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Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

Ichiro Shimada
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Abstract

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 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 $T\subset X$ of certain codimension. In some cases, 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 $X$ by means of Gröbner bases. Hence a consequence of the generalized Hodge conjecture is verified in these cases.

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Ichiro Shimada. 2003-11-12. Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases. https://arxiv.org/abs/math/0311180

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