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Non-pluripolar products on vector bundles and Chern–Weil formulae

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In this paper, we develop several pluripotential-theoretic techniques for singular metrics on vector bundles. We first introduce the theory of non-pluripolar products on holomorphic vector bundles on complex manifolds. Then we define and study a special class of singularities of Hermitian metrics on vector bundles, called \(\mathcal {I}\)-good singularities, partially extending Mumford’s notion of good singularities. Next, we derive a Chern–Weil type formula expressing the Chern numbers of Hermitian vector bundles with \(\mathcal {I}\)-good singularities in terms of the associated b-divisors. We also define an intersection theory on the Riemann–Zariski space and apply it to reformulate our Chern–Weil formula.

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References

  1. Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S.: Smooth compactifications of locally symmetric varieties. Second. Cambridge Mathematical Library. With the collaboration of Peter Scholze. Cambridge University Press, Cambridge, pp. x+230 (2010). https://doi.org/10.1017/CBO9780511674693

  2. Baily, W.L., Jr., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. 2(84), 442–528 (1966). https://doi.org/10.2307/1970457

    Article  MathSciNet  Google Scholar 

  3. Botero, A., Burgos Gil, J.I., Holmes, D., de Jong, R.: Chern–Weil and Hilbert– Samuel formulae for singular Hermitian line bundles (2021). arXiv:2112.09007 [math.AG]

  4. Botero, A., Burgos Gil, J.I., Holmes, D., de Jong, R.: Rings of Siegel–Jacobi forms of bounded relative index are not finitely generated (2022). arXiv:2203.14583 [math.AG]

  5. Berman, R.J., Boucksom, S., Jonsson, M.: A variational approach to the Yau–Tian–Donaldson conjecture. J. Am. Math. Soc. 34(3), 605–652 (2021). https://doi.org/10.1090/jams/964

    Article  MathSciNet  Google Scholar 

  6. Berthelot, P., Grothendieck, A., Illusie, L.: Théorie des intersections et Théorème de Riemann–Roch: Séminaire de Géométrie Algébrique du Bois Marie 1966/67, vol. 225. Springer (2006)

  7. Boucksom, S., Favre, C., Jonsson, M.: Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci. 44(2), 449–494 (2008). https://doi.org/10.2977/prims/1210167334

    Article  MathSciNet  Google Scholar 

  8. Boucksom, S., Favre, C., Jonsson, M.: Differentiability of volumes of divisors and a problem of Teissier. J. Algebraic Geom. 18(2), 279–308 (2009). https://doi.org/10.1090/S1056-3911-08-00490-6

    Article  MathSciNet  Google Scholar 

  9. Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Monge–Ampère equations in big cohomology classes. Acta Math. 205(2), 199–262 (2010). https://doi.org/10.1007/s11511-010-0054-7

    Article  MathSciNet  Google Scholar 

  10. Boucksom, S., Demailly, J.-P., Păun, M., Peternell, T.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebraic Geom. 22(2), 201–248 (2013). https://doi.org/10.1090/S1056-3911-2012-00574-8

    Article  MathSciNet  Google Scholar 

  11. Boucksom, S., Jonsson, M.: Global pluripotential theory over a trivially valued field (2021). arXiv:1801.08229 [math.AG]

  12. Burgos Gil, J.I., Kramer, J., Kühn, U.: Arithmetic characteristic classes of automorphic vector bundles. Doc. Math. 10, 619–716 (2005)

    Article  MathSciNet  Google Scholar 

  13. Burgos Gil, J.I., Kramer, J., Kühn, U.: The singularities of the invariant metric on the Jacobi line bundle. In: Recent advances in Hodge theory. London Math. Soc. Lecture Note Ser., vol. 427, pp. 45–77. Cambridge Univ. Press, Cambridge (2016)

  14. Bonavero, L.: Inégalités de morse holomorphes singulières. J. Geom. Anal. 8(3), 409–425 (1998). https://doi.org/10.1007/BF02921793

    Article  MathSciNet  Google Scholar 

  15. Berndtsson, B., Păun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145(2), 341–378 (2008). https://doi.org/10.1215/00127094-2008-054

