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Ricci curvatures on Hermitian manifolds


Authors: Kefeng Liu and Xiaokui Yang
Journal: Trans. Amer. Math. Soc. 369 (2017), 5157-5196
MSC (2010): Primary 53C55, 32Q25; Secondary 32Q20
DOI: https://doi.org/10.1090/tran/7000
Published electronically: March 17, 2017
MathSciNet review: 3632564
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Abstract: In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $ (1,1)$-component of the curvature $ 2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-Kähler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $ \mathbb{S}^{2n-1}\times \mathbb{S}^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and non-negative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with non-negative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.


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  • [1] Bogdan Alexandrov and Stefan Ivanov, Vanishing theorems on Hermitian manifolds, Differential Geom. Appl. 14 (2001), no. no. 3, 251-265. MR 1836272
  • [2] Daniele Angella and Adriano Tomassini, On the $ \partial \overline {\partial }$-lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. no. 1, 71-81. MR 3032326
  • [3] Vestislav Apostolov and Tedi Draghici, Hermitian conformal classes and almost Kähler structures on $ 4$-manifolds, Differential Geom. Appl. 11 (1999), no. no. 2, 179-195. MR 1712115
  • [4] Vestislav Apostolov and Tedi Draghici, The curvature and the integrability of almost-Kähler manifolds: a survey, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001) Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, pp. 25-53. MR 1969266
  • [5] Gil Bor and Luis Hernández-Lamoneda, The canonical bundle of a Hermitian manifold, Bol. Soc. Mat. Mexicana (3) 5 (1999), no. no. 1, 187-198. MR 1692502
  • [6] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225
  • [7] Ionuå Chiose, Obstructions to the existence of Kähler structures on compact complex manifolds, Proc. Amer. Math. Soc. 142 (2014), no. no. 10, 3561-3568. MR 3238431
  • [8] J.-C. Chu, V. Tosatti, and B. Weinkove, The Monge-Ampère equation for non-integrable almost complex structures, arXiv:1603.00706.
  • [9] Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245-274. MR 0382702
  • [10] Jean-Pierre Demailly and Mihai Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2) 159 (2004), no. no. 3, 1247-1274. MR 2113021
  • [11] Anna Fino and Gueo Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy, Adv. Math. 189 (2004), no. no. 2, 439-450. MR 2101226
  • [12] Anna Fino, Maurizio Parton, and Simon Salamon, Families of strong KT structures in six dimensions, Comment. Math. Helv. 79 (2004), no. no. 2, 317-340. MR 2059435
  • [13] Anna Fino and Luis Ugarte, On generalized Gauduchon metrics, Proc. Edinb. Math. Soc. (2) 56 (2013), no. no. 3, 733-753. MR 3109756
  • [14] Jixiang Fu, Specific non-Kähler Hermitian metrics on compact complex manifolds, Recent developments in geometry and analysis, Adv. Lect. Math. (ALM), vol. 23, Int. Press, Somerville, MA, 2012, pp. 79-90. MR 3077200
  • [15] Jixiang Fu, Balanced metrics, Su Buqing memorial lectures. No. 1, Tohoku Math. Publ., vol. 35, Tohoku Univ., Sendai, 2011, pp. 1-29. MR 2893920
  • [16] Jixiang Fu, On non-Kähler Calabi-Yau threefolds with balanced metrics, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 705-716. MR 2827815
  • [17] Ji-Xiang Fu and Shing-Tung Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation, J. Differential Geom. 78 (2008), no. no. 3, 369-428. MR 2396248
  • [18] Jixiang Fu, Jun Li, and Shing-Tung Yau, Balanced metrics on non-Kähler Calabi-Yau threefolds, J. Differential Geom. 90 (2012), no. no. 1, 81-129. MR 2891478
  • [19] Jixiang Fu, Zhizhang Wang, and Damin Wu, Form-type Calabi-Yau equations, Math. Res. Lett. 17 (2010), no. no. 5, 887-903. MR 2727616
  • [20] Jixiang Fu, Zhizhang Wang, and Damin Wu, Semilinear equations, the $ \gamma _k$ function, and generalized Gauduchon metrics, J. Eur. Math. Soc. (JEMS) 15 (2013), no. no. 2, 659-680. MR 3017048
