Relative Kahler-Ricci flows and their quantization - Mathematics > Differential GeometryReport as inadecuate

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Abstract: Let X be a complex manifold fibered over the base S and let L be a relativelyample line bundle over X. We define relative Kahler-Ricci flows on the space ofall Hermitian metrics on L with relatively positive curvature. Mainly threedifferent settings are investigated: the case when the fibers are Calabi-Yaumanifolds and the case when L is the relative anti- canonical line bundle.The main theme studied is whether posivity in families is preserved under theflows and its relation to the variation of the moduli of the complex structuresof the fibres. The quantization of this setting is also studied, where the roleof the Kahler-Ricci flow is played by Donaldson-s iteration on the space of allHermitian metrics on the finite rank vector bundle over S defined as the zerothdirect image induced by the fibration. Applications to the construction ofcanonical metrics on relative canonical bumdles and Weil-Petersson geometry aregiven. Some of the main results are a parabolic analogue of a recent ellipticequation of Schumacher and the convergence towards the K\-ahler-Ricci flow ofDonaldson-s iteration in a certain double scaling limit.

Author: Robert J. Berman


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