Complete Intersections with S^1-action

TL;DR

This paper classifies 6-dimensional complete intersections with S^1-symmetry, showing only specific manifolds admit such actions, and proves finiteness of torus actions in odd complex dimensions.

Contribution

It provides a complete diffeomorphism classification of low-dimensional complete intersections with S^1-actions and establishes finiteness results for higher-dimensional torus actions.

Findings

6-dimensional complete intersections with S^1-action are diffeomorphic to complex projective space or the quadric

Only finitely many complete intersections in odd complex dimensions admit higher-rank torus actions

Classification results are specific to dimensions ≤ 6

Abstract

We give the diffeomorphism classification of complete intersections with S^1-symmetry in dimension less than or equal to 6. In particular, we show that a 6-dimensional complete intersection admits a smooth non-trivial S^1-action if and only if it is diffeomorphic to the complex projective space or the quadric. We also prove that in any odd complex dimension only finitely many complete intersections can carry a smooth effective action by a torus of rank $>1$.