## 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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### References:

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