Skip to main content
SpringerLink
Log in

Constant Scalar Curvature Sasaki Metrics and Projective Bundles

  • Conference paper
  • First Online:
Birational Geometry, Kähler–Einstein Metrics and Degenerations (BGKEMD 2019, BGKEMD 2019, BGKEMD 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 409))

Abstract

In this paper we consider the Boothby-Wang construction over twist 1 stage 3 Bott orbifolds given in terms of the log pair \((S_\textbf{n},\Delta _\textbf{m})\). We give explicit constant scalar curvature (CSC) Sasaki metrics either directly from CSC Kähler orbifold metrics or by using the weighted extremal approach of Apostolov and Calderbank. The Sasaki 7-manifolds (orbifolds) are finitely covered by compact simply connected manifolds (orbifolds) with the rational homology of the 2-fold connected sum of \(S^2\times S^5\).

The authors were partially supported by grants from the Simons Foundation, CPB by (#519432), and CWT-F by (#422410).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    See Remark 3.10 below.

  2. 2.

    We refer to [20] for details and a more general description of the Yamazaki fiber joins.

References

  1. Apostolov, V., Calderbank, D.M.J.: The CR geometry of weighted extremal Kähler and Sasaki metrics. Math. Ann. 379(3–4), 1047–1088 (2021). MR 4238260

    Google Scholar 

  2. Araujo, C., Castravet, A.-M., Cheltsov, I., Fujita, K., Kaloghiros, A.-S., Martinez-Garcia, J., Shramov, C., Süss, H., Viswanathan, N.: The Calabi problem for Fano threefolds (2021)

    Google Scholar 

  3. Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.W.: Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability. Invent. Math. 173(3), 547–601 (2008). MR 2425136 (2009m:32043)

    Google Scholar 

  4. Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.W.: Hamiltonian 2-forms in Kähler geometry. IV. Weakly Bochner-flat Kähler manifolds. Commun. Anal. Geom. 16(1), 91–126 (2008). MR 2411469 (2010c:32043)

    Google Scholar 

  5. Apostolov, V., Calderbank, D.M.J., Legendre, E.: Weighted K-stability of polarized varieties and extremality of Sasaki manifolds. Adv. Math. 391 (2021). Paper No. 107969, 63. MR 4300921

    Google Scholar 

  6. Apostolov, V., Jubert, S., Lahdili, A.: Weighted \(K\)-Stability and coercivity with applications to Extremal Kähler and Sasaki Metrics (2021). arXiv: 2104.09709

  7. Apostolov, V., Maschler, G., Tønnesen-Friedman, C.W.: Weighted Extremal Kähler metrics and the Einstein-Maxwell geometry of Projective Bundles. Comm. Anal. Geom. 30(4), 689–744 (2022)

    Google Scholar 

  8. Aguilar, M.A., Prieto, C.: Transfers for ramified covering maps in homology and cohomology. Int. J. Math. Math. Sci. (2006). Art. ID 94651, 28. MR 2251716

    Google Scholar 

  9. Boyer, C.P., Calderbank, D.M.J., Tønnesen-Friedman, C.W.: The Kähler geometry of Bott manifolds. Adv. Math. 350, 1–62 (2019). MR 3945589

    Google Scholar 

  10. Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 124. B.G. Teubner Verlagsgesellschaft mbH, Stuttgart (1991). With German, French and Russian summaries. MR 94a:53077

    Google Scholar 

  11. Boyer, C.P., Galicki, K.: Sasakian geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008). MR MR2382957 (2009c:53058)

    Google Scholar 

  12. Boyer, C., Huang, H., Legendre, E.: An application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometry. Geom. Topol. 22(7), 4205–4234 (2018). MR 3890775

    Google Scholar 

  13. Boyer, C.P., Huang, H., Legendre, E., Tønnesen-Friedman, C.W: Existence and non-existence of constant scalar curvature and extremal Sasaki metrics. arXiv:2210.12203

  14. Boyer, C.P., Huang, H., Legendre, E., Tønnesen-Friedman, C.W.: Reducibility in Sasakian geometry. Trans. Am. Math. Soc. 370(10), 6825–6869 (2018). MR 3841834

