Fano Manifolds of Index n−1 and the Cone Conjecture

FANO MANIFOLDS OF INDEX n− 1 AND THE CONE CONJECTURE

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair (X,∆) have finite, rational polyhedral fundamental domains. Let Z be an n-dimensional Fano manifold of index n − 1 such that −KZ = (n − 1)H for an ample divisor H. Let Γ be the base locus of a...

Cone-Manifolds and the Density Conjecture

We give an expository account of our proof that each cusp-free hyperbolic 3-manifold with finitely generated fundamental group and incompressible ends is an algebraic limit of geometrically finite ones.

Mirror Symmetry and Fano Manifolds

The Α-invariant on Toric Fano Manifolds

The global holomorphic invariant αG(X) introduced by Tian [?], Tian and Yau [?] is closely related to the existence of Kähler-Einstein metrics. In his solution to the Calabi conjecture, Yau [?] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with nonpositive first Chern class. Kähler-Einstein metrics do not always exist in the case when the first Chern class is posi...

The Α-invariants on Toric Fano Manifolds

The global holomorphic invariant αG(M) introduced by Tian[14], Tian and Yau[13] is closely related to the existence of Kähler-Einstein metrics. In his solution to the Calabi conjecture, Yau[19] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with nonpositive first Chern class. Kähler-Einstein metrics do not always exist in the case when the first Chern class is posi...