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Complete intersections with \(S^{1}\)-action. (English) Zbl 1379.57040

A complete intersection is a smooth \(2n\)-dimensional manifold given by a transversal intersection of \(r\) non-singular hypersurfaces in the complex projective space \({\mathbb{C}}P^{n+r}\). In this paper the authors consider the question of which complete intersections admit a smooth non-trivial \({\mathbb{S}}^1\)-action. They classify complete intersections with such an action in real dimension \(\leq 6\).
It follows from the Lefschetz fixed point formula for the Euler characteristic and the classification of surfaces that the only two-dimensional complete intersections with \({\mathbb{S}}^1\)-symmetry are diffeomorphic to the sphere or the torus. These complete intersections admit holomorphic \({\mathbb{S}}^1\)-actions with respect to their natural complex structure.
In dimension four the classifications of complete intersections with holomorphic and smooth \({\mathbb{S}}^1\)-symmetries do not coincide. The authors use Seiberg-Witten theory to show that a four-dimensional complete intersection admits a smooth non-trivial \({\mathbb{S}}^1\)-action if and only if it is diffeomorphic to a complex projective plane, a quadric, a cubic or an intersection of two quadrics.
To study the \(6\)-dimensional case, the authors use methods from equivariant cohomology and equivariant index theory. They show that a \(6\)-dimensional complete intersection admits a smooth non-trivial \({\mathbb{S}}^1\)-action if and only if it is diffeomorphic to a complex projective space or a quadric.
The authors also consider complete intersections in odd complex dimensions. They prove that in any odd complex dimension there are only finitely many complete intersections which admit a smooth effective action of a \(2\)-dimensional torus.

MSC:

57S15 Compact Lie groups of differentiable transformations
14M10 Complete intersections
32M05 Complex Lie groups, group actions on complex spaces
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