The Mukai conjecture for log Fano manifolds. (English) Zbl 1408.14126

Summary: For a log Fano manifold \((X,D)\) with \(D\neq 0\) and of the log Fano pseudoindex \(\geq 2\), we prove that the restriction homomorphism \(\mathrm{Pic}(X)\to\mathrm{Pic}(D_1)\) 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 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).


14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
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