Plurisigned hermitian metrics

Angella, Daniele; Guedj, Vincent; Lu, Chinh

doi:10.1090/tran/8916

Plurisigned hermitian metrics
HTML articles powered by AMS MathViewer

by Daniele Angella, Vincent Guedj and Chinh H. Lu HTML | PDF

Trans. Amer. Math. Soc. 376 (2023), 4631-4659 Request permission

Abstract:

Let $(X,\omega )$ be a compact hermitian manifold of dimension $n$. We study the asymptotic behavior of Monge-Ampère volumes $\int _X (\omega +dd^c \varphi )^n$, when $\omega +dd^c \varphi$ varies in the set of hermitian forms that are $dd^c$-cohomologous to $\omega$. We show that these Monge-Ampère volumes are uniformly bounded if $\omega$ is “strongly pluripositive”, and that they are uniformly positive if $\omega$ is “strongly plurinegative”. This motivates the study of the existence of such plurisigned hermitian metrics.

We analyze several classes of examples (complex parallelisable manifolds, twistor spaces, Vaisman manifolds) admitting such metrics, showing that they cannot coexist. We take a close look at $6$-dimensional nilmanifolds which admit a left-invariant complex structure, showing that each of them admit a plurisigned hermitian metric, while only few of them admit a pluriclosed metric. We also study $6$-dimensional solvmanifolds with trivial canonical bundle.