    Google Scholar 

  15. Boyer, C.P., Huang, H., Legendre, E., Tønnesen-Friedman, C.W.: Some Open Problems in Sasaki Geometry. Differential Geometry in the Large, pp. 143–168. Cambridge University Press, Cambridge, UK (2021)

    Google Scholar 

  16. Boyer, C.P., Tønnesen-Friedman, C.W.: Iterated \({S}^3\) Sasaki joins and Bott orbifolds. Ann. Fac. Sci. Toulouse Math. 31(3), 837–860 (2022) (6)

    Google Scholar 

  17. Boyer, C.P., Tønnesen-Friedman, C.W.: The Sasaki join, Hamiltonian 2-forms, and constant scalar curvature. J. Geom. Anal. 26(2), 1023–1060 (2016). MR 3472828

    Google Scholar 

  18. Boyer, C.P., Tønnesen-Friedman, C.W.: Sasaki-Einstein metrics on a class of 7-manifolds. J. Geom. Phys. 140, 111–124 (2019). MR 3923473

    Google Scholar 

  19. Boyer C.P., Tønnesen-Friedman, C.W.: The \({S}^3\) Sasaki join construction (2021). J. Math. Soc. Japan 74(4), 1335–1371 (2022)

    Google Scholar 

  20. Boyer, C.P., Tønnesen-Friedman, C.W.: Sasakian geometry on sphere bundles. Differ. Geom. Appl. 77, 101765, 37 (2021). MR 4253896

    Google Scholar 

  21. Boothby, W.M., Wang, H.C.: On contact manifolds. Ann. of Math. (2) 68, 721–734 (1958). MR 22 #3015

    Google Scholar 

  22. Calabi, E.: Extremal Kähler metrics,. In: Seminar on Differential Geometry. Annals of Mathematics Studies, vol. 102, pp. 259–290. Princeton University Press, Princeton, N.J. (1982) MR 83i:53088

    Google Scholar 

  23. Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics (I)—A priori estimates. J. Am. Math. Soc. 34(4), 909–936 (2021). MR 4301557

    Google Scholar 

  24. Chen, X., Cheng, J.: On the constant scalar curvature Kähler metrics (II)—Existence results. J. Am. Math. Soc. 34(4), 937–1009 (2021). MR 4301558

    Google Scholar 

  25. Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. I: approximation of metrics with cone singularities. J. Am. Math. Soc. 28(1), 183–197 (2015). MR 3264766

    Google Scholar 

  26. Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \(2\pi \). J. Am. Math. Soc. 28(1), 199–234 (2015). MR 3264767

    Google Scholar 

  27. Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \(2\pi \) and completion of the main proof. J. Amer. Math. Soc. 28(1), 235–278 (2015). MR 3264768

    Google Scholar 

  28. Collins, T.C., Székelyhidi, G.: K-semistability for irregular Sasakian manifolds. J. Differ. Geom. 109(1), 81–109 (2018). MR 3798716

    Google Scholar 

  29. Collins, T.C., Székelyhidi, G.: Sasaki-Einstein metrics and K-stability. Geom. Topol. 23(3), 1339–1413 (2019). MR 3956894

    Google Scholar 

  30. Donaldson, S.K.: Extremal metrics on toric surfaces: a continuity method. J. Differ. Geom. 79(3), 389–432 (2008). MR MR2433928 (2009j:58018)

    Google Scholar 

  31. Dancer, A.S., Wang, M.Y.: On Ricci solitons of cohomogeneity one. Ann. Global Anal. Geom. 39(3), 259–292 (2011). MR 2769300

    Google Scholar 

  32. Futaki, A., Ono, H., Wang, G.: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differ. Geom. 83(3), 585–635 (2009). MR MR2581358

    Google Scholar 

  33. Grossberg, M., Karshon, Y.: Bott towers, complete integrability, and the extended character of representations. Duke Math. J 76(1), 23–58 (1994). MR 1301185 (96i:22030)

    Google Scholar 

  34. Guan, D.: Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends. Trans. Am. Math. Soc. 347(6), 2255–2262 (1995). MR1285992 (96a:58059)

