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Classification of Mukai pairs with corank $$3$$. (Classification des paires de Mukai de corang $$3$$.) (English. French summary) Zbl 1419.14066
In this paper, the author studies Mukai pairs $$(X,\mathcal{E})$$ as introduced in [S. Mukai, “Problems on characterization of complex projective space”, in: Birational Geometry of Algebraic Varieties, Open Problems, Katata, the 23rd Int’l Symp., Taniguchi Foundation. 57–60 (1988)]. A Mukai pair $$(X, \mathcal{E})$$ consists of a smooth Fano $$n$$-fold $$X$$ and an ample vector bundle $$\mathcal{E}$$ on $$X$$ of rank $$r$$ with $$c_1(X) = c_1(\mathcal{E})$$. The corank of a Mukai pair $$(X,\mathcal{E})$$ of dimensin $$n$$ and rank $$r$$ is defined as the integer $$c = n - r + 1$$. The cases $$c = 0$$ and $$1$$ are treated in [T. Fujita, Lect. Notes Math. 1507, 105–112 (1992; Zbl 0782.14018)]; T. Peternell, Int. J. Math. 2, No. 3, 311–322 (1991; Zbl 0744.14009); Y.-G. Ye and Q. Zhang, Duke Math. J. 60, No. 3, 671–687 (1990; Zbl 0709.14011)] and for the case $$c = 2$$ see [T. Peternell et al., Math. Ann. 294, No. 1, 151–165 (1992; Zbl 0786.14027)]. In the present paper the author extends the above results to the case $$c = 3$$ and gives a complete classification of such Mukai pairs with $$n \geq 5$$ and $$r = n-2$$. Furthermore, as a corollary to his main theorem, he points out that given $$(X,\mathcal{E})$$ a generalized polarized pair of dimension $$n\geq5$$ and rank $$n - 2$$, if there exists a $$K3$$ surface in $$X$$ which is the zero locus of a section of $$\mathcal{E}$$ then $$(X,\mathcal{E})$$ is one of the pairs as stated in his main theorem.

##### MSC:
 14J45 Fano varieties 14J40 $$n$$-folds ($$n>4$$) 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
##### Keywords:
Fano manifold; vector bundle
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##### References:
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