Transversely projective structures on a transversely holomorphic foliation, II

Author:
Indranil Biswas

Journal:
Conform. Geom. Dyn. **6** (2002), 61-73

MSC (2000):
Primary 37F75; Secondary 53B10

Published electronically:
August 7, 2002

MathSciNet review:
1948849

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Abstract | References | Similar Articles | Additional Information

Abstract: Given a transversely projective foliation on a manifold and a nonnegative integer , a transversal differential operator of order from to is constructed, where denotes the normal bundle for the foliation. There is a natural homomorphism from the space of all infinitesimal deformations of the transversely projective foliation to the first cohomology of the locally constant sheaf over defined by the kernel of the operator . On the other hand, from this first cohomology there is a homomorphism to the first cohomology of the sheaf of holomorphic sections of . 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.

**1.**I. Biswas, Transversely projective structures on a transversely holomorphic foliation,*Conform. Geom. Dyn.***5**(2001), 74-80.**2.**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**, 10.1016/S0021-7824(99)80007-5**3.**B. Azevedo Scárdua and C. Camacho,*Holomorphic foliations and Kupka singular sets*, Comm. Anal. Geom.**7**(1999), no. 3, 623–640. MR**1698391**, 10.4310/CAG.1999.v7.n3.a6**4.**T. Duchamp and M. Kalka,*Deformation theory for holomorphic foliations*, J. Differential Geom.**14**(1979), no. 3, 317–337 (1980). MR**594704****5.**J. Girbau, A. Haefliger, and D. Sundararaman,*On deformations of transversely holomorphic foliations*, J. Reine Angew. Math.**345**(1983), 122–147. MR**717890**, 10.1515/crll.1983.345.122**6.**Xavier Gómez-Mont,*Transversal holomorphic structures*, J. Differential Geom.**15**(1980), no. 2, 161–185 (1981). MR**614365****7.**Xavier Gómez-Mont,*The transverse dynamics of a holomorphic flow*, Ann. of Math. (2)**127**(1988), no. 1, 49–92. MR**924673**, 10.2307/1971416**8.**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****9.**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****10.**B. Azevedo Scárdua,*Transversely affine and transversely projective holomorphic foliations*, Ann. Sci. École Norm. Sup. (4)**30**(1997), no. 2, 169–204. MR**1432053**, 10.1016/S0012-9593(97)89918-1

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Additional Information

**Indranil Biswas**

Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

Email:
indranil@math.tifr.res.in

DOI:
https://doi.org/10.1090/S1088-4173-02-00085-1

Received by editor(s):
October 22, 2001

Received by editor(s) in revised form:
June 24, 2002

Published electronically:
August 7, 2002

Article copyright:
© Copyright 2002
American Mathematical Society