Related papers: Fano generalized Bott manifolds

The Manin-Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano…

This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc.

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 paper, we give a partial affirmative answer to the BAB conjecture for $3$-folds in characteristic $p>5$. Specifically, we prove that a set $\mathcal{D}$ of weak Fano $3$-folds over an uncountable algebraically closed field is…

In dimension two, we reduce the classification problem for asymptotically log Fano pairs to the problem of determining generality conditions on certain blow-ups. In any dimension, we prove the rationality of the body of ample angles of an…

In this short note, we show that K-semistable Fano manifolds with the smallest alpha invariant are projective spaces. Singular cases are also investigated.

We study the K-stability of singular Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system is base-point-free but not very ample.

Let X be a compact K\"ahler manifold such that the anticanonical bundle $-K_X$ is nef. A classical conjecture claims that the Albanese map is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is…

We construct smooth varieties admitting small contractions from arbitrary smooth projective varieties. This construction generalizes Kawamata's four-dimensional example. We also give sufficient conditions for divisors on these varieties to…

Fano varieties are 'atomic pieces' of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and classification of Fano varieties, with an…

In this note, we classify smooth equivariant compactifications of $\mathbb{G}_a^n$ which are Fano manifolds with index $\geq n-2$.

By Delzant's theorem, closed symplectic toric manifolds are classified by the images of moment maps. In the case of a generalized Bott manifold, this image is a polytope $P$ combinatorially equivalent to the product of simplices. We compute…

Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This…

We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine…

Let X be a Fano manifold with Picard number one such that the tangent bundle T_X is big. If X admits a rational curve with trivial normal bundle, we show that X is isomorphic to the del Pezzo threefold of degree five.

Let (M,E) be a generalized polarized manifold, i.e., a pair of an n-dimensional smooth projective variety and an ample vector bundle E of rank r on M. Let t be the nef value of a polarized manifold (M, det E), i.e., the minimum of the set…

In this paper, we provide examples of Sarkisov links of type II between complex projective Fano threefolds $X$ with $\rho(X) = 1$. To show examples of these links, we study smooth weak Fano threefolds with extremal rays of type $E$. We…

This paper is a survey about cylinders in Fano varieties and related problems.

We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine the mutation-equivalence classes of such…

In this paper, we prove that any polarized K-stable manifold is CM-stable. This extends what I did for Fano manifolds in my 2012 paper.