Invariant volume and finite minimal classes

On a smooth $d$-dimensional group scheme the cotangent module at the identity is free of rank $d$ over the DVR. Choose a generator of its top exterior power and translate it by the group law; the translation trivialization gives an invariant top form which generates at every point, hence is nowhere vanishing. For a smooth model with irreducible special fibre, let $\mathrm{ord}$ be the valuation of its nowhere-vanishing generic invariant form $\omega$ at the special generic point, and write $\omega ={\pi }^{\mathrm{ord}}{\omega }_0$ there. The normalized form ${\pi }^{-\mathrm{ord}}\omega$ has neither zeros nor poles along the special component, and none along horizontal prime divisors since the generic invariant form is nowhere zero. Hartogs therefore extends it over the regular total space. Its zero locus would have a prime-divisor component by [F1], but no such divisor is available, so it is a generator everywhere. This proves (a). For an abelian variety, left and right invariance agree by commutativity, so the general bounded modular-character argument is unnecessary.

Let $\phi :X⇢Y$ be $R$-rational and generically identical between smooth models with irreducible special fibres. Pulling back the normalized invariant form of $Y$ along $\phi$ gives a rational form on $X$ whose scalar coefficient relative to the normalized form of $X$ is ${\pi }^{\mathrm{ord}X-\mathrm{ord}Y}$ times a unit; invariance under the generic identification and the divisor computation of step 1.1 give $\mathrm{ord}X\ge \mathrm{ord}Y$. If equality holds, the relative differential determinant of $\phi$ is a unit on its domain, so $\phi$ is etale there by [F2]; a generically identical etale separated morphism with reduced source onto a normal target is an open immersion by the Zariski Main Theorem argument in [F2]. This proves (b).

Split a finite weak model collection into its finitely many models with irreducible special fibre. By [F3], every smooth model $X$ with irreducible special fibre has an $R$-rational map, generically the identity, into some member $V_i$. Step 2.1 gives $\mathrm{ord}(X)\ge \mathrm{ord}(V_i)$, so the finite minimum of the collection's orders is a lower bound for all model orders. Conversely each $V_i$ is itself a model, so a member attaining that minimum is minimal among all models. A minimal $X$ maps into a $V_i$ with the same order; step 2.1 makes that map an open immersion on a fibre-dense domain, giving equivalence with one of the finitely many minimal collection members. After the smooth-DVR base change $R\to {\mathcal{O}}_{Z,\eta }$, [F3] keeps the collection weak, and the uniformizer and the normalized volume orders of its split components are unchanged. The same lower-bound argument therefore preserves the minimum and covers all minimal classes by the base-changed representatives, proving (c).