Subsubsection [04WE]

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(4.2.1) In this section, we will extend Theorems 3.2.8, 4.1.4, 4.1.7 and 4.1.10 to the case where $X$ is a Calabi-Yau variety over $K=k(\negthinspace(t)\negthinspace)$ instead of over the curve $C$ . The crucial point is that the skeleton of a Calabi-Yau variety can be computed from the logarithmic structure on the special fiber of any $snc$ -model.

Proposition 4.2.2.

Let $\mathscr{Y}$ be a connected regular flat proper $R$ -scheme such that $\mathscr{Y}_{k}$ is a strict normal crossings divisor. Then for every connected flat proper $R$ -scheme $\mathscr{Z}$ and every isomorphism of $R/(t^{2})$ -schemes

f:\mathscr{Y}\times_{R}R/(t^{2})\to\mathscr{Z}\times_{R}R/(t^{2}),

the following properties hold.

(1)

The scheme $\mathscr{Z}$ is regular and $\mathscr{Z}_{k}$ is a divisor with strict normal crossings.
(2)

Denote by $S^{+}$ the log scheme associated to $R\setminus\{0\}\to R$ and by $\mathscr{Y}^{+}$ and $\mathscr{Z}^{+}$ the schemes $\mathscr{Y}$ and $\mathscr{Z}$ endowed with the divisorial log structures associated to their special fibers. For every integer $d>0$ we denote by $s^{+}_{d}$ the standard log point $(\mathrm{Spec}\,k,k^{*}\oplus\mathbb{N})$ viewed as a log scheme over $S^{+}$ via the morphism of charts $\mathbb{N}\to\mathbb{N}:n\mapsto dn$ . If we denote by $e$ the least common multiple of the multiplicities of the components of $\mathscr{Y}_{k}$ , then there exists an isomorphism of log schemes

$g:\mathscr{Y}^{+}\times_{S^{+}}s^{+}_{e}\to\mathscr{Z}^{+}\times_{S^{+}}s^{+}_{e}$

over $s^{+}_{e}$ , such that $g$ is compatible with the reduction of $f$ modulo $t$ (meaning that the obvious square in the category of $k$ -schemes commutes).

Proof.

It is easy to see that (1) holds, since we can detect regularity by looking at the dimensions of the Zariski tangent spaces at the points of

\mathscr{Y}\times_{R}R/(t^{2})\cong\mathscr{Z}\times_{R}R/(t^{2}).

Moreover, the special fibers of $\mathscr{Y}$ and $\mathscr{Z}$ are isomorphic so that $\mathscr{Z}_{k}$ is a divisor with strict normal crossings. Point (2) is more subtle and follows from [Ki03, 2.6(2)]. ∎

Proposition 4.2.3.

Let $\mathscr{Y}$ be a connected regular flat proper $R$ -scheme such that $\mathscr{Y}_{K}$ has trivial canonical sheaf and $\mathscr{Y}_{k}$ is a strict normal crossings divisor. Then the skeleta $\mathrm{Sk}(\mathscr{Y})$ and $\mathrm{Sk}(\mathscr{Y}_{K})$ only depend on $\mathscr{Y}\times_{R}R/(t^{2}),$ in the following sense. Assume that $\mathscr{Z}$ is a regular flat proper $R$ -scheme such that $\mathscr{Z}_{K}$ has trivial canonical sheaf and there exists an isomorphism of $R/(t^{2})$ -schemes

f:\mathscr{Y}\times_{R}R/(t^{2})\to\mathscr{Z}\times_{R}R/(t^{2}).

Then there exists an isomorphism of simplicial spaces $\mathrm{Sk}(\mathscr{Y})\to\mathrm{Sk}(\mathscr{Z})$ that maps $\mathrm{Sk}(\mathscr{Y}_{K})$ onto $\mathrm{Sk}(\mathscr{Z}_{K})$ .

Proof.

