Infinitesimal derivative of the Bott class and the Schwarzian derivatives
Taro Asuke
Source: Tohoku Math. J. (2) Volume 61, Number 3 (2009), 393-416.
Abstract
An infinitesimal derivative of the Bott class is defined by generalizing Heitsch'es construction. We prove a formula relating the infinitesimal derivative to the Schwarzian derivatives, which gives a generalization of the Maszczyk formula for the Godbillon-Vey class of real codimension-one foliations. As an application, a residue of infinitesimal derivatives with respect to the Julia set in the sense of Ghys, Gomez-Mont and Saludes is introduced.
Primary Subjects: 58H10
Secondary Subjects: 32S65, 53B10
Keywords: Infinitesimal deformations; Bott class; Schwarzian derivatives
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Permanent link to this document: http://projecteuclid.org/euclid.tmj/1255700201
Digital Object Identifier: doi:10.2748/tmj/1255700201
References
T. Asuke, On the real secondary classes of transversely holomorphic foliations, Ann. Inst. Fourier, Grenoble 50 (2000), 995--1017. %
Mathematical Reviews (MathSciNet):
MR1779903
T. Asuke, A remark on the Bott class, Ann. Fac. Sci. Toulouse X (2001), 5--21. %
Mathematical Reviews (MathSciNet):
MR1928986
T. Asuke, Residues of the Bott class and an application to the Futaki invariant, Asian J. Math. 7 (2003), 239--268. %
Mathematical Reviews (MathSciNet):
MR2014966
Zentralblatt MATH:
1086.32027
T. Asuke, On quasiconformal deformations of transversely holomorphic foliations, Jour. Math. Soc. Japan 57 (2005), 725--734. %
Mathematical Reviews (MathSciNet):
MR2139731
Zentralblatt MATH:
1160.37374
Digital Object Identifier: doi:10.2969/jmsj/1158241932
Project Euclid: euclid.jmsj/1158241932
T. Asuke, Godbillon-Vey class of transversely holomorphic foliations, preprint. %
C. Benson and D. B. Ellis, Characteristic classes of transversely homogeneous foliations, Trans. Amer. Math. Soc. 289 (1985), 849--859. %
Mathematical Reviews (MathSciNet):
MR784016
Zentralblatt MATH:
0573.57013
Digital Object Identifier: doi:10.2307/2000265
JSTOR: links.jstor.org
S. Bouarroudj and V. Yu. Ovsienko, Schwarzian derivative related to modules of differential operators on a locally projective manifold, Poisson geometry (Warsaw, 1998), 15--23, Banach Center Publ., 51, Polish Acad. Sci., Warsaw, 2000. %
Mathematical Reviews (MathSciNet):
MR1764429
Zentralblatt MATH:
1024.17016
R. Bott, Lectures on characteristic classes and foliations, Notes by Lawrence Conlon, with two appendices by J. Stasheff, Lectures on algebraic and differential topology (Second Latin American School in Math., Mexico City, 1971), 1--94, Lecture Notes in Math. 279, Springer, Berlin, 1972. %
Mathematical Reviews (MathSciNet):
MR362335
Zentralblatt MATH:
0241.57010
T. Duchamp and M. Kalka, Deformation theory for holomorphic foliations, J. Differential Geom. 14 (1979), 317--337. %
Mathematical Reviews (MathSciNet):
MR594704
Zentralblatt MATH:
0451.57015
Project Euclid: euclid.jdg/1214435099
H. Flanders, The Schwarzian as a curvature, J. Differential Geom. 4 (1970), 515--519. %
Mathematical Reviews (MathSciNet):
MR276879
Zentralblatt MATH:
0232.53005
Project Euclid: euclid.jdg/1214429647
A. Futaki and S. Morita, Invariant polynomials of the automorphism group of a compact complex manifold, J. Differential Geom. 21 (1985), 135--142. %
Mathematical Reviews (MathSciNet):
MR806707
Zentralblatt MATH:
0598.53055
Project Euclid: euclid.jdg/1214439469
A. Futaki, On a character of the automorphism group of a compact complex manifold, Invent. Math. 87 (1987), 655--660. %
Mathematical Reviews (MathSciNet):
MR874041
Zentralblatt MATH:
0612.53044
Digital Object Identifier: doi:10.1007/BF01389247
É. Ghys, X. Gómez-Mont and J. Saludes, Fatou and Julia components of transversely holomorphic foliations, Essays on geometry and related topics: Memoires dediés à André Haefliger (É. Ghys, P. de la Harpe, V. F. R. Jones, V. Sergiescu and T. Tsuboi, eds.), 287--319, Monog. Enseign. Math. 38, 2001. %
Mathematical Reviews (MathSciNet):
MR1929331
Zentralblatt MATH:
1013.37043
J. Heitsch, A cohomology for foliated manifolds, Comment. Math. Helv. 15 (1975), 197--218. %
Mathematical Reviews (MathSciNet):
MR372877
Zentralblatt MATH:
0311.57014
Digital Object Identifier: doi:10.1007/BF02565746
J. Heitsch, Derivatives of secondary characteristic classes, J. Differential Geom. 13 (1978), 311--339. %
Mathematical Reviews (MathSciNet):
MR551563
Zentralblatt MATH:
0407.57024
Project Euclid: euclid.jdg/1214434602
J. Heitsch, Independent variation of secondary classes, Ann. of Math. 108 (1978), 421--460. %
Mathematical Reviews (MathSciNet):
MR512428
Digital Object Identifier: doi:10.2307/1971183
JSTOR: links.jstor.org
J. Heitsch, A residue formula for holomorphic foliations, Michigan Math. J. 27 (1980), 181--194. %
Mathematical Reviews (MathSciNet):
MR568640
Digital Object Identifier: doi:10.1307/mmj/1029002356
Project Euclid: euclid.mmj/1029002356
S. Kobayashi and T. Nagano, On projective connections, J. Math. Mech. 13 (1964), 215--235. %
Mathematical Reviews (MathSciNet):
MR159284
Zentralblatt MATH:
0117.39101
T. Maszczyk, Foliations with rigid Godbillon-Vey class, Math. Z. 230 (1999), 329--344. %
Mathematical Reviews (MathSciNet):
MR1676718
Zentralblatt MATH:
0940.57030
Digital Object Identifier: doi:10.1007/PL00004695
R. Molzon and K. P. Mortensen, Differential operators associated with holomorphic mappings, Ann. Global Anal. Geom. 12 (1994), 291--304. %
Mathematical Reviews (MathSciNet):
MR1295104
Zentralblatt MATH:
0830.53009
Digital Object Identifier: doi:10.1007/BF02108302
T. Oda, On Schwarzian derivatives in several variables (Japanese), Res. Inst. Math. Sci., Kyoto Univ., Kyoto, Kôkyûroku 226 (1974), 82--85.
V. Ovsienko and S. Tabachnikov, Projective differential geometry old and new, from the Schwarzian derivative to the cohomology of diffeomorphism groups, Cambridge Tracts in Math. 165, Cambridge University Press, Cambridge, 2005. %
Mathematical Reviews (MathSciNet):
MR2177471
M. Yoshida, Canonical forms of some systems of linear partial differential equations, Proc. Japan Acad. 52 (1976), 473--476.
Mathematical Reviews (MathSciNet):
MR426013
Digital Object Identifier: doi:10.3792/pja/1195518208
Project Euclid: euclid.pja/1195518208
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