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2013 Relative Kähler–Ricci flows and their quantization
Robert Berman
Anal. PDE 6(1): 131-180 (2013). DOI: 10.2140/apde.2013.6.131


Let π:XS be a holomorphic fibration and let be a relatively ample line bundle over X. We define relative Kähler–Ricci flows on the space of all Hermitian metrics on with relatively positive curvature and study their convergence properties. Mainly three different settings are investigated: the case when the fibers are Calabi–Yau manifolds and the case when =±KXS is the relative (anti)canonical line bundle. The main theme studied is whether “positivity in families” is preserved under the flows and its relation to the variation of the moduli of the complex structures of the fibers. The “quantization” of this setting is also studied, where the role of the Kähler–Ricci flow is played by Donaldson’s iteration on the space of all Hermitian metrics on the finite rank vector bundle πS. Applications to the construction of canonical metrics on the relative canonical bundles of canonically polarized families and Weil–Petersson geometry are given. Some of the main results are a parabolic analogue of a recent elliptic equation of Schumacher and the convergence towards the Kähler–Ricci flow of Donaldson’s iteration in a certain double scaling limit.


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Robert Berman. "Relative Kähler–Ricci flows and their quantization." Anal. PDE 6 (1) 131 - 180, 2013.


Received: 21 August 2011; Revised: 15 November 2011; Accepted: 20 December 2011; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1282.14069
MathSciNet: MR3068542
Digital Object Identifier: 10.2140/apde.2013.6.131

Primary: 14J32 , 32G05 , 32Q20 , 53C55

Keywords: balanced metric , Kähler–Einstein metric , Kähler–Ricci flow , positivity , Weil–Petersson metric

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.6 • No. 1 • 2013
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