8. Log structures [0304]

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8. Log structures

We review the notion of log structures of Fontaine-Illusie and Kato ([44], [49]). These play a key role in trying to understand mirror symmetry via degenerations.

Definition 8.1.

A log structure on a scheme (or analytic space) $X$ is a (unital) homomorphism

\alpha_{X}:\mathcal{M}_{X}\rightarrow\mathcal{O}_{X}

of sheaves of (multiplicative and commutative) monoids inducing an isomorphism $\alpha_{X}^{-1}(\mathcal{O}_{X}^{\times})\rightarrow\mathcal{O}_{X}^{\times}$ . The monoid structure on $\mathcal{O}_{X}$ is given by multiplication. The triple $(X,\mathcal{M}_{X},\alpha_{X})$ is then called a log space. We often write the whole package as $X^{\dagger}$ .

A morphism of log spaces $F:X^{\dagger}\rightarrow Y^{\dagger}$ consists of a morphism $\underline{F}:X\rightarrow Y$ of underlying spaces together with a homomorphism $F^{\#}:\underline{F}^{-1}(\mathcal{M}_{Y})\rightarrow\mathcal{M}_{X}$ commuting with the structure homomorphisms:

\alpha_{X}\circ F^{\#}=\underline{F}^{*}\circ\alpha_{Y}.

The key examples:

Examples 8.2.

(1) Let $X$ be a scheme and $Y\subseteq X$ a closed subset of codimension one. Denote by $j:X\setminus Y\rightarrow X$ the inclusion. Then the inclusion

\alpha_{X}:\mathcal{M}_{X}=j_{*}(\mathcal{O}_{X\setminus Y}^{\times})\cap\mathcal{O}_{X}\rightarrow\mathcal{O}_{X}

of the sheaf of regular functions invertible off of $Y$ is a log structure on $X$ . This is called a divisorial log structure on $X$ .

(2) A prelog structure, i.e., an arbitrary homomorphism of sheaves of monoids $\varphi:\mathcal{P}\rightarrow\mathcal{O}_{X}$ , defines an associated log structure $\mathcal{M}_{X}$ by

\mathcal{M}_{X}=(\mathcal{P}\oplus\mathcal{O}_{X}^{\times})/\{(p,\varphi(p)^{-1})\,|\,p\in\varphi^{-1}(\mathcal{O}_{X}^{\times})\}

and $\alpha_{X}(p,h)=h\cdot\varphi(p)$ .

(3) If $f:X\rightarrow Y$ is a morphism of schemes and $\alpha_{Y}:\mathcal{M}_{Y}\rightarrow\mathcal{O}_{Y}$ is a log structure on $Y$ , then the prelog structure $f^{-1}(\mathcal{M}_{Y})\rightarrow\mathcal{O}_{X}$ given as the composition of $\alpha_{Y}:f^{-1}(\mathcal{M}_{Y})\rightarrow f^{-1}\mathcal{O}_{Y}$ and $f^{*}:f^{-1}\mathcal{O}_{Y}\rightarrow\mathcal{O}_{X}$ defines an associated log structure on $X$ , the pull-back log structure.

(4) In (1) we can pull back the log structure on $X$ to $Y$ using (3). Thus in particular, if $\mathcal{X}\rightarrow D$ is a toric degeneration, the inclusion $\mathcal{X}_{0}\subseteq\mathcal{X}$ gives a log structure on $\mathcal{X}$ and an induced log structure on $\mathcal{X}_{0}$ . Similarly the inclusion $0\in D$ gives a log structure on $D$ and an induced one on $0$ . Here $\mathcal{M}_{0}=\mathbb{C}^{\times}\oplus\mathbb{N}$ , where $\mathbb{N}$ is the (additive) monoid of natural (non-negative) numbers, and

\alpha_{0}(h,n)=\begin{cases}h&n=0\\ 0&n\not=0.\end{cases}

$0^{\dagger}$ is usually called the standard log point.

