71 research outputs found
Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
Let 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 is Jordan. This means that there
exists a constant such that any finite subgroup of has an
abelian subgroup whose index in is at most . 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 acts algebraically on a projective
manifold with generically trivial stabilizers. Then the Zariski closure of
the set of pairs
defines a nonzero equivariant cohomology class . We give an analogue of this construction in the case where
is a compact symplectic manifold endowed with a hamiltonian action of a torus,
whose complexification plays the role of . We also prove that the Kirwan map
sends the class 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 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 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
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
For any symplectic form on we construct infinitely
many nonisomorphic finite groups which admit effective smooth actions on
that are trivial in cohomology but which do not admit any
effective symplectic action on . We also prove that for
any there is another symplectic form on and
a finite group acting symplectically and effectively on which does not admit any effective symplectic action on
.
A basic ingredient in our arguments is the study of the Jordan property of
the symplectomorphism groups of . A group is Jordan if there
exists a constant such that any finite subgroup of contains an
abelian subgroup whose index in is at most . Csik\'os, Pyber and
Szab\'o proved recently that the diffeomorphism group of is not
Jordan. We prove that, in contrast, for any symplectic form on
the group of symplectomorphisms is
Jordan. We also give upper and lower bounds for the optimal value of the
constant in Jordan's property for depending on
the cohomology class represented by . Our bounds are sharp for a large
class of symplectic forms on .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 be a compact symplectic manifold of dimension and let
be its group of Hamiltonian diffeomorphisms. We prove the
existence of a constant , depending on but not on , such that
any finite subgroup has an abelian subgroup
satisfying , and can be generated by
elements or fewer. If we prove an analogous statement for the entire
group of symplectomorphisms of . If we prove the
existence of a constant depending only on such that any finite
subgroup has a subgroup which is
either abelian or -step nilpotent and which satisfies .
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 be a complex vector bundle over
a compact, connected, smooth and oriented manifold ; suppose that the real
rank of is equal to the dimension of , and that , where is the Euler class of ; then there
exists a constant such that, for any prime and any finite -group
acting on by vector bundle automorphisms preserving an almost complex
structure on , there is a subgroup satisfying
and .Comment: 42 pages; v2 substantial revision incorporating referee's comments;
proof of Theorem 1.6 correcte
A Hitchin-Kobayashi correspondence for Kaehler fibrations
Let be a compact Kaehler manifold and a principal bundle,
where is a compact connected Lie group. Let be the set of
connections on whose curvature lies in , where is the Lie algebra of . Endow with a
nondegenerate biinvariant bilinear pairing. This allows to identify \{\frak
k}\simeq{\frak k}^*. Let be a Kaehler left -manifold and suppose that
there exists a moment map for the action of on . Let . In this paper we study the equation for and a section ,
where is a fixed central element. We study which orbits of the
action of the complex gauge group on contain
solutions of the equation, and we define a positive functional on 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 is a compact, oriented, connected -dimensional smooth
manifold, possibly with boundary, satisfying , then there exists
an integer such that any finite group acting smoothly and
effectively on has an abelian subgroup satisfying ,
, and can be generated by at most elements.
Furthermore, if then is cyclic. This proves, for any such ,
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
A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves
that the diffeomorphism group of 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
there exist manifolds admitting effective actions of arbitrarily large
-groups all of whose abelian subgroups have at most
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 be a compact and connected smooth manifold endowed with a smooth
action of a finite group , and let be a -invariant Morse
function on . We prove that the space of -invariant Riemannian
metrics on contains a residual subset with the following
property. Let and let be the gradient vector
field of with respect to . For any diffeomorphism of
preserving there exists some real number and some
such that for every we have
, where is the time- flow of the
vector field .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
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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