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

Let $X$ 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 $Diff(X)$ is Jordan. This means that there exists a constant $C$ such that any finite subgroup $G$ of $Diff(X)$ has an abelian subgroup whose index in $G$ is at most $C$. 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

Suppose that an algebraic torus $G$ acts algebraically on a projective manifold $X$ with generically trivial stabilizers. Then the Zariski closure of the set of pairs $\{(x,y)\in X\times X\mid y=gx \text{for some}g\in G\}$ defines a nonzero equivariant cohomology class $[\Delta_G]\in H^*_{G\times G}(X\times X)$. We give an analogue of this construction in the case where $X$ is a compact symplectic manifold endowed with a hamiltonian action of a torus, whose complexification plays the role of $G$. We also prove that the Kirwan map sends the class $[\Delta_G]$ 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

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let $L\to X$ 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 $L^d$ 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

For any symplectic form $\omega$ on $T^2\times S^2$ we construct infinitely many nonisomorphic finite groups which admit effective smooth actions on $T^2\times S^2$ that are trivial in cohomology but which do not admit any effective symplectic action on $(T^2\times S^2,\omega)$. We also prove that for any $\omega$ there is another symplectic form $\omega'$ on $T^2\times S^2$ and a finite group acting symplectically and effectively on $(T^2\times S^2,\omega')$ which does not admit any effective symplectic action on $(T^2\times S^2,\omega)$. A basic ingredient in our arguments is the study of the Jordan property of the symplectomorphism groups of $T^2\times S^2$. A group $G$ is Jordan if there exists a constant $C$ such that any finite subgroup $\Gamma$ of $G$ contains an abelian subgroup whose index in $\Gamma$ is at most $C$. Csik\'os, Pyber and Szab\'o proved recently that the diffeomorphism group of $T^2\times S^2$ is not Jordan. We prove that, in contrast, for any symplectic form $\omega$ on $T^2\times S^2$ the group of symplectomorphisms $Symp(T^2\times S^2,\omega)$ is Jordan. We also give upper and lower bounds for the optimal value of the constant $C$ in Jordan's property for $Symp(T^2\times S^2,\omega)$ depending on the cohomology class represented by $\omega$. Our bounds are sharp for a large class of symplectic forms on $T^2\times S^2$.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

Let $(X,\omega)$ be a compact symplectic manifold of dimension $2n$ and let $Ham(X,\omega)$ be its group of Hamiltonian diffeomorphisms. We prove the existence of a constant $C$, depending on $X$ but not on $\omega$, such that any finite subgroup $G\subset Ham(X,\omega)$ has an abelian subgroup $A\subseteq G$ satisfying $[G:A]\leq C$, and $A$ can be generated by $n$ elements or fewer. If $b_1(X)=0$ we prove an analogous statement for the entire group of symplectomorphisms of $(X,\omega)$. If $b_1(X)\neq 0$ we prove the existence of a constant $C'$ depending only on $X$ such that any finite subgroup $G\subset Symp(X,\omega)$ has a subgroup $N\subseteq G$ which is either abelian or $2$-step nilpotent and which satisfies $[G:N]\leq C'$. 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 $E$ be a complex vector bundle over a compact, connected, smooth and oriented manifold $M$; suppose that the real rank of $E$ is equal to the dimension of $M$, and that $\langle e(E),[M]\rangle\neq 0$, where $e(E)$ is the Euler class of $E$; then there exists a constant $C"$ such that, for any prime $p$ and any finite $p$-group $G$ acting on $E$ by vector bundle automorphisms preserving an almost complex structure on $M$, there is a subgroup $G_0\subseteq G$ satisfying $M^{G_0}\neq\emptyset$ and $[G:G_0]\leq C"$.Comment: 42 pages; v2 substantial revision incorporating referee's comments; proof of Theorem 1.6 correcte

A Hitchin-Kobayashi correspondence for Kaehler fibrations

Let $X$ be a compact Kaehler manifold and $E\to X$ a principal $K$ bundle, where $K$ is a compact connected Lie group. Let ${\cal A}^{1,1}$ be the set of connections on $E$ whose curvature lies in $\Omega^{1,1}(E\times_{Ad} {\frak k})$, where ${\frak k}$ is the Lie algebra of $K$. Endow $\frak k$ with a nondegenerate biinvariant bilinear pairing. This allows to identify \{\frak k}\simeq{\frak k}^*. Let $F$ be a Kaehler left $K$-manifold and suppose that there exists a moment map $\mu$ for the action of $K$ on $F$. Let ${\cal S}=\Gamma(E\times_K F)$. In this paper we study the equation $$\Lambda F_A+\mu(\Phi)=c$$ for $A\in {\cal A}^{1,1}$ and a section $\Phi\in {\cal S}$, where $c\in{\frak k}$ is a fixed central element. We study which orbits of the action of the complex gauge group on ${cal A}^{1,1}\times{\cal S}$ contain solutions of the equation, and we define a positive functional on ${cal A}^{1,1}\times{\cal S}$ 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

We prove that if $X$ is a compact, oriented, connected $4$-dimensional smooth manifold, possibly with boundary, satisfying $\chi(X)\neq 0$, then there exists an integer $C\geq 1$ such that any finite group $G$ acting smoothly and effectively on $X$ has an abelian subgroup $A$ satisfying $[G:A]\leq C$, $\chi(X^A)=\chi(X)$, and $A$ can be generated by at most $2$ elements. Furthermore, if $\chi(X)<0$ then $A$ is cyclic. This proves, for any such $X$, 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

Hamiltonian Gromov-Witten invariants

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

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

A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ 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 $\epsilon>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $\Gamma$ all of whose abelian subgroups have at most $|\Gamma|^{\epsilon}$ 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

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $\Gamma$, and let $f$ be a $\Gamma$-invariant Morse function on $M$. We prove that the space of $\Gamma$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal Met}_f$ with the following property. Let $g\in{\mathcal Met}_f$ and let $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $\phi$ of $M$ preserving $\nabla^gf$ there exists some real number $t$ and some $\gamma\in\Gamma$ such that for every $x\in M$ we have $\phi(x)=\gamma\,\Phi_t^g(x)$, where $\Phi_t^g$ is the time-$t$ flow of the vector field $\nabla^gf$.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