Coupled equations for Kähler metrics and Yang–Mills connections

Álvarez-Cónsul, Luis¹ ; García-Fernández, Mario² ; García-Prada, Oscar³

¹ Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera 13–15, Campus Cantoblanco, 28049 Madrid, Spain
² MATHGEOM, École Polytechnique Fédéral de Lausanne, MA B1 437, Station 8, CH-1015 Lausanne, Switzerland
³ Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera 13–15, Campus Cantoblanco, 28049 Madrid, Spain

Geometry & topology, Tome 17 (2013) no. 5, pp. 2731-2812.

Résumé

We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kähler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kähler metric and Hermite–Yang–Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K–energy and geodesic stability. We finish by giving some examples of solutions.

Détail

DOI : 10.2140/gt.2013.17.2731

Classification : 32Q20, 53C07
Keywords: coupled Kähler–Yang–Mills equation, Kähler metric, Hermitian–Yang–Mills connection, generalized Futaki invariant, generalized Mabuchi energy

Bibliographie

[1] M C Abbati, R Cirelli, A Manià, P Michor, The Lie group of automorphisms of a principal bundle, J. Geom. Phys. 6 (1989) 215

[2] L Álvarez-Cónsul, M García-Fernández, O García-Prada, Gravitating vortices, in preparation

[3] B Anchouche, I Biswas, Einstein–Hermitian connections on polystable principal bundles over a compact Kähler manifold, Amer. J. Math. 123 (2001) 207

[4] M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523

[5] T Aubin, Some nonlinear problems in Riemannian geometry, Springer (1998)

[6] A L Besse, Einstein manifolds, 10, Springer (1987)

[7] J P Bourguignon, Invariants intégraux fonctionnels pour des équations aux dérivées partielles d’origine géométrique, from: "Partial differential equations, Part 1, 2" (editors B Bojarski, W Zajączkowski, B Ziemian), Banach Center Publ. 27, Polish Acad. Sci. (1992) 65

[8] E Calabi, Extremal Kähler metrics, from: "Seminar on Differential Geometry" (editor S T Yau), Ann. of Math. Stud. 102, Princeton Univ. Press (1982) 259

[9] L Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994) 589

[10] X X Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000) 189

[11] X X Chen, Space of Kähler metrics, III : On the lower bound of the Calabi energy and geodesic distance, Invent. Math. 175 (2009) 453

[12] X X Chen, S K Donaldson, S Sun, Kähler–Einstein metrics and stability

[13] X X Chen, G Tian, Geometry of Kähler metrics and foliations by holomorphic discs, Publ. Math. Inst. Hautes Études Sci. (2008) 1

[14] S K Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983) 269

[15] S K Donaldson, Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985) 1

[16] S K Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231

[17] S K Donaldson, Remarks on gauge theory, complex geometry and 4–manifold topology, from: "Fields Medallists’ lectures" (editors M F Atiyah, D Iagolnitzer), World Sci. Ser. 20th Century Math. 5, World Sci. Publ. (1997) 384

[18] S K Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, from: "Northern California Symplectic Geometry Seminar" (editors Y Eliashberg, D Fuchs, T Ratiu, A Weinstein), Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc. (1999) 13

[19] S K Donaldson, Scalar curvature and projective embeddings, I, J. Differential Geom. 59 (2001) 479

[20] S K Donaldson, P B Kronheimer, The geometry of four-manifolds, The Clarendon Press (1990)

[21] A Fujiki, Moduli space of polarized algebraic manifolds and Kähler metrics, Sugaku Expositions 5 (1992) 173

[22] A Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983) 437

[23] A Futaki, Asymptotic Chow semi-stability and integral invariants, Internat. J. Math. 15 (2004) 967

[24] A Futaki, H Ono, Einstein metrics and GIT stability, Sūgaku 60 (2008) 175

[25] M García-Fernández, Coupled equations for Kähler metrics and Yang–Mills connections, PhD thesis, Instituto de Ciencias Matemáticas, Madrid (2009)

[26] M García-Fernández, C Tipler, Deformation of complex structures and the coupled Kähler–Yang–Mills equations

[27] D Gieseker, I Morrison, Hilbert stability of rank-two bundles on curves, J. Differential Geom. 19 (1984) 1

