Proof. The proof is a routine application of the arithmetic Riemann–Roch theorem of Gillet–Soulé [Reference Gillet and SouléGS92, Thm. 7]. We give the details for the convenience of the reader. Consider the virtual vector bundle $\sum (-1)^{p}p\Omega _{{\mathcal X}/S}^{p}$ , with virtual hermitian structure deduced from the metric h and denoted $h^{\bullet }$ . Its determinant of cohomology $\lambda _{{\scriptscriptstyle {\mathrm {BCOV}}}}$ carries the Quillen metric $h_{{\scriptscriptstyle {\mathrm {Q}}}}$ . The theorem of Gillet–Soulé provides an equality in $\widehat {\operatorname {CH}}^{1}(S)_{{\mathbb Q}}$

(2.11) $$ \begin{align} \widehat{\operatorname{c}}_{1}(\lambda_{{\scriptscriptstyle{\mathrm{BCOV}}}}, h_{{\scriptscriptstyle{\mathrm{Q}}}}) &=f_{\ast}\left(\widehat{\operatorname{ch}}(\sum(-1)^{p}p\Omega_{{\mathcal X}/S}^{p}),h^{\bullet})\widehat{\operatorname{Td}}(T_{{\mathcal X}/S},h)\right)^{(1)}\nonumber \\ &\quad -a\left(\operatorname{ch}(\sum(-1)^{p}p\Omega_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}}^{p})\operatorname{Td} (T_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}}) R(T_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}})\right)^{(1)}\nonumber \\ &=\frac{1}{12}f_{\ast}\left(\widehat{\operatorname{c}}_{1}(K_{{\mathcal X}/S},h^{\ast})\widehat{\operatorname{c}}_{n}(T_{{\mathcal X}/S},h)\right), \end{align} $$

where $h^{\ast }=(\det h)^{-1}$ is the hermitian metric on $K_{{\mathcal X}/S}$ induced from h. Notice that the topological factor containing the R-genus in (2.11) vanishes in our situation since

$$ \begin{align*} \operatorname{ch}\left(\sum(-1)^{p}p\Omega_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}}^{p}\right)\operatorname{Td}(T_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}})=-\mathrm{c}_{n-1}+\frac{n}{2}\mathrm{c}_n-\frac{1}{12}\mathrm{c}_1 \mathrm{c}_n +\textrm{higher-degree terms}, \end{align*} $$

and R has only odd degree terms and $\mathrm {c}_1(T_{{\mathcal X}^{\hspace{1pt}\operatorname {\mathrm {an}}}/S^{\operatorname {\mathrm {an}}}})=0$ . Now the evaluation map $f^{\ast }f_{\ast }K_{{\mathcal X}/S}\to K_{{\mathcal X}/S}$ is an isomorphism, but it is in general not an isometry if we equip $f_{\ast }K_{{\mathcal X}/S}$ with the $L^{2}$ metric and $K_{{\mathcal X}/S}$ with the metric $h^{\ast }$ . Comparing both metrics yields a relation in $\widehat {\operatorname {CH}}^{1}({\mathcal X})$

(2.12) $$ \begin{align} \widehat{\operatorname{c}}_{1}(K_{{\mathcal X}/S},h^{\ast})=f^{\ast}\widehat{\operatorname{c}}_{1}(f_{\ast}K_{{\mathcal X}/S},h_{{\scriptscriptstyle{\mathrm{L}^2}}})+[(0,-\log\varphi)]. \end{align} $$

Here $\varphi $ is the smooth function on $X^{\operatorname {\mathrm {an}}}$ given by

$$ \begin{align*} \varphi=\frac{i^{n^{2}}\eta\wedge\overline{\eta}}{\|\eta\|_{{\scriptscriptstyle{\mathrm{L}^2}}}^{2}}\frac{n!}{(2\pi\omega)^{n}}, \end{align*} $$

where $\eta $ denotes a local trivialisation of $f_{\ast }K_{{\mathcal X}^{\hspace{1pt}\operatorname {\mathrm {an}}}/S^{\operatorname {\mathrm {an}}}}$ , thought of as a section of $K_{{\mathcal X}^{\hspace{1pt}\operatorname {\mathrm {an}}}/S^{\operatorname {\mathrm {an}}}}$ via the evaluation map. Multiplying (2.12) by $\widehat {\operatorname {c}}_{n}(T_{{\mathcal X}/S},h)$ and applying $f_{\ast }$ and the projection formula for arithmetic Chow groups, we find

$$ \begin{align*} \begin{split} f_{\ast}\left(\widehat{\operatorname{c}}_{1}(K_{{\mathcal X}/S},h^{\ast})\widehat{\operatorname{c}}_{n}(T_{{\mathcal X}/S},h)\right)&=f_{\ast}\left(f^{\ast}\widehat{\operatorname{c}}_{1}(f_{\ast}K_{{\mathcal X}/S},h_{{\scriptscriptstyle{\mathrm{L}^2}}})\widehat{\operatorname{c}}_{n}(T_{{\mathcal X}/S},h)\right)+f_{\ast}\left([(0,-\log\varphi)]\widehat{\operatorname{c}}_{n}(T_{{\mathcal X}/S},h)\right)\\ &=\chi(X_{\infty})\widehat{\operatorname{c}}_{1}(f_{\ast}K_{{\mathcal X}/S},h_{{\scriptscriptstyle{\mathrm{L}^2}}}) +\left[\left(0,-\int_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}}(\log\varphi)\mathrm{c}_{n}(T_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}},h)\right) \right], \end{split} \end{align*} $$

where $\mathrm {c}_{n}(T_{{\mathcal X}/S},h)$ is the nth Chern–Weil differential form of $(T_{{\mathcal X}^{\hspace{1pt}\operatorname {\mathrm {an}}}/S^{\operatorname {\mathrm {an}}}},h)$ . Together with (2.11), this shows that the metric

$$ \begin{align*} h_{{\scriptscriptstyle{\mathrm{Q}}},{\scriptscriptstyle{\mathrm{BCOV}}}}=h_{{\scriptscriptstyle{\mathrm{Q}}}}\cdot\exp\left(-\frac{1}{12}\int_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}}(\log\varphi)\mathrm{c}_{n}(T_{{\mathcal X}^{\hspace{1pt}\operatorname{\mathrm{an}}}/S^{\operatorname{\mathrm{an}}}},h)\right) \end{align*} $$

indeed satisfies (2.9).

The outcome of (2.10) is a translation of the meaning of the equality in (2.9) in $\widehat {\operatorname {CH}}^{1}(S)_{{\mathbb Q}}$ in terms of the constructions in (2.8) and (2.7).

By [Reference Eriksson, Freixas i Montplet and MourouganeEFiMM21, Prop. 4.2], the normalising factor B is constant on each connected manifold $S_{\sigma }^{\operatorname {\mathrm {an}}}$ and would be rational if the $L^2$ inner products on cohomology groups were computed with $h/2\pi $ .