Reducing $f$ modulo $t$ , we obtain an isomorphism of $k$ -schemes $\mathscr{Y}_{k}\to\mathscr{Z}_{k}$ and, by taking the dual intersection complexes, an isomorphism of simplicial spaces with piecewise $\mathbb{Z}$ -affine structure $\mathrm{Sk}(\mathscr{Y})\to\mathrm{Sk}(\mathscr{Z})$ . We will prove that this isomorphism maps $\mathrm{Sk}(\mathscr{Y}_{K})$ onto $\mathrm{Sk}(\mathscr{Z}_{K})$ .

We use the notations from Proposition 4.2.2(2) and we set $s^{+}=s_{1}^{+}$ . We denote by $\mathscr{Y}^{+}_{k}$ the log scheme $\mathscr{Y}^{+}\times_{S^{+}}s^{+}$ obtained by restricting the log structure on $\mathscr{Y}^{+}$ to the special fiber $\mathscr{Y}_{k}$ of $\mathscr{Y}$ . It follows from [IKN05, 7.1] that

\Omega:=H^{0}(\mathscr{Y},\omega_{\mathscr{Y}^{+}/S^{+}})

is a free $R$ -module of rank one and that the reduction map

\Omega\otimes_{R}k\to\Omega_{k}:=H^{0}(\mathscr{Y}_{k},\omega_{\mathscr{Y}^{+}_{k}/s^{+}})

is an isomorphism. Let $\omega$ be a generator of the $R$ -module $\Omega$ and denote by $\omega_{k}$ its image in $\Omega_{k}$ . By (3.2), the generic point $\xi$ of an irreducible component $E$ of $\mathscr{Y}_{k}$ is $\omega$ -essential in the sense of [MN13, 4.5.4] if and only if $\omega_{k}$ generates $\omega_{\mathscr{Y}^{+}_{k}/s^{+}}$ at the point $\xi$ . Moreover, the skeleton $\mathrm{Sk}(\mathscr{Y}_{K})=\mathrm{Sk}(\mathscr{Y}_{K},\omega)$ is the simplicial subspace of $\mathrm{Sk}(\mathscr{Y})$ spanned by the vertices corresponding to such points $\xi$ [MN13, 4.5.5]. However, for every integer $d>0$ , the stalk of $\omega_{\mathscr{Y}^{+}_{k}/s^{+}}$ at $\xi$ is generated by global sections if and only if $\omega_{\mathscr{Y}^{+}\times_{S^{+}}s_{d}^{+}/s_{d}^{+}}$ is generated by global sections at any point lying above $\xi$ , by the base change property in [IKN05, 7.1]. The analogous statements hold for $\mathscr{Z}$ . Thus it follows from Proposition 4.2.2(2) that the isomorphism $\mathrm{Sk}(\mathscr{Y})\to\mathrm{Sk}(\mathscr{Z})$ maps $\mathrm{Sk}(\mathscr{Y}_{K})$ onto $\mathrm{Sk}(\mathscr{Z}_{K})$ . ∎

Theorem 4.2.4.

Let $X$ be a geometrically connected, smooth and proper $K$ -variety with trivial canonical sheaf. Then the following properties hold.

(1)

The essential skeleton $\mathrm{Sk}(X)$ is a strong deformation retract of $X^{\mathrm{an}}$ .
(2)

If $\mathscr{X}$ is a proper $snc$ -model of $X$ over $R$ , then $\mathrm{Sk}(X)$ is contained in $\mathrm{Sk}(\mathscr{X})$ and can be obtained from $\mathrm{Sk}(\mathscr{X})$ (as a topological subspace of $\mathrm{Sk}(\mathscr{X})$ with piecewise affine structure) by a finite number of elementary collapses.
(3)

The essential skeleton $\mathrm{Sk}(X)$ is a pseudo-manifold with boundary. If $k$ is algebraically closed and $\mathrm{Sk}(X)$ has dimension $\mathrm{dim}(X)$ , then it is a closed pseudo-manifold.
(4)

