Arithmetically Cohen-Macaulay bundles on complete intersection varieties of sufficiently high multidegree. (English) Zbl 1200.14082
Recall that if \((X,\mathcal{O}_X(1))\) is a polarized projective variety, a vector bundle \(E\) is said to be Arithmetically Cohen-Macaulay (ACM) with respect to this polarization if \(H^i(E(d))=0\) for all \(d\) and \(0<i<\dim X\). This article generalizes [N. Mohan Kumar, A. P. Rao and G. V. Ravindra, Comment. Math. Helv. 82, No. 4, 829–843 (2007; Zbl 1131.14047) and Int. Math. Res. Not. 2007, No. 8, Article ID rnm025, 10 p. (2007; Zbl 1132.14040)] by the reviewer, Rao and one of the authors. In the above mentioned articles it was proved that any rank two ACM bundle on a general hypersurface in projective space is a direct sum of line bundles if either dimension is four and degree is at least 3 or the dimension is three and degree is at least 6. The remaining dimensions were well understood earlier. The article under review proves similar theorems when the varieties are general complete intersections of sufficiently large multidegree. The methods are quite different and appeals to Griffith’s definition of normal functions and its infinitesimal versions [P. A. Griffiths, Ann. Math. (2) 90, 460–495, 496–541 (1969; Zbl 0215.08103); J. Carlson, M. Green, P. Griffiths and J. Harris, Compos. Math. 50, 109–205 (1983; Zbl 0531.14006)]. One of the crucial facts used in the earlier papers is that for any ACM bundle on the projective space, its exterior powers are also ACM, since by Horrock’s criterion, such bundles are direct sum of line bundles and hence so are their exterior powers. This fact is unavilable on more general varieties and thus the complete intersection varieties can not be studied as successive hypersurfaces in any obvious way.
Reviewer: N. Mohan Kumar (St. Louis)
MSC:
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
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