A curvature formula associated to a family of pseudoconvex domains
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, p. 269-313
We shall give a definition of the curvature operator for a family of weighted Bergman spaces { t } associated to a smooth family of smoothly bounded strongly pseudoconvex domains {D t }. In order to study the “boundary term” in the curvature operator, we shall introduce the notion of geodesic curvature for the associated family of boundaries {D t }. As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for { t } and its applications to triviality of fibrations are also given in this paper.
Nous définissons l’opérateur de courbure pour une famille d’espaces de Bergman pondérés { t } associés à une famille lisse de domaines lisses bornés strictement pseudoconvexes {D t }. Afin d’étudier le “terme au bord” dans l’opérateur de courbure, nous introduisons la notion de courbure géodésique pour la famille des bords associés. Comme application, nous obtenons une formule de variation pour les normes de projections de Bergman des courants à support compact. Un critère de platitude pour { t } et ses applications à la trivialité des fibrations sont également données dans cet article.
Received : 2015-10-06
Revised : 2015-12-20
Accepted : 2016-06-14
Published online : 2017-01-10
Classification:  32A25,  32L25,  32G05
Keywords: Brunn–Minkowski theory, Prekopa theorem, ¯-equation, Hörmander theory.
@article{AIF_2017__67_1_269_0,
     author = {Wang, Xu},
     title = {A curvature formula associated to a family of pseudoconvex domains},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     pages = {269-313},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_1_269_0}
}
A curvature formula associated to a family of pseudoconvex domains. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 269-313. https://aif.centre-mersenne.org/item/AIF_2017__67_1_269_0/

[1] Berndtsson, B. The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman, Ann. Inst. Fourier, Tome 46 (1996) no. 4, pp. 1083-1094

[2] Berndtsson, B. Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions, Math. Ann., Tome 312 (1998) no. 4, pp. 785-792

[3] Berndtsson, B. Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier, Tome 56 (2006) no. 6, pp. 1633-1662

[4] Berndtsson, B. Curvature of vector bundles associated to holomorphic fibrations, Ann. Math., Tome 169 (2009) no. 2, pp. 531-560

[5] Berndtsson, B. Positivity of direct image bundles and convexity on the space of Kähler metrics, J. Diff. Geom., Tome 81 (2009) no. 3, pp. 457-482

[6] Berndtsson, B. An introduction to things ¯, Analytic and algebraic geometry, Providence, RI (IAS/Park City Math.) Tome 17 (2010), pp. 7-76

[7] Berndtsson, B. Strict and nonstrict positivity of direct image bundles, Math. Z., Tome 269 (2011) no. 3-4, pp. 1201-1218

[8] Berndtsson, B. Convexity on the space of Kähler metrics, Ann. Fac. Sci. Toulouse, Tome 22 (2013) no. 4, pp. 713-746

[9] Berndtsson, B. The openness conjecture for plurisubharmonic functions (2013) (http://arxiv.org/abs/1305.5781 )

[10] Berndtsson, B. A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math., Tome 200 (2014) no. 1, pp. 149-200

[11] Berndtsson, B. A comparison principle for Bergman kernels (2015) (http://arxiv.org/abs/1501.02440 )

[12] Berndtsson, B. The Openness Conjecture and Complex Brunn-Minkowski Inequalities, Complex Geometry and Dynamics, Springer (Abel Symposia) Tome 10 (2015), pp. 29-44

[13] Berndtsson, B.; Berman, R. J. Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics (2015) (http://arxiv.org/abs/1405.0401 )

[14] Berndtsson, B.; Lempert, L. A proof of the Ohsawa-Takegoshi theorem with sharp estimates (2014) (http://arxiv.org/abs/1407.4946 )

[15] Berndtsson, B.; Paun, M. Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J., Tome 145 (2008) no. 2, pp. 341-378

[16] Brascamp, H. J.; Lieb, E. H. On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal., Tome 22 (1976), pp. 366-389

[17] Choi, Y. J. A study of variations of pseudoconvex domains via Kähler-Einstein metrics, Math. Z., Tome 281 (2015) no. 1-2, pp. 299-314

[18] Demailly, J.-P. Complex analytic and differential geometry (Book available from the author’s homepage https://www-fourier.ujf-grenoble.fr/~demailly/documents.html)

[19] Demailly, J.-P. Estimations L 2 pour l’opérateur ¯ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Éc Norm. Sup., Tome 15 (1982), pp. 457-511

[20] Donaldson, S. K. Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. (Amer. Math. Soc. Transl. Ser. 2) Tome 196 (1999), pp. 13-33

[21] Donnelly, H.; Fefferman, C. L 2 -cohomology and index theorem for the Bergman metric, Ann. Math., Tome 118 (1983), pp. 593-618

[22] Folland, G. B.; Kohn, J. J. The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Ann. Math. Studies, Tome 75 (1972), viii+146 pages

[23] Geiger, T.; Schumacher, G. Curvature of higher direct image sheaves (2015) (http://arxiv.org/abs/1501.07070 )

