On moderate degenerations of polarized Ricci-flat Kähler manifolds (Q2788598)

Let \(\mathfrak{X}\) be an \(n+1\) dimensional complex manifold with a hololorphic map \(\pi:\mathfrak{X}\to \Delta\) which is projective, flat and smooth over \(\Delta^*\), where \(\Delta\subset\mathbb{C}\) is the unit disc. Assume that \(X_t=\pi^{-1}(t)\) all have trivial canonical bundle for \(t\in\Delta^*\), and fix a holomorphic never-vanishing section \(\Omega\) of \(H^0(\Delta,\pi_*K_{\mathfrak{X}})\), which induces never-vanishing holomorphic \(n\)-forms \(\Omega_t=\Omega|_{X_t}\). Fix also a relatively ample line bundle \(L\) over \(\mathfrak{X}\), and let \(L_t=L|_{X_t}\).NEWLINENEWLINEWith these we can define a Weil-Petersson semipositive real \((1,1)\) form NEWLINE\[NEWLINE\omega_{WP}=-i\partial\overline{\partial}\log\int_{X_t}(-1)^{n^2/2}\Omega_t\wedge\overline{\Omega}_t,NEWLINE\]NEWLINE on \(\Delta^*\). \textit{C.-L. Wang} [Math. Res. Lett. 4, No. 1, 157--171 (1997; Zbl 0881.32017)] conjectured that \(0\) is a finite distance from \(\Delta^*\) with respect to \(\omega_{WP}\) if and only if up to modifying \(\mathfrak{X}\) by a finite base change and by a birational transformation, the central fiber \(X_0\) has at worst canonical singularities. This conjecture was proved by \textit{V. Tosatti} [Int. Math. Res. Not. 2015, No. 20, 10586--10594 (2015; Zbl 1338.32022)], who also observed that these conditions imply that the Ricci-flat Kähler metric \(\omega_t\) on \(X_t\) in \(c_1(L_t)\), for \(t\in\Delta^*\) satisfies a uniform diameter upper bound as \(t\) goes to \(0\).NEWLINENEWLINEThe main result of this paper proves that conversely, if the diameter bound holds then \(0\) is at finite Weil-Petersson distance.