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2020•Transformation GroupsOpen access

ADJUNCTION FOR VARIETIES WITH A ℂ* ACTION

Eleonora A. Romano, Jarosław A. Wiśniewski

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Abstract

Abstract Let X be a complex projective manifold, L an ample line bundle on X , and assume that we have a ℂ* action on ( X ; L ). We classify such triples ( X ; L ;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

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Abstract Let X be a complex projective manifold, L an ample line bundle on X , and assume that we have a ℂ* action on ( X ; L ). We classify such triples ( X ; L ;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

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Abstract Let X be a complex projective manifold, L an ample line bundle on X , and assume that we have a ℂ* action on ( X ; L ). We classify such triples ( X ; L ;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

Key concepts: Mathematics, Adjunction, Automorphism, Fixed point, Fano plane, Pure mathematics, Group action, Rank (graph theory)

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