    Google Scholar 

  35. Haefliger, A.: Groupoïdes d’holonomie et classifiants. Astérisque 116, 70–97 (1984). Transversal structure of foliations (Toulouse, 1982). MR 86c:57026a

    Google Scholar 

  36. He, W.: Scalar curvature and properness on Sasaki manifolds (2018). arXiv:1802.03841

  37. He, W., Li, J.: Geometric pluripotential theory on Sasaki manifolds. J. Geom. Anal. 31(2), 1093–1179 (2021). MR 4215260

    Google Scholar 

  38. Hwang, A.D., Singer, M.A.: A momentum construction for circle-invariant Kähler metrics. Trans. Am. Math. Soc. 354(6), 2285–2325 (2002). (electronic). MR 1885653 (2002m:53057)

    Google Scholar 

  39. Hwang, A.D.: On existence of Kähler metrics with constant scalar curvature. Osaka J. Math. 31(3), 561–595 (1994). MR 1309403 (96a:53061)

    Google Scholar 

  40. Kegel, M., Lange, C.: A Boothby-Wang theorem for Besse contact manifolds. Arnold Math. J. 7(2), 225–241 (2021). textbfMR 4260074

    Google Scholar 

  41. Koiso, N.: On rotationally symmetric Hamilton’s equation for Kähler-Einstein metrics. In: Recent Topics in Differential and Analytic Geometry. Advance Studies Pure Mathematics, vol. 18, pp. 327–337. Academic, Boston, MA (1990) MR 1145263 (93d:53057)

    Google Scholar 

  42. Kollár, J.: Seifert \(G_m\)-bundles (2004). arXiv:math.AG/0404386

  43. Kollár, J.: Einstein metrics on connected sums of \(S^2\times S^3\) . J. Differential. Geom. 77(2), 259–272 (2007). MR 2269781

    Google Scholar 

  44. Koiso, N., Sakane, Y.: Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds. In: Curvature and Topology of Riemannian Manifolds (Katata, 1985). Lecture Notes in Mathematics, vol. 1201, pp. 165–179. Springer, Berlin (1986) MR 859583 (88c:53047)

    Google Scholar 

  45. Legendre, E.: Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics. Compos. Math. 147(5), 1613–1634 (2011). MR 2834736

    Google Scholar 

  46. Legendre, E., Tønnesen-Friedman, C.W.: Toric generalized Kähler-Ricci solitons with Hamiltonian 2-form. Math. Z 274(3–4), 1177–1209 (2013). MR 3078263

    Google Scholar 

  47. Mabuchi, T., Nakagawa, Y.: New examples of Sasaki-Einstein manifolds. Tohoku Math. J. (2) 65(2), 243–252 (2013). MR 3079287

    Google Scholar 

  48. Orlik, P., Wagreich, P.: Seifert \(n\)-manifolds. Invent. Math. 28, 137–159 (1975). MR 361150

    Google Scholar 

  49. Quillen, D.: he spectrum of an equivariant cohomology ring. I, II. Ann. Math. (2) 94, 549–572 (1971). ibid. (2) 94 (1971), 573–602. MR 298694

    Article  MathSciNet  MATH  Google Scholar 

  50. Smith, L.: Transfer and ramified coverings. Math. Proc. Camb. Philos. Soc. 93(3), 485–493 (1983). MR 698352

    Article  MathSciNet  MATH  Google Scholar 

  51. Shi, Y., Zhu, X.: Kähler-Ricci solitons on toric Fano orbifolds. Math. Z. 271(3–4), 1241–1251 (2012). MR 3665915

    Article  MathSciNet  MATH  Google Scholar 

  52. Tian, G., Zhu, X.: Uniqueness of Kähler-Ricci solitons. Acta Math. 184(2), 271–305 (2000). MR 1768112

    Article  MathSciNet  MATH  Google Scholar 

  53. Wang, X.-J., Zhu, X.: Kähler-Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188(1), 87–103 (2004). MR MR2084775 (2005d:53074)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yamazaki, T.: A construction of \(K\)-contact manifolds by a fiber join. Tohoku Math. J. (2) 51(4), 433–446 (1999). MR 2001e:53094

