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Ressayre’s pairs in the Kähler setting. (English) Zbl 1496.53089

As the title indicates, this paper develops an analogue in Kähler geometry of a theory in algebraic geometry due to N. Ressayre [Invent. Math. 180, No. 2, 389–441 (2010; Zbl 1197.14051)]. More precisely, given a connected reductive group \(G\) acting on a normal projective variety \(X\), Ressayre has shown that the facets of particular polyhedral cones in \(\operatorname{Pic}^G(X)\) can be parametrised by (in)equalities satisfied by so-called well-covering pairs. In the present work, analogous notions of Ressayre’s pairs are introduced to parametrise the facets of the Kirwan polytope, which is defined using a Hamiltonian action of a connected compact Lie group \(K\) on a Kähler manifold \(M\). (To build the analogy, one uses the holomorphic action of the complexification \(K_{\mathbb{C}}\) on \(M\) seen as a complex manifold.) The author’s main idea behind this parametrisation consists in considering several embeddings of polytopes in order to prove that they coincide and that they are nothing else than the Kirwan polytope. Interestingly, these (a priori different) polytopes are defined from the same set of inequalities which depend on various notions of Ressayre’s pairs, see the beginning of Section 4. Two specific examples of this parametrisation are worked out in Section 6. They concern actions of \(\tilde{K}_{\mathbb{C}}\times K_{\mathbb{C}}\) on \(\tilde{K}_{\mathbb{C}}\) and on \(\tilde{K}_{\mathbb{C}} \times V\), where \(K\hookrightarrow \tilde{K}\) is a closed connected Lie subgroup and \(V\) is a vector space with \(K_{\mathbb{C}}\)-linear action. The introduction is particularly well written and it gives a very precise outline of the objects at stake in this paper.

MSC:

53D20 Momentum maps; symplectic reduction
32M05 Complex Lie groups, group actions on complex spaces
32Q15 Kähler manifolds
14M15 Grassmannians, Schubert varieties, flag manifolds

Citations:

Zbl 1197.14051
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References:

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