## Transversely projective structures on a transversely holomorphic foliation, II

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- by Indranil Biswas
- Conform. Geom. Dyn.
**6**(2002), 61-73 - DOI: https://doi.org/10.1090/S1088-4173-02-00085-1
- Published electronically: August 7, 2002
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## Abstract:

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.## References

- Bi1 I. Biswas, Transversely projective structures on a transversely holomorphic foliation,
- I. Biswas,
*Differential operators on complex manifolds with a flat projective structure*, J. Math. Pures Appl. (9)**78**(1999), no. 1, 1–26. MR**1671218**, DOI 10.1016/S0021-7824(99)80007-5 - B. Azevedo Scárdua and C. Camacho,
*Holomorphic foliations and Kupka singular sets*, Comm. Anal. Geom.**7**(1999), no. 3, 623–640. MR**1698391**, DOI 10.4310/CAG.1999.v7.n3.a6 - T. Duchamp and M. Kalka,
*Deformation theory for holomorphic foliations*, J. Differential Geometry**14**(1979), no. 3, 317–337 (1980). MR**594704**, DOI 10.4310/jdg/1214435099 - J. Girbau, A. Haefliger, and D. Sundararaman,
*On deformations of transversely holomorphic foliations*, J. Reine Angew. Math.**345**(1983), 122–147. MR**717890**, DOI 10.1515/crll.1983.345.122 - Xavier Gómez-Mont,
*Transversal holomorphic structures*, J. Differential Geometry**15**(1980), no. 2, 161–185 (1981). MR**614365** - Xavier Gómez-Mont,
*The transverse dynamics of a holomorphic flow*, Ann. of Math. (2)**127**(1988), no. 1, 49–92. MR**924673**, DOI 10.2307/1971416 - R. C. Gunning,
*Affine and projective structures on Riemann surfaces*, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 225–244. MR**624816** - André Haefliger,
*Homotopy and integrability*, Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 133–163. MR**0285027** - B. Azevedo Scárdua,
*Transversely affine and transversely projective holomorphic foliations*, Ann. Sci. École Norm. Sup. (4)**30**(1997), no. 2, 169–204. MR**1432053**, DOI 10.1016/S0012-9593(97)89918-1

*Conform. Geom. Dyn.*

**5**(2001), 74–80.

## Bibliographic Information

**Indranil Biswas**- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
- MR Author ID: 340073
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**6**(2002), 61-73 - MSC (2000): Primary 37F75; Secondary 53B10
- DOI: https://doi.org/10.1090/S1088-4173-02-00085-1
- MathSciNet review: 1948849