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On the projective classification of smooth n-folds with n even. (English) Zbl 0703.14005

For H a nef and big line bundle on the normal projective variety X, the spectral value \(\sigma\) (X,H) is the smallest \(\tau\in {\mathbb{R}}\), such that \(H^ 0(X,(K_ X\otimes H^{(n+1-p/q)})^ N)=0\) for all \(N>0\) and all (p/q)\(\in {\mathbb{Q}}\) such that \(p/q>\tau\) and \(q| N.\)
For M smooth projective of dimension n, and L very ample on M, let (X,L) (respectively \((X',H))\) be the first (respectively second) reduction of (M,L). - If \(\sigma (M,L)<4-3/(n+1)\), the pair (M,L) is already classified.
The main result of this paper is the following: if \(n\geq 4\) is even, \(either:\)
(K\({}_{X'}\otimes H^{n-1})\) is nef and big, \(\sigma (M,L)\geq (4- (n+3)/(n^ 2+1))\) and \(H^ 0(M,K_ M^{n^ 2+1}\otimes L^{n(n- 1)(n-2)})\neq 0\), \(or:\)
\(\sigma\) (X\({}',L')\) is either 10/3 or 7/2, and \((X',L')\) belongs to a certain explicit list, where \(L'\) is 2-Cartier, defined by: \(H=K_{X'}\otimes L^{'(n-2)}.\)
The classification of even-dimensional polarized manifolds is thus reduced to the case: \(\sigma (M,L)\geq (4-(n+3)/(n^ 2+1))\). - Applications are given to the odd-dimensional case, and to manifolds containing many lines.
Reviewer: F.Campana

MSC:

14C20 Divisors, linear systems, invertible sheaves
14J10 Families, moduli, classification: algebraic theory
14J40 \(n\)-folds (\(n>4\))
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