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1. CJM 2013 (vol 66 pp. 31)
Symplectic Foliations and Generalized Complex Structures We answer the natural question: when is a transversely holomorphic
symplectic foliation induced by a generalized complex structure? The
leafwise symplectic form and transverse complex structure determine an
obstruction class in a certain cohomology, which vanishes if and only
if our question has an affirmative answer. We first study a component
of this obstruction, which gives the condition that the leafwise
cohomology class of the symplectic form must be transversely
pluriharmonic. As a consequence, under certain topological
hypotheses, we infer that we actually have a symplectic fibre bundle
over a complex base. We then show how to compute the full obstruction
via a spectral sequence. We give various concrete necessary and
sufficient conditions for the vanishing of the obstruction.
Throughout, we give examples to test the sharpness of these
conditions, including a symplectic fibre bundle over a complex base
which does not come from a generalized complex structure, and a
regular generalized complex structure which is very unlike a
symplectic fibre bundle, i.e., for which nearby leaves are not
symplectomorphic.
Keywords:differential geometry, symplectic geometry, mathematical physics Category:53D18 |
2. CJM 2012 (vol 65 pp. 1255)
Variations of Integrals in Diffeology We establish the formula for the variation of
integrals of differential forms on cubic chains, in the
context of diffeological spaces. Then, we establish the diffeological version of Stoke's
theorem, and we apply that to get the diffeological variant of the
Cartan-Lie formula. Still in the context of Cartan-De-Rham calculus
in diffeology, we
construct a Chain-Homotopy Operator $\mathbf K$ we apply it here to
get the homotopic invariance of De Rham cohomology for
diffeological spaces. This is the Chain-Homotopy Operator which used in
symplectic diffeology to construct the Moment Map.
Keywords:diffeology, differential geometry, Cartan-De-Rham calculus Categories:58A10, 58A12, 58A40 |