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Weil Class

A Weil class is a Hodge cycle arising from the determinant construction for an abelian variety with a balanced imaginary-quadratic action. If A has dimension 2n and an imaginary quadratic field E acts compatibly with a polarization with signature (n,n), the determinant space ^ _E^(2n)H^1(A,Q) gives a two-dimensional rational space of classes of type (n,n) in H^(2n)(A,Q). These classes need not be generated by divisor classes.

The associated Hermitian form is hyperbolic when it contains a totally isotropic n-dimensional E-subspace.

See also

Abelian Variety, Hodge Conjecture, Hodge Cycle, Hodge Structure, Imaginary Quadratic Field

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References

Milne, J. S. "Hodge Classes on Abelian Varieties." Version 1.1, July 12, 2022. https://www.jmilne.org/math/articles/HAV.pdf.Moonen, B. J. J. and Zarhin, Yu. G. "Weil Classes on Abelian Varieties." J. reine angew. Math. 496, 83-92, 1998. https://doi.org/10.1515/crll.1998.034.

Cite this as:

Weisstein, Eric W. "Weil Class." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeilClass.html

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