Riv. Mat. Univ. Parma, Vol. 13, No. 2, 2022

Ryushi Goto [a]

Moduli spaces of Einstein-Hermitian generalized connections over generalized Kähler manifolds of symplectic type

Pages: 611-649
Received: 12 January 2022
Accepted in revised form: 30 May 2022
Mathematics Subject Classification: 53D18, 53D20, 53D17, 53C26.
Keywords: Generalized complex structure, generalized Kähler structure, moment map, generalized Einstein-Hermitian metric, Poisson structure, co-Higgs bundles.
Author address:
[a]: Osaka University, Department of Mathematics, Graduate School of Science Toyonaka, Osaka, Japan.

Full Text (PDF)

Abstract: From a view point of the moment map, we shall introduce the notion of Einstein-Hermitian generalized connections over a generalized Kähler manifold of symplectic type. We show that moduli spaces of Einstein-Hermitian generalized connections arise as the Kähler quotients. The deformation complex of Einstein-Hermitian generalized connections is an elliptic complex and it turns out that the smooth part of the moduli space is a finite dimensional Kähler manifold. The canonical line bundle over a generalized Kähler manifold of symplectic type has the canonical generalized connection and its curvature coincides with "the scalar curvature as the moment map" which is defined in the previous paper [10]. Kähler-Ricci solitons provide examples of Einstein-Hermitian generalized connections and Einstein Hermitian co-Higgs bundles are also discussed.

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523-615. MR0702806
V. Apostolov, P. Gauduchon and G. Grantcharov, Bi-Hermitian structures on complex surfaces, Proc. London Math. Soc. 79 (1999), 414-429, and Erratum in Proc. London Math. Soc. 92 (2006), 200-202. MR1702248 | MR2192389
V. Apostolov and J. Streets, The nondegenerate generalized Kähler Calabi-Yau problem, J. Reine Angew. Math. 777 (2021), 1-48. MR4292863
S. K. Donaldson, Remarks on gauge theory, complex geometry and 4-manifold topology, Fields Medallists' lectures, 384-403, World Sci. Ser. 20th Century Math., 5, World Sci. Publ., River Edge, NJ, 1997. MR1622931
A. Fujiki, Moduli space of polarized algebraic manifolds and Kähler metrics, Sugaku Expositions 5 (1992), no. 2, 173-191, translation from Japanese of Sugaku 42 (1990), no. 3, 231-243. MR1207204
R. Goto, Poisson structures and generalized Kähler structures, J. Math. Soc. Japan 61 (2009), no. 1, 107-132. MR2272873
R. Goto, Deformations of generalized complex and generalized Kähler structures, J. Differential Geom. 84 (2010), no. 3, 525-560. MR2669364
R. Goto, Unobstructed K-deformations of generalized complex structures and bi-hermitian structures, Adv. Math. 231 (2012), 1041-1067. MR2955201
R. Goto, Unobstructed deformations of generalized complex structures induced by \(C^\infty\) logarithmic symplectic structures and logarithmic Poisson structures, in ''Geometry and topology of manifolds'', Proc. 10th China-Japan Geometry Conference in Shanghai 2014, Springer Proc. Math. Stat., 154, Springer, Tokyo, 2016, 159-183. MR3555982
R. Goto, Scalar curvature as moment map in generalized Kähler geometry, J. Symplectic Geom. 18 (2020), no. 1, 147-190. MR4088750
R. Goto, Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kähler manifolds of symplectic type, Int. Math. Res. Not. IMRN 2023, DOI: 10.1093/imrn/rnad038.
R. Goto, Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type, Math. Ann. 384 (2022), no. 1-2, 805-855. MR4476242
M. Gualtieri, Branes on Poisson varieties, The many facets of geometry, Oxford Univ. Press, Oxford, 2010, 368-394. MR2681704
M. Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75-123. MR2811595
N. J. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281-308. MR2013140
N. J. Hitchin, Instantons, Poisson structures and generalized Kähler geometry, Comm. Math. Phys. 265 (2006), 131-164. MR2217300
N. J. Hitchin, Bihermitian metrics on Del Pezzo surfaces, J. Symplectic Geom. 5 (2007), 1-8. MR2371181
N. J. Hitchin, Generalized holomorphic bundles and the B-field action, J. Geom. Phys. 61 (2011), no. 1, 352-362. MR2747007
N. J. Hitchin, Poisson modules and generalized geometry, in ''Geometry and analysis. No. 1'', Adv. Lect. Math. (ALM), 17, Int. Press, Somerville, MA, 2011, 403-417. MR2882431
K. Kodaira and D. C. Spencer, On deformations of complex analytic structures, III, Stability theorems for complex structures, Ann. of Math. (2) 71 (1960), 43-76. MR0115189
Y. Lin and S. Tolman, Symmetries in generalized Kähler geometry, Comm. Math. Phys. 268 (2006), 199-122. MR2249799
J. Streets, Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces, Adv. Math. 316 (2017), 187-215. MR3672905
S. Hu, R. Moraru and R. Seyyedali, A Kobayashi-Hitchin correspondence for \(I_\pm\)-holomorphic bundles, Adv. Math. 287 (2016), 519-566. MR3422685
S. Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Kanô Memorial Lectures, 5, Princeton University Press, Princeton, NJ, 1987. MR0909698
S. Rayan, Co-Higgs bundles on \(\Bbb P^1\), New York J. Math. 19 (2013), 925-945. MR3158239
S. Rayan, Constructing co-Higgs bundles on \(\Bbb C \Bbb P^2\), Q. J. Math. 65 (2014), no. 4, 1437-1460. MR3285779

Home Riv.Mat.Univ.Parma