Related papers: The generalised Mukai conjecture for spherical var…

We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp(X)$, which we conjecture to satisfy the inequality $\wp(X) \le \operatorname{dim} X - \operatorname{rank} X$ with equality holding if and only if…

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 show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal…

In the case of spherical varieties with reductive general isotropy group we prove a conjecture of G. Gagliardi and J. Hofscheier, which implies the generalized Mukai conjecture of L. Bonavero, C. Casagrande, O. Debarre and S. Druel for…

In this paper, we generalise the theory of complements to log canonical log fano varieties and prove boundedness of complements for them in dimension less than or equal to 3. We also prove some boundedness results for the canonical index of…

In this note, we reduce various conjectures in birational geometry, including Shokurov conjecture on singularities of the base of log Calabi-Yau fibrations of Fano type and boundedness conjecture for rationally connected Calabi-Yau…

A Mukai variety is a Fano n-fold of index n-2. In this paper we study the fundamental divisor of a Mukai variety with at worst log terminal singularities. The main result is a complete classification of log terminal Mukai varieties which…

Let X be a Fano variety of dimension n, pseudoindex i_X and Picard number \rho_X. A generalization of a conjecture of Mukai says that \rho_X(i_X-1)\le n. We prove that the conjecture holds if: a) X has pseudoindex i_X \ge \frac{n+3}{3} and…

In this article we give the definitions of log Fano varieties and log Calabi-Yau varieties in the framework of theory of log schemes of Fontain-Illusie-Kato and give congruences of the cardinalities of rational points of them over the log…

Let X be a complex Fano manifold of dimension n. Let s(X) be the sum of l(R)-1 for all the extremal rays of X, the edges of the cone NE(X) of curves of X, where l(R) denotes the minimum of (-K_X \cdot C) for all rational curves C whose…

I state a conjecture asserting that for all generic klt Fano varieties X, there exists a generalised cluster variety U and a surjection from the set of torus charts on U to the set of toric specializations of X. I prove the conjecture in…

We introduce some invariants of Fano varieties and propose a Mukai-type conjecture which characterizes the product of projective spaces. Moreover, we prove that the Ambro--Kawamata effective non-vanishing conjecture implies the Mukai-type…

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a…

We verify the Mukai conjecture for Fano quiver moduli spaces associated to dimension vectors in the interior of the fundamental domain.

We prove the metric version of the SYZ conjecture for a class of Calabi-Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge-Amp\`ere…

In this article, we introduce the generalized complexity of a generalized Calabi--Yau pair $(X,B,\textbf{M})$. This invariant compares the dimension of $X$ and Picard rank of $X$ with the sum of the coefficients of $B$ and $\textbf{M}$. It…

For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete…

We prove some general results on syzygies of smooth projective varieties with numerically trivial canonical line bundle. This allows to confirm several cases of Mukai's syzygies conjecture for finite quotients of abelian varieties in any…

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…

We formulate a conjecture on the number of integral points of bounded height on log Fano varieties in analogy with Manin's conjecture on the number of rational points of bounded height on Fano varieties. We also give a prediction for the…