TY - JOUR

T1 - Hamiltonian circle actions with minimal fixed sets

AU - Li, Hui

AU - Tolman, Susan

N1 - Funding Information:
The first author would like to thank the University of Luxembourg, and particularly the University of Illinois at Urbana-Champaign for financial support while she was visiting the second author. We thank Professor Ono for suggesting several references. The second author was partially supported by National Science Foundation Grant DMS #07-07122.

PY - 2012

Y1 - 2012

N2 - Consider an effective Hamiltonian circle action on a compact symplectic 2n-dimensional manifold (M, ω). Assume that the fixed set MS1 is minimal, in two senses: It has exactly two components, X and Y , and dim(X) + dim(Y ) = dim(M) - 2. We prove that the integral cohomology ring and Chern classes of M are isomorphic to either those of CPn or (if n ≠ 1 is odd) to those of G2(Rn+2), the Grassmannian of oriented two-planes in Rn+2. In particular, Hi(M; Z) = H i(CPn; Z) for all i, and the Chern classes of M are determined by the integral cohomology ring. We also prove that the fixed set data of M agrees exactly with the fixed set data for one of the standard circle actions on one of these two manifolds. In particular, we show that there are no points with stabilizer Zk for any k > 2. The same conclusions hold when MS1 has exactly two components and the even Betti numbers of M are minimal, that is, b2i(M) = 1 for all i ∈ {0, . . . , 1/2 dim(M)}. This provides additional evidence that very few symplectic manifolds with minimal even Betti numbers admit Hamiltonian actions.

AB - Consider an effective Hamiltonian circle action on a compact symplectic 2n-dimensional manifold (M, ω). Assume that the fixed set MS1 is minimal, in two senses: It has exactly two components, X and Y , and dim(X) + dim(Y ) = dim(M) - 2. We prove that the integral cohomology ring and Chern classes of M are isomorphic to either those of CPn or (if n ≠ 1 is odd) to those of G2(Rn+2), the Grassmannian of oriented two-planes in Rn+2. In particular, Hi(M; Z) = H i(CPn; Z) for all i, and the Chern classes of M are determined by the integral cohomology ring. We also prove that the fixed set data of M agrees exactly with the fixed set data for one of the standard circle actions on one of these two manifolds. In particular, we show that there are no points with stabilizer Zk for any k > 2. The same conclusions hold when MS1 has exactly two components and the even Betti numbers of M are minimal, that is, b2i(M) = 1 for all i ∈ {0, . . . , 1/2 dim(M)}. This provides additional evidence that very few symplectic manifolds with minimal even Betti numbers admit Hamiltonian actions.

KW - Chern classes

KW - Equivariant cohomology

KW - Hamiltonian circle action

KW - Moment map

KW - Symplectic manifold

KW - Symplectic quotient

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U2 - 10.1142/S0129167X12500711

DO - 10.1142/S0129167X12500711

M3 - Article

AN - SCOPUS:84876246103

VL - 23

JO - International Journal of Mathematics

JF - International Journal of Mathematics

SN - 0129-167X

IS - 8

ER -