Details

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.

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.

References

[1]: 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
[2]: 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
[3]: V. Apostolov and J. Streets, The nondegenerate generalized Kähler Calabi-Yau problem, J. Reine Angew. Math. 777 (2021), 1-48. MR4292863
[4]: 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
[5]: 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
[6]: R. Goto, Poisson structures and generalized Kähler structures, J. Math. Soc. Japan 61 (2009), no. 1, 107-132. MR2272873
[7]: R. Goto, Deformations of generalized complex and generalized Kähler structures, J. Differential Geom. 84 (2010), no. 3, 525-560. MR2669364
[8]: R. Goto, Unobstructed K-deformations of generalized complex structures and bi-hermitian structures, Adv. Math. 231 (2012), 1041-1067. MR2955201
[9]: 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
[10]: R. Goto, Scalar curvature as moment map in generalized Kähler geometry, J. Symplectic Geom. 18 (2020), no. 1, 147-190. MR4088750
[11]: 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.
[12]: 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
[13]: M. Gualtieri, Branes on Poisson varieties, The many facets of geometry, Oxford Univ. Press, Oxford, 2010, 368-394. MR2681704
[14]: M. Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75-123. MR2811595
[15]: N. J. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281-308. MR2013140
[16]: N. J. Hitchin, Instantons, Poisson structures and generalized Kähler geometry, Comm. Math. Phys. 265 (2006), 131-164. MR2217300
[17]: N. J. Hitchin, Bihermitian metrics on Del Pezzo surfaces, J. Symplectic Geom. 5 (2007), 1-8. MR2371181
[18]: N. J. Hitchin, Generalized holomorphic bundles and the B-field action, J. Geom. Phys. 61 (2011), no. 1, 352-362. MR2747007
[19]: 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
[20]: 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
[21]: Y. Lin and S. Tolman, Symmetries in generalized Kähler geometry, Comm. Math. Phys. 268 (2006), 199-122. MR2249799
[22]: J. Streets, Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces, Adv. Math. 316 (2017), 187-215. MR3672905
[23]: S. Hu, R. Moraru and R. Seyyedali, A Kobayashi-Hitchin correspondence for \(I_\pm\)-holomorphic bundles, Adv. Math. 287 (2016), 519-566. MR3422685
[24]: 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
[25]: S. Rayan, Co-Higgs bundles on \(\Bbb P^1\), New York J. Math. 19 (2013), 925-945. MR3158239
[26]: S. Rayan, Constructing co-Higgs bundles on \(\Bbb C \Bbb P^2\), Q. J. Math. 65 (2014), no. 4, 1437-1460. MR3285779