Conformal Geometry and Dynamics
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Transversely projective structures on a transversely holomorphic foliation, II
by Indranil Biswas PDF
Conform. Geom. Dyn. 6 (2002), 61-73 Request permission
Given a transversely projective foliation $\mathcal F$ on a $C^\infty$ manifold $M$ and a nonnegative integer $k$, a transversal differential operator ${\mathcal D}_{\mathcal F}(2k+1)$ of order $2k+1$ from $N^{\otimes k}$ to $N^{\otimes (-k-1)}$ is constructed, where $N$ denotes the normal bundle for the foliation. There is a natural homomorphism from the space of all infinitesimal deformations of the transversely projective foliation $\mathcal F$ to the first cohomology of the locally constant sheaf over $M$ defined by the kernel of the operator ${\mathcal D}_{\mathcal F}(3)$. On the other hand, from this first cohomology there is a homomorphism to the first cohomology of the sheaf of holomorphic sections of $N$. The composition of these two homomorphisms coincide with the infinitesimal version of the forgetful map that sends a transversely projective foliation to the underlying transversely holomorphic foliation.
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Indranil Biswas
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
Email: indranil@math.tifr.res.in
Received by editor(s): October 22, 2001
Received by editor(s) in revised form: June 24, 2002
Published electronically: August 7, 2002
Journal: Conform. Geom. Dyn. 6 (2002), 61-73
MSC (2000): Primary 37F75; Secondary 53B10