Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

Tom 65 / 2004

Ichiro Shimada Banach Center Publications 65 (2004), 227-259 MSC: Primary 14C30, 14M10; Secondary 14C05, 14J45, 14J05. DOI: 10.4064/bc65-0-15


Let $X$ be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of $X$ is ample. Using the cylinder homomorphism associated with the family of complete intersections of a smaller multi-degree contained in $X$, we prove that the vanishing cycles in the middle homology group of $X$ are represented by topological cycles whose support is contained in a proper Zariski closed subset $T$ of $X$ with certain codimension. In some cases, by means of Gröbner bases, we can find such a Zariski closed subset $T$ with codimension equal to the upper bound obtained from the Hodge structure of the middle cohomology group of $X$. Hence a consequence of the generalized Hodge conjecture is verified in these cases.


  • Ichiro ShimadaDivision of Mathematics
    Graduate School of Science
    Hokkaido University
    Sapporo 060-0810, Japan

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