References

E. Abbena and A. Grassi, Hermitian left invariant metrics on complex Lie groups and cosymplectic Hermitian manifolds, Boll. Un. Mat. Ital. A (6) 5 (1986), no. 3, 371–379 (English, with Italian summary). MR 866545
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
A. Andrada, M. L. Barberis, and I. Dotti, Classification of abelian complex structures on 6-dimensional Lie algebras, J. Lond. Math. Soc. (2) 83 (2011), no. 1, 232–255. MR 2763953, DOI 10.1112/jlms/jdq071
D. Angella, A. Dubickas, A. Otiman, and J. Stelzig, On metric and cohomological properties of Oeljeklaus-Toma manifolds, Publicacions Matemàtiques, to appear. arXiv:2201.06377.
Daniele Angella and Hisashi Kasuya, Bott-Chern cohomology of solvmanifolds, Ann. Global Anal. Geom. 52 (2017), no. 4, 363–411. MR 3735904, DOI 10.1007/s10455-017-9560-6
Daniele Angella, Antonio Otal, Luis Ugarte, and Raquel Villacampa, Complex structures of splitting type, Rev. Mat. Iberoam. 33 (2017), no. 4, 1309–1350. MR 3729601, DOI 10.4171/RMI/973
D. Angella, A. Otal, L. Ugarte, and R. Villacampa, On Gauduchon connections with Kähler-like curvature, to appear in Commun. Anal. Geom.
M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
María L. Barberis, Isabel G. Dotti, and Misha Verbitsky, Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry, Math. Res. Lett. 16 (2009), no. 2, 331–347. MR 2496748, DOI 10.4310/MRL.2009.v16.n2.a10
Giovanni Bazzoni, Locally conformally symplectic and Kähler geometry, EMS Surv. Math. Sci. 5 (2018), no. 1-2, 129–154. MR 3880223, DOI 10.4171/EMSS/29
Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165, DOI 10.1007/BF02392348
Florin Alexandru Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), no. 1, 1–40. MR 1760667, DOI 10.1007/s002080050357
Chal Benson and Carolyn S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518. MR 976592, DOI 10.1016/0040-9383(88)90029-8
M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa, Invariant complex structures on 6-nilmanifolds: classification, Frölicher spectral sequence and special Hermitian metrics, J. Geom. Anal. 26 (2016), no. 1, 252–286. MR 3441513, DOI 10.1007/s12220-014-9548-4
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
Ionuţ Chiose, The Kähler rank of compact complex manifolds, J. Geom. Anal. 26 (2016), no. 1, 603–615. MR 3441529, DOI 10.1007/s12220-015-9564-z
I. Chiose, On the invariance of the total Monge-Ampère volume of hermitian metrics, Preprint, arXiv:1609.05945.
Dan Coman, Vincent Guedj, and Ahmed Zeriahi, Extension of plurisubharmonic functions with growth control, J. Reine Angew. Math. 676 (2013), 33–49. MR 3028754, DOI 10.1515/crelle.2011.185
Jean-Pierre Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361–409. MR 1158622
Jean-Pierre Demailly, Analytic methods in algebraic geometry, Surveys of Modern Mathematics, vol. 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012. MR 2978333
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
Guillaume Deschamps, Noël Le Du, and Christophe Mourougane, Hessian of the natural Hermitian form on twistor spaces, Bull. Soc. Math. France 145 (2017), no. 1, 1–7 (English, with English and French summaries). MR 3636749, DOI 10.24033/bsmf.2729
Sławomir Dinew, Pluripotential theory on compact Hermitian manifolds, Ann. Fac. Sci. Toulouse Math. (6) 25 (2016), no. 1, 91–139 (English, with English and French summaries). MR 3485292, DOI 10.5802/afst.1488
Sławomir Dinew and Sławomir Kołodziej, Pluripotential estimates on compact Hermitian manifolds, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 69–86. MR 3077248
Sorin Dragomir and Liviu Ornea, Locally conformal Kähler geometry, Progress in Mathematics, vol. 155, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1481969, DOI 10.1007/978-1-4612-2026-8
Nadaniela Egidi, Special metrics on compact complex manifolds, Differential Geom. Appl. 14 (2001), no. 3, 217–234. MR 1836270, DOI 10.1016/S0926-2245(01)00041-9
Nicola Enrietti, Anna Fino, and Luigi Vezzoni, Tamed symplectic forms and strong Kähler with torsion metrics, J. Symplectic Geom. 10 (2012), no. 2, 203–223. MR 2926995, DOI 10.4310/JSG.2012.v10.n2.a3
Marisa Fernández and Alfred Gray, The Iwasawa manifold, Differential geometry, Peñíscola 1985, Lecture Notes in Math., vol. 1209, Springer, Berlin, 1986, pp. 157–159. MR 863753, DOI 10.1007/BFb0076628
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, Gueo Grantcharov, and Luigi Vezzoni, Astheno-Kähler and balanced structures on fibrations, Int. Math. Res. Not. IMRN 22 (2019), 7093–7117. MR 4032184, DOI 10.1093/imrn/rnx337
Anna Fino, Antonio Otal, and Luis Ugarte, Six-dimensional solvmanifolds with holomorphically trivial canonical bundle, Int. Math. Res. Not. IMRN 24 (2015), 13757–13799. MR 3436163, DOI 10.1093/imrn/rnv112
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 Adriano Tomassini, Blow-ups and resolutions of strong Kähler with torsion metrics, Adv. Math. 221 (2009), no. 3, 914–935. MR 2511043, DOI 10.1016/j.aim.2009.02.001
Anna Fino and Luigi Vezzoni, Special Hermitian metrics on compact solvmanifolds, J. Geom. Phys. 91 (2015), 40–53. MR 3327047, DOI 10.1016/j.geomphys.2014.12.010
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
Bo Guan and Qun Li, Complex Monge-Ampère equations and totally real submanifolds, Adv. Math. 225 (2010), no. 3, 1185–1223. MR 2673728, DOI 10.1016/j.aim.2010.03.019
V. Guedj and C. H. Lu, Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, Preprint, arXiv:2106.04273.
Vincent Guedj and Chinh H. Lu, Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes, Algebr. Geom. 9 (2022), no. 6, 688–713. MR 4518244, DOI 10.14231/ag-2022-021
