Related papers: Higher Fano Manifolds
In this paper, we investigate Fano manifolds whose Chern characters satisfy some positivity conditions. We prove that such manifolds admit long chains of higher order minimal families of rational curves and are covered by higher rational…
In this paper, we investigate higher order minimal families $H_i$ of rational curves associated to Fano manifolds $X$. We prove that $H_i$ is also a Fano manifold if the Chern characters of $X$ satisfy some positivity conditions. We also…
In this paper we classify n-dimensional Fano manifolds with index >=n-2 and positive second Chern character.
In this paper we investigate Fano manifolds $X$ whose Chern characters $ch_k(X)$ satisfy some positivity conditions. Our approach is via the study of polarized minimal families of rational curves $(H_x,L_x)$ through a general point $x\in…
Let $X$ be an $n$-dimensional complex Fano manifolds $(n\geq 3)$. Assume that $X$ contains a divisor $A$, which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle $\mathscr{N}^*_{A/X}$ is…
We show that the second Chern character of any projective toric manifold of Picard number three is not ample. In connection with this result, we give various examples of the positivity of higher Chern characters of projective toric…
We show that any toric Fano manifold of dimension at most eight with the positive second Chern character is isomorphic to the projective space by using polymake.
This note outlines some first steps in the classification of Fano manifolds for which $c_1^2-2c_2$ is positive or nef.
We prove that Fano n-folds with nef tangent bundle and Picard number greater than n-5 are rational homogeneous manifolds.
In this paper, we prove a conjecture by T. Suzuki, which says if a smooth Fano manifold satisfies some positivity condition on its Chern character, then it can be covered by rational $N$-folds. We prove this conjecture by using purely…
Let $X$ be a Fano manifold of Picard number one. We establish a lower bound for the second Chern class of $X$ in terms of its index and degree. As an application, if $Y$ is a $n$-dimensional Fano manifold with $-K_Y=(n-3)H$ for some ample…
We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.
We classify primitive Fano threefolds in positive characteristic whose Picard numbers are at least two. We also classify Fano theefolds of Picard rank two.
Over an algebraically closed field of positive characteristic, we classify smooth Fano threefolds of Picard number one whose anti-canonical linear systems are not very ample. Furthermore, we also prove that an anti-canonically embedded Fano…
In this notes we classify toric Fano 4-folds having positive second Chern Character.
For a Fano manifold of pseudo-index at least 3 and $c_1^2-2c_2$ nef, we show irreducibility of certain spaces of curves on the Fano manifold implies the manifold is a union of rational surfaces.
A simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold $G^C/P$ is provided in terms of symplectic data.
We study smooth complex projective polarized varieties $(X,H)$ of dimension $ n \ge 2$ which admit a dominating family $V$ of rational curves of $H$-degree $3$, such that two general points of $X$ may be joined by a curve parametrized by…
For $n\geq 4$, let $X$ be a complex smooth Fano $n$-fold whose minimal anticanonical degree of non-free rational curves on $X$ is at least $n-2$. We classify extremal contractions of such varieties. As an application, we obtain a…
We classify smooth Fano manifolds X with the Picard number $\rho_X \geq 3$ such that there exists an extremal ray which has a birational contraction that maps a divisor to a point.