Related papers: Varieties with free tangent sheaves

We give a complete classification of smooth quotients of abelian varieties by finite groups that fix the origin. In the particular case where the action of the group $G$ on the tangent space at the origin of the abelian variety $A$ is…

The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves…

We prove the Lipman-Zariski conjecture for complex surface singularities with $p_g - g - b \le 2$. Here $p_g$ is the geometric genus, $g$ is the sum of the genera of the exceptional curves and $b$ is the first Betti number of the dual…

Let $(A,\Theta)$ be a principally polarised abelian variety, and let Y be a subvariety. Pareschi and Popa conjectured that Y has minimal cohomology class if and only if the structure sheaf of Y satisfies a property that they call…

We study slope-stable vector bundles and Bridgeland stability conditions on varieties which are a quotient of a smooth projective variety by a finite abelian group $G$ acting freely. We show there is an analytic isomorphism between…

Let f:X->Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map of relative dimension no more than three, then X has a good minimal model.

We formulate the notion of \emph{typical boundedness} of torsion on a family of abelian varieties defined over number fields. This means that the torsion subgroups of elements in the family can be made uniformly bounded by removing from the…

If A and B are abelian varieties over a number field K such that there are non-trivial geometric homomorphisms of abelian varieties between reductions of A and B at most primes of K, then there exists a non-trivial (geometric) homomorphism…

G. Pareschi and M. Popa give criterions for global generations and surjectivity of multiplication maps of global sections of coherent sheaves on abelian varieties in the theory of M-regularity. In this paper, we generalize some of their…

The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of algebraically trivial…

A celebrated theorem of Bogomolov asserts that the $\ell$-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic $p$: a…

Let $A_t$ be a family of abelian varieties over a number field $k$ parametrized by a rational coordinate $t$, and suppose the generic fiber of $A_t$ is geometrically simple. For example, we may take $A_t$ to be the Jacobian of the…

We classify almost homogeneous normal varieties of Albanese codimension $1$, defined over an arbitrary field. We prove that such a variety has a unique normal equivariant completion. Over a perfect field, the group scheme of automorphisms…

We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial P(T) modulo l. As an application we find the proportion of isogeny classes of abelian varieties with a…

Two abelian varieties $A$ and $B$ over a number field $K$ are said to be strongly locally quadratic twists if they are quadratic twists at every completion of $K$. While it was known that this does not imply that $A$ and $B$ are quadratic…

We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for…

We show that, for any prime $p$, there exist absolutely simple abelian varieties over $\mathbb{Q}$ with arbitrarily large $p$-torsion in their Tate-Shafarevich group. To prove this, we construct explicit $\mu_p$-covers of Jacobians of the…

Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$ of $L$ over $K$. Under the following three…

In this paper we construct abelian varieties of large Mordell-Weil rank over function fields. We achieve this by using a generalization of the notion of Prym variety to higher dimensions and a structure theorem for the Mordell-Weil group of…

Let $X$ be a smooth proper variety over an algebraically closed field of characteristic zero, and let $\mathcal{A} \subset D^{b}_{\mathrm{coh}}(X)$ be an admissible subcategory. Let $Z \subset X$ be the union of set-theoretical supports of…