Relative Kähler-Ricci flows and their quantization (Q371158)

Consider \(\pi: \mathcal{X}\rightarrow S\) a proper holomophic submersion of relative dimension \(n\) over a connected complex base and a relative ample line bundle \(\mathcal{L}\rightarrow \mathcal{X}\). This is a classical setting that arises in the key study of certain natural metrics on certain moduli spaces in complex geometry (typically like the Teichmüller space of Riemann surfaces). Given an initial metric (of potential \(\phi_0\)) with fiberwise positive curvature on \(\mathcal{L}\), the author is interested in the relative Kähler-Ricci flows of the form NEWLINE\[NEWLINE\frac{\partial \phi_t}{\partial t}= \log \frac{(dd^c \phi_t)^n}{\mu(\phi_t)}NEWLINE\]NEWLINE where \(\mu(\phi_t)\) takes values in the space of volume forms. Depending on the geometric setup, this function \(\mu\) has various natural expressions and the properties of the corresponding relative Kähler-Ricci flows have been studied for a while (e.g long time existence, uniqueness).NEWLINENEWLINEIn the Calabi-Yau setting, the measure \(\mu\) is independent of \(\phi\). Then, the author shows that the positivity (resp. semipositivity) of the curvature of the induced metric on \(\mathcal{L}-tK_{\mathcal{X}/S}\) is preserved along the flow. This is done by introducing a parabolic evolution equation along the flow for the function on \(\mathcal{X}\) NEWLINE\[NEWLINE\frac{(\partial\bar\partial \phi_t)^{n+1}}{ (\partial_X\bar\partial_X \phi_t)^{n}\wedge ds \wedge d\bar s},NEWLINE\]NEWLINE for \(\mathcal{X}\simeq_{\text{loc}} X\times \mathbb{C}\) (compare with the [\textit{G. Schumacher}, Invent. Math. 190, No. 1, 1--56 (2012; Zbl 1258.32005)]). He also considers the quantization of this setting and provides an extension of Donaldson's iteration scheme [\textit{S. K. Donaldson}, Pure Appl. Math. Q. 5, No. 2, 571--618 (2009; Zbl 1178.32018)] on the space of all Hermitian metrics on \(\pi_* \mathcal{L}\rightarrow S\). In particular, the author shows the following nice result. Assume \(L\) is an ample line bundle on the projective manifold \(X\) and \(\phi_{m}^{(k)}\) is the \(m\)-th iterate of the Bergman iteration on the space of metrics of \(L^k\). Then \(\phi_{m_k}^{(k)}\rightarrow \phi_t\) when \(m_k/k \rightarrow t\). In other words, one can approximate the usual Kähler-Ricci flow at any given time using algebraic metrics. This has to be compared with \textit{H.-D. Cao} and \textit{J. Keller} [J. Eur. Math. Soc. (JEMS) 15, No. 3, 1033--1065 (2013; Zbl 1308.32025)].NEWLINENEWLINEIn the anticanonical and canonical setting, the author derives similar results as above but the study is more tricky as there may not be Kähler-Einstein metric (or a balanced metric) at least on a Fano manifold. Among other things, the authors obtains that the balanced metric on \(K_{\mathcal{X}/S}\) at level \(k\) is smooth with semipositive curvature on \(\mathcal{X}\). Furthermore when \(k\rightarrow +\infty\), the balanced metrics induce metrics on \(S\) that converge towards the Weil-Petersson metric (see [the reviewer, ``Numerical Weil-Petersson metrics on moduli spaces of Calabi-Yau manifolds'', \url{arXiv:0907.1387} (2009)] where another construction of the Weil-Petersson metric is provided).NEWLINENEWLINEThe paper is well written.