By flat base change we may assume that $k$ is algebraically closed. All fibers of $f:\mathcal{X}\to S$ are Cohen-Macaulay since $\mathcal{X}$ is regular, and the desired result is equivalent to $R^{q}f_{*}(\omega_{\mathcal{X}}\otimes\mathcal{L})=0$ for $q\geq 1$ since $S$ is affine. We may therefore use relative duality for $f$, which shows that the desired result is equivalent to $R^{q}f_{*}\mathcal{L}^{-1}=0$ for $q<n=\dim X$.

Let $d\in\mathbf{N}^{*}$ be a common multiple of the multiplicities of $\mathcal{X}_{0}$, set $S_{d}:=\spec k[[t^{1/d}]]$ and let $\mathcal{Y}$ be the normalization of $\mathcal{X}\times_{S}S_{d}$, with structure map $g:\mathcal{Y}\to S$. The pull-back $\mathcal{M}$ of $\mathcal{L}$ to $\mathcal{Y}$ is still ample since $\mathcal{Y}\to\mathcal{X}$ is finite. By [KKMS, pp.200–201] the $S$-scheme $\mathcal{Y}$ is toroidal and its special fiber $\mathcal{Y}_{0}$ is reduced.

The relative trace $\tr_{\mathcal{Y}/\mathcal{X}}$ shows that $R^{q}g_{*}\mathcal{M}^{-1}$ contains $R^{q}f_{*}\mathcal{L}^{-1}$ as a direct summand, and it is therefore enough to show by semicontinuity that $H^{q}(\mathcal{Y}_{0},\mathcal{M}^{-1})=0$ for $q<n$. Like any toroidal $S$-scheme, $\mathcal{Y}$ is Cohen-Macaulay. As a consequence, the Cartier divisor $\mathcal{Y}_{0}$ is Cohen-Macaulay as well. By another application of duality, this time on $\mathcal{Y}_{0}$, we are reduced to showing that $H^{q}(\mathcal{Y}_{0},\omega_{\mathcal{Y}_{0}}\otimes\mathcal{M})=0$ for $q\geq 1$.

By [KKMS] we may choose a toroidal vertical blow-up $\pi:\mathcal{Y}^{\prime}\to\mathcal{Y}$ such that $\mathcal{Y}_{0}$ has simple normal crossing support. A toric computation (compare [Kol97, Proposition 3.7]) shows that

Since $\mathcal{Y}_{0}$ and $\mathcal{Y}^{\prime}_{0,\mathrm{red}}$ are Cartier divisors on $\mathcal{Y}$ and $\mathcal{Y}^{\prime}$ respectively, adjunction applies (see for instance [KM98, Proposition 5.73]) and we get $\omega_{\mathcal{Y}^{\prime}_{0,\mathrm{red}}}\simeq\pi^{*}\omega_{\mathcal{Y}_{0}}$. On the other hand the projective reduced (but a priori reducible) $k$-scheme $\mathcal{Y}^{\prime}_{0,\mathrm{red}}$ has embedded SNC singularities. It is indeed an SNC divisor in $\mathcal{Y}^{\prime}$, and [Art69] implies that the existence of an algebraic $k$-variety containing $\mathcal{Y}^{\prime}_{0}$ as a divisor. Since $\pi:\mathcal{Y}^{\prime}_{0,\mathrm{red}}\to\mathcal{Y}_{0}$ is projective and $\mathcal{M}$ is ample on $\mathcal{Y}_{0}$, we may therefore apply a vanishing theorem originally due to Kawamata and Ambro and corrected by Fujino ([Kaw85, Theorem 4.4], [Amb03, Theorem 3.2] and [Fuj09, Theorem 2.39]) to get that

is acyclic on $\mathcal{Y}_{0}$. ∎

Writing $\mathcal{L}=(\mathcal{A}+\mathcal{L})-\mathcal{A}$ for some sufficiently ample $\mathcal{A}\in\Pic(\mathcal{X})$ reduces us to the case where $\mathcal{L}$ is very ample. We then get a closed embedding $i:\mathcal{X}\hookrightarrow\mathbf{P}^{N}_{k}\times_{k}S$ over $S$ such that $\mathcal{L}$ coincides with the restriction of $\mathcal{O}(1)$. Let $\pi:\mathbf{P}^{N}_{k}\to\mathbf{P}^{N}_{k}$ be the morphism $[X_{0}:\dots:X_{N}]\mapsto[X_{0}^{m}:\dots:X_{N}^{m}]$, which satisfies $\pi^{*}\mathcal{O}(1)=\mathcal{O}(m)$. For each $\sigma\in\mathrm{PGL}(N+1,k)$ set $i_{\sigma}:=\sigma\circ i$ and consider $\mathcal{X}^{\prime}:=\mathcal{X}\times_{i_{\sigma}}\pi$ with the finite surjective morphism $\rho:\mathcal{X}^{\prime}\to\mathcal{X}$, so that $\rho^{*}\mathcal{L}=\mathcal{O}(m)|_{\mathcal{Y}}$ is divisible by $m$ in $\Pic(\mathcal{Y})$.