[24] Greene, R. E.; Krantz, S. G. Deformation of complex structures, estimates for the ¯-equation, and stability of the Bergman kernel, Adv. Math., Tome 43 (1982), pp. 1-86

[25] Griffiths, P. Topics in transcendental algebraic geometry, Princeton University Press, Annals of Mathematics Studies, Tome 106 (1984), viii+316 pages

[26] Gromov, M. Kähler hyperbolicity and L 2 -Hodge theory, J. Diff. Geom., Tome 33 (1991) no. 1, pp. 263-292

[27] Hamilton, R. S. Deformation of complex structures on manifolds with boundary. I. The stable case, J. Diff. Geom., Tome 12 (1977), pp. 1-45

[28] Hamilton, R. S. Deformation of complex structures on manifolds with boundary. II. Families of noncoercive boundary value problems, J. Diff. Geom., Tome 14 (1979), pp. 409-473

[29] Hörmander, L. An introduction to complex analysis in several variables, Elsevier, North-Holland Mathematical Library, Tome 7 (1973), 213 pages

[30] Kodaira, K. Complex manifolds and deformation of complex structures, Springer-Verlag, Grundlehren der Mathematischen Wissenschaften, Tome 283 (1986), x+465 pages

[31] Kodaira, K.; Spencer, D. C. On deformations of complex analytic structures, III. Stability theorems for complex structures, Ann. Math., Tome 71 (1960), pp. 43-76

[32] Komatsu, G. Hadamard’s variational formula for the Bergman kernel, Proc. Japan Acad., Tome 58 (1982), pp. 345-348

[33] Lempert, L.; Szőke, R. Direct Images, Fields of Hilbert Spaces, and Geometric Quantization, Commun. Math. Phys., Tome 327 (2014) no. 1, pp. 49-99

[34] Liu, K.; Yang, X. Curvatures of direct image sheaves of vector bundles and applications, J. Diff. Geom., Tome 98 (2014) no. 1, pp. 117-145

[35] Liu, R. S. A property of the graph of a holomorphic motion, J. Fudan Univ., Tome 47 (2008) no. 2, pp. 172-176 (in Chinese)

[36] Màñé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Éc. Norm. Supér., Tome 16 (1983), pp. 193-217

[37] Mabuchi, T. K-energy maps integrating Futaki invariants, Tohoku Math. J., Tome 38 (1986) no. 1-2, pp. 575-593

[38] Maitani, F. Variations of meromorphic differentials under quasiconformal deformations, J. Math. Kyoto Univ., Tome 24 (1984), pp. 49-66

[39] Maitani, F.; Yamaguchi, H. Variation of Bergman metrics on Riemann surfaces, Math. Ann., Tome 330 (2004) no. 3, pp. 477-489

[40] Mourougane, C.; Takayama, S. Hodge metrics and positivity of direct images, J. Reine Angew. Math., Tome 606 (2007), pp. 167-178

[41] Mourougane, C.; Takayama, S. Hodge metrics and the curvature of higher direct images, Ann. Éc. Norm. Supér., Tome 41 (2008) no. 6, pp. 905-924

[42] Ohsawa, T.; Takegoshi, K. On the extension of L 2 holomorphic functions, Math. Z., Tome 195 (1987), pp. 197-204

[43] Prekopa, A. On logarithmic concave measures and functions, Acad. Sci. Math., Tome 34 (1973), pp. 335-343

[44] Schumacher, G. Positivity of relative canonical bundles and applications, Invent. Math., Tome 190 (2012) no. 1, pp. 1-56

[45] Semmes, S. Interpolation of Banach spaces, differential geometry and differential equations, Rev. Mat. Iberoam., Tome 4 (1988) no. 1, pp. 155-176

[46] Siu, Y. T. Every Stein subvariety admits a Stein neighborhood, Invent. Math., Tome 38 (1976), pp. 89-100

[47] Siu, Y. T. Curvature of the Weil-Petersson metric in the moduli space of compact Kähler-Einstein manifolds of negative first Chern class, Contributions to Several Complex Variables, Proc. Conf. Complex Analysis, Notre Dame/Indiana, 1984, Vieweg (Aspects Math.) Tome E9 (1986), pp. 261-298

[48] Siu, Y. T. Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, Complex Geometry, Springer, Berlin (2002), pp. 223-277

[49] Slodkowski, Z. Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc., Tome 111 (1991) no. 2, pp. 347-355

[50] Sullivan, D.-P.; Thurston, W.-P. Extending holomorphic motions, Acta Math., Tome 157 (1986), pp. 243-286

[51] Tran, D. V. Direct images as Hilbert fields and their curvatures, Purdue University, USA (2014) (Ph. D. Thesis)

[52] Tsuji, H. Variation of Bergman kernels of adjoint line bundles (2005) (http://arxiv.org/abs/math/0511342 )

[53] Wang, X. Variation of the Bergman kernels under deformations of complex structures (2013) (http://arxiv.org/abs/1307.5660 )