    Article  MathSciNet  Google Scholar 

  16. Cao, J.: Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compact Kähler manifolds. Compos. Math. 150(11), 1869–1902 (2014). https://doi.org/10.1112/S0010437X14007398

    Article  MathSciNet  Google Scholar 

  17. Chatzistamatiou, A., Rülling, K.: Vanishing of the higher direct images of the structure sheaf. Compos. Math. 151(11), 2131–2144 (2015). https://doi.org/10.1112/S0010437X15007435

    Article  MathSciNet  Google Scholar 

  18. Dahlhausen, C.: K-theory of admissible Zariski-Riemann spaces. Ann. K-Theory 8(1), 1–23 (2023). https://doi.org/10.2140/akt.2023.8.1

    Article  MathSciNet  Google Scholar 

  19. Darvas, T., Di Nezza, E., Lu, C.H.: Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity. Anal. PDE 11(8), 2049–2087 (2018). https://doi.org/10.2140/apde.2018.11.2049

    Article  MathSciNet  Google Scholar 

  20. Darvas, T., Di Nezza, E., Lu, C.H.: On the singularity type of full mass currents in big cohomology classes. Compos. Math. 154(2), 380–409 (2018). https://doi.org/10.1112/S0010437X1700759X

    Article  MathSciNet  Google Scholar 

  21. Darvas, T., Di Nezza, E., Lu, H.-C.: The metric geometry of singularity types. J. Reine Angew. Math. 771, 137–170 (2021). https://doi.org/10.1515/crelle-2020-0019

    Article  MathSciNet  Google Scholar 

  22. Demailly, J.-P.: On the cohomology of pseudoeffective line bundles. Complex geometry and dynamics. Abel Symp., vol. 10, pp. 51–99. Springer, Cham (2015)

  23. Dang, N.-B., Favre, C.: Intersection theory of nef b-divisor classes. Compos. Math. 158(7), 1563–1594 (2022). https://doi.org/10.1112/s0010437x22007515

    Article  MathSciNet  Google Scholar 

  24. Demailly, J.-P., Peternell, T., Schneider, M.: Pseudo-effective line bundles on compact Kähler manifolds. Int. J. Math. 12(6), 689–741 (2001). https://doi.org/10.1142/S0129167X01000861

    Article  Google Scholar 

  25. Dinh, T.-C., Sibony, N.: Pull-back of currents by holomorphic maps. Manuscr. Math. 123(3), 357–371 (2007). https://doi.org/10.1007/s00229-007-0103-5

    Article  MathSciNet  Google Scholar 

  26. Darvas, T., Wu, K.-R.: Griffiths extremality, interpolation of norms, and Kähler quantization. J. Geom. Anal. 32(7), Paper No. 203, 27 (2022). https://doi.org/10.1007/s12220-022-00940-0

  27. Darvas, T., Xia, M.: The volume of pseudoeffective line bundles and partial equilibrium. Geom. Topol. (2021). arXiv:2112.03827 [math.DG] (to appear)

  28. Darvas, T., Xia, M.: The closures of test configurations and algebraic singularity types. Adv. Math. 397, Paper No. 108198, 56 (2022). https://doi.org/10.1016/j.aim.2022.108198

  29. Ein, L., Lazarsfeld, R., Mustaţǎ, M., Nakamaye, M., Popa, M.: Asymptotic invariants of line bundles. Pure Appl. Math. Q. 1(2), 379–403 (2005). https://doi.org/10.4310/PAMQ.2005.v1.n2.a8

    Article  MathSciNet  Google Scholar 

  30. Fujiwara, K., Kato, F.: Foundations of rigid geometry. I. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, pp. xxxiv+829 (2018)

  31. Fulger, M., Lehmann, B.: Zariski decompositions of numerical cycle classes. J. Algebraic Geom. 26(1), 43–106 (2017). https://doi.org/10.1090/jag/677

    Article  MathSciNet  Google Scholar 

  32. Fujino, O.: Cone and contraction theorem for projective morphisms between complex analytic spaces (2022). https://www.math.kyoto-u.ac.jp/~fujino/analytic-lc9.pdf

