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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
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