Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases

• Autorzy: Ichiro Shimada
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 14M10: Complete intersections
• 14C30: Transcendental methods, Hodge theory , Hodge conjecture
• 14C05: Parametrization (Chow and Hilbert schemes)
• 14J45: Fano varieties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2004
• Tom: 65
• Numer: 1
• Strony: 227-259

Daty

wydano

2004

Twórcy

autor

Ichiro Shimada

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

Identyfikatory

DOI

10.4064/bc65-0-15