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Higher cohomology of divisors on a projective variety. (English) Zbl 1127.14012

Let \(X\) be an irreducible complex projective variety of dimension \(d\geq 1\), and let \(L\) be a Cartier divisor on \(X\). It is known that the dimension of the cohomology groups \(H^{i}(\mathcal{O}_{X}(mL))\) grow at most like \(m^{d}\). In this paper, the authors consider the problem of when one or more of the higher cohomology groups grows maximally and they obtained the following result: Fix any very ample divisor \(A\) on \(X\). If \(L\) is not ample, then there exists an integer \(i>0\) such that for any sufficiently small \(t>0\), \(\dim H^{i}(X,\mathcal{O}_{X}(m(L-tA)))\geq cm^{d}\) for some constant \(c>0\) which depends on \(L\), \(A\) and \(t\), and arbitrary large values of \(m\) clearing the denominator of \(t\). As a corollary the authors also prove a result which can be regarded as an asymptotic analogue of Serre’s criterion for amplitude.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14F99 (Co)homology theory in algebraic geometry
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References:

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