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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).

MSC:

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