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In this paper we extend to the singular setting the theory of Fano foliations developed in our previous paper. A Q-Fano foliation on a complex projective variety X is a foliation F whose anti-canonical class is an ample Q-Cartier divisor.…

In this paper we study the algebraic ranks of foliations on $\mathbb{Q}$-factorial normal projective varieties. We start by establishing a Kobayashi-Ochiai's theorem for Fano foliations in terms of algebraic rank. We then investigate the…

The cotangent bundle of a non-uniruled projective manifold is generically nef, due to a theorem of Miyaoka. We show that the cotangent bundle is actually generically ample, if the manifold is of general type and study in detail the case of…

We prove the generalized Obata theorem on foliations. Let M be a complete Riemannian manifold with a foliation F of codimension $q>1$ and a bundle-like metric. Then $(M, F)$ is transversally isometric to the q-sphere of radius 1/c in…

Let $\mathcal{F}$ be a foliation on a projective manifold $X$ with $-K_{\mathcal{F}}$ nef. Assume that either $\mathcal{F}$ is regular, or it has a compact leaf. We prove that there is a locally trivial fibration $f\colon X\to Y$, and a…

In this paper, we extend the structure theorem for smooth projective varieties with nef tangent bundle to projective klt varieties whose tangent sheaf is either positively curved or almost nef. Specifically, we show that such a variety $X$,…

This paper generalises Mori's famous theorem about "Projective manifolds with ample tangent bundles" to normal projective varieties in the following way: A normal projective variety over $\mathbb{C}$ with ample tangent sheaf is isomorphic…

In this paper we aim at the description of foliations having tangent sheaf $T\mathcal F$ with $c_1(T\mathcal F)=c_2(T\mathcal F)=0$ on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as…

Let $X$ be a complex smooth projective variety such that the exterior power of the tangent bundle $\bigwedge^{r} T_X$ is nef for some $1\leq r<\dim X$. We prove that, up to an \'etale cover, $X$ is a Fano fiber space over an Abelian…

Let $X$ be a $\mathbb Q$-Fano variety admitting a K\"ahler-Einstein metric. We prove that up to a finite quasi-\'etale cover, $X$ splits isometrically as a product of K\"ahler-Einstein $\mathbb Q$-Fano varieties whose tangent sheaf is…

In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally…

The purpose of this paper is to prove the following theorem. Let $X$ be a projective normal variety defined over an algebraically closed field of characteristic zero and let $\Omega_{X}^{1}\to L$ be a one-dimensional foliation on $X$. If…

Due to a theorem by Orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of Fourier-Mukai type. We extend this result to the case of bounded derived categories…

A foliation is said to admit a foliated contact structure if there is a codimension 1 distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture in the…

We prove a new restriction theorem for semistable sheaves on varieties in all characteristics strengthening previous results. We also prove restriction theorem for strong semistability for varieties with some non-negativity constrains on…

We show that the set of singular holomorphic foliations of the projective spaces with split tangent sheaf and with good singular set is open in the space of holomorphic foliations. As applications we present a generalization of a result by…

A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V \otimes O_X by the sheaf of differentials \Omega_X, given by the inclusion of a linear space V in Ext^1(O_X,\Omega_X). For…

Let $X$ be a scheme, proper over a commutative noetherian ring $A$. We introduce the concept of an ample filter of invertible sheaves on $X$ and generalize the most important equivalent criteria for ampleness of an invertible sheaf. We also…

Let $X$ be a normal algebraic variety over an algebraically closed field $k$ of characteristic zero. We prove that the K\"{a}hler differential sheaf of $X$ is torsion-free if and only if any regular section of the ideal sheaf of the first…

By analogy with algebraic geometry, we define a category of non-linear sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising…