71 research outputs found

    Finite group actions on homology spheres and manifolds with nonzero Euler characteristic

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    Let XX be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that Diff(X)Diff(X) is Jordan. This means that there exists a constant CC such that any finite subgroup GG of Diff(X)Diff(X) has an abelian subgroup whose index in GG is at most CC. Using a result of Randall and Petrie we deduce that the automorphism groups of connected, non necessarily compact, smooth real affine varieties with nonzero Euler characteristic are Jordan.Comment: 17 pages; v4: the previous version v3 has been substantially revised and split in two parts (roughly coinciding with arXiv:1403.0383v2 and arXiv:1310.6565); this is one of the two parts; a corollary on algebraic actions on smooth real affine manifolds has been added; v5: final version, accepted for publication by Journal of Topolog

    The biinvariant diagonal class for Hamiltonian torus actions

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    Suppose that an algebraic torus GG acts algebraically on a projective manifold XX with generically trivial stabilizers. Then the Zariski closure of the set of pairs {(x,y)∈X×X∣y=gxfor someg∈G}\{(x,y)\in X\times X\mid y=gx \text{for some}g\in G\} defines a nonzero equivariant cohomology class [ΔG]∈HG×G∗(X×X)[\Delta_G]\in H^*_{G\times G}(X\times X). We give an analogue of this construction in the case where XX is a compact symplectic manifold endowed with a hamiltonian action of a torus, whose complexification plays the role of GG. We also prove that the Kirwan map sends the class [ΔG][\Delta_G] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.Comment: A substatially revised version of the paper "A right inverse to the Kirwan map". Improved exposition. Singular quotients are also considered in the new version. 23 pages. Accepted for publication in Adv. in Mat

    Lifts of smooth group actions to line bundles

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    Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L→XL\to X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c\sb 1(L) of L can be lifted to an integral equivariant cohomology class in H\sp 2\sb G(X;\ZZ), and that the different lifts of the action are classified by the lifts of c\sb 1(L) to H\sp 2\sb G(X;\ZZ). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇\nabla is a connection on L which is unitary for some metric on L and whose curvature is G-invariant, then there is a lift of the action to a certain power L\sp d (where d is independent of L) which leaves fixed the induced metric on LdL^d and the connection \nabla\sp{\otimes d}. This generalises to symplectic geometry a well known result in Geometric Invariant Theory.Comment: 12 page

    Finite groups acting symplectically on T2×S2T^2\times S^2

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    For any symplectic form ω\omega on T2×S2T^2\times S^2 we construct infinitely many nonisomorphic finite groups which admit effective smooth actions on T2×S2T^2\times S^2 that are trivial in cohomology but which do not admit any effective symplectic action on (T2×S2,ω)(T^2\times S^2,\omega). We also prove that for any ω\omega there is another symplectic form ω′\omega' on T2×S2T^2\times S^2 and a finite group acting symplectically and effectively on (T2×S2,ω′)(T^2\times S^2,\omega') which does not admit any effective symplectic action on (T2×S2,ω)(T^2\times S^2,\omega). A basic ingredient in our arguments is the study of the Jordan property of the symplectomorphism groups of T2×S2T^2\times S^2. A group GG is Jordan if there exists a constant CC such that any finite subgroup Γ\Gamma of GG contains an abelian subgroup whose index in Γ\Gamma is at most CC. Csik\'os, Pyber and Szab\'o proved recently that the diffeomorphism group of T2×S2T^2\times S^2 is not Jordan. We prove that, in contrast, for any symplectic form ω\omega on T2×S2T^2\times S^2 the group of symplectomorphisms Symp(T2×S2,ω)Symp(T^2\times S^2,\omega) is Jordan. We also give upper and lower bounds for the optimal value of the constant CC in Jordan's property for Symp(T2×S2,ω)Symp(T^2\times S^2,\omega) depending on the cohomology class represented by ω\omega. Our bounds are sharp for a large class of symplectic forms on T2×S2T^2\times S^2.Comment: 24 pages; v2: substantial revision; results improved: we give concrete (often sharp) values for the constants in the estimates in the main theorems; v3: title and abstract changed, included corrections and improvements suggested by the referee, added an appendix with a geometric interpretation of the automorphisms of the Heisenberg group; to appear in Trans. AM

    Finite subgroups of Ham and Symp

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    Let (X,ω)(X,\omega) be a compact symplectic manifold of dimension 2n2n and let Ham(X,ω)Ham(X,\omega) be its group of Hamiltonian diffeomorphisms. We prove the existence of a constant CC, depending on XX but not on ω\omega, such that any finite subgroup G⊂Ham(X,ω)G\subset Ham(X,\omega) has an abelian subgroup A⊆GA\subseteq G satisfying [G:A]≤C[G:A]\leq C, and AA can be generated by nn elements or fewer. If b1(X)=0b_1(X)=0 we prove an analogous statement for the entire group of symplectomorphisms of (X,ω)(X,\omega). If b1(X)≠0b_1(X)\neq 0 we prove the existence of a constant C′C' depending only on XX such that any finite subgroup G⊂Symp(X,ω)G\subset Symp(X,\omega) has a subgroup N⊆GN\subseteq G which is either abelian or 22-step nilpotent and which satisfies [G:N]≤C′[G:N]\leq C'. These results are deduced from the classification of the finite simple groups, the topological rigidity of hamiltonian loops, and the following theorem, which we prove in this paper. Let EE be a complex vector bundle over a compact, connected, smooth and oriented manifold MM; suppose that the real rank of EE is equal to the dimension of MM, and that ⟨e(E),[M]⟩≠0\langle e(E),[M]\rangle\neq 0, where e(E)e(E) is the Euler class of EE; then there exists a constant C"C" such that, for any prime pp and any finite pp-group GG acting on EE by vector bundle automorphisms preserving an almost complex structure on MM, there is a subgroup G0⊆GG_0\subseteq G satisfying MG0≠∅M^{G_0}\neq\emptyset and [G:G0]≤C"[G:G_0]\leq C".Comment: 42 pages; v2 substantial revision incorporating referee's comments; proof of Theorem 1.6 correcte

    A Hitchin-Kobayashi correspondence for Kaehler fibrations

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    Let XX be a compact Kaehler manifold and E→XE\to X a principal KK bundle, where KK is a compact connected Lie group. Let A1,1{\cal A}^{1,1} be the set of connections on EE whose curvature lies in Ω1,1(E×Adk)\Omega^{1,1}(E\times_{Ad} {\frak k}), where k{\frak k} is the Lie algebra of KK. Endow k\frak k with a nondegenerate biinvariant bilinear pairing. This allows to identify \{\frak k}\simeq{\frak k}^*. Let FF be a Kaehler left KK-manifold and suppose that there exists a moment map μ\mu for the action of KK on FF. Let S=Γ(E×KF){\cal S}=\Gamma(E\times_K F). In this paper we study the equation ΛFA+μ(Φ)=c\Lambda F_A+\mu(\Phi)=c for A∈A1,1A\in {\cal A}^{1,1} and a section Φ∈S\Phi\in {\cal S}, where c∈kc\in{\frak k} is a fixed central element. We study which orbits of the action of the complex gauge group on calA1,1×S{cal A}^{1,1}\times{\cal S} contain solutions of the equation, and we define a positive functional on calA1,1×S{cal A}^{1,1}\times{\cal S} which generalises the Yang-Mills-Higgs functional and whose local minima coincide with the solutions of the equation.Comment: 41 pages, no figures, Latex2

    Finite group actions on 4-manifolds with nonzero Euler characteristic

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    We prove that if XX is a compact, oriented, connected 44-dimensional smooth manifold, possibly with boundary, satisfying χ(X)≠0\chi(X)\neq 0, then there exists an integer C≥1C\geq 1 such that any finite group GG acting smoothly and effectively on XX has an abelian subgroup AA satisfying [G:A]≤C[G:A]\leq C, χ(XA)=χ(X)\chi(X^A)=\chi(X), and AA can be generated by at most 22 elements. Furthermore, if χ(X)<0\chi(X)<0 then AA is cyclic. This proves, for any such XX, a conjecture of Ghys. We also prove an analogous result for manifolds of arbitrary dimension and non-vanishing Euler characteristic, but restricted to pseudofree actions.Comment: 18 pages, v2: the main theorem has been strengthened for manifolds with negative Euler characteristic; a gap has been corrected in the proof of Lemma 6.1 of v1, which in v2 has been split in Lemmas 6.1 and 6.2; part of the introduction has been rewritten; some other minor changes; v3: final version, substantial revision of v2, to appear in Mathematische Zeitschrif

    Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds

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    A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves that the diffeomorphism group of T2×S2T^2\times S^2 is not Jordan. The purpose of this paper is to generalize the arguments of Csik\'os, Pyber and Szab\'o in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any ϵ>0\epsilon>0 there exist manifolds admitting effective actions of arbitrarily large pp-groups Γ\Gamma all of whose abelian subgroups have at most ∣Γ∣ϵ|\Gamma|^{\epsilon} elements. Finally, we also recover some results on nonexistence of effective actions of compact connected semisimple Lie group on manifolds.Comment: 14 page

    Automorphisms of generic gradient vector fields with prescribed finite symmetries

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    Let MM be a compact and connected smooth manifold endowed with a smooth action of a finite group Γ\Gamma, and let ff be a Γ\Gamma-invariant Morse function on MM. We prove that the space of Γ\Gamma-invariant Riemannian metrics on MM contains a residual subset Metf{\mathcal Met}_f with the following property. Let g∈Metfg\in{\mathcal Met}_f and let ∇gf\nabla^gf be the gradient vector field of ff with respect to gg. For any diffeomorphism ϕ\phi of MM preserving ∇gf\nabla^gf there exists some real number tt and some γ∈Γ\gamma\in\Gamma such that for every x∈Mx\in M we have ϕ(x)=γ Φtg(x)\phi(x)=\gamma\,\Phi_t^g(x), where Φtg\Phi_t^g is the time-tt flow of the vector field ∇gf\nabla^gf.Comment: 37 pages. Comments welcome; v2: one reference added, cosmetic changes in the introduction; v3: substantial simplification of the proof following the referee's suggestion, to appear in the Revista Matem\'atica Iberoamerican

    Hamiltonian Gromov-Witten invariants

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    In this paper we introduce invariants of semi-free Hamiltonian actions of S\sp 1 on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the holomorphicity equation used in Gromov-Witten theory. In the definition of the invariants we combine ideas coming from gauge theory and the ideas underlying the construction of Gromov-Witten invariants. This paper is based on a part of my PhD Thesis (see math/9912150).Comment: 36 page
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