of the Hecke orbit of subvarieties of Hodge type Shimura varieties. By investigating the monodromy of $p$-adic Galois representations associated with points on such Shimura varieties,…

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our…

We prove that certain Severi varieties of nodal curves of positive genus on general blow-ups of the twofold symmetric product of a general elliptic curve are non-empty and smooth of the expected dimension. This result, besides its intrinsic…

We generalize the classical approach of describing the infinitesimal Torelli map in terms of multiplication in a Jacobi ring to the case of quasi-smooth complete intersections in weighted projective space. As an application, we prove the…

Merkurjev's theorem--the statement that the 2-torsion of the Brauer group is represented by Clifford algebras of quadratic forms--is in general false when the base is no longer a field. The work of Parimala, Scharlau, and Sridharan proves…

We generalize the torsion vanishing results of Caraiani-Scholze and Koshikawa. Our results apply to the cohomology of general Shimura varieties $(\mathbf{G},X)$ of PEL type $A$ or $C$, localized at a suitable maximal ideal $\mathfrak{m}$ in…

We give restrictions on the existence of families of curves on smooth projective surfaces $S$ of nonnegative Kodaira dimension all having constant geometric genus $g \geq 2$ and hyperelliptic normalizations. In particular, we prove a…

We study the geometry of unitary Shimura varieties without assuming the existence of an ordinary locus. We prove, by a simple argument, the existence of canonical subgroups on a strict neighborhood of the $\mu$-ordinary locus (with an…

We determine non hyper elliptic curves of genus $g(C)\geq 9$, such that for some very ample line bundle on them and for some integers d and r with some prescribed assumptions, the dimension of secant loci, attains one less than its maximum…

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism…

Let $X$ be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor $K$ with every very general curve is positive…

We consider the locus of $r$-tuples of homogeneous forms of some fixed degree whose common vanishing locus in $\mathbb{P}^r$ is positive dimensional. We show that any component of maximal dimension of that locus either consists of…

I give a simplified proof of one of the main results in the recent preprint arXiv:1311.5858 by X. Luo and K. Zuo. Furthermore, I discuss the relation with the Coleman-Oort conjecture on special curves in the Torelli locus.

We derive a formula for reconstructing a generic complex canonical curve $C$ of genus 6 and 7 in terms of the theta hyperplanes of $C$. Hence, we get a generic inverse to the Torelli map, as well as a complete description of the Schottky…