On a symplectic generalization of Petrie’s conjecture

By S. Tolman. 2009

64 citations recorded by the source corpus.

Motivated by the Petrie conjecture, we consider the following questions: Let a circle act in a Hamiltonian fashion on a compact symplectic manifold (M, ω) which satisfies H 2i (M ; ℝ) = H 2i (ℂℙ n ,ℝ) for all i. Is H j (M ; ℤ) = H j (ℂℙ n ;ℤ) for all j? Is the total Chern class of M determined by the cohomology ring H * (M; Z)? We answer these questions in the six-dimensional case by showing that H j (M ; ℤ) is equal to H j (ℂℙ 3 ; ℤ) for all j, by proving that only four cohomology rings can arise, and by computing the total Chern class in each case. We also prove that there are no exotic actions. More precisely, if H*(M; ℤ) is isomorphic to H*(ℂℙ 3 ; ℤ) or H*(G 2 (ℝ 5 ); ℤ), then the representations at the fixed components are compatible with one of the standard actions; in the remaining two cases, the representation is strictly determined by the cohomology ring. Finally, our results suggest a natural question: Do the remaining two cohomology rings actually arise? This question is closely related to some interesting problems in symplectic topology, such as embeddings of ellipsoids.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c69f997c9e59d547f1e64edcd3d1c867c20575ba

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Citations in the corpus

Symplectic embeddings of 4‐dimensional ellipsoids (2008, unknown): “I also thank Tolman for giving me an advance copy of her paper [25], Schlenk for his encouragement and useful comments, and the referree for pointing out various small inaccuracies.”
Hilbert schemes of lines and conics and automorphism groups of Fano threefolds (2016, result): “3, and to some extent can be used to recover it; namely, the results of [Tol10] imply that an automorphism group of a smooth Fano threefold X of Picard rank 1 can contain a subgroup isomorphic to Gm only if h(1,2)(X) = 0.”