## 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.

### MSC:

 14C25 Algebraic cycles 14F17 Vanishing theorems in algebraic geometry 14C20 Divisors, linear systems, invertible sheaves
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### References:

 [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
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