Related papers: Varieties with $P_2(X)=2$ and $q(X)=\dim X$

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety X with maximal Albanese dimension and…

We study the quantum invariants of projective varieties over the number fields. Namely, explicit formulas for a functor $\mathscr{Q}$ on such varieties are proved. The case of abelian varieties with complex multiplication is treated in…

A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to…

Semisimple representations of the free product Z_p*Z_q determine \theta-semistable representations of a specific quiver Q_pq. The dimension vectors of \theta-stable representations of this quiver were classified by Le Bruyn and…

Let $p\neq 2$, and let $R$ be a smooth affine algebra of dimension $3$ over $\overline{F}_p$ and $P, Q$ be projective $R$-modules of rank $2$, each with trivial determinant. We prove: $P$ is isomorphic to $Q$ if and only if there is an…

The following conjecture arose out of discussions between B. Harbourne, J. Ro\'e, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal…

Let $i\colon X\to \Pk^N$ be a projective manifold of dimension $n$ embedded in projective space $\Pk^N$, and let $L$ be the pull-back to $X$ of the line bundle $\Ok_{\Pk^N}(1)$. We construct global explicit Koppelman formulas on $X$ for…

In the first part of this paper, we show that the assertion "T_pH_k(X,Q)=G_pH_k(X,Q)" (which is called the Friedlander-Mazur conjecture) is a birationally invariant statement for smooth projective varieties X when p=dim(X)-2 and when p=1.…

We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with…

We study the problem of classifying the irreducible projective varieties $X$ of dimension $n\ge 2$ in $\Bbb P^N$ which contain an algebraic family $\Cal F$ of dimension $h+1$ ($h<n$) of subvarieties $Y$ of dimension $n-h$, each one…

For a $G$-variety $X$ with an open orbit, we define its boundary $\partial X$ as the complement of the open orbit. The action sheaf $S_X$ is the subsheaf of the tangent sheaf made of vector fields tangent to $\partial X$. We prove, for a…

Let $Y$ be a smooth projective variety of dimension $n \geq 2$ endowed with a finite morphism $\phi:Y \to \mathbb P^n$ of degree $3$, and suppose that $Y$, polarized by some ample line bundle, is a scroll over a smooth variety $X$ of…

Consider an absolutely simple abelian variety X over a number field K. If the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then the reduction X_p is absolutely simple for almost all p. Conversely,…

We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long standing open question if a smooth complex projective rational surface has…

We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of…

We establish the Hodge conjecture for the top dimensional cohomology group with integer coefficients of any $q$-complete complex manifold $X$ with $q<\dim X$. This holds in particular for the complement $X=\mathbb{C}\mathbb{P}^n\setminus A$…

Let $X$ be an $n$-dimensional normal $\mathbb{Q}$-factorial projective variety with canonical singularities and Picard number one such that $X$ is smooth in codimension two, $-K_X$ is ample and $n\geq 2$. We prove that $X$ satisfies the…

Let X be a smooth projective complex variety, of dimension 3, whose Hodge numbers h^{3,0}(X), h^{1,0}(X) both vanish. Let f: X--> X be a birational map that induces an isomorphism on (dense) open subvarieties U,V of X. Then we show that the…

Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from…