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Simple normal crossing Fano varieties and log Fano manifolds. (English) Zbl 1297.14047

A log pair \((X,D)\) is called a log Fano manifold if \(X\) is a smooth projective variety, \(D\) a reduced simple normal crossing divisor and \(-(K_X+D)\) is ample. For a log Fano manifold \((X,D)\), the log Fano index is the maximum of integer \(r\) such that \(\mathcal{O}_X (-(K_X+D)) \cong \mathcal{L}^{\otimes r}\) for some line bundle \(\mathcal{L}\). Also, for a log Fano manifold, the log Fano pseudoindex of \((X,D)\) is the minimum of \((-(K_X + D) \cdot C)\), where \(C\) runs over the rational curves on \(X\).
The aim of the paper under review is to classify log Fano manifolds with non-zero boundary in terms of their log Fano (pseudo)index. The consideration of the (pseudo)index is natural in the context of the (generalized) Mukai conjecture. In the main theorems, the author classifies
(1) \(n\)-dimensional log Fano manifolds with non-zero boundary and with log Fano pseudoindex \(\iota\) such that \(2 \iota \geq n\) and \(\rho (X) \geq 2\), and also classifies
(2) \(2 r\)-dimensional log Fano manifolds with log Fano index \(r \geq 2\) such that \(\rho (X) \geq 2\).
Combining these results with H. Maeda’s result [Compos. Math. 57, 81–125 (1986; Zbl 0658.14019)], the classification of \(n\)-dimensional log Fano manifolds with log Fano indices \(r \geq n-2\) and with non-zero boundaries is completed.

MSC:

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

Citations:

Zbl 0658.14019
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References:

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