Kähler manifolds and mixed curvature

Chu, Jianchun; Lee, Man-Chun; Tam, Luen-Fai

doi:10.1090/tran/8735

Kähler manifolds and mixed curvature
HTML articles powered by AMS MathViewer

by Jianchun Chu, Man-Chun Lee and Luen-Fai Tam PDF

Trans. Amer. Math. Soc. 375 (2022), 7925-7944 Request permission

Abstract:

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a “convex combination” of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni [Comm. Pure Appl. Math. 74 (2021), pp. 1100–1126] concerning manifolds with negative $k$-Ricci curvature and generalize a result of Wu-Yau [Comm. Anal. Geom. 24 (2016), pp. 901–912] and Diverio-Trapani [J. Differential Geom. 111 (2019), pp. 303–314] to the conformally Kähler case. We also show that the compact Kähler manifold is projective and simply connected if the mixed curvature is positive.

References

Thierry Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), no. 1, 63–95 (French, with English summary). MR 494932
F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 539–545 (French). MR 1191735, DOI 10.24033/asens.1658
Jianchun Chu, $C^{2,\alpha }$ regularities and estimates for nonlinear elliptic and parabolic equations in geometry, Calc. Var. Partial Differential Equations 55 (2016), no. 1, Art. 8, 20. MR 3447154, DOI 10.1007/s00526-015-0948-5
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
Simone Diverio and Stefano Trapani, Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle, J. Differential Geom. 111 (2019), no. 2, 303–314. MR 3909909, DOI 10.4310/jdg/1549422103
Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303
Vincent Guedj and Ahmed Zeriahi, Regularizing properties of the twisted Kähler-Ricci flow, J. Reine Angew. Math. 729 (2017), 275–304. MR 3680377, DOI 10.1515/crelle-2014-0105
Gordon Heier and Bun Wong, On projective Kähler manifolds of partially positive curvature and rational connectedness, Doc. Math. 25 (2020), 219–238. MR 4106891
Gordon Heier, Steven S. Y. Lu, and Bun Wong, On the canonical line bundle and negative holomorphic sectional curvature, Math. Res. Lett. 17 (2010), no. 6, 1101–1110. MR 2729634, DOI 10.4310/MRL.2010.v17.n6.a9
Gordon Heier, Steven S. Y. Lu, and Bun Wong, Kähler manifolds of semi-negative holomorphic sectional curvature, J. Differential Geom. 104 (2016), no. 3, 419–441. MR 3568627
Gordon Heier, Steven S. Y. Lu, Bun Wong, and Fangyang Zheng, Reduction of manifolds with semi-negative holomorphic sectional curvature, Math. Ann. 372 (2018), no. 3-4, 951–962. MR 3880288, DOI 10.1007/s00208-017-1638-8
S.-C Huang, M.-C. Lee, L.-F. Tam, and F. Tong, Longtime existence of Kähler- Ricci flow and holomorphic sectional curvature, Preprint, arXiv:1805.12328, appear in Comm. Anal. Geom.
Shoshichi Kobayashi, On compact Kähler manifolds with positive definite Ricci tensor, Ann. of Math. (2) 74 (1961), 570–574. MR 133086, DOI 10.2307/1970298
János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992), no. 3, 765–779. MR 1189503
N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670 (Russian). MR 661144
Man-Chun Lee and Luen-Fai Tam, Chern-Ricci flows on noncompact complex manifolds, J. Differential Geom. 115 (2020), no. 3, 529–564. MR 4120818, DOI 10.4310/jdg/1594260018
Man-Chun Lee and Jeffrey Streets, Complex manifolds with negative curvature operator, Int. Math. Res. Not. IMRN 24 (2021), 18520–18528. MR 4365995, DOI 10.1093/imrn/rnz331
Chang Li, Lei Ni, and Xiaohua Zhu, An application of a $C^2$-estimate for a complex Monge-Ampère equation, Internat. J. Math. 32 (2021), no. 12, Paper No. 2140007, 13. MR 4349966, DOI 10.1142/S0129167X21400073
Xiaonan Ma and George Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007. MR 2339952, DOI 10.1007/978-3-7643-8115-8
S. Matsumura, On morphisms of compact Kähler manifolds with semi-positive holomorphic sectional curvature, Preprint, arXiv:1809.08859.