  33. Fulton, W.: Intersection theory. Second. Vol. 2. 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], pp. xiv+470. Springer, Berlin (1998). https://doi.org/10.1007/978-1-4612-1700-8

  34. Grauert, H., Remmert, R.: Plurisubharmonische Funktionen in komplexen Räumen. Math. Z. 65, 175–194 (1956). https://doi.org/10.1007/BF01473877

    Article  MathSciNet  Google Scholar 

  35. Grothendieck, A., Raynaud, M.: Revêtements Étales et Groupe Fondamental: Séminaire de Géométrie Algébrique du Bois Marie 1960/61, vol. 224. Springer, (2006)

  36. Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with Hermitian metric. I. Ann. Math. (2) 131(1), 163–203 (1990). https://doi.org/10.2307/1971512

    Article  MathSciNet  Google Scholar 

  37. Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with Hermitian metric. II. Ann. Math. (2) 131(2), 205–238 (1990). https://doi.org/10.2307/1971493

    Article  MathSciNet  Google Scholar 

  38. Gillet, H., Soulé, C.: Arithmetic intersection theory. Inst. Hautes Études Sci. Publ. Math. 72(1990), 93–174 (1991). http://www.numdam.org/item?id=PMIHES_1990__72__93_0

  39. Guedj, V., Zeriahi, A.: Degenerate complex Monge-Ampère equations. EMS Tracts in Mathematics, vol. 26, pp. xxiv+472. European Mathematical Society (EMS), Zürich (2017). https://doi.org/10.4171/167

  40. Hirzebruch, F.: Automorphe Formen und der Satz von Riemann-Roch. In: Symposium internacional de topologia algebraica International symposium on algebraic topology, pp. 129–144. Universidad Nacional Autónoma de México and UNESCO, Mexico City (1958)

  41. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, vol. 134. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  42. Kobayashi, S.: Negative vector bundles and complex Finsler structures. Nagoya Math. J. 57, 153–166 (1975). http://projecteuclid.org/euclid.nmj/1118795367

  43. Kudla, S.S., Rapoport, M., Yang, T.: Modular forms and special cycles on Shimura curves. Annals of Mathematics Studies, vol. 161, pp. x+373. Princeton University Press, Princeton (2006). https://doi.org/10.1515/9781400837168

  44. Kerz, M., Strunk, F., Tamme, G.: Algebraic K-theory and descent for blow-ups. Invent. Math. 211(2), 523–577 (2018). https://doi.org/10.1007/s00222-017-0752-2

    Article  ADS  MathSciNet  Google Scholar 

  45. Kudla, S.S.: Central derivatives of Eisenstein series and height pairings. Ann. Math. (2) 146(3), 545–646 (1997). https://doi.org/10.2307/2952456

    Article  MathSciNet  Google Scholar 

  46. Lipman, J., Neeman, A.: Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor. Ill. J. Math. 51(1), 209–236 (2007). http://projecteuclid.org/euclid.ijm/1258735333

  47. Lärkäng, R., Raufi, H., Ruppenthal, J., Sera, M.: Chern forms of singular metrics on vector bundles. Adv. Math. 326, 465–489 (2018). https://doi.org/10.1016/j.aim.2017.12.009

    Article  MathSciNet  Google Scholar 

  48. Lärkäng, R., Raufi, H., Sera, M., Wulcan, E.: Chern forms of Hermitian metrics with analytic singularities on vector bundles. Indiana Univ. Math. J. 71(1), 153–189 (2022). https://doi.org/10.1512/iumj.2022.71.8834

    Article  MathSciNet  Google Scholar 

  49. Liu, K., Sun, X., Yang, X.: Positivity and vanishing theorems for ample vector bundles. J. Algebraic Geom. 22(2), 303–331 (2013). https://doi.org/10.1090/S1056-3911-2012-00588-8

    Article  MathSciNet  Google Scholar 

  50. Ma, X., Marinescu, G.: Holomorphic Morse inequalities and Bergman kernels. Progress in Mathematics, vol. 254, pp. xiv+422. Birkhäuser Verlag, Basel (2007)