  • [21] Jixiang Fu and Xianchao Zhou, Twistor geometry of Riemannian 4-manifolds by moving frames, Comm. Anal. Geom. 23 (2015), no. no. 4, 819-839. MR 3385780
  • [22] Georgi Ganchev and Stefan Ivanov, Holomorphic and Killing vector fields on compact balanced Hermitian manifolds, Internat. J. Math. 11 (2000), no. no. 1, 15-28. MR 1757889
  • [23] Georgi Ganchev and Stefan Ivanov, Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds, Differential Geom. Appl. 14 (2001), no. no. 1, 79-93. MR 1812526
  • [24] Paul Gauduchon, Fibrés hermitiens à endomorphisme de Ricci non négatif, Bull. Soc. Math. France 105 (1977), no. 2, 113-140 (French). MR 0486672
  • [25] Paul Gauduchon, La $ 1$-forme de torsion d'une variété hermitienne compacte, Math. Ann. 267 (1984), no. no. 4, 495-518 (French). MR 742896
  • [26] Matt Gill, Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds, Comm. Anal. Geom. 19 (2011), no. no. 2, 277-303. MR 2835881
  • [27] M. Gill, The Chern-Ricci flow on smooth minimal models of general type, arXiv:1307.0066
  • [28] Matthew Gill and Daniel Smith, The behavior of the Chern scalar curvature under the Chern-Ricci flow, Proc. Amer. Math. Soc. 143 (2015), no. no. 11, 4875-4883. MR 3391045
  • [29] D. Grantcharov, G. Grantcharov, and Y. S. Poon, Calabi-Yau connections with torsion on toric bundles, J. Differential Geom. 78 (2008), no. no. 1, 13-32. MR 2406264
  • [30] Alfred Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tôhoku Math. J. (2) 28 (1976), no. 4, 601-612. MR 0436054
  • [31] Gordon Heier and Bun Wong, Scalar curvature and uniruledness on projective manifolds, Comm. Anal. Geom. 20 (2012), no. no. 4, 751-764. MR 2981838
  • [32] Alan T. Huckleberry, Stefan Kebekus, and Thomas Peternell, Group actions on $ S^6$ and complex structures on $ \mathbf {P}_3$, Duke Math. J. 102 (2000), no. no. 1, 101-124. MR 1741779
  • [33] S. Ivanov and G. Papadopoulos, Vanishing theorems and string backgrounds, Classical Quantum Gravity 18 (2001), no. no. 6, 1089-1110. MR 1822270
  • [34] S. Ivanov and G. Papadopoulos, Vanishing theorems on $ (\ell \vert k)$-strong Kähler manifolds with torsion, Adv. Math. 237 (2013), 147-164. MR 3028575
  • [35] Claude LeBrun, Orthogonal complex structures on $ S^6$, Proc. Amer. Math. Soc. 101 (1987), no. no. 1, 136-138. MR 897084
  • [36] Claude LeBrun, On Einstein, Hermitian 4-manifolds, J. Differential Geom. 90 (2012), no. no. 2, 277-302. MR 2899877
  • [37] Tian-Jun Li and Weiyi Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom. 17 (2009), no. no. 4, 651-683. MR 2601348
  • [38] Yi Li, A priori estimates for Donaldson's equation over compact Hermitian manifolds, Calc. Var. Partial Differential Equations 50 (2014), no. no. 3-4, 867-882. MR 3216837
  • [39] Kefeng Liu, Xiaofeng Sun, and Xiaokui Yang, Positivity and vanishing theorems for ample vector bundles, J. Algebraic Geom. 22 (2013), no. no. 2, 303-331. MR 3019451
  • [40] Kefeng Liu and Xiaokui Yang, Hermitian harmonic maps and non-degenerate curvatures, Math. Res. Lett. 21 (2014), no. no. 4, 831-862. MR 3275649
  • [41] Ke-Feng Liu and Xiao-Kui Yang, Geometry of Hermitian manifolds, Internat. J. Math. 23 (2012), no. no. 6, 1250055, 40. MR 2925476
  • [42] Francisco Martín Cabrera and Andrew Swann, Curvature of special almost Hermitian manifolds, Pacific J. Math. 228 (2006), no. no. 1, 165-184. MR 2263028
  • [43] M. L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 149 (1982), no. 3-4, 261-295. MR 688351
  • [44] Dan Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bull. Soc. Math. France 143 (2015), no. no. 4, 763-800 (English, with English and French summaries). MR 3450501
  • [45] D. Popovici, Holomorphic deformations of balanced Calabi-Yau $ \partial \bar \partial $-manifolds, arXiv:1304.0331
  • [46] Dan Popovici, Deformation openness and closedness of various classes of compact complex manifolds; examples, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. no. 2, 255-305. MR 3235516