[28] M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307

[29] Y J Hong, Constant Hermitian scalar curvature equations on ruled manifolds, J. Differential Geom. 53 (1999) 465

[30] Y J Hong, Stability and existence of critical Kaehler metrics on ruled manifolds, J. Math. Soc. Japan 60 (2008) 265

[31] L Huang, On joint moduli spaces, Math. Ann. 302 (1995) 61

[32] J Keller, C W Tønnesen-Friedman, Nontrivial examples of coupled equations for Kähler metrics and Yang–Mills connections, Cent. Eur. J. Math. 10 (2012) 1673

[33] G Kempf, L Ness, The length of vectors in representation spaces, from: "Algebraic geometry" (editor K Lønsted), Lecture Notes in Math. 732, Springer (1979) 233

[34] S Kobayashi, Differential geometry of complex vector bundles, 15, Princeton Univ. Press (1987)

[35] S Kobayashi, K Nomizu, Foundations of differential geometry, Vol. I, John Wiley (1963)

[36] S Kobayashi, K Nomizu, Foundations of differential geometry, Vol. II, 15, John Wiley (1969)

[37] C Lebrun, The Einstein–Maxwell equations, extremal Kähler metrics, and Seiberg–Witten theory, from: "The many facets of geometry" (editors O García-Prada, J P Bourguignon, S Salamon), Oxford Univ. Press (2010) 17

[38] C Lebrun, S R Simanca, On the Kähler classes of extremal metrics, from: "Geometry and global analysis" (editors T Kotake, S Nishikawa, R Schoen), Tohoku Univ. (1993) 255

[39] C Lebrun, S R Simanca, Extremal Kähler metrics and complex deformation theory, Geom. Funct. Anal. 4 (1994) 298

[40] J Li, S T Yau, The existence of supersymmetric string theory with torsion, J. Differential Geom. 70 (2005) 143

[41] T Mabuchi, K–energy maps integrating Futaki invariants, Tohoku Math. J. 38 (1986) 575

[42] T Mabuchi, Some symplectic geometry on compact Kähler manifolds, I, Osaka J. Math. 24 (1987) 227

[43] J E Marsden, G Misiołek, J P Ortega, M Perlmutter, T S Ratiu, Hamiltonian reduction by stages, 1913, Springer (2007)

[44] M Maruyama, Moduli of stable sheaves, II, J. Math. Kyoto Univ. 18 (1978) 557

[45] D Mcduff, D Salamon, Introduction to symplectic topology, The Clarendon Press (1998)

[46] R Pandharipande, A compactification over Mg of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996) 425

[47] A Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975) 129

[48] A Ramanathan, S Subramanian, Einstein–Hermitian connections on principal bundles and stability, J. Reine Angew. Math. 390 (1988) 21

[49] I Mundet I Riera, A Hitchin–Kobayashi correspondence for Kähler fibrations, J. Reine Angew. Math. 528 (2000) 41

[50] G Schumacher, M Toma, Moduli of Kähler manifolds equipped with Hermite–Einstein vector bundles, Rev. Roumaine Math. Pures Appl. 38 (1993) 703

[51] S Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992) 495

[52] I M Singer, The geometric interpretation of a special connection, Pacific J. Math. 9 (1959) 585

[53] J Stoppa, Twisted constant scalar curvature Kähler metrics and Kähler slope stability, J. Differential Geom. 83 (2009) 663

[54] A Teleman, Symplectic stability, analytic stability in non-algebraic complex geometry, Internat. J. Math. 15 (2004) 183

[55] G Tian, K–stability and Kähler–Einstein metrics

[56] G Tian, Canonical metrics in Kähler geometry, Birkhäuser (2000)

[57] G Tian, S T Yau, Kähler–Einstein metrics on complex surfaces with C1 > 0, Comm. Math. Phys. 112 (1987) 175

[58] K Uhlenbeck, S T Yau, On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986)

[59] X Wang, Moment map, Futaki invariant and stability of projective manifolds, Comm. Anal. Geom. 12 (2004) 1009

[60] Y S Yang, Prescribing topological defects for the coupled Einstein and abelian Higgs equations, Comm. Math. Phys. 170 (1995) 541

[61] S T Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978) 339