Ricci curvatures on Hermitian manifolds

Liu, Kefeng; Yang, Xiaokui

doi:10.1090/tran/7000

Ricci curvatures on Hermitian manifolds
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by Kefeng Liu and Xiaokui Yang PDF

Trans. Amer. Math. Soc. 369 (2017), 5157-5196 Request permission

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.

References

Bogdan Alexandrov and Stefan Ivanov, Vanishing theorems on Hermitian manifolds, Differential Geom. Appl. 14 (2001), no. 3, 251–265. MR 1836272, DOI 10.1016/S0926-2245(01)00044-4
Daniele Angella and Adriano Tomassini, On the $\partial \overline {\partial }$-lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. 1, 71–81. MR 3032326, DOI 10.1007/s00222-012-0406-3
Vestislav Apostolov and Tedi Drăghici, Hermitian conformal classes and almost Kähler structures on $4$-manifolds, Differential Geom. Appl. 11 (1999), no. 2, 179–195. MR 1712115, DOI 10.1016/S0926-2245(99)00033-9
Vestislav Apostolov and Tedi Drăghici, 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
Gil Bor and Luis Hernández-Lamoneda, The canonical bundle of a Hermitian manifold, Bol. Soc. Mat. Mexicana (3) 5 (1999), no. 1, 187–198. MR 1692502
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, DOI 10.1007/978-3-642-57739-0
Ionuţ Chiose, Obstructions to the existence of Kähler structures on compact complex manifolds, Proc. Amer. Math. Soc. 142 (2014), no. 10, 3561–3568. MR 3238431, DOI 10.1090/S0002-9939-2014-12128-9
J.-C. Chu, V. Tosatti, and B. Weinkove, The Monge-Ampère equation for non-integrable almost complex structures, arXiv:1603.00706.
Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 382702, DOI 10.1007/BF01389853
Jean-Pierre Demailly and Mihai Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247–1274. MR 2113021, DOI 10.4007/annals.2004.159.1247
Anna Fino and Gueo Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy, Adv. Math. 189 (2004), no. 2, 439–450. MR 2101226, DOI 10.1016/j.aim.2003.10.009
Anna Fino, Maurizio Parton, and Simon Salamon, Families of strong KT structures in six dimensions, Comment. Math. Helv. 79 (2004), no. 2, 317–340. MR 2059435, DOI 10.1007/s00014-004-0803-3
Anna Fino and Luis Ugarte, On generalized Gauduchon metrics, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 3, 733–753. MR 3109756, DOI 10.1017/S0013091512000405
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
Jixiang Fu, Balanced metrics, Su Buqing memorial lectures. No. 1, Tohoku Math. Publ., vol. 35, Tohoku Univ., Sendai, 2011, pp. 1–29. MR 2893920
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
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. 3, 369–428. MR 2396248
Jixiang Fu, Jun Li, and Shing-Tung Yau, Balanced metrics on non-Kähler Calabi-Yau threefolds, J. Differential Geom. 90 (2012), no. 1, 81–129. MR 2891478
Jixiang Fu, Zhizhang Wang, and Damin Wu, Form-type Calabi-Yau equations, Math. Res. Lett. 17 (2010), no. 5, 887–903. MR 2727616, DOI 10.4310/MRL.2010.v17.n5.a7
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. 2, 659–680. MR 3017048, DOI 10.4171/JEMS/370
Jixiang Fu and Xianchao Zhou, Twistor geometry of Riemannian 4-manifolds by moving frames, Comm. Anal. Geom. 23 (2015), no. 4, 819–839. MR 3385780, DOI 10.4310/CAG.2015.v23.n4.a4
Georgi Ganchev and Stefan Ivanov, Holomorphic and Killing vector fields on compact balanced Hermitian manifolds, Internat. J. Math. 11 (2000), no. 1, 15–28. MR 1757889, DOI 10.1142/S0129167X00000039
Georgi Ganchev and Stefan Ivanov, Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds, Differential Geom. Appl. 14 (2001), no. 1, 79–93. MR 1812526, DOI 10.1016/S0926-2245(00)00039-5
Paul Gauduchon, Fibrés hermitiens à endomorphisme de Ricci non négatif, Bull. Soc. Math. France 105 (1977), no. 2, 113–140 (French). MR 486672, DOI 10.24033/bsmf.1846
Paul Gauduchon, La $1$-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), no. 4, 495–518 (French). MR 742896, DOI 10.1007/BF01455968
Matt Gill, Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds, Comm. Anal. Geom. 19 (2011), no. 2, 277–303. MR 2835881, DOI 10.4310/CAG.2011.v19.n2.a2
M. Gill, The Chern-Ricci flow on smooth minimal models of general type, arXiv:1307.0066
Matthew Gill and Daniel Smith, The behavior of the Chern scalar curvature under the Chern-Ricci flow, Proc. Amer. Math. Soc. 143 (2015), no. 11, 4875–4883. MR 3391045, DOI 10.1090/proc/12745
D. Grantcharov, G. Grantcharov, and Y. S. Poon, Calabi-Yau connections with torsion on toric bundles, J. Differential Geom. 78 (2008), no. 1, 13–32. MR 2406264, DOI 10.4310/jdg/1197320602
Alfred Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tohoku Math. J. (2) 28 (1976), no. 4, 601–612. MR 436054, DOI 10.2748/tmj/1178240746
Gordon Heier and Bun Wong, Scalar curvature and uniruledness on projective manifolds, Comm. Anal. Geom. 20 (2012), no. 4, 751–764. MR 2981838, DOI 10.4310/CAG.2012.v20.n4.a2