We then have log morphisms $\mathcal{X}^{\dagger}\rightarrow D^{\dagger}$ and $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$ .

(5) If $\sigma\subseteq M_{\mathbb{R}}=\mathbb{R}^{n}$ is a strictly convex rational polyhedral cone, ${\sigma}^{\scriptscriptstyle\vee}\subseteq N_{\mathbb{R}}$ the dual cone, let $P={\sigma}^{\scriptscriptstyle\vee}\cap N$ : this is a monoid under addition. The affine toric variety defined by $\sigma$ can be written as $X=\operatorname{Spec}\mathbb{C}[P]$ . We then have a pre-log structure induced by the homomorphism of monoids

P\rightarrow\mathbb{C}[P]

given by $p\mapsto z^{p}$ . There is then an associated log structure on $X$ . This is in fact the same as the log structure induced by $\partial X\subseteq X$ , where $\partial X$ is the toric boundary of $X$ , i.e., the union of toric divisors of $X$ .

If $p\in P$ , then the monomial $z^{p}$ defines a map $f:X\rightarrow\operatorname{Spec}\mathbb{C}[\mathbb{N}]=\mathbb{A}^{1}$ which is a log morphism with the log structure on $\operatorname{Spec}\mathbb{C}[\mathbb{N}]$ induced similarly by $\mathbb{N}\rightarrow\mathbb{C}[\mathbb{N}]$ . The fibre $X_{0}=\operatorname{Spec}\mathbb{C}[P]/(z^{p})$ is a subscheme of $X$ , there is an induced log structure on $X_{0}$ , and a map $X_{0}^{\dagger}\rightarrow 0^{\dagger}$ as in (4). The log morphism $f$ is an example of a log smooth morphism. Essentially all log smooth morphisms are étale locally of this form (if $\mathbb{N}$ is replaced by a more general monoid). See [48] for details.

Condition (4) of Definition 7.1 in fact implies that locally, away from $Z$ , $\mathcal{X}^{\dagger}$ and $\mathcal{X}_{0}^{\dagger}$ are of the above form. So we should view $\mathcal{X}^{\dagger}\rightarrow D^{\dagger}$ as log smooth away from $Z$ , and from the log point of view, $\mathcal{X}_{0}^{\dagger}$ can be treated much like a non-singular scheme away from $Z$ .

(6) Given a monoid $P$ as in (5) and a morphism $X\rightarrow\operatorname{Spec}\mathbb{C}[P]$ , we can pull back the log structure defined above on $\operatorname{Spec}\mathbb{C}[P]$ to $X$ . If $X^{\dagger}$ is a log scheme which étale locally can be described in this way, we say $X^{\dagger}$ is a fine saturated log scheme. The adjective “fine” tells us it is locally described via maps to schemes of the form $\operatorname{Spec}\mathbb{C}[P]$ where $P$ is a finitely generated integral monoid, i.e., the canonical homomorphism $P\rightarrow P^{{\operatorname{gp}}}$ is an injection. The adjective “saturated” tells us the monoid $P$ is saturated. This means that $P$ is integral and whenever $p\in P^{{\operatorname{gp}}}$ satisfies $mp\in P$ for some $m>0$ , $p\in P$ . Such monoids arise, e.g., as the intersection of a rational polyhedral cone with a lattice.

Most of the literature on log geometry tends to apply only to fine log structures. In the key example of $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$ , the log structure is fine saturated away from the set $Z$ . However, it is not in general fine along $Z$ , and this tends to cause many technical problems as new techniques have to be developed to deal properly with the log structure along $Z$ . ∎

On a log scheme $X^{\dagger}$ there is always an exact sequence

1\smash{\mathop{\longrightarrow}\limits}\mathcal{O}_{X}^{\times}\smash{\mathop{\longrightarrow}\limits^{\alpha^{-1}}}\mathcal{M}_{X}\smash{\mathop{\longrightarrow}\limits}\overline{\mathcal{M}}_{X}\smash{\mathop{\longrightarrow}\limits}0,

where we write the quotient sheaf of monoids $\overline{\mathcal{M}}_{X}$ additively. We call $\overline{\mathcal{M}}_{X}$ the ghost sheaf of the log structure. I like to view $\overline{\mathcal{M}}_{X}$ as specifying the combinatorial information associated to the log structure. For example, if $X^{\dagger}$ is induced by the Cartier divisor $Y\subseteq X$ with $X$ normal, then the stalk $\overline{\mathcal{M}}_{X,x}$ at $x\in X$ is the monoid of effective Cartier divisors on a neighbourhood of $x$ supported on $Y$ .

