Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski–West Construction

D. M. J. Calderbank

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

I present a construction of real or complex selfdual conformal $4$-manifolds (of signature $(2,2)$ in the real case) from a natural gauge field equation on a real or complex projective surface, the gauge group being the group of diffeomorphisms of a real or complex $2$-manifold. The $4$-manifolds obtained are characterized by the existence of a foliation by selfdual null surfaces of a special kind. The classification by Dunajski and West of selfdual conformal $4$-manifolds with a null conformal vector field is the special case in which the gauge group reduces to the group of diffeomorphisms commuting with a vector field, and I analyse the presence of compatible scalar-flat Kähler, hypercomplex and hyperkähler structures from a gauge-theoretic point of view. In an appendix, I discuss the twistor theory of projective surfaces, which is used in the body of the paper, but is also of independent interest.

Détail

Keywords: selfduality; twistor theory; integrable systems; projective geometry.

Bibliographie

[1] Bailey T. N., Eastwood M. G., Gover A. R., Mason L. J., “The {F}unk transform as a {P}enrose transform”, Math. Proc. Cambridge Philos. Soc., 125 (1999), 67–81 | DOI | MR | Zbl

[2] Baston R. J., Eastwood M. G., The {P}enrose transform: its interaction with representation theory, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1989 | MR | Zbl

[3] Calderbank D. M. J., Selfdual {E}instein metrics and conformal submersions, arXiv: math.DG/0001041

[4] Calderbank D. M. J., “Integrable background geometries”, SIGMA, 10 (2014), 034, 51 pp., arXiv: 1403.3471 | DOI | Zbl

[5] Calderbank D. M. J., Pedersen H., “Selfdual spaces with complex structures, {E}instein–{W}eyl geometry and geodesics”, Ann. Inst. Fourier (Grenoble), 50 (2000), 921–963, arXiv: math.DG/9911117 | DOI | MR | Zbl

[6] Čap A., “Correspondence spaces and twistor spaces for parabolic geometries”, J. Reine Angew. Math., 582 (2005), 143–172, arXiv: math.DG/0102097 | DOI | MR | Zbl

[7] Čap A., Slovák J., Souček V., “Bernstein–{G}elfand–{G}elfand sequences”, Ann. of Math., 154 (2001), 97–113, arXiv: math.DG/0001164 | DOI | MR | Zbl

[8] Dubois-Violette M., “Structures complexes au-dessus des variétés, applications”, Mathématiques et Physiques ({P}aris, 1979/1982), Progr. Math., 37, Birkhäuser Boston, Boston, MA, 1983, 1–42 | MR

[9] Dunajski M., “Anti-self-dual four-manifolds with a parallel real spinor”, Proc. Roy. Soc. London Ser. A, 458 (2002), 1205–1222, arXiv: math.DG/0102225 | DOI | MR | Zbl

[10] Dunajski M., West S., “Anti-self-dual conformal structures with null {K}illing vectors from projective structures”, Comm. Math. Phys., 272 (2007), 85–118, arXiv: math.DG/0601419 | DOI | MR | Zbl

[11] Dunajski M., West S., “Anti-self-dual conformal structures in neutral signature”, Recent Developments in Pseudo-{R}iemannian Geometry, ESI Lect. Math. Phys., eds. D. V. Alekseevsky, H. Baum, Eur. Math. Soc., Zürich, 2008, 113–148, arXiv: math.DG/0610280 | DOI | MR | Zbl

[12] Hitchin N. J., “Complex manifolds and {E}instein's equations”, Twistor Geometry and Nonlinear Systems ({P}rimorsko, 1980), Lecture Notes in Math., 970, eds. H. D. Doebner, T. D. Palev, Springer, Berlin, 1982, 73–99 | DOI | MR

[13] Hitchin N. J., “Geometrical aspects of {S}chlesinger's equation”, J. Geom. Phys., 23 (1997), 287–300 | DOI | MR | Zbl

[14] LeBrun C., Spaces of complex geodesics and related structures, Ph.D. Thesis, University of Oxford, 1980

[15] LeBrun C., Mason L. J., “Zoll manifolds and complex surfaces”, J. Differential Geom., 61 (2002), 453–535, arXiv: math.DG/0211021 | MR | Zbl

[16] LeBrun C., Mason L. J., “Nonlinear gravitons, null geodesics, and holomorphic disks”, Duke Math. J., 136 (2007), 205–273, arXiv: math.DG/0504582 | DOI | MR | Zbl

[17] Mason L. J., Woodhouse N. M. J., “Integrability, self-duality, and twistor theory”, London Mathematical Society Monographs. New Series, 15, The Clarendon Press, Oxford University Press, New York, 1996 | MR

[18] Nakata F., “Self-dual {Z}ollfrei conformal structures with {$\alpha$}-surface foliation”, J. Geom. Phys., 57 (2007), 2077–2097, arXiv: math.DG/0701116 | DOI | MR | Zbl

[19] Pedersen H., Tod P., “Einstein metrics and hyperbolic monopoles”, Classical Quantum Gravity, 8 (1991), 751–760 | DOI | MR | Zbl

[20] Penrose R., “Nonlinear gravitons and curved twistor theory”, Gen. Relativity Gravitation, 7 (1976), 31–52 | DOI | MR | Zbl

[21] Przanowski M., “Killing vector fields in self-dual, {E}uclidean {E}instein spaces with {$\Lambda\neq 0$}”, J. Math. Phys., 32 (1991), 1004–1010 | DOI | MR | Zbl

[22] Tafel J., “Two-dimensional reductions of the self-dual {Y}ang–{M}ills equations in self-dual spaces”, J. Math. Phys., 34 (1993), 1892–1907 | DOI | MR | Zbl

[23] Tafel J., Wójcik D., “Null {K}illing vectors and reductions of the self-duality equations”, Nonlinearity, 11 (1998), 835–844 | DOI | MR | Zbl

[24] Tod K. P., “The {${\rm SU}(\infty)$}-{T}oda field equation and special four-dimensional metrics”, Geometry and Physics ({A}arhus, 1995), Lecture Notes in Pure and Appl. Math., 184, eds. J. E. Andersen, J. Dupont, H. Pedersen, A. Swann, Dekker, New York, 1997, 307–312 | MR | Zbl

[25] Ward R. S., “On self-dual gauge fields”, Phys. Lett. A, 61 (1977), 81–82 | DOI | MR | Zbl