The Mukai conjecture for log Fano manifolds

TL;DR

This paper proves a key property of log Fano manifolds with pseudoindex at least 2, showing the injectivity of the Picard group restriction and linking the Mukai conjecture to its log version.

Contribution

It establishes the injectivity of Picard group restriction for certain log Fano manifolds and connects the Mukai conjecture to its log analogue.

Findings

Picard group restriction homomorphism is injective for log Fano manifolds with pseudoindex ≥ 2.

The Mukai conjecture implies the log Mukai conjecture.

The proof uses a minimal model program similar to Casagrande's approach.

Abstract

For a log Fano manifold (X, D) with D\neq 0 and with the log Fano pseudoindex \geq 2, we prove that the restriction homomorphism Pic(X)\to Pic(D_1) of Picard groups is injective for any irreducible component D_1\subset D.The strategy of our proof is to run a certain minimal model program and is similar to the argument of Casagrande's one. 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).