The Log-Concavity Conjecture for the Duistermaat-Heckman Measure Revisited

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Abstract

Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat–Heckman measure: a Hamiltonian six-manifold whose fixed-points set is the disjoint union of two copies of T 4. In this article, for any closed symplectic four-manifold N with b + > 1, we show that there is a Hamiltonian six-manifold M such that its fixed-points set is the disjoint union of two copies of N and such that its Duistermaat–Heckman function is not log-concave. On the other hand, we prove that if there is a torus action of complexity 2 such that all the symplectic reduced spaces taken at regular values satisfy the condition b + = 1, then its Duistermaat–Heckman function has to be log-concave. As a consequence, we prove the log-concavity conjecture for Hamiltonian circle actions on six manifolds such that the fixed-points sets have no 4-dimensional components, or only have 4-dimensional pieces with b + = 1.
Original languageAmerican English
JournalInternational Mathematics Research Notices
Volume2008
DOIs
StatePublished - Jan 1 2008

Disciplines

  • Mathematics

Keywords

  • Duistermaat-Heckman measure
  • Symplectic manifolds

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