With this understood, we find

(2.13) $$ \begin{align} \operatorname{vol}_{{\scriptscriptstyle{\mathrm{L}^2}}}(H^{k}(X_{s},{\mathbb Z}),\omega) \in (2\pi)^{-kb_{k}}{\mathbb Q}^{\times}_{>0} \end{align} $$

for any $s\in S_{\sigma }^{\operatorname {\mathrm {an}}}$ . Together with the definition of B in (2.4), this is responsible for the constants $C_{\sigma }$ .

Proof. A standard computation shows that $\mathrm {Pic}(U) = {\mathbb Z}/\deg \Delta $ , providing the first claim of the lemma. For the second assertion, for any of the line bundles under consideration, let $\theta $ and $\theta ^{\prime }$ be two trivialisations on U. Therefore, $\theta =h\theta ^{\prime }$ for some invertible function h on U. The previous description of $\mathrm {Pic}(U)$ shows that the divisor of h, as a rational function on $\check {{\mathbb P}}$ , is supported on $\Delta $ . As $\Delta $ is irreducible, in the projective space $\check {{\mathbb P}}$ this is only possible if the divisor vanishes. We conclude that h is necessarily constant.

Proof. We apply Theorem 2.3 to $f\colon {\mathcal X}\to U$ (over ${\mathbb C}$ ), which in terms of $\beta $ and $\eta $ becomes

$$ \begin{align*} m\log \tau_{{\scriptscriptstyle{\mathrm{BCOV}}}}{( X_H})= \log|g|^2+\frac{\chi}{12}\log\|\eta\|^2_{{\scriptscriptstyle{\mathrm{L}^2}}}-\log \|\beta\|^2_{{\scriptscriptstyle{\mathrm{L}^2}}}+\log C \end{align*} $$

for some regular invertible function g on U and some constant C. By construction, as a rational function on $\check {{\mathbb P}}$ , g must have its zeros or poles along $\Delta $ . Since $\Delta $ is irreducible, this forces g to be constant.

Proof. Outside of the singular points, there is nothing to modify. The points corresponding to ordinary double point singularities are provided by the affine equations $x_{0}=1$ and $x_j^{n+1} = 1$ for $j\geq 1$ , and blowing up along the corresponding locus of ${{\mathcal X}}$ provides a normal crossings model. The locus of ordinary double points is defined over ${\mathbb Q}$ and is G-equivariant and thus so is also the blowup.

We next consider our family at the point at infinity. Introduce the divisor $D_0$ in ${\mathbb P}^{n}$ given by $\sum _{j} x_j^{n+1}=0$ and the divisor $D_{\infty }=\sum _{j} H_{j}$ , where $H_{j}$ is the hyperplane cut out by $x_{j}=0$ . The axis of the pencil ${\mathcal X} \to {\mathbb P}^1$ is $D_0 \cap D_{\infty }$ and hence ${\mathcal X} = \mathrm {Bl}_{D_0 \cap D_{\infty }}({\mathbb P}^n)$ . We construct another model by blowing up ${\mathbb P}^n$ in $D_0 \cap H_{0}$ to get ${\mathcal X}_1$ . Continue to blow up the strict transform of $D_0$ in ${\mathcal X}_1$ intersected by the strict transform of $H_1$ , and so on. Each such blowup is a blowup in a smooth centre that is G-equivariant. The final result is a G-equivariant $\widetilde {{\mathcal X}}$ projective manifold with an equivariant morphism $\nu : \widetilde {{\mathcal X}} \to {\mathbb P}^{n}$ . Denote by $\widetilde {D_0}$ (respectively, $\widetilde {D_{\infty }}$ ) the strict transforms of $D_0$ (respectively, $D_{\infty }$ ). By construction they are disjoint, and computation shows that $\nu ^* D_0 \sim \widetilde {D_0} + \sum E_i$ and $\nu ^* (D_{\infty }) \sim \widetilde {D_{\infty }} + \sum E_i,$ where the $E_i$ denote the strict transforms of the exceptional divisors. Since $D_0-D_{\infty }$ is the divisor of a rational function, hence linearly equivalent to zero, and $\widetilde {D_{\infty }}$ is disjoint from $\widetilde {D_0}$ , we find a morphism $p: \widetilde {{\mathcal X}} \to {\mathbb P}^1$ such that $p^{-1}(\infty )=\widetilde {D_{\infty }}$ and $p^{-1}(0)= \widetilde {D_0}$ . This is the searched for semi-stable model at infinity. From the local description, we also see that $\nu ^{-1}(D_0 \cap D_{\infty })= \sum E_i$ that is principal, so that $\widetilde {{\mathcal X}} \to {\mathbb P}^1$ factors over ${\mathcal X} \to {\mathbb P}^1$ .

All of the above constructions can be defined over ${\mathbb Q}$ , and taking them together with the previous considerations with the ordinary double points provides a model ${\mathcal X}^{\hspace{1pt}\prime } \to {\mathbb P}^1$ as in the statement of the proposition.

Proof. To lighten notations, let us write in this proof ${\mathcal X}$ and $\mathcal Y$ for the corresponding restrictions to $\mathbb A^{1}$ . The total space ${\mathcal X}$ is nonsingular, and $\mathcal Y$ is a quotient of it by the action of a finite group. Therefore, $\mathcal Y$ has rational singularities. In particular, it is normal and Cohen–Macaulay. Consequently, if $\mathcal Y^{\hspace{1pt}ns}$ is the nonsingular locus of $\mathcal Y$ and $j\colon \mathcal Y^{\hspace{1pt}ns}\hookrightarrow \mathcal Y$ the open immersion, then we have a relation between relative dualising sheaves $j_{\ast }\omega _{\mathcal Y^{ns}/\mathbb A^{1}}=\omega _{\mathcal Y/\mathbb A^{1}}$ . We will use this below.

Now for the Gorenstein property and the descent claim. Notice that since $\mathbb A^{1}$ is nonsingular, $\mathcal Y$ is Gorenstein if and only if the fibres of $\mathcal Y\to \mathbb A^{1}$ are Gorenstein. We will implicitly confound both the absolute and relative points of view. We introduce ${\mathcal X}^{\hspace{1pt}\circ }$ the complement of the fixed locus of G and ${\mathcal X}^{\hspace{1pt}\ast }$ the smooth locus of ${\mathcal X}\to \mathbb A^{1}$ . These are G-invariant open subschemes of ${\mathcal X}$ and constitute an open cover because the ordinary double points in the fibres of ${\mathcal X}\to \mathbb A^{1}$ are disjoint from the fixed point locus of G. Then $\mathcal Y^{\circ }={\mathcal X}^{\hspace{1pt}\circ }/G$ and $\mathcal Y^{\ast }={\mathcal X}^{\hspace{1pt}\ast }/G$ form an open cover of $\mathcal Y$ , and it is enough to proceed for each one separately.

