Geometry & Topology Volume 17, issue 5 (2013)

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Coupled equations for Kähler metrics and Yang–Mills connections

Luis Álvarez-Cónsul, Mario García-Fernández and Oscar García-Prada

Geometry & Topology 17 (2013) 2731–2812

DOI: 10.2140/gt.2013.17.2731

Bibliography

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 MR1040392
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 MR1828221
4	M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523 MR702806
5	T Aubin, Some nonlinear problems in Riemannian geometry, Springer (1998) MR1636569
6	A L Besse, Einstein manifolds, 10, Springer (1987) MR867684
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 MR1205812
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 MR645743
9	L Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994) 589 MR1254134
10	X X Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000) 189 MR1863016
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 MR2471594
12	X X Chen, S K Donaldson, S Sun, Kähler–Einstein metrics and stability arXiv:1210.7494
13	X X Chen, G Tian, Geometry of Kähler metrics and foliations by holomorphic discs, Publ. Math. Inst. Hautes Études Sci. (2008) 1 MR2434691
14	S K Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983) 269 MR710055
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 MR765366
16	S K Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231 MR885784
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 MR1622931
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 MR1736211
19	S K Donaldson, Scalar curvature and projective embeddings, I, J. Differential Geom. 59 (2001) 479 MR1916953
20	S K Donaldson, P B Kronheimer, The geometry of four-manifolds, The Clarendon Press (1990) MR1079726
21	A Fujiki, Moduli space of polarized algebraic manifolds and Kähler metrics, Sugaku Expositions 5 (1992) 173 MR1207204
22	A Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math. 73 (1983) 437 MR718940
23	A Futaki, Asymptotic Chow semi-stability and integral invariants, Internat. J. Math. 15 (2004) 967 MR2106156
24	A Futaki, H Ono, Einstein metrics and GIT stability, Sūgaku 60 (2008) 175 MR2422701
25	M García-Fernández, Coupled equations for Kähler metrics and Yang–Mills connections, PhD thesis, Instituto de Ciencias Matemáticas, Madrid (2009) arXiv:1102.0985
26	M García-Fernández, C Tipler, Deformation of complex structures and the coupled Kähler–Yang–Mills equations arXiv:1301.4480
27	D Gieseker, I Morrison, Hilbert stability of rank-two bundles on curves, J. Differential Geom. 19 (1984) 1 MR739780
28	M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 MR809718
29	Y J Hong, Constant Hermitian scalar curvature equations on ruled manifolds, J. Differential Geom. 53 (1999) 465 MR1806068
30	Y J Hong, Stability and existence of critical Kaehler metrics on ruled manifolds, J. Math. Soc. Japan 60 (2008) 265 MR2392011
31	L Huang, On joint moduli spaces, Math. Ann. 302 (1995) 61 MR1329447
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 MR2949645
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 MR555701
34	S Kobayashi, Differential geometry of complex vector bundles, 15, Princeton Univ. Press (1987) MR909698
35	S Kobayashi, K Nomizu, Foundations of differential geometry, Vol. I, John Wiley (1963) MR0152974
36	S Kobayashi, K Nomizu, Foundations of differential geometry, Vol. II, 15, John Wiley (1969) MR0238225
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 MR2681684
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 MR1361191
39	C LeBrun, S R Simanca, Extremal Kähler metrics and complex deformation theory, Geom. Funct. Anal. 4 (1994) 298 MR1274118
40	J Li, S T Yau, The existence of supersymmetric string theory with torsion, J. Differential Geom. 70 (2005) 143 MR2192064
41	T Mabuchi, K–energy maps integrating Futaki invariants, Tohoku Math. J. 38 (1986) 575 MR867064
42	T Mabuchi, Some symplectic geometry on compact Kähler manifolds, I, Osaka J. Math. 24 (1987) 227 MR909015
43	J E Marsden, G Misiołek, J P Ortega, M Perlmutter, T S Ratiu, Hamiltonian reduction by stages, 1913, Springer (2007) MR2337886
44	M Maruyama, Moduli of stable sheaves, II, J. Math. Kyoto Univ. 18 (1978) 557 MR509499
45	D McDuff, D Salamon, Introduction to symplectic topology, The Clarendon Press (1998) MR1698616
46	R Pandharipande, A compactification over M_g of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996) 425 MR1308406
47	A Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975) 129 MR0369747
48	A Ramanathan, S Subramanian, Einstein–Hermitian connections on principal bundles and stability, J. Reine Angew. Math. 390 (1988) 21 MR953674
49	I Mundet i Riera, A Hitchin–Kobayashi correspondence for Kähler fibrations, J. Reine Angew. Math. 528 (2000) 41 MR1801657
50	G Schumacher, M Toma, Moduli of Kähler manifolds equipped with Hermite–Einstein vector bundles, Rev. Roumaine Math. Pures Appl. 38 (1993) 703 MR1263214
51	S Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992) 495 MR1165352
52	I M Singer, The geometric interpretation of a special connection, Pacific J. Math. 9 (1959) 585 MR0111062
53	J Stoppa, Twisted constant scalar curvature Kähler metrics and Kähler slope stability, J. Differential Geom. 83 (2009) 663 MR2581360
54	A Teleman, Symplectic stability, analytic stability in non-algebraic complex geometry, Internat. J. Math. 15 (2004) 183 MR2055369
55	G Tian, K–stability and Kähler–Einstein metrics arXiv:1211.4669
56	G Tian, Canonical metrics in Kähler geometry, Birkhäuser (2000) MR1787650
57	G Tian, S T Yau, Kähler–Einstein metrics on complex surfaces with C₁ > 0, Comm. Math. Phys. 112 (1987) 175 MR904143
58	K Uhlenbeck, S T Yau, On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986) MR861491
59	X Wang, Moment map, Futaki invariant and stability of projective manifolds, Comm. Anal. Geom. 12 (2004) 1009 MR2103309
60	Y S Yang, Prescribing topological defects for the coupled Einstein and abelian Higgs equations, Comm. Math. Phys. 170 (1995) 541 MR1337133
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 MR480350