Related papers: Projective varieties with nef tangent bundle in po…

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 smooth complex projective variety with nef $\bigwedge^2 T_X$ and $\dim X \geq 3$. We prove that, up to a finite \'etale cover $\tilde{X} \to X$, the Albanese map $\tilde{X} \to {\rm Alb}(\tilde{X})$ is a locally trivial…

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 $X$,…

In this paper, we study smooth complex projective varieties $X$ such that some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two…

We prove that a smooth projective variety $X$ of dimension $n$ with strictly nef third, fourth or $(n-1)$-th exterior power of the tangent bundle is a Fano variety. Moreover, in the first two cases, we provide a classification for $X$ under…

In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety $X^n$ $(n>4)$ has strictly nef $\Lambda^2 TX$, then it is isomorphic to…

We show how to construct tilting bundles for a class of smooth projective varieties using characteristic $p$ methods. Given such a variety $X$, reduce it modulo a prime number and consider the direct image of the structure sheaf under the…

Up to finite \'etale cover, any smooth complex projective variety $X$ with nef anti-canonical bundle is a holomorphic fibre bundle over a $K$-trivial variety with locally constant transition functions. We show that this result is optimal by…

We define the nef complexity of a projective variety $X$. This invariant compares $\dim X+\rho(X)$ with the sum of the coefficients of nef partitions of $-K_X$. We prove that the nef complexity is non-negative and it is zero precisely for…

Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group…

In this paper we prove the following abundance-type result: for any smooth Fano variety $X$, the tangent bundle $T_X$ is nef if and only if it is big and semiample in the sense that the tautological line bundle…

We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite \'etale cover, admit a toric fibration over…

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…

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability of…

In this paper, we prove that a smooth projective variety $X$ of characteristic $p>0$ is an ordinary abelian variety if and only if $K_X$ is pseudo-effective and $F^e_*\mathcal O_X$ splits into a direct sum of line bundles for an integer $e$…

We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.

In this paper, we prove that a smooth projective globally $F$-split variety with numerically flat tangent bundle is an \'etale quotient of an ordinary abelian variety. We also show its logarithmic analog, which contains a characterization…

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ of $\dim X\geq 4$ and Picard number $\rho(X)=1$. Suppose that $X$ satisfies $H^i(X,F^{m*}_X(\Omg^j_X)\otimes\Ls^{-1})=0$ for any ample…

We study complex projective manifolds X that admit surjective endomorphisms f:X->X of degree at least two. In case f is etale, we prove structure theorems that describe X. In particular, a rather detailed description is given if X is a…

Let $X$ be a smooth projective variety with a nef anticanonical divisor over an algebraically closed field of characteristic $p>0$. In this paper, we establish a precise structure of $X$ under the condition that $a_X: X \to {\rm Alb}(X)$ is…