V. Guedj and C. H. Lu, Quasi-plurisubharmonic envelopes 3: Solving Monge-Ampère equations on hermitian manifolds, Preprint, arXiv:2107.01938.
Vincent Guedj and Ahmed Zeriahi, Degenerate complex Monge-Ampère equations, EMS Tracts in Mathematics, vol. 26, European Mathematical Society (EMS), Zürich, 2017. MR 3617346, DOI 10.4171/167
Reese Harvey and H. Blaine Lawson Jr., An intrinsic characterization of Kähler manifolds, Invent. Math. 74 (1983), no. 2, 169–198. MR 723213, DOI 10.1007/BF01394312
Keizo Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71. MR 946638, DOI 10.1090/S0002-9939-1989-0946638-X
Heisuke Hironaka, ON THE THEORY OF BIRATIONAL BLOWING-UP, ProQuest LLC, Ann Arbor, MI, 1960. Thesis (Ph.D.)–Harvard University. MR 2939137
N. J. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981), no. 1, 133–150. MR 623721, DOI 10.1112/plms/s3-43.1.133
S. Ivashkovich, Extension properties of meromorphic mappings with values in non-Kähler complex manifolds, Ann. of Math. (2) 160 (2004), no. 3, 795–837. MR 2144969, DOI 10.4007/annals.2004.160.795
D. Kaledin and M. Verbitsky, Non-Hermitian Yang-Mills connections, Selecta Math. (N.S.) 4 (1998), no. 2, 279–320. MR 1669956, DOI 10.1007/s000290050033
Hisashi Kasuya, Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds, Bull. Lond. Math. Soc. 45 (2013), no. 1, 15–26. MR 3033950, DOI 10.1112/blms/bds057
Sławomir Kołodziej and Ngoc Cuong Nguyen, Weak solutions to the complex Monge-Ampère equation on Hermitian manifolds, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, Contemp. Math., vol. 644, Amer. Math. Soc., Providence, RI, 2015, pp. 141–158. MR 3372464, DOI 10.1090/conm/644/12775
Sławomir Kołodziej and Ngoc Cuong Nguyen, Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds, Adv. Math. 346 (2019), 264–304. MR 3910489, DOI 10.1016/j.aim.2019.02.004
A. Latorre, L. Ugarte, and R. Villacampa, On generalized Gauduchon nilmanifolds. part A, Differential Geom. Appl. 54 (2017), no. part A, 150–164. MR 3693921, DOI 10.1016/j.difgeo.2017.03.016
Chinh H. Lu, Trong-Thuc Phung, and Tât-Dat Tô, Stability and Hölder regularity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds, Ann. Inst. Fourier (Grenoble) 71 (2021), no. 5, 2019–2045 (English, with English and French summaries). MR 4398254, DOI 10.5802/aif.3436
L. Magnin, Sur les algèbres de Lie nilpotentes de dimension $\leq 7$, J. Geom. Phys. 3 (1986), no. 1, 119–144 (French, with English summary). MR 855573, DOI 10.1016/0393-0440(86)90005-7
A. Malcev, On solvable Lie algebras, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 9 (1945), 329–356 (Russian, with English summary). MR 0022217
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
John Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293–329. MR 425012, DOI 10.1016/S0001-8708(76)80002-3
V. V. Morozov, Classification of nilpotent Lie algebras of sixth order, Izv. Vysš. Učebn. Zaved. Matematika 1958 (1958), no. 4 (5), 161–171 (Russian). MR 0130326
Iku Nakamura, Complex parallelisable manifolds and their small deformations, J. Differential Geometry 10 (1975), 85–112. MR 393580
Liviu Ornea and Misha Verbitsky, Locally conformal Kähler manifolds with potential, Math. Ann. 348 (2010), no. 1, 25–33. MR 2657432, DOI 10.1007/s00208-009-0463-0
L. Ornea and M. Verbitsky, A report on locally conformally Kähler manifolds, Harmonic maps and differential geometry, Contemp. Math., vol. 542, Amer. Math. Soc., Providence, RI, 2011, pp. 135–149. MR 2796645, DOI 10.1090/conm/542/10703
Liviu Ornea and Misha Verbitsky, Hopf surfaces in locally conformally Kähler manifolds with potential, Geom. Dedicata 207 (2020), 219–226. MR 4117569, DOI 10.1007/s10711-019-00495-5
L.Ornea and M.Verbistky, Principles of locally conformally Kähler geometry. arXiv:2208.07188.
A. Otal, Solvmanifolds with holomorphically trivial canonical bundle, PhD Thesis, Universidad de Zaragoza, 2014.
Alexandra Otiman, Special Hermitian metrics on Oeljeklaus-Toma manifolds, Bull. Lond. Math. Soc. 54 (2022), no. 2, 655–667. MR 4453697, DOI 10.1112/blms.12590
Dan Popovici, Volume and self-intersection of differences of two nef classes, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 4, 1255–1299. MR 3752527
Simon Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), no. 1, 143–171. MR 664330, DOI 10.1007/BF01393378
S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2-3, 311–333. MR 1812058, DOI 10.1016/S0022-4049(00)00033-5
Gábor Székelyhidi, Valentino Tosatti, and Ben Weinkove, Gauduchon metrics with prescribed volume form, Acta Math. 219 (2017), no. 1, 181–211. MR 3765661, DOI 10.4310/ACTA.2017.v219.n1.a6
Clifford Henry Taubes, The existence of anti-self-dual conformal structures, J. Differential Geom. 36 (1992), no. 1, 163–253. MR 1168984
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
Luis Ugarte, Hermitian structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), no. 1, 175–202. MR 2308035, DOI 10.1007/s00031-005-1134-1
Luis Ugarte and Raquel Villacampa, Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry, Asian J. Math. 18 (2014), no. 2, 229–246. MR 3217635, DOI 10.4310/AJM.2014.v18.n2.a3
Izu Vaisman, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), no. 3, 231–255. MR 690671, DOI 10.1007/BF00148231
Misha Verbitsky, Rational curves and special metrics on twistor spaces, Geom. Topol. 18 (2014), no. 2, 897–909. MR 3190604, DOI 10.2140/gt.2014.18.897
Hsien-Chung Wang, Closed manifolds with homogeneous complex structure, Amer. J. Math. 76 (1954), 1–32. MR 66011, DOI 10.2307/2372397
A. Yachou, Sur les variétés semi-kählériennes, PhD Thesis, Université de Lille, 1998.
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