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhu, X.: Kähler-Ricci soliton typed equations on compact complex manifolds with\(C_1(M)>0\). J. Geom. Anal. 10(4), 759–774 (2000). MR 1817785

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors of this paper have benefitted from conversations with Vestislav Apostolov, David Calderbank, Hongnian Huang, Eveline Legendre, and Carlos Prieto.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131, USA

    Charles P. Boyer

  2. Department of Mathematics, Union College, Schenectady, NY, 12308, USA

    Christina W. Tønnesen-Friedman

Authors
  1. Charles P. Boyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christina W. Tønnesen-Friedman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charles P. Boyer .

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, UK

    Ivan Cheltsov

  2. Mathematics Department, Stony Brook University, Stony Brook, NY, USA

    Xiuxiong Chen

  3. Department of Mathematics, University of Miami, Coral Gables, FL, USA

    Ludmil Katzarkov

  4. Center for Geometry and Physics, Institute for Basic Science, Pohang, Korea (Republic of)

    Jihun Park

Appendix

Appendix

\(p_1\)

\(q_1\)

\(p_2\)

\(q_2\)

\(n_1\)

\(n_2\)

\(m_0\)

\(m_\infty \)

m

\(v_0\)

\(v_\infty \)

\({\mathcal I}_{\textbf{n},\textbf{m}}\)

1

2

−1

15

5124072

−740316

6438801

4797538

126251

51

38

89

1

2

−1

14

775675

−120061

972325

726685

10235

95

71

83

1

2

−1

13

48

−8

60

45

15

4

3

7

1

2

−1

12

2080161

−375516

2591676

1951756

31996

81

61

71

1

2

−1

11

1462832

−288008

1815479

1373876

49067

37

28

65

1

2

−1

10

110483

−23919

136479

103887

2037

67

51

59

1

2

−1

9

129720

−31188

159330

122153

5311

30

23

53

1

2

−1

8

80401

−21730

98050

75850

1850

53

41

47

1

2

−1

7

25944

−8004

31349

24534

1363

23

18

41

1

2

−1

6

124527

−44733

148629

118141

3811

39

31

35

1

2

−1

5

59072

−25376

69296

56303

4331

16

13

29

1

2

−1

4

525

−280

600

504

24

25

21

23

1

2

−1

3

1440

−1008

1575

1400

175

9

8

17

1

2

−1

2

1

−1

1

1

1

1

1

1

1

2

1

2

51

51

85

45

5

17

9

13

1

2

1

3

10416

7224

15996

9331

1333

12

7

19

1

2

1

4

31217

16492

46004

28196

1484

31

19

25

1

2

1

5

8208

3496

11799

7452

621

19

12

31

1

2

1

6

30015

10701

42435

27347

943

45

29

37

1

2

1

7

54808

16796

76570

50065

2945

26

17

43

1

2

1

8

51389

13806

71154

47034

1206

59

39

49

1

2

1

9

552

132

759

506

253

3

2

5

1

2

1

10

1112447

239659

1521101

1021013

20837

73

49

61

1

2

1

11

36000

7056

49000

33075

1225

40

27

67

1

2

1

12

456837

82128

619440

420080

7120

87

59

73

1

2

1

13

3134336

520384

4236251

2884256

90133

47

32

79

1

2

1

14

466923

72013

629331

429939

6231

101

69

85

1

2

1

15

5522472

795204

7425486

5087833

137509

54

37

91

TABLE: This gives a sample of Kähler−Einstein orbifold solutions. See Example 4.3.

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Boyer, C.P., Tønnesen-Friedman, C.W. (2023). Constant Scalar Curvature Sasaki Metrics and Projective Bundles. In: Cheltsov, I., Chen, X., Katzarkov, L., Park, J. (eds) Birational Geometry, Kähler–Einstein Metrics and Degenerations. BGKEMD BGKEMD BGKEMD 2019 2019 2019. Springer Proceedings in Mathematics & Statistics, vol 409. Springer, Cham. https://doi.org/10.1007/978-3-031-17859-7_5

Download citation

Publish with us

Policies and ethics

Search

Navigation