Assume that $k$ is algebraically closed and $X$ is projective. Let $\sigma$ be a topological generator of the absolute Galois group $G(K^{a}/K)$ and let $\ell$ be a prime. Then $\mathrm{Sk}(X_{K})$ has dimension $n=\dim(X)$ if and only if the action of $\sigma$ on

$H^{n}_{\mathrm{\acute{e}t}}(X\times_{K}K^{a},\mathbb{Q}_{\ell})$

has a Jordan block of size $n+1$ . If this holds, and $h^{i,0}(X)=0$ for $0<i<n$ , then $\mathrm{Sk}(X_{K})$ is a $\mathbb{Q}$ -homology sphere.

Proof.

Let $\mathscr{X}$ be a proper $snc$ -model of $X$ over $R$ . By a standard argument based on spreading out and Greenberg Approximation (as explained in [MN13, 5.1.2], for instance) we can find a connected smooth $k$ -curve $\mathscr{C}$ , a $k$ -rational point $s$ on $\mathscr{C}$ , a uniformizer $t$ in $\mathcal{O}_{\mathscr{C},s}$ and a smooth and proper $\mathscr{C}$ -scheme $\mathscr{X}^{\prime}$ with geometrically connected fibers such that there exists an isomorphism

\mathscr{X}\times_{R}R/(t^{2})\to\mathscr{X}^{\prime}\times_{\mathscr{C}}\mathrm{Spec}\,\mathcal{O}_{\mathscr{C},s}/(t^{2})

over $R/(t^{2})\cong{O}_{\mathscr{C},s}/(t^{2})$ . Inspecting the proof of [MN13, 5.1.2], we see that we can also assume that the relative canonical sheaf of $\mathscr{X}^{\prime}$ is trivial over $C=\mathscr{C}\setminus\{s\}$ (if the generic fiber of a smooth and proper family over an integral scheme has trivial canonical sheaf, then this holds for all fibers over some dense open subscheme of the base).

By (3.1) we know that $\mathrm{Sk}(\mathscr{X})$ is a strong deformation retract of $X_{K}^{\mathrm{an}}$ . Thus by Proposition 4.2.3, it suffices to prove assertions (1)–(3) for $\mathscr{X}^{\prime}\times_{\mathscr{C}}\mathrm{Spec}\,K$ instead of $X$ . In this case, they follow from Corollary 3.3.6 and Theorems 3.2.8, 3.3.4, 4.1.4 and 4.1.7.

It remains to prove (4). Invoking the Lefschetz Principle, we may assume that $k=\mathbb{C}$ . Taking for $\mathscr{X}$ a projective $snc$ -model over $R$ , we can arrange that $\mathscr{X}^{\prime}$ is projective over $\mathscr{C}$ . By Proposition 4.2.2 and the theory of logarithmic nearby cycles [Na98, 3.3] the action of $\sigma$ on

H^{n}_{\mathrm{\acute{e}t}}(X\times_{K}K^{a},\mathbb{Q}_{\ell})

has a Jordan block of size $n+1$ if and only if the corresponding statement holds for $\mathscr{X}^{\prime}_{K}$ . By Deligne’s comparison theorem for étale and complex analytic nearby cycles in [SGA7b, Exp.XIV], it is also equivalent to the property that the monodromy action on the degree $n$ singular cohomology of a general fiber of $\mathscr{X}^{\prime}$ has a Jordan block of size $n+1$ . If $h^{i,0}(X)=0$ for $0<i<n$ , then we can assume that this also holds for a general fiber of $\mathscr{X}^{\prime}$ , by the proof of [MN13, 5.1.2] (if this property is satisfied by the generic fiber of a smooth and proper family over an integral scheme, then it holds for all fibers over a dense open subscheme of the base, by semi-continuity). Thus the assertion (4) follows from Theorem 4.1.10. ∎