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. 1, 101–124. MR 1741779, DOI 10.1215/S0012-7094-00-10214-1
S. Ivanov and G. Papadopoulos, Vanishing theorems and string backgrounds, Classical Quantum Gravity 18 (2001), no. 6, 1089–1110. MR 1822270, DOI 10.1088/0264-9381/18/6/309
S. Ivanov and G. Papadopoulos, Vanishing theorems on $(\ell |k)$-strong Kähler manifolds with torsion, Adv. Math. 237 (2013), 147–164. MR 3028575, DOI 10.1016/j.aim.2012.12.019
Claude LeBrun, Orthogonal complex structures on $S^6$, Proc. Amer. Math. Soc. 101 (1987), no. 1, 136–138. MR 897084, DOI 10.1090/S0002-9939-1987-0897084-7
Claude LeBrun, On Einstein, Hermitian 4-manifolds, J. Differential Geom. 90 (2012), no. 2, 277–302. MR 2899877
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. 4, 651–683. MR 2601348, DOI 10.4310/CAG.2009.v17.n4.a4
Yi Li, A priori estimates for Donaldson’s equation over compact Hermitian manifolds, Calc. Var. Partial Differential Equations 50 (2014), no. 3-4, 867–882. MR 3216837, DOI 10.1007/s00526-013-0659-8
Kefeng Liu, Xiaofeng Sun, and Xiaokui Yang, Positivity and vanishing theorems for ample vector bundles, J. Algebraic Geom. 22 (2013), no. 2, 303–331. MR 3019451, DOI 10.1090/S1056-3911-2012-00588-8
Kefeng Liu and Xiaokui Yang, Hermitian harmonic maps and non-degenerate curvatures, Math. Res. Lett. 21 (2014), no. 4, 831–862. MR 3275649, DOI 10.4310/MRL.2014.v21.n4.a12
Ke-Feng Liu and Xiao-Kui Yang, Geometry of Hermitian manifolds, Internat. J. Math. 23 (2012), no. 6, 1250055, 40. MR 2925476, DOI 10.1142/S0129167X12500553
Francisco Martín Cabrera and Andrew Swann, Curvature of special almost Hermitian manifolds, Pacific J. Math. 228 (2006), no. 1, 165–184. MR 2263028, DOI 10.2140/pjm.2006.228.165
M. L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 149 (1982), no. 3-4, 261–295. MR 688351, DOI 10.1007/BF02392356
Dan Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bull. Soc. Math. France 143 (2015), no. 4, 763–800 (English, with English and French summaries). MR 3450501, DOI 10.24033/bsmf.2704
D. Popovici, Holomorphic deformations of balanced Calabi-Yau $\partial \bar \partial$-manifolds, arXiv:1304.0331
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. 2, 255–305. MR 3235516, DOI 10.2422/2036-2145.201110_{0}08
Jeffrey Streets and Gang Tian, Hermitian curvature flow, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 601–634. MR 2781927, DOI 10.4171/JEMS/262
Jeffrey Streets and Gang Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN 16 (2010), 3101–3133. MR 2673720, DOI 10.1093/imrn/rnp237
Jeffrey Streets and Gang Tian, Regularity results for pluriclosed flow, Geom. Topol. 17 (2013), no. 4, 2389–2429. MR 3110582, DOI 10.2140/gt.2013.17.2389
Zizhou Tang, Curvature and integrability of an almost Hermitian structure, Internat. J. Math. 17 (2006), no. 1, 97–105. MR 2204841, DOI 10.1142/S0129167X0600331X
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, DOI 10.1090/conm/644/12770
Valentino Tosatti and Ben Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), no. 4, 1187–1195. MR 2669712, DOI 10.1090/S0894-0347-2010-00673-X
Valentino Tosatti and Ben Weinkove, The Chern-Ricci flow on complex surfaces, Compos. Math. 149 (2013), no. 12, 2101–2138. MR 3143707, DOI 10.1112/S0010437X13007471
Valentino Tosatti and Ben Weinkove, On the evolution of a Hermitian metric by its Chern-Ricci form, J. Differential Geom. 99 (2015), no. 1, 125–163. MR 3299824
V. Tosatti and B. Weinkove, The Monge-Ampère equation for $(n-1)$-plurisubharmonic functions on a compact Kähler manifold, arXiv:1305.7511.
V. Tosatti and B. Weinkove, Hermitian metrics, $(n-1, n-1)$ forms and Monge-Ampère equations, arXiv:1310.6326
Valentino Tosatti, Ben Weinkove, and Xiaokui Yang, Collapsing of the Chern-Ricci flow on elliptic surfaces, Math. Ann. 362 (2015), no. 3-4, 1223–1271. MR 3368098, DOI 10.1007/s00208-014-1160-1
Franco Tricerri and Lieven Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), no. 2, 365–397. MR 626479, DOI 10.1090/S0002-9947-1981-0626479-0
Izu Vaisman, Some curvature properties of complex surfaces, Ann. Mat. Pura Appl. (4) 132 (1982), 1–18 (1983). MR 696036, DOI 10.1007/BF01760974
Claire Voisin, 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, DOI 10.1017/CBO9780511615344
Xiaokui Yang, Hermitian manifolds with semi-positive holomorphic sectional curvature, Math. Res. Lett. 23 (2016), no. 3, 939–952. MR 3533202, DOI 10.4310/MRL.2016.v23.n3.a17
X.-K. Yang, Big vector bundles and compact complex manifolds with semi-positive holomorphic bisectional curvature, arXiv:1412.5156. To appear in Math. Ann.
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, DOI 10.1002/cpa.3160310304
Shing Tung Yau, On the curvature of compact Hermitian manifolds, Invent. Math. 25 (1974), 213–239. MR 382706, DOI 10.1007/BF01389728
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, DOI 10.1142/9789812386588
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, DOI 10.1090/amsip/018