Michigan Mathematical Journal

$L^{2}$ Estimates and Vanishing Theorems for Holomorphic Vector Bundles Equipped with Singular Hermitian Metrics

Takahiro Inayama

Abstract

We investigate singular hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^{2}$ estimates and vanishing theorems usually require an assumption that the vector bundles are Nakano positive. However, there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^{2}$ estimates and vanishing theorems by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.

Article information

Source
Michigan Math. J., Volume 69, Issue 1 (2020), 79-96.

Dates
Revised: 21 October 2019
First available in Project Euclid: 14 November 2019

https://projecteuclid.org/euclid.mmj/1573700740

Digital Object Identifier
doi:10.1307/mmj/1573700740

Mathematical Reviews number (MathSciNet)
MR4071346

Zentralblatt MATH identifier
07208926

Subjects
Secondary: 32L20: Vanishing theorems

Citation

Inayama, Takahiro. $L^{2}$ Estimates and Vanishing Theorems for Holomorphic Vector Bundles Equipped with Singular Hermitian Metrics. Michigan Math. J. 69 (2020), no. 1, 79--96. doi:10.1307/mmj/1573700740. https://projecteuclid.org/euclid.mmj/1573700740

References

• [1] B. Berndtsson, An introduction to things $\bar{\partial}$, analytic and algebraic geometry, IAS/Park City Math. Ser., 17, pp. 7–76, Amer. Math. Soc., Providence, RI, 2010.
• [2] B. Berndtsson and M. Păun, Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J. 145 (2008), no. 2, 341–378.
• [3] M. A. A. de Cataldo, Singular Hermitian metrics on vector bundles, J. Reine Angew. Math. 502 (1998), 93–122.
• [4] J.-P. Demailly, Complex analytic and differential geometry, http://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf.
• [5] J.-P. Demailly, Analytic methods in algebraic geometry, Surv. Mod. Math., 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012.
• [6] J.-P. Demailly and H. Skoda, Relations entre les notions de positivité de P. A. Griffiths et de S. Nakano, Séminaire P. Lelong-H. Skoda (Analyse), année 1978/79, Lecture Notes in Math., 822, pp. 304–309, Springer-Verlag, Berlin, 1980.
• [7] C. Hacon, M. Popa, and C. Schnell, Algebraic fiber spaces over Abelian varieties: around a recent theorem by Cao and Paun, arXiv:1611.08768.
• [8] T. Ohsawa, Isomorphism theorems for cohomology groups of weakly $1$-complete manifolds, Publ. RIMS, Kyoto Univ. 18 (1982), 191–232.
• [9] M. Păun and S. Takayama, Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebraic Geom. 27 (2018), 211–272.
• [10] H. Raufi, Singular Hermitian metrics on holomorphic vector bundles, Ark. Mat. 53 (2015), no. 2, 359–382.
• [11] H. Raufi, The Nakano vanishing theorem and a vanishing theorem of Demailly–Nadel type for holomorphic vector bundles, arXiv:1212.4417.
• [12] Y. T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976/77), no. 1, 89–100.
• [13] H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans $\mathbb{C}^{n}$, Bull. Soc. Math. France 100 (1972), 353–408.