Applying Kleiman’s Bertini-type theorem (cf. [Har77, III.10.8]) to the smooth $k$-varieties $E_{J}=\bigcap_{j\in J}E_{j}$ for all subsets $J\subset I$ shows that we may choose $\sigma\in\mathrm{PGL}(N+1,k)$ such that each $\rho^{*}E_{i}$ is smooth over $k$ and $\sum_{i}\rho^{*}E_{i}$ has simple normal crossings. This implies in particular that $\mathcal{X}^{\prime}$ is an SNC model. ∎

As in Theorem B.1 the desired result is equivalent to $H^{q}(\mathcal{X},\mathcal{L}^{-1})=0$ for $q<n$ by relative duality. By flat base change we may assume that $k$ is algebraically closed.

Since $\rho^{*}{E_{i}}$ is smooth for each $i$ and $\sum_{i=2}^{N}\rho^{*}E_{i}$ has normal crossings with $\mathrm{div}(s)=\rho^{*}E_{1}$, one sees as in [KM98, Claim 2.65] that $E_{i}^{(1)}:=\tau^{*}\rho^{*}E_{i}$ is smooth for each $i$ and $\mathcal{X}_{1,0}$ has SNC support, so that $\mathcal{X}_{1}$ is an SNC $S$-scheme. Finally $\mathcal{L}_{1}-\sum_{i=2}^{N}a_{i}E_{i}^{(1)}$ is $\mathbf{Q}$-linearly equivalent to $\tau^{*}\rho^{*}(\mathcal{L}-D)$, hence is ample.

We now use Lemma B.2 to find $c_{1}:\mathcal{X}_{1}^{\prime}\to\mathcal{X}_{1}$ such that $c_{1}^{*}E_{i}^{(1)}$ is smooth for all $i$, $\sum_{i}c_{1}^{*}E_{i}^{(1)}$ is SNC and $c_{1}^{*}E_{2}^{(1)}$ is divisible in $\Pic(\mathcal{X}_{1}^{\prime})$ by the denominator of $a_{2}$. We then perform the same cyclic cover construction as above. Iterating the whole process finally yields an SNC $S$-variety $\mathcal{X}_{N}$ with an ample line bundle $\mathcal{L}_{N}$ such that $H^{q}(\mathcal{X},\mathcal{L}^{-1})$ is a direct summand of $H^{q}(\mathcal{X}_{N},\mathcal{L}_{N}^{-1})$, and we conclude by Theorem B.1.

We are next going to show that $R^{q}\pi_{*}(\omega_{\mathcal{Y}}\otimes\pi^{*}\mathcal{L})=0$ for each $q\geq 1$. Since we have $\pi_{*}\omega_{\mathcal{Y}}=\omega_{\mathcal{X}}$ (the relative canonical bundle $K_{\mathcal{Y}/\mathcal{X}}$ is $\pi$-exceptional and effective since $\mathcal{X}$ is regular), the degeneration of the Leray spectral sequence of $\pi$ will then yield as desired

for $q\geq 1$. Let us now prove the claim. Given $q\geq 1$ choose $\mathcal{A}\in\Pic(\mathcal{X})$ sufficiently ample to guarantee that $\mathcal{A}\otimes R^{q}\pi_{*}(\omega_{\mathcal{Y}}\otimes\pi^{*}\mathcal{L})$ is globally generated on $\mathcal{X}$ and

(note that we are only imposing finitely many non-trivial conditions). The degeneration of the Leray spectral sequence yields

for $q\geq 1$ by Step 1 again, since $\pi^{*}(\mathcal{L}\otimes\mathcal{A})-\varepsilon E$ is also ample. It follows that $\mathcal{A}\otimes R^{q}\pi_{*}(\omega_{\mathcal{Y}}\otimes\pi^{*}\mathcal{L})=0$ by global generation, which proves the claim since $\mathcal{A}$ is invertible. ∎

Let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be an SNC model dominating the blow-up of $\mathcal{X}$ along $\mathfrak{a}_{m}$, so that we have $\mathfrak{a}_{m}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(-D)$ for some effective divisor $D\in\Div_{0}(\mathcal{X}^{\prime})$. By the projection formula we have

Now $\pi^{*}\mathcal{L}-\tfrac{1}{m}D$ is nef and $c\,D-\lfloor c\,D\rfloor$ has coefficients in $[0,1[$. Lemma B.6 below together with the projection formula yields

The Leray spectral sequence is thus degenerate and we conclude using Theorem B.3. ∎

We argue as in the last part of the proof of Theorem B.3. Let $\mathcal{A}\in\Pic(\mathcal{X})$ be sufficiently ample to guarantee:

• (i)

  $\pi^{*}\mathcal{A}-cD$ is nef.

• (ii)

  $\mathcal{A}\otimes R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)$ is globally generated on $\mathcal{X}$.

• (iii)

  $H^{p}\left(\mathcal{X},\mathcal{A}\otimes R^{m}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)\right)=0\,\,\text{ for all }p\geq 1\text{ and }m\geq 0$.

Note that the first condition can be achieved since $-D$ is $\pi$-globally generated. The degeneration of the Leray spectral sequence shows that

which vanishes by Theorem B.3. It follows that $\mathcal{A}\otimes R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)=0$ by global generation, whence the result. ∎

This is proved exactly as in [Laz, Theorem 9.5.20] using local vanishing. (See also [JM11, Theorem A.2] for a different proof.) ∎

Let $\mathcal{B}$ be a given very ample line bundle such that $\mathcal{A}:=\omega_{\mathcal{X}}\otimes\mathcal{B}^{n+1}$ is ample. By the Castelnuovo-Mumford criterion it is enough to check that

for $q=1,\dots,n$, and this is a consequence of Theorem B.5. ∎