  51. Mourougane, C.: Computations of Bott–Chern classes on P(E). Duke Math. J. 124(2), 389–420 (2004). https://doi.org/10.1215/S0012-7094-04-12425-X

    Article  MathSciNet  Google Scholar 

  52. Mumford, D.: Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ. 9, 195–204 (1968). https://doi.org/10.1215/kjm/1250523940

    Article  MathSciNet  Google Scholar 

  53. Mumford, D.: Hirzebruch’s proportionality theorem in the noncompact case. Invent. Math. 42, 239–272 (1977). https://doi.org/10.1007/BF01389790

    Article  ADS  MathSciNet  Google Scholar 

  54. Păun, M., Takayama, S.: Positivity of twisted relative pluricanonical bundles and their direct images. J. Algebraic Geom. 27(2), 211–272 (2018). https://doi.org/10.1090/jag/702

    Article  MathSciNet  Google Scholar 

  55. Raufi, H.: Singular hermitian metrics on holomorphic vector bundles. Ark. Mat. 53(2), 359–382 (2015). https://doi.org/10.1007/s11512-015-0212-4

    Article  MathSciNet  Google Scholar 

  56. Ross, J., Witt Nyström, D.: Analytic test configurations and geodesic rays. J. Symplectic Geom. 12(1), 125–169 (2014). https://doi.org/10.4310/JSG.2014.v12.n1.a5

    Article  MathSciNet  Google Scholar 

  57. T. Stacks Project Authors.: Stacks Project. (2020). http://stacks.math.columbia.edu

  58. Temkin, M.: Relative Riemann–Zariski spaces. Isr. J. Math. 185, 1–42 (2011). https://doi.org/10.1007/s11856-011-0099-0

    Article  MathSciNet  Google Scholar 

  59. Temkin, M., Tyomkin, I.: On relative birational geometry and Nagata’s compactification for algebraic spaces. Int. Math. Res. Not. IMRN 11, 3342–3387 (2018). https://doi.org/10.1093/imrn/rnw339

    Article  MathSciNet  Google Scholar 

  60. Trusiani, A.: A relative Yau–Tian–Donaldson conjecture and stability thresholds (2023). arXiv:2302.07213 [math.AG]

  61. Vu, D.-V.: Relative non-pluripolar product of currents. Ann. Glob. Anal. Geom. 60(2), 269–311 (2021). https://doi.org/10.1007/s10455-021-09780-7

    Article  MathSciNet  Google Scholar 

  62. Weibel, C.A.: The K-Book: An Introduction to Algebraic K-Theory, vol. 145. American Mathematical Society, Providence (2013)

    Google Scholar 

  63. Witt Nyström, D.: Monotonicity of non-pluripolar Monge–Ampère masses. Indiana Univ. Math. J. 68(2), 579–591 (2019). https://doi.org/10.1512/iumj.2019.68.7630

    Article  MathSciNet  Google Scholar 

  64. Xia, M.: Partial Okounkov bodies and Duistermaat–Heckman measures of non-Archimedean metrics (2021). arXiv:2112.04290 [math.AG]

  65. Xia, M.: Operations on transcendental non-Archimedean metrics (2023). arXiv:2312.17150 [math.AG]

  66. Xia, M.: Pluripotential-theoretic stability thresholds. Int. Math. Res. Not. IMRN 14, 12324–12382 (2023). https://doi.org/10.1093/imrn/rnac186

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I would like to thank Elizabeth Wulcan, Yanbo Fang, Yaxiong Liu, Richard Lärkäng, José Burgos Gil, Tamás Darvas, David Witt Nyström, Yu Zhao, Dennis Eriksson, Moritz Kerz and Osamu Fujino for discussions. I am grateful to the referees for their valuable suggestions.The author is supported by Knut och Alice Wallenbergs Stiftelse grant KAW 2021.0231.

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    Mingchen Xia

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Xia, M. Non-pluripolar products on vector bundles and Chern–Weil formulae. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02838-4

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