MUM quintic degeneration and normalized Ricci-flat metrics

A one-parameter quintic degeneration is a proper holomorphic map $f:\mathcal{X}\to \mathrm{\Delta }$ from a complex analytic space, where $\mathrm{\Delta }=\{t\in \mathbb{C}:\mathrm{\mid }t\mathrm{\mid }<\epsilon \}$ and ${\mathrm{\Delta }}^{\ast }=\mathrm{\Delta }\setminus \{0\}$. Require $f^{-1}({\mathrm{\Delta }}^{\ast })$ to be a complex manifold and $f{\mathrm{\mid }}_{f^{-1}({\mathrm{\Delta }}^{\ast })}:f^{-1}({\mathrm{\Delta }}^{\ast })\to {\mathrm{\Delta }}^{\ast }$ to be a holomorphic submersion. For every $t\in {\mathrm{\Delta }}^{\ast }$, require $X_t=f^{-1}(t)$ to be biholomorphic to a smooth degree-five hypersurface in ${\mathbb{P}}_{\mathbb{C}}^4$; here smooth means the defining homogeneous polynomial has no singular point on its zero locus. Fix $t_0\in {\mathrm{\Delta }}^{\ast }$. Proper smoothness makes $R^3{f}_{\ast }\mathbb{Z}$ a local system on ${\mathrm{\Delta }}^{\ast }$; its monodromy $T:H^3(X_{t_0};\mathbb{Z})\to H^3(X_{t_0};\mathbb{Z})$ is parallel transport around the positively oriented generator of ${\pi }_1({\mathrm{\Delta }}^{\ast },t_0)$. It is maximally unipotent (MUM) when $T$ is unipotent and $N=\log T:={\sum }_{j=1}^{s-1}(-1{)}^{j+1}(T-\mathrm{id}{)}^j\mathrm{/}j$ satisfies $N^4=0$ and $N^3\mathrm{\ne }0$, where $s$ is any positive integer with $(T-\mathrm{id}{)}^s=0$.

A normalized Ricci-flat metric family is a smooth family of real vertical two-forms ${\omega }_t$ on the fibres $X_t$. More precisely, put $V=\ker (d{f}_{\mathbb{R}}:T_{\mathbb{R}}{f}^{-1}({\mathrm{\Delta }}^{\ast })\to f^{\ast }{T}_{\mathbb{R}}{\mathrm{\Delta }}^{\ast })$, the real vertical tangent bundle, and take a smooth section of ${\mathrm{\Lambda }}_{\mathbb{R}}^2{V}_{\mathbb{R}}^{\ast }$ whose restriction to $X_t$ is ${\omega }_t$. Require each ${\omega }_t$ to be closed, of type $(1,1)$, and positive, hence to define a Kähler metric $g_t$ by ${\omega }_t(v,w)=g_t(J_{t}v,w)$. Ricci-flat means that in every holomorphic coordinate chart on $X_t$, if ${\omega }_t=i{g}_{j\bar{k}}d{z}^j\wedge d{\bar{z}}^k$, its Ricci form ${\rho }_t=-i\mathrm{\partial }\bar{\mathrm{\partial }}\log \det (g_{j\bar{k}})$ vanishes. Require ${\int }_{X_t}\frac{{\omega }_t^3}{3!}=1.$ The Kähler classes and the scale normalization are part of the data. This definition does not prove that the family exists; existence is conditional on the precise Calabi–Yau metric inputs.

Compact Calabi–Yau manifold conventions