An effective version of Nori’s theorem. (English) Zbl 0790.14005

Recently, M. V. Nori [Invent. Math. 111, No. 2, 349-373 (1993)] has shown that a homologically non-trivial cycle on a projective variety \(X\) is algebraically nontrivial on \(Y=X \cap D_ 1\cap \cdots \cap D_ h\), where the \(D_ i\) are divisors on \(X\) of large enough degree. We give a sufficient condition on the positivity of the divisors \(D_ i\), for Nori’s theorem to be applicable. - The main ingredient is an effective vanishing theorem for some Koszul type cohomology groups. We also prove an extension of our vanishing theorem which addresses a conjecture of Nori.


14C25 Algebraic cycles
14F17 Vanishing theorems in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
Full Text: DOI EuDML


[1] [EL] Ein, L., Lazarsfeld, R.: Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension. Invent. Math.111, 51–67 (1993) · Zbl 0814.14040
[2] [G] Green, M.: Koszul cohomology and Geometry. In: Cornalba, M. et al. (eds.) Lectures on Riemann Surfaces. Singapore: World Scientific 1989 · Zbl 0800.14004
[3] [N] Nori, M.: Algebraic cycles and Hodge theoretic connectivity results. Invent. Math.111, 349–373 (1993) · Zbl 0822.14008
[4] [SS] Shiffman, B., Sommese, A.: Vanishing Theorems on Complex Manifolds. (Prog. Math. vol.56) Boston Basel Stuttgart: Birkhäuser 1985 · Zbl 0578.32055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.