Since G acts freely on ${\mathcal X}^{\hspace{1pt}\circ }$ , the quotient $\mathcal Y^{\circ }$ is nonsingular and is therefore Gorenstein. The morphism ${\mathcal X}^{\hspace{1pt}\circ }\to \mathcal Y^{\circ }$ is étale, and hence $K_{{\mathcal X}^{\hspace{1pt}\circ }/\mathbb A^{1}}$ descends to $K_{\mathcal Y^{\circ }/\mathbb A^{1}}$ .

For $\mathcal Y^{\ast }$ , we observe that G preserves a relative holomorphic volume form on ${\mathcal X}^{\hspace{1pt}\ast }$ . Indeed, in affine coordinates $z_k = \frac {x_k}{x_j}$ on the open set $x_j \neq 0$ , and where $\partial F_{\psi }/\partial z_i \neq 0$ , the expression

(3.1) $$ \begin{align} \theta_{0}=\frac{(-1)^{i-1} dz_0 \wedge \ldots \widehat{dz_i} \wedge \ldots \wedge \widehat{dz_j} \wedge \ldots \wedge dz_n}{\partial F_{\psi}/\partial z_i} \Big|_{F_{\psi}=0} \end{align} $$

provides such an invariant relative volume form. This entails that $K_{{\mathcal X}^{\hspace{1pt}\ast }/\mathbb A^{1}}$ descends to an invertible sheaf $\mathcal K$ on $\mathcal Y^{\ast }$ . Now the singular locus of $\mathcal Y^{\ast }$ is contained in the image of the fixed point set of G on ${\mathcal X}^{\hspace{1pt}\ast }$ . We infer that $\mathcal K$ is an invertible extension of the relative canonical bundle of $(\mathcal Y^{\ast })^{ns}\to \mathbb A^{1}$ . But $\mathcal Y^{\ast }$ is normal so that $\mathcal K \simeq j_{\ast } j^{\ast } \mathcal K$ . Then as mentioned at the beginning of the proof, $j_{\ast } \omega _{\mathcal Y^{ns}/\mathbb A^{1}}=\omega _{\mathcal Y/\mathbb A^{1}}$ , and we conclude, since $\mathcal K$ is also an extension of $\omega _{\mathcal Y^{ns}/\mathbb A^{1}}$ .

Proof. The proof of (1)–(4) is based on [Reference Dais, Henk and ZieglerDHZ98, Sec. 8 (v)], [Reference Dais, Henk and ZieglerDHZ06] and [Reference Bini and GarbagnatiBG14, Prop. 3.1], together with Hironaka’s resolution of singularities. We recall the strategy to justify the existence of a model over ${\mathbb Q}$ .

Introduce $W={\mathbb P}^{n}/G$ . We claim this is a split toric variety over ${\mathbb Q}$ . First of all, it can be realised as the hypersurface in ${\mathbb P}^{n+1}_{{\mathbb Q}}$ of equation

$$ \begin{align*} W \colon y_{0}^{n+1}=\prod_{j=1}^{n+1}y_{j}. \end{align*} $$

Second, the associated torus is split over ${\mathbb Q}$ . It is actually given by $\mathbb G_{\mathrm {m}\ {\mathbb Q}}\times \mathbb T$ , where $\mathbb T$ is the kernel of the multiplication map $\mathbb G_{\mathrm {m}\ {\mathbb Q}}^{n+1}\to \mathbb G_{\mathrm {m}\ {\mathbb Q}}$ . Finally, the action of the torus on W is defined over ${\mathbb Q}$ :

$$ \begin{align*} \left((t_{0},t_{1},\ldots,t_{n+1}), (y_{0},y_{1},\ldots,y_{n+1})\right)\mapsto (t_{0}y_{0},t_{0}t_{1}y_{1},\ldots, t_{0}t_{n+1}y_{n+1}). \end{align*} $$

Once we know that W is a split toric variety over ${\mathbb Q}$ with the same equation as in [Reference Dais, Henk and ZieglerDHZ06, Application 5.5], the toric and crepant projective resolution exhibited in [Reference Dais, Henk and ZieglerDHZ06] automatically works over ${\mathbb Q}$ as well. We write $\widetilde {W}$ for this resolution of W.

We now consider $\mathcal Y$ as a closed integral ${\mathbb Q}$ -subscheme of $W \times {\mathbb P}^{1}$ . Let $\widetilde {\mathcal Y}$ be the strict transform of $\mathcal Y$ in $\widetilde {W}\times {\mathbb P}^{1}$ . By [Reference Dais, Henk and ZieglerDHZ98, Sec. 8 (v)], the fibres of $\widetilde {\mathcal Y}$ at $\psi \in {\mathbb C}\setminus \mu _{n+1}$ are projective crepant resolutions of the fibres $Y_{\psi }$ . In particular, $\widetilde {\mathcal Y}$ is smooth over ${\mathbb C}\setminus \mu _{n+1}$ , and in turn this implies smoothness over the complement U of the closed subscheme $V(\psi ^{n+1}-1)$ of $\mathbb A^{1}_{{\mathbb Q}}$ . Necessarily, the fibres of $\widetilde {\mathcal Y}$ over U have trivial canonical bundle as well. For the fibres at $\psi ^{n+1}=1$ , the claim of the lemma requires two observations:

• • The ordinary double points of $X_{\psi }$ are permuted freely and transitively by G and are identified to a single point in the quotient $Y_{\psi }$ . This entails that the total space $\mathcal Y$ is nonsingular in a neighbourhood of these points and that they remain ordinary double points of $\mathcal Y\to {\mathbb P}^{1}$ .

• • The centre of the toric resolution is disjoint from the ordinary double points since it is contained in the locus of ${\mathbb P}^{n}/G$ where two or more projective coordinates vanish. Therefore, the morphism $\widetilde {\mathcal Y}\to \mathcal Y$ is an isomorphism in a neighbourhood of these points. Finally, on the complement, $\widetilde {\mathcal Y}_{\psi }$ is a resolution of singularities of $Y_{\psi }$ . Indeed, this is a local question in a neighbourhood of the fixed points of G so that the above references [Reference Dais, Henk and ZieglerDHZ98, Reference Dais, Henk and ZieglerDHZ06] still apply.

Finally, $\widetilde {\mathcal Y}$ is by construction smooth on the complement of the fibre $\psi =\infty $ . After a resolution of singularities given by blowups with smooth centres in $\widetilde {Y}_{\infty }$ (defined over ${\mathbb Q}$ ), we obtain a smooth algebraic variety $\mathcal Z$ over ${\mathbb Q}$ such that $Z_{\infty }$ is a simple normal crossings divisor in $\mathcal Z$ . This sets (1–4).

