Relative Kähler–Ricci flows and their quantization

Abstract

Let $\pi :\mathcal{X}\to S$ be a holomorphic fibration and let $\mathcal{ℒ}$ be a relatively ample line bundle over $\mathcal{X}$. We define relative Kähler–Ricci flows on the space of all Hermitian metrics on $\mathcal{ℒ}$ 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 $\mathcal{ℒ}=\pm K_{\mathcal{X}∕S}$ 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 ${\pi }_{\ast }\mathcal{ℒ}\to S$. 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.