We study the evolution of pluri-anticanonical line bundles $K_M^{-\nu }$ along the Kähler Ricci flow on a Fano manifold $M$. Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of $M$. For example, the Kähler Ricci flow on $M$ converges when $M$ is a Fano surface satisfying $c_1^2\left(M\right)=1$ or $c_1^2\left(M\right)=3$. Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups....

For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D 1) of Picard groups is injective for any irreducible component D 1 ⊂ D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).

Let $X$ be a Gorenstein, $\mathbb{Q}$-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of $X$ in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.

In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K X big and nef but not ample, Picard number γ(X) = 2, and whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop X + are not both birational.