For (5), we refer for instance to [Reference Batyrev and DaisBD96, Thm. 6.9, Conj. 7.5 & Ex. 8.7]. This is specific to the Dwork pencil. More generally, we can cite work of Yasuda, who proves an invariance property of orbifold Hodge structures (and hence orbifold Hodge numbers) under crepant resolutions, for quotient Gorenstein singularities [Reference YasudaYas04, Thm. 1.5]. Orbifold Hodge numbers coincide with stringy Hodge numbers of global (finite) quotient orbifolds, whose underlying group respects a holomorphic volume form [Reference Batyrev and DaisBD96, Thm. 6.14]. Finally, by [Reference Batyrev and BorisovBB96, Thm. 4.15], stringy Hodge numbers satisfy the expected mirror symmetry property for the mirror pairs constructed by Batyrev [Reference BatyrevBat94].

Proof. The items are a consequence of the mirror symmetry property for the Hodge numbers in Lemma 3.3 and the computation of the cohomology of a hypersurface in projective space (see [Reference Batyrev and DaisBD96, Ex. 8.7]).

Proof. We first prove that the local system $R^{2p}f_{\ast }{\mathbb Q}$ is trivial. Take a base point $b\in U^{\operatorname {\mathrm {an}}}$ , and let $\rho \colon \pi _{1}(U^{\operatorname {\mathrm {an}}},b)\to \mathsf {GL}(H^{2p}(Z_{b},{\mathbb Q}))$ be the monodromy representation determining the local system. The fundamental group $\pi _{1}(U^{\operatorname {\mathrm {an}}},b)$ is generated by loops $\gamma _{\xi }$ circling around $\xi \in \mu _{n+1}$ and a loop $\gamma _{\infty }$ circling around $\infty $ , with a relation $\prod _{\xi }\gamma _{\xi }=\gamma _{\infty }$ . Because the singularities of $\mathcal Z\to {\mathbb P}^{1}$ at the points $\xi $ are ordinary double points, and $2p\neq n-1$ , the local monodromies $\rho (\gamma _{\xi })$ are trivial. Therefore $\rho (\gamma _{\infty })$ is trivial as well, and so is $\rho $ .

Now the first claim implies the triviality of $R^{p}f_{\ast }^{\prime }\Omega _{\mathcal Z^{\prime }/{\mathbb P}^{1}}^{p}(\log )=R^{2p}f_{\ast }^{\prime }\Omega _{\mathcal Z^{\prime }/{\mathbb P}^{1}}^{\bullet }(\log )$ in the analytic category, since the latter realises the Deligne extension of $R^{2p}f_{\ast }\Omega _{\mathcal Z/U}^{\bullet }$ . By the GAGA principle, $R^{p}f_{\ast }^{\prime }\Omega _{\mathcal Z^{\prime }/{\mathbb P}^{1}}^{p}(\log )$ is algebraically trivial as a complex vector bundle. This already implies the second claim. Indeed, let E be a vector bundle over ${\mathbb P}^{1}_{{\mathbb Q}}$ , which is trivial after base change to ${\mathbb C}$ . Then the natural morphism $H^{0}({\mathbb P}^{1}_{{\mathbb Q}},E)\otimes \mathcal O_{{\mathbb P}^{1}_{{\mathbb Q}}}\to E$ is necessarily an isomorphism, since it is an isomorphism after a flat base change.

Proof. We first show that if the local monodromy of $(R^{n-1}f_{\ast } {\mathbb C})_{\min }$ around one ODP point is trivial, then it is so around all the ODP points. Since $(R^{n-1}f_{\ast } {\mathbb C})_{\min }$ is isomorphic to $(R^{n-1}h_{\ast }{\mathbb C})^{G}_{\mathrm {prim}}$ as a local system, it is enough to show that the latter descends along the natural projection $(U-\{0\})\to (U-\{0\})/\mu _{n+1}$ , where $\mu _{n+1}$ acts by multiplication on $(U-\{0\})\subset {\mathbb P}^{1}$ . Notice that for any $\zeta \in \mu _{n+1}$ , the automorphism $\psi \mapsto \zeta \cdot \psi $ lifts to an automorphism of the family $g\colon \mathcal Y \to (U-\{0\})$ via the formula $[x_0:x_1:\ldots :x_n] \mapsto [x_0':x_1':\ldots :x_n']$ , where $x_i'=x_i$ except for one i, for which $x_i' = \zeta ^{-1} \cdot x_i$ . Since we work in the quotient by the group G, all the choices of i correspond to the same action. We conclude that the local systems $(R^{k}h_{\ast }{\mathbb C})^{G}\simeq R^{k}g_{\ast }{\mathbb C}$ descend for all k. Observe that $R^{2}h_{\ast }{\mathbb C}$ is actually constant with fibre $H^{2}({\mathbb P}^{n},{\mathbb C})$ , by Lefschetz, with G acting trivially. Therefore, the polarisation necessarily descends. We conclude that $(R^{n-1}h_{\ast }{\mathbb C})^{G}_{\mathrm {prim}}=\ker \left (L\colon (R^{n-1}h_{\ast }{\mathbb C})^{G}\to (R^{n+1}h_{\ast }{\mathbb C})^{G}\right )$ descends too, as was to be shown.

We now show that $\mathbb V$ is a trivial local system. It is enough to argue for $\mathbb V_{{\mathbb C}}$ . In the odd-dimensional case, there is nothing to prove. In the even-dimensional case, we first recall that by the Picard–Lefschetz formula, the local monodromies on $(R^{n-1}f_{\ast }{\mathbb C})_{\mathrm {prim}}$ around the ODP points are semi-simple with a single nontrivial eigenvalue $-1$ of multiplicity one. It follows that around each ODP point, exactly one of the sublocal systems $(R^{n-1}f_{\ast } {\mathbb C})_{\min }$ and $\mathbb V_{{\mathbb C}}$ has trivial local monodromy. By the argument in the previous lemma, if the monodromies around one and hence all the ODP points on $(R^{n-1}f_{\ast } {\mathbb C})_{\min }$ were trivial, it would follow that the monodromy around $\infty $ would also be trivial. As this is excluded by Lemma 4.3 below, we infer that $\mathbb V$ is a trivial local system.

