Classification conjecture for generalized polarized Fano manifolds with l(R_1)=n

Let MM be a Fano manifold of Picard number one and let E\mathcal{E} be an ample vector bundle of rank rr on MM. Let LL, the extremal ray R1R_1, and a minimal rational curve C1C_1 be as in the common setup, and set l=L.C1l=L.C_1. If the length of R1R_1 is l(R1)=nl(R_1)=n, then the classification conjecture. (M,E)(M,\mathcal{E}) is one of the following:

(Qn,O(l)r);(\mathbb{Q}^n,\mathcal{O}(l)^{\oplus r}); (Pn,O(l)(r1)O(l+1));(\mathbb{P}^n,\mathcal{O}(l)^{\oplus (r-1)}\oplus \mathcal{O}(l+1)); (Pn,TPn(l1)O(l)(rn))(rn).(\mathbb{P}^n,T_{\mathbb{P}^n}(l-1)\oplus \mathcal{O}(l)^{\oplus (r-n)})\qquad (r\geq n).

This conjecture predicts a complete classification of the generalized polarized manifolds satisfying the extremal-length condition l(R1)=nl(R_1)=n. The preceding argument establishes the listed possibilities in the cases treated there, while the full classification is posed as an open conjecture.

Sources & referencesView supporting material

Primary source

Masahiro Ohno, “Classification of generalized polarized manifolds by their nef values”, arXiv:math/0503119 (2005).

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