Related papers: Sur une conjecture de Mukai

For an embedded Fano manifold $X$, we introduce a new invariant $S_X$ related to the dimension of covering linear spaces. The aim of this paper is to classify Fano manifolds $X$ which have large $S_X$.

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 view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration and the image of any elementary contraction also satisfies the same property. In this…

In this paper, we prove a special case of Campana--Peternell's conjecture in dimension 4. Specifically, we show that a projective smooth fourfold $X$ with $c^2_1(X)\cdot c_2(X)\neq 0$ and strictly nef anti-canonical divisor $-K_X$ is a Fano…

We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We construct two two-dimensional components…

Let X be an $n$-dimensional Fano manifold of Picard number 1. We study how many different ways X can compactify the complex vector group C^n equivariantly. Hassett and Tschinkel showed that when X = P^n with n \geq 2, there are many…

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify…

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We show that if rho(X)>12, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f: X->Y such that the…

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…

In this note we consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high…

We propose an analogue of Dubrovin's conjecture for the case where Fano manifolds have quantum connections of exponential type. It includes the case where the quantum cohomology rings are not necessarily semisimple. The conjecture is…

Ross and Thomas introduced the concept of slope stability to study K-stability, which has conjectural relation with the existence of constant scalar curvature K\"ahler metric. This paper presents a study of slope stability of Fano manifolds…

We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a…

We prove the Shafarevich conjecture for Fano threefolds of Picard rank 1, index 1 and degree 4.

A V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two…

Property $\mathcal{O}$ for an arbitrary complex, Fano manifold $X$, is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of $X$. Conjecture $\mathcal{O}$ is a…

We prove the generalised Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how…

A conjecture of Pukhlikov states that a smooth Fano variety of dimension at least four and index one is birationally rigid. We show that a general member of the linear system given by the ample generator of the Picard group of the moduli…

We prove the Mukai conjecture on the characterisation of products of projective spaces among Fano varieties for a class of locally factorial Fano varieties defined in terms of their Cox rings. The Fano varieties of this class are…

We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective,…