We next address the triviality of $\mathcal V$ asserted by the second point. We will now make use of the G-equivariant normal crossings model $h^{\prime }\colon {\mathcal X}^{\hspace{1pt}\prime }\to {\mathbb P}^{1}$ of Proposition 3.1. We summarise the current geometric setting in the following diagram, which builds upon (3.2):

We first argue analytically. By the minimal decomposition (Proposition 3.13), the morphism in (3.13) induces a morphism between Deligne extensions

(4.1) $$ \begin{align} ^{\ell}(R^{n-1}h_{\ast}\Omega_{{\mathcal X}/U}^{\bullet})^{G}_{\mathrm{prim}}{\hookrightarrow}\ ^{\ell}(R^{n-1}f_{\ast}\Omega_{\mathcal Z/U}^{\bullet})_{\mathrm{prim}}. \end{align} $$

We can reformulate (4.1) as a morphism

$$ \begin{align*} \psi\colon(R^{n-1}h_{\ast}^{\prime}\Omega_{{\mathcal X}^{\hspace{1pt}\prime}/{\mathbb P}^{1}}^{\bullet}(\log))^{G}_{\mathrm{prim}}{\hookrightarrow} (R^{n-1}f_{\ast}^{\prime}\Omega_{\mathcal Z^{\prime}/{\mathbb P}^{1}}^{\bullet}(\log))_{\mathrm{prim}}, \end{align*} $$

extending (3.14) to ${\mathbb P}^{1}$ . By the GAGA principle, this morphism is algebraic. Notice that the coherent sheaves involved in $\psi $ are locally free and defined over ${\mathbb Q}$ . By Lemma 3.11 (2), $\psi _{\mid U}$ is defined over ${\mathbb Q}$ . Therefore, $\psi _{\mid U}$ is invariant under the action of $\operatorname {\mathrm {Aut}}({\mathbb C}/{\mathbb Q})$ . Because U is a nonempty Zariski open subset of ${\mathbb P}^{1}$ , which is an integral scheme, we infer that $\psi $ is invariant under $\operatorname {\mathrm {Aut}}({\mathbb C}/{\mathbb Q})$ . Therefore, $\psi $ is defined over ${\mathbb Q}$ , and so is its cokernel. We denote by $\widetilde {\mathcal V}$ this cokernel of $\psi $ modulo its torsion part. Then $\widetilde {\mathcal V}$ is a vector bundle.

By Proposition 3.13 (2), $\widetilde {\mathcal V}_{\mid U}$ is canonically isomorphic to $\mathcal V$ , over ${\mathbb Q}$ , and in particular inherits a connection from $\mathcal V$ . Also, by the same proposition, we know that the analytification of (3.16) is canonically identified with the tensor product of (3.15) with $\mathcal O_{U^{\operatorname {\mathrm {an}}}}$ . By taking Deligne extensions, we deduce that $\widetilde {\mathcal V}^{\operatorname {\mathrm {an}}}$ is a vector bundle with regular singular connection, canonically isomorphic to the Deligne extension of $\mathbb V\otimes \mathcal O_{U^{\operatorname {\mathrm {an}}}}$ . By the first part of the lemma, we thus infer that $\widetilde {\mathcal V}^{\operatorname {\mathrm {an}}}$ is a trivial vector bundle with connection, and in particular any trivialisation over ${\mathbb P}^{1}$ is flat. As in the proof of Lemma 4.1 (2), we deduce that $\widetilde {\mathcal V}$ is a trivial vector bundle over ${\mathbb P}^{1}$ , defined over ${\mathbb Q}$ . From all the above, we conclude that the restriction to U of any trivialisation of $\widetilde {\mathcal V}$ , defined over ${\mathbb Q}$ , induces a flat trivialisation of $\mathcal V\simeq \widetilde {\mathcal V}_{\mid U}$ , defined over ${\mathbb Q}$ . This concludes the proof.

Proof. The maximally unipotent property for $(R^{n-1}f_{\ast }{\mathbb Q})_{\min }\simeq (R^{n-1}h_{\ast }{\mathbb Q})^{G}_{\mathrm {prim}}$ is proven in odd relative dimension in [Reference Harris, Shepherd-Barron and TaylorHSBT10, Cor. 1.7]. Exactly the same argument yields the claim in even relative dimension. In particular, $N^{n-1}\neq 0$ . This settles the first point. Because moreover $N^{n-1}$ induces an isomorphism $\operatorname {Gr}^{W}_{2(n-1)}H^{n-1}_{\lim }\overset {\sim }{\to } \operatorname {Gr}^{W}_{0}H^{n-1}_{\lim }$ , we deduce that $\operatorname {Gr}^{W}_{0}H^{n-1}_{\lim }\neq 0$ . By Lemma 3.9, $H^{n-1}_{\lim }$ is n-dimensional, and the second item follows for dimension reasons. Finally, we use that $\operatorname {Gr}_{F_{\infty }}^{p}H^{n-1}_{\lim }$ is one-dimensional again by Lemma 3.9 and then necessarily $\operatorname {Gr}_{F_{\infty }}^{p}H^{n-1}_{\lim }=\operatorname {Gr}_{F_{\infty }}^{p}\operatorname {Gr}^{W}_{2p}H^{n-1}_{\lim }=\operatorname {Gr}^{W}_{2p}H^{n-1}_{\lim }$ . Hence the second point implies the third.

Proof. The argument is well-known – see, for example, [Reference MorrisonMor93, Prop.] and [Reference VoisinVoi99, Lemma 3.10] – but we sketch it due to its relevance.

The pairing $\langle \gamma ,\eta \rangle =Q(\gamma ^{\prime },\eta )$ is clearly a holomorphic function on ${\mathbb D}^{\times }_{\infty }$ since both $\gamma ^{\prime }$ and $\eta $ are holomorphic sections of $(R^{n-1}f_{\ast }{\mathbb C})\otimes \mathcal O_{{\mathbb D}^{\times }_{\infty }}$ . Moreover, they are both global sections of the Deligne extension. This ensures that $|Q(\gamma ^{\prime },\eta )|$ has at most a logarithmic singularity at $0$ . It follows that $Q(\gamma ^{\prime },\eta )$ is actually a holomorphic function.

For the nonvanishing property, we make use of the interplay between the intersection pairing seen on $H^{n-1}_{\lim }$ and the monodromy weight filtration [Reference SchmidSch73, Lemma 6.4], together with Lemma 4.3. Let $\eta ^{\prime }\in H^{n-1}_{\lim }$ be the fibre of $\eta $ at $0$ . We need to show that $Q(\gamma ^{\prime },\eta ^{\prime })\neq 0$ . Suppose the contrary. Since $\gamma ^{\prime }$ is a basis of $W_{0}=\ker N=\operatorname {\mathrm {Im}} N^{n-1}$ , we have $\eta ^{\prime }\in (\operatorname {\mathrm {Im}} N^{n-1})^{\perp }$ . The intersection pairing is nondegenerate and satisfies $Q(Nx,y)+Q(x,Ny)=0$ . Therefore, we find that $\eta ^{\prime }\in (\operatorname {\mathrm {Im}} N^{n-1})^{\perp }=\ker N^{n-1}=W_{2n-3}$ . But $\eta ^{\prime }$ is a basis of $F^{n-1}H^{n-1}_{\lim }=F^{n-1}\operatorname {Gr}^{W}_{2n-2}H^{n-1}_{\lim }$ , and therefore $\eta ^{\prime }\not \in W_{2n-3}$ . We thus have reached a contradiction.