It is useful for understanding pull-backs of log structures to note that if $f:Y\rightarrow X$ is a morphism with $X$ carrying a log structure, and $Y$ is given the pull-back log structure, then $\overline{\mathcal{M}}_{Y}=f^{-1}\overline{\mathcal{M}}_{X}$ . In the case that $\mathcal{M}_{X}$ is induced by an inclusion of $Y\subseteq X$ , $\overline{\mathcal{M}}_{X}$ is supported on $Y$ , so we can equate $\overline{\mathcal{M}}_{X}$ and $\overline{\mathcal{M}}_{Y}$ , the ghost sheaves for the divisorial log structure on $X$ and its restriction to $Y$ .

Exercise 8.3.

Show that in Example 8.2, (5), $\overline{\mathcal{M}}_{X,x}=P$ if $\dim\sigma=\dim M_{\mathbb{R}}$ and $x$ is the unique zero-dimensional torus orbit of $X$ . More generally,

\overline{\mathcal{M}}_{X,x}={{\tau}^{\scriptscriptstyle\vee}\cap N\over\tau^{\perp}\cap N}=\operatorname{Hom}_{monoid}(\tau\cap M,\mathbb{N}),

when $x\in X$ is in the torus orbit corresponding to a face $\tau$ of $\sigma$ . In particular, $\tau$ can be recovered as $\operatorname{Hom}_{monoid}(\overline{\mathcal{M}}_{X,x},\mathbb{R}_{\geq 0})$ , where $\mathbb{R}_{\geq 0}$ is the additive monoid of non-negative real numbers. ∎

In the sections which follow, the key logarithmic spaces we consider will be those arising from toric degenerations $\mathcal{X}\rightarrow D$ . As above, the central fibre $\mathcal{X}_{0}\subseteq\mathcal{X}$ induces a divisorial log structure on $\mathcal{X}$ , and restricting gives a log scheme $\mathcal{X}_{0}^{\dagger}$ along with a morphism $\mathcal{X}_{0}^{\dagger}\rightarrow 0^{\dagger}$ which is log smooth off of the bad set $Z\subseteq\mathcal{X}_{0}$ .

We can now elaborate on the philosophy we wish to take with the following diagram:

There are two sides to mirror symmetry. The $A$ -model side involves counting curves: we wish to count curves in the general fibre of a toric degeneration $\mathcal{X}\rightarrow D$ . There are good reasons to believe that this count can in fact be performed on $\mathcal{X}_{0}^{\dagger}$ , using a theory of logarithmic Gromov-Witten invariants: see §9. The hope is that $\mathcal{X}_{0}$ is a sufficiently combinatorial object so that such a count can be carried out in a combinatorial manner.

The $B$ -side involves deformations of complex structure. The idea is that to understand deformations of complex structure, we should start with the central fibre $\mathcal{X}_{0}^{\dagger}$ and try to construct smoothings, i.e., construct a toric degeneration with this central fibre. The log structure is necessary to find a unique smoothing. If this smoothing can be described sufficiently explicitly, then again one should be able to extract the necessary periods for the $B$ -model calculations purely in terms of combinatorics.

So log geometry will play an important role on both sides of mirror symmetry, but as the above suggests, there should be some combinatorial objects underlying both calculations.

In fact, log geometry is closely related to tropical geometry. We will explore in the following sections how tropical geometry controls both the $A$ - and $B$ -model sