Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Asymptotic bounds for Nori's connectivity theorem


Author: Ania Otwinowska
Journal: J. Algebraic Geom. 14 (2005), 643-661
DOI: https://doi.org/10.1090/S1056-3911-05-00404-2
Published electronically: June 9, 2005
MathSciNet review: 2147354
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Abstract: Let $Y$ be a smooth complex projective variety. I study the cohomology of smooth families of hypersurfaces $\mathcal{X}\to B$ for $B\subset\mathbb{P}\mathrm{H}^0(Y,\mathcal{O}(d))$ a codimension $c$ subvariety. I give an asymptotically optimal bound on $c$ and $k$ when $d\to\infty$ for the space $\mathrm{H}^k(Y\times B,\mathcal{X},\mathbb{Q} )$ to vanish, thus extending the validity of the Lefschetz Hyperplane Section Theorem and Nori's Connectivity Theorem (1993). Next, I construct in the limit case explicit families of higher Chow groups whose class does not vanish in $\mathrm{H}^k(Y\times B,\mathcal{X},\mathbb{Q} )$. Some of them are indecomposable. This suggests that in the limit case the space $\mathrm{H}^k(Y\times B,\mathcal{X},\mathbb{Q} )$ should be spanned by higher Chow groups, containing Nori's and Otwinowska's results as special cases.


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Ania Otwinowska
Affiliation: Départment de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay, Cedex, France

DOI: https://doi.org/10.1090/S1056-3911-05-00404-2
Received by editor(s): March 11, 2004
Received by editor(s) in revised form: November 8, 2004
Published electronically: June 9, 2005

American Mathematical Society