Proof. First of all, we prove that $\eta _{0}$ is a meromorphic section of $f_{\ast }K_{\mathcal Z/{\mathbb D}_{\infty }}(\log )$ . Indeed, $\eta _{0}$ is an algebraic section of $f_{\ast }K_{\mathcal Z/U}$ (see Lemma 3.17), hence a rational section of $f_{\ast }K_{\mathcal Z^{\prime }/{\mathbb P}^{1}}(\log )$ and thus a meromorphic section of $f_{\ast }K_{\mathcal Z/{\mathbb D}_{\infty }}(\log )$ .

Second, we establish the claim of the theorem for $\eta _{0}$ . By Lemma 4.4, we need to show that the holomorphic function $\langle \gamma ,\eta _{0}\rangle $ on ${\mathbb D}^{\times }_{\infty }$ extends holomorphically to ${\mathbb D}_{\infty }$ and does not vanish at the origin. This property can be checked by a standard explicit computation reproduced below (5.8).

Finally, for the sections $\eta _{k}$ , we use the recurrence in (3.19) and the logarithmic extension of the Kodaira–Spencer maps in (4.2). It follows that the sections $\eta _{k}$ are global sections of the sheaves $R^{k}f_{\ast }\Omega ^{n-1-k}_{\mathcal Z/{\mathbb D}_{\infty }}(\log )_{\min }$ . Let us denote by $\eta ^{\prime }_{k}$ the fibre at 0 of the sections $\eta _{k}$ . Specialising (3.19) at $0$ , we find $(\operatorname {Gr}_{F_{\infty }}^{n-1-k} N)\eta _{k}^{\prime }=\eta _{k+1}^{\prime }$ . By Lemma 4.3 (3), and because $\eta _{0}^{\prime }\neq 0$ , we see that $\eta _{k}^{\prime }\neq 0$ for all k. This concludes the proof.

Proof. The proof follows from [Reference SteenbrinkSte77, Ex. 2.15], noticing that $(R^{n-1}\widetilde {f}_{\ast }{\mathbb Q})_{\min }=\rho ^{\ast }(R^{n-1}f_{\ast }{\mathbb Q})_{\min }$ is a direct factor of $R^{n-1}\widetilde {f}_{\ast }{\mathbb Q}$ , whose complement is a trivial variation of Hodge structures by Lemma 4.1 and Lemma 4.2. For the case $n-1$ is even, we moreover recall that $\mathbb V$ as in Proposition 3.13 has pure bidegree $((n-1)/2,(n-1)/2)$ .

Proof. The results in [Reference Eriksson, Freixas i Montplet and MourouganeEFiMM21, Sec. 2 & Prop. 3.10] are explicitly stated for the whole Hodge bundles and describe the cokernels in terms of the semi-simple part of the monodromy acting on the limiting Hodge structure. For their minimal components, however, see Remark 2.7 (iii) in [Reference Eriksson, Freixas i Montplet and MourouganeEFiMM21] together with Proposition 3.13 and Lemma 4.2.

Proof. Throughout the proof, we write ${\mathcal X}$ , $\mathcal Y$ and $\mathcal Z$ for the respective total spaces over $\mathbb A^{1}$ . We begin by showing that $\eta _{0}$ extends to a global section of $f^{\prime }_{\ast }K_{\mathcal Z^{\prime }/\mathbb A^{1}}(\log )$ , nonvanishing at $\xi $ . Since the singular fibres of $\mathcal Z\to \mathbb A^{1}$ present only ordinary double points, there is an equality

$$ \begin{align*} f_{\ast}K_{\mathcal Z/\mathbb A^{1}}=f^{\prime}_{\ast}K_{\mathcal Z^{\prime}/\mathbb A^{1}}(\log). \end{align*} $$

This can be seen as the coincidence of the upper and lower extensions of $f_{\ast }K_{\mathcal Z/U}$ to $\mathbb A^{1}$ (apply [Reference Eriksson, Freixas i Montplet and MourouganeEFiMM21, Cor. 2.8 & Prop. 2.10] and the Picard–Lefschetz formula for the monodromy). Since $\mathcal Y$ has rational singularities (see Lemma 3.2), the natural morphism $g_{\ast }K_{\mathcal Y/\mathbb A^{1}}\to f_{\ast }K_{\mathcal Z/\mathbb A^{1}}$ is an isomorphism. Also $g_{\ast }K_{\mathcal Y/\mathbb A^{1}}=(h_{\ast }K_{{\mathcal X}/\mathbb A^{1}})^{G}$ . Indeed, let ${\mathcal X}^{\hspace{1pt}\circ }$ be the complement of the fixed point locus of G in ${\mathcal X}$ and similarly for $\mathcal Y^{\circ }$ , so that $\mathcal Y\setminus \mathcal Y^{\circ }$ has codimension $\geq 2$ . Then because $\mathcal Y$ is normal Gorenstein and $\mathcal Y^{\circ }={\mathcal X}^{\hspace{1pt}\circ }/G$ is an étale quotient and ${\mathcal X}$ is nonsingular, we find

$$ \begin{align*} g_{\ast}K_{\mathcal Y/\mathbb A^{1}}=g_{\ast}K_{\mathcal Y^{\circ}/\mathbb A^{1}}=(h_{\ast}K_{{\mathcal X}^{\hspace{1pt}\circ}/\mathbb A^{1}})^{G} =(h_{\ast}K_{{\mathcal X}/\mathbb A^{1}})^{G}. \end{align*} $$

By construction of $\eta _{0}$ (see Definition 3.15), it is enough to prove that $\theta _{0}$ defines a trivialisation of $h_{\ast }K_{{\mathcal X}/\mathbb A^{1}}$ around $\xi $ . Denote by ${\mathcal X}^{\hspace{1pt}\ast }$ the complement in ${\mathcal X}$ of the ordinary double points, so that ${\mathcal X}\setminus {\mathcal X}^{\hspace{1pt}\ast }$ has codimension $\geq 2$ . Because ${\mathcal X}$ is nonsingular, we have $h_{\ast }K_{{\mathcal X}/\mathbb A^{1}}=h_{\ast }K_{{\mathcal X}^{\hspace{1pt}\ast }/\mathbb A^{1}}$ . Now the expression in (3.1) for $\theta _{0}$ defines a relative holomorphic volume form on the whole ${\mathcal X}^{\hspace{1pt}\ast }$ and hence a trivialisation of $h_{\ast }K_{{\mathcal X}^{\hspace{1pt}\ast }/\mathbb A^{1}}$ as desired.

That the sections $\eta _{k}$ define meromorphic sections of the sheaves $R^{k}f^{\prime }_{\ast }\Omega ^{n-1-k}_{\mathcal Z^{\prime }/\mathbb A^{1}}(\log )_{{\min }}$ follows from the corresponding property for $\eta _{0}$ plus the recurrence in (3.19) and the existence of the logarithmic extension of the Kodaira–Spencer maps in (4.2). From the same recurrence, we reduce the computation of $\operatorname {\mathrm {ord}}_{\xi }\eta _{k}$ to the computation of the orders at $\xi $ of the rational morphisms $\mathrm {KS}^{(j)}(\psi d/d\psi )$ , with respect to the logarithmic extension of the Hodge bundles

$$ \begin{align*} \operatorname{\mathrm{ord}}_{\xi} \eta_k =\operatorname{\mathrm{ord}}_{\xi} \eta_0 + \sum_{j=0}^{k-1} \operatorname{\mathrm{ord}}_{\xi}\mathrm{KS}^{(j)}\left(\psi \frac{d}{d\psi}\right) =\sum_{j=0}^{k-1} \operatorname{\mathrm{ord}}_{\xi}\mathrm{KS}^{(j)}\left(\psi \frac{d}{d\psi}\right). \end{align*} $$

Let us define $M^{(j)}=\operatorname {\mathrm {ord}}_{\xi }\mathrm {KS}^{(j)}\left (\psi \frac {d}{d\psi }\right )$ . Because $\eta _{0}$ trivialises $f_{\ast }K_{\mathcal Z/\mathbb A^{1}}$ at $\xi $ , formula (3.11) shows that

(4.5) $$ \begin{align} \sum_{j=0}^{n-2}M^{(j)}=\operatorname{\mathrm{ord}}_{\xi}Y(\psi)=-1. \end{align} $$

We argue that all but one of the $M^{(j)}$ are zero. For this, we relate $M^{(j)}$ to the action of the nilpotent operator N on the limiting mixed Hodge structure at $\xi $ . Recall that we defined the coordinate $t=\psi -\xi $ on a disc neighbourhood ${\mathbb D}_{\xi }$ of $\xi $ . The first observation is

(4.6) $$ \begin{align} \operatorname{\mathrm{ord}}_{t=0}\mathrm{KS}^{(j)}\left(t\frac{d}{dt}\right)=\operatorname{\mathrm{ord}}_{\xi}\mathrm{KS}^{(j)}\left((\psi-\xi)\frac{d}{d\psi}\right)=M^{(j)}+1\geq 0, \end{align} $$

since the Kodaira–Spencer maps along logarithmic tangent vectors preserve the logarithmic Hodge bundles (see (4.2)). Hence, we see that $M^{(j)}\geq -1$ . We now need to distinguish two cases, depending on the parity of $n-1$ .

Odd case: If $n-1$ is odd, then the monodromy is unipotent and the fibre of $\mathrm {KS}^{(p)}(td/dt)$ at $t=0$ is already $\operatorname {Gr}^{p}_{F_{\infty }}N\colon \operatorname {Gr}_{F_{\infty }}^p H^{n-1}_{\lim }\to \operatorname {Gr}_{F_{\infty }}^{p-1} H^{n-1}_{\lim }$ . From Lemma 4.6, we deduce that unless $p=n/2$ , $\operatorname {Gr}^p_{F_{\infty }}N=0$ so that $\operatorname {\mathrm {ord}}_{t=0} \mathrm {KS}^{(p)}(td/dt)> 0$ and hence $M^{(p)} \geq 0.$ By (4.5), we necessarily have $M^{(n/2)}=-1$ and the other $M^{(j)}=0$ .

Even case: If $n-1$ is even, the nilpotent operator N is trivial, but the monodromy is no longer unipotent. The construction of the limiting mixed Hodge structure thus involves a semi-stable reduction. Choose a square root u of t as in (4.3). Then since $ u \frac {d}{du} = 2 t \frac {d}{dt}$ , we get, comparing the Gauss-Manin connection before and after semi-stable reduction, a commutative diagram of maps of line bundles

Together with $\operatorname {\mathrm {ord}}_{u=0}= 2 \operatorname {\mathrm {ord}}_{t=0}$ , we conclude that

(4.7) $$ \begin{align} \operatorname{\mathrm{ord}}_{u=0} \varphi^{p,q}+\operatorname{\mathrm{ord}}_{u=0} \mathrm{KS}^{(q)}\left(u \frac{d}{du}\right) = \operatorname{\mathrm{ord}}_{u=0} (\varphi^{p-1,q+1})+2 \operatorname{\mathrm{ord}}_{t=0} \mathrm{ KS}^{(q)}\left(t \frac{d}{dt}\right). \end{align} $$

By Lemma 4.7, $\operatorname {\mathrm {ord}}_{u=0} (\varphi ^{p,q})=0$ except for the case $(p,q)=((n-1)/2, (n-1)/2),$ where $\operatorname {\mathrm {ord}}_{u=0} (\varphi ^{p,q})=1$ . From (4.7), we then conclude that

(4.8) $$ \begin{align} \operatorname{\mathrm{ord}}_{u=0} \mathrm{KS}^{((n-3)/2)}\left(u \frac{d}{du}\right) = 1+2 \operatorname{\mathrm{ord}}_{t=0} \mathrm{KS}^{((n-3)/2)}\left(t \frac{d}{dt}\right) \end{align} $$

(4.9) $$ \begin{align} 1+\operatorname{\mathrm{ord}}_{u=0} \mathrm{KS}^{((n-1)/2)}\left(u \frac{d}{du}\right) =2 \operatorname{\mathrm{ord}}_{t=0} \mathrm{KS}^{((n-1)/2)}\left(t \frac{d}{dt}\right). \end{align} $$

In both (4.8) and (4.9), the order of vanishing of Kodaira–Spencer along the vector field $u\frac {d}{du}$ is strictly positive, since the restriction to 0 is the nilpotent operator $N=0$ . It follows that

$$ \begin{align*} \operatorname{\mathrm{ord}}_{t=0} \mathrm{KS}^{((n-3)/2)}\left(t \frac{d}{dt}\right) \geq 0,\quad \text{i.e.,}\quad M^{((n-3)/2)} \geq -1, \end{align*} $$

and

$$ \begin{align*} \operatorname{\mathrm{ord}}_{t=0} \mathrm{KS}^{((n-1)/2)}\left(t \frac{d}{dt}\right) \geq 1, \quad \text{i.e.,}\quad M^{((n-1)/2)} \geq 0. \end{align*} $$

Since all other $M^{(j)} \geq 0$ as in the odd case, we conclude from (4.5) that all these inequalities are equalities.

Proof. We apply the version of the arithmetic Riemann–Roch theorem formulated in Theorem 2.3 to the family $f\colon \mathcal Z\to U$ as being defined over ${\mathbb Q}$ .