  • [47] Jeffrey Streets and Gang Tian, Hermitian curvature flow, J. Eur. Math. Soc. (JEMS) 13 (2011), no. no. 3, 601-634. MR 2781927
  • [48] Jeffrey Streets and Gang Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN no. 16 (2010), 3101-3133. MR 2673720
  • [49] Jeffrey Streets and Gang Tian, Regularity results for pluriclosed flow, Geom. Topol. 17 (2013), no. no. 4, 2389-2429. MR 3110582
  • [50] Zizhou Tang, Curvature and integrability of an almost Hermitian structure, Internat. J. Math. 17 (2006), no. no. 1, 97-105. MR 2204841
  • [51] Valentino Tosatti, Non-Kähler Calabi-Yau manifolds, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, Contemp. Math., vol. 644, Amer. Math. Soc., Providence, RI, 2015, pp. 261-277. MR 3372471
  • [52] Valentino Tosatti and Ben Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), no. no. 4, 1187-1195. MR 2669712
  • [53] Valentino Tosatti and Ben Weinkove, The Chern-Ricci flow on complex surfaces, Compos. Math. 149 (2013), no. no. 12, 2101-2138. MR 3143707
  • [54] Valentino Tosatti and Ben Weinkove, On the evolution of a Hermitian metric by its Chern-Ricci form, J. Differential Geom. 99 (2015), no. no. 1, 125-163. MR 3299824
  • [55] V. Tosatti and B. Weinkove, The Monge-Ampère equation for $ (n-1)$-plurisubharmonic functions on a compact Kähler manifold, arXiv:1305.7511.
  • [56] V. Tosatti and B. Weinkove, Hermitian metrics, $ (n-1, n-1)$ forms and Monge-Ampère equations, arXiv:1310.6326
  • [57] Valentino Tosatti, Ben Weinkove, and Xiaokui Yang, Collapsing of the Chern-Ricci flow on elliptic surfaces, Math. Ann. 362 (2015), no. no. 3-4, 1223-1271. MR 3368098
  • [58] Franco Tricerri and Lieven Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), no. 2, 365-397. MR 626479
  • [59] Izu Vaisman, Some curvature properties of complex surfaces, Ann. Mat. Pura Appl. (4) 132 (1982), 1-18 (1983). MR 696036
  • [60] Claire Voisin and Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002. Translated from the French original by Leila Schneps. MR 1967689
  • [61] Xiaokui Yang, Hermitian manifolds with semi-positive holomorphic sectional curvature, Math. Res. Lett. 23 (2016), no. no. 3, 939-952. MR 3533202
  • [62] X.-K. Yang, Big vector bundles and compact complex manifolds with semi-positive holomorphic bisectional curvature, arXiv:1412.5156. To appear in Math. Ann.
  • [63] Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. MR 480350
  • [64] Shing Tung Yau, On the curvature of compact Hermitian manifolds, Invent. Math. 25 (1974), 213-239. MR 0382706
  • [65] Weiping Zhang, Lectures on Chern-Weil theory and Witten deformations, Nankai Tracts in Mathematics, vol. 4, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1864735
  • [66] Fangyang Zheng, Complex differential geometry, AMS/IP Studies in Advanced Mathematics, vol. 18, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000. MR 1777835

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Additional Information

Kefeng Liu
Affiliation: Department of Mathematics, Capital Normal University, Beijing, 100048, People’s Republic of China — and — Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095
Email: liu@math.ucla.edu

Xiaokui Yang
Affiliation: Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China — and — Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing,100190, People’s Republic of China
Email: xkyang@amss.ac.cn

DOI: https://doi.org/10.1090/tran/7000
Received by editor(s): April 23, 2015
Received by editor(s) in revised form: April 16, 2016, and May 20, 2016
Published electronically: March 17, 2017
Additional Notes: The first author was supported in part by an NSF Grant.
The second author was partially supported by China’s Recruitment Program of Global Experts and National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences
Article copyright: © Copyright 2017 American Mathematical Society

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