Fano manifolds of index n − 2 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.

On Classification of Q-fano 3-folds with Gorenstein Index 2 and Fano Index

We generalize the theory developed by K. Takeuchi in [T1] and restrict the birational type of a Q-factorial Q-Fano 3-fold X with the following properties: (1) the Picard number of X is 1; (2) the Gorenstein index of X is 2; (3) the Fano index of X is 1 2 ; (4) h(−KX) ≥ 4; (5) there exists an index 2 point P such that (X, P ) ≃ ({xy + f(z, u) = 0}/Z2(1, 1, 1, 0), o) with ordf(Z, U) = 1. This giv...

Mirror Symmetry and Fano Manifolds

THEOREMS ABOUT GOOD DIVISORS ON LOG FANO VARIETIES ( CASE OF INDEX r > n - 2 )