Choices of sections. We need to specify the section $\eta $ and the sections $\eta _{p,q}$ in (2.10). The section $\eta $ is chosen to be $\eta _0$ , as defined in Definition 3.15. We next describe our choices of $\eta _{p,q}$ :

• • If $p+q \neq n-1$ and $p\neq q$ , then the corresponding Hodge bundle vanishes by Lemma 3.4 and thus gives no contribution.

• • For $2p\neq n-1$ , Lemma 4.1 guarantees that $\det R^{p}f_{\ast }^{\prime }\Omega _{\mathcal Z^{\prime }/{\mathbb P}^{1}}^{p}(\log )=\det R^{2p}f_{\ast }^{\prime }\Omega _{\mathcal Z^{\prime }/{\mathbb P}^{1}}^{\bullet }(\log )$ is trivial, in the algebraic category over ${\mathbb Q}$ , and any trivialisation is flat for the Gauss–Manin. We choose $\eta _{p,p}$ to be any trivialisation defined over ${\mathbb Q}$ and then restrict it to U. Notice that the $L^{2}$ norm $\|\eta _{p,p}\|_{{\scriptscriptstyle {\mathrm {L}^2}}}$ is then constant.

• • For $p+q=n-1$ and $p\neq q$ , the $(p,q)$ Hodge bundle is primitive and has rank one. Then we take $\eta _{p,q}=\eta _{{\scriptscriptstyle {\mathrm {Q}}}}$ in Definition 3.15. By Lemma 3.17, $\eta _{{\scriptscriptstyle {\mathrm {Q}}}}$ is defined over ${\mathbb Q}$ .

• • For $p+q=n-1$ and $p=q$ , which can only occur when $n-1$ is even, the $(p,q)$ Hodge bundle is no longer primitive of rank one. We employ first the algebraic primitive decomposition and then the minimal decomposition of Proposition 3.13 (2)

  (5.1) $$ \begin{align} \begin{aligned} \det R^{p}f_{\ast}\Omega^{p}_{\mathcal Z/U}&=\det (R^{p}f_{\ast}\Omega^{p}_{\mathcal Z/U})_{\mathrm{prim}}\otimes\det L R^{p-1}f_{\ast}\Omega^{p-1}_{\mathcal Z/U}\\ &\simeq \det (R^{p}f_{\ast}\Omega^{p}_{\mathcal Z/U})_{\mathrm{prim}}\otimes\det R^{p-1}f_{\ast}\Omega^{p-1}_{\mathcal Z/U} \\ &\simeq \det (R^{p}f_{\ast}\Omega^{p}_{\mathcal Z/U})_{\min} \otimes \det \mathcal V \otimes\det R^{p-1}f_{\ast}\Omega^{p-1}_{\mathcal Z/U}. \end{aligned} \end{align} $$
  We define $\eta _{\frac {n-1}{2},\frac {n-1}{2}}$ as the element corresponding to $\eta _{\frac {n-1}{2}} \otimes v \otimes \eta _{\frac {n-3}{2},\frac {n-3}{2}}$ under this isomorphism, for any algebraic flat trivialisation $v \in \det \mathcal V$ , defined over ${\mathbb Q}$ , provided by Lemma 4.2. We claim that
  (5.2) $$ \begin{align} \|\eta_{\frac{n-1}{2},\frac{n-1}{2}}\|_{{\scriptscriptstyle{\mathrm{L}^2}}}^{2}\sim_{{\mathbb Q}^{\times}} \|\eta_{\frac{n-1}{2}}\|_{{\scriptscriptstyle{\mathrm{L}^2}}}^{2} \|v\|_{{\scriptscriptstyle{\mathrm{L}^2}}}^{2}\|\eta_{\frac{n-3}{2},\frac{n-3}{2}}\|_{{\scriptscriptstyle{\mathrm{L}^2}}}^{2}, \end{align} $$
  where $\sim _{{\mathbb Q}^{\times }}$ denotes equality up to a rational number. For this, we bring together several facts. The first is that the Lefschetz decomposition is orthogonal for the $L^{2}$ metrics regardless of the normalisation of the Kähler forms. The second is that the algebraic cycle class of L corresponds to $(2\pi i)[\omega ]$ in analytic de Rham cohomology. The third fact is that the operator $[2\pi \omega ]\wedge \cdot $ is an isometry up to a rational constant, since $2\pi \omega $ is the Hodge theoretic Kähler form (see, for instance, [Reference HuybrechtsHuy05, Prop. 1.2.31]). The last fact is that the minimal component decomposition of Proposition 3.13 is also orthogonal for the $L^{2}$ norm, since it is orthogonal for the intersection form by construction. This settles (5.2). Furthermore, we notice that as for $\eta _{\frac {n-3}{2},\frac {n-3}{2}}$ , the $L^{2}$ norm of v is constant, since it is flat by construction and it is the wedge product of a collection of sections of pure Hodge bidegree $((n-1)/2,(n-1)/2)$ . Therefore, the norm $\|\eta _{\frac {n-1}{2},\frac {n-1}{2}}\|_{{\scriptscriptstyle {\mathrm {L}^2}}}^{2}$ equals $\|\eta _{\frac {n-1}{2}}\|_{{\scriptscriptstyle {\mathrm {L}^2}}}^{2}$ up to a constant.

Determining the rational function $\Delta $ . To establish the theorem, we need to specify the element $\Delta \in {\mathbb Q}(\psi )^{\times }\otimes {\mathbb Q}$ in (2.10) (formal rational power of a rational function), which satisfies

(5.3) $$ \begin{align} \log \tau_{{\scriptscriptstyle{\mathrm{BCOV}}}}= \log|\Delta|^{2}+\frac{\chi}{12}\log\|\eta\|_{{\scriptscriptstyle{\mathrm{L}^2}}}^{2} -\sum_{p,q} (-1)^{p+q}p\log \|\eta_{p,q}\|_{{\scriptscriptstyle{\mathrm{L}^2}}}^{2} +\log C_{\sigma}. \end{align} $$

We will determine $\Delta $ up to an algebraic number. To this end, it suffices to know its divisor. Unless $\psi = 0$ or $\psi = \xi $ , where $\xi ^{n+1}=1$ , $\Delta $ has no zeroes or poles by construction, since the sections $\eta _{p,q}$ are holomorphic and nonvanishing, and $\log \tau _{{\scriptscriptstyle {\mathrm {BCOV}}}}$ is smooth. Hence we are led to consider the logarithmic behaviour of the right-hand side of (5.3) at these points. Since for $2p\neq n-1$ the sections $\eta _{p,p}$ have constant $L^{2}$ norm, we only need to examine the functions $\log \|\eta _{p,q}\|_{{\scriptscriptstyle {\mathrm {L}^2}}}$ with $p+q=n-1$ .