Let \(f: X\rightarrow S\) be a family of compact Riemann surfaces endowed with its canonical \(L^2\)-metric or a Quillen metric on its determinant bundle.
In the semi-stable case, the singularities and the curvature current of the Quillen metric on the determinant of the Hodge bundle \(f_\ast \omega_{X/S}\) was described in [J. M. Bismut and J. B. Bost, Acta Math. 165, No. 1–2, 1–103 (1990; Zbl 0709.32019)].
In the special case, where \(S\) is the unit disk and there is a unique singular fiber \(X_0\) over \(0 \in S\), the principal part of the curvature current has the form \(\frac1{12} \# \mathrm{sing}(X_0) \delta_0\), where \(\delta_0\) is the Dirac current at \(0\) and \(\# \mathrm{sing}(X_0)\) is the number of singular points in the fiber \(X_0\).
The authors study analogues of this phenomenon for \(L^2\)-metrics on Hodge bundles for Calabi-Yau families, for Quillen metrics on determinant bundles and for the so-called BCOV metric for Calabi-Yau 3-folds (see [H. Fang et al., J. Differ. Geom. 80, No. 2, 175–259 (2008; Zbl 1167.32016)]).
It turns out that the dominant and subdominant terms in the expansions of the metrics close to nonsmooth fibers are related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds. The corresponding invariants for more general degenerating families in the case of the Quillen metric are also described.
The authors emphasize that, in fact, similar results partially appeared in other works, in slightly different terms, conditions and settings [C.-L. Wang, Math. Res. Lett. 4, No. 1, 157–171 (1997; Zbl 0881.32017); D. Eriksson, Int. Math. Res. Not. 2013, No. 2, 347–361 (2013; Zbl 1314.30087); K.-I. Yoshikawa, Math. Ann. 337, No. 1, 61–89 (2007; Zbl 1126.32029)].