an:07067403
Zbl 1419.14066
Kanemitsu, Akihiro
Classification of Mukai pairs with corank $3$
EN
Ann. Inst. Fourier 69, No. 1, 231-282 (2019).
00434116
2019
j
14J45 14J40 14J60
Fano manifold; vector bundle
In this paper, the author studies Mukai pairs \((X,\mathcal{E})\) as introduced in [\textit{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 [\textit{T. Fujita}, Lect. Notes Math. 1507, 105--112 (1992; Zbl 0782.14018)]; \textit{T. Peternell}, Int. J. Math. 2, No. 3, 311--322 (1991; Zbl 0744.14009); \textit{Y.-G. Ye} and \textit{Q. Zhang}, Duke Math. J. 60, No. 3, 671--687 (1990; Zbl 0709.14011)] and for the case \(c = 2\) see [\textit{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.
Aigli Papantonopoulou (Ewing)
Zbl 0782.14018; Zbl 0744.14009; Zbl 0709.14011; Zbl 0786.14027