S. Matsumura, On projective manifolds with semi-positive holomorphic sectional curvature, Preprint, arXiv:1811.04182.
Shin-ichi Matsumura, On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature, Pure Appl. Math. Q. 16 (2020), no. 5, 1419–1439. MR 4221001, DOI 10.4310/PAMQ.2020.v16.n5.a3
Lei Ni, The fundamental group, rational connectedness and the positivity of Kähler manifolds, J. Reine Angew. Math. 774 (2021), 267–299. MR 4250473, DOI 10.1515/crelle-2020-0040
Lei Ni, Liouville theorems and a Schwarz lemma for holomorphic mappings between Kähler manifolds, Comm. Pure Appl. Math. 74 (2021), no. 5, 1100–1126. MR 4230067, DOI 10.1002/cpa.21987
Lei Ni and Fangyang Zheng, Comparison and vanishing theorems for Kähler manifolds, Calc. Var. Partial Differential Equations 57 (2018), no. 6, Paper No. 151, 31. MR 3858834, DOI 10.1007/s00526-018-1431-x
L. Ni and F. Zheng, Positivity and Kodaira embedding theorem, Preprint, arXiv:1804.09696.
Ryosuke Nomura, Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach, Int. Math. Res. Not. IMRN 21 (2018), 6611–6616. MR 3873539, DOI 10.1093/imrn/rnx075
H. L. Royden, The Ahlfors-Schwarz lemma in several complex variables, Comment. Math. Helv. 55 (1980), no. 4, 547–558. MR 604712, DOI 10.1007/BF02566705
Morgan Sherman and Ben Weinkove, Interior derivative estimates for the Kähler-Ricci flow, Pacific J. Math. 257 (2012), no. 2, 491–501. MR 2972475, DOI 10.2140/pjm.2012.257.491
Jian Song and Gang Tian, The Kähler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), no. 3, 609–653. MR 2357504, DOI 10.1007/s00222-007-0076-8
Jian Song and Ben Weinkove, An introduction to the Kähler-Ricci flow, An introduction to the Kähler-Ricci flow, Lecture Notes in Math., vol. 2086, Springer, Cham, 2013, pp. 89–188. MR 3185333, DOI 10.1007/978-3-319-00819-6_{3}
Gang Tian and Zhou Zhang, On the Kähler-Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179–192. MR 2243679, DOI 10.1007/s11401-005-0533-x
Valentino Tosatti, KAWA lecture notes on the Kähler-Ricci flow, Ann. Fac. Sci. Toulouse Math. (6) 27 (2018), no. 2, 285–376 (English, with English and French summaries). MR 3831026, DOI 10.5802/afst.1571
Valentino Tosatti, Yu Wang, Ben Weinkove, and Xiaokui Yang, $C^{2,\alpha }$ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 431–453. MR 3385166, DOI 10.1007/s00526-014-0791-0
Valentino Tosatti and Xiaokui Yang, An extension of a theorem of Wu-Yau, J. Differential Geom. 107 (2017), no. 3, 573–579. MR 3715350, DOI 10.4310/jdg/1508551226
Hajime Tsuji, Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type, Math. Ann. 281 (1988), no. 1, 123–133. MR 944606, DOI 10.1007/BF01449219
Damin Wu and Shing-Tung Yau, Negative holomorphic curvature and positive canonical bundle, Invent. Math. 204 (2016), no. 2, 595–604. MR 3489705, DOI 10.1007/s00222-015-0621-9
Damin Wu and Shing-Tung Yau, A remark on our paper “Negative holomorphic curvature and positive canonical bundle” [ MR3489705], Comm. Anal. Geom. 24 (2016), no. 4, 901–912. MR 3570421, DOI 10.4310/CAG.2016.v24.n4.a9
Xiaokui Yang, RC-positivity, rational connectedness and Yau’s conjecture, Camb. J. Math. 6 (2018), no. 2, 183–212. MR 3811235, DOI 10.4310/CJM.2018.v6.n2.a2
Xiaokui Yang, RC-positive metrics on rationally connected manifolds, Forum Math. Sigma 8 (2020), Paper No. e53, 19. MR 4176757, DOI 10.1017/fms.2020.32
X. Yang, Compact Kähler manifolds with quasi-positive second Chern-Ricci curvature, Preprint, arXiv:2006.13884.
Xiaokui Yang and Fangyang Zheng, On real bisectional curvature for Hermitian manifolds, Trans. Amer. Math. Soc. 371 (2019), no. 4, 2703–2718. MR 3896094, DOI 10.1090/tran/7445
Shing Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), no. 1, 197–203. MR 486659, DOI 10.2307/2373880
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, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. MR 645762