Transversely projective structures on a transversely holomorphic foliation, II
Author:
Indranil Biswas
Journal:
Conform. Geom. Dyn. 6 (2002), 6173
MSC (2000):
Primary 37F75; Secondary 53B10
Published electronically:
August 7, 2002
MathSciNet review:
1948849
Fulltext PDF Free Access
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), 7480.
 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
(99m:14034), http://dx.doi.org/10.1016/S00217824(99)800075
 3.
B.
Azevedo Scárdua and C.
Camacho, Holomorphic foliations and Kupka singular sets, Comm.
Anal. Geom. 7 (1999), no. 3, 623–640. MR 1698391
(2000f:32043)
 4.
T.
Duchamp and M.
Kalka, Deformation theory for holomorphic foliations, J.
Differential Geom. 14 (1979), no. 3, 317–337
(1980). MR
594704 (82b:57019)
 5.
J.
Girbau, A.
Haefliger, and D.
Sundararaman, On deformations of transversely holomorphic
foliations, J. Reine Angew. Math. 345 (1983),
122–147. MR
717890 (84j:32026), http://dx.doi.org/10.1515/crll.1983.345.122
 6.
Xavier
GómezMont, Transversal holomorphic structures, J.
Differential Geom. 15 (1980), no. 2, 161–185
(1981). MR
614365 (82j:53065)
 7.
Xavier
GómezMont, The transverse dynamics of a holomorphic
flow, Ann. of Math. (2) 127 (1988), no. 1,
49–92. MR
924673 (89d:32049), http://dx.doi.org/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
(83g:30054)
 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 (44 #2251)
 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
(97k:32049), http://dx.doi.org/10.1016/S00129593(97)899181
 1.
 I. Biswas, Transversely projective structures on a transversely holomorphic foliation, Conform. Geom. Dyn. 5 (2001), 7480.
 2.
 I. Biswas, Differential operators on complex manifolds with a flat projective structure, J. Math. Pures Appl. 78 (1999), 126. MR 99m:14034
 3.
 C. Camacho and B. A. Scárdua, Holomorphic foliations and Kupka singular sets, Comm. Anal. Geom. 7 (1999), 623640. MR 2000f:32043
 4.
 T. Duchamp and M. Kalka, Deformation theory for holomorphic foliations, J. Differential Geom. 14 (1979), 317337. MR 82b:57019
 5.
 J. Girbau, A. Haefliger and D. Sundararaman, On deformations of transversely holomorphic foliations, J. Reine Angew. Math. 345 (1983), 122147. MR 84j:32026
 6.
 X. GómezMont, Transversal holomorphic structures, J. Differential Geom. 15 (1980), 161185. MR 82j:53065
 7.
 X. GómezMont, The transverse dynamics of a holomorphic flow, Ann. of Math. 127 (1988), 4992. MR 89d:32049
 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), pp. 225244, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981. MR 83g:30054
 9.
 A. Haefliger, Homotopy and integrability. ManifoldsAmsterdam 1970 (Proc. Nuffic Summer School) pp. 133163, Lecture Notes in Mathematics, Vol. 197 Springer, Berlin, 1971. MR 44:2251
 10.
 B. A. Scárdua, Transversely affine and transversely projective holomorphic foliations, Ann. Sci. École Norm. Sup. 30 (1997), 169204. MR 97k:32049
Similar Articles
Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society
with MSC (2000):
37F75,
53B10
Retrieve articles in all journals
with MSC (2000):
37F75,
53B10
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:
http://dx.doi.org/10.1090/S1088417302000851
PII:
S 10884173(02)000851
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
