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

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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)⊕(r−1)⊕O(l+1));(\mathbb{P}^n,\mathcal{O}(l)^{\oplus (r-1)}\oplus \mathcal{O}(l+1)); (Pn,TPn(l−1)⊕O(l)⊕(r−n))(r≥n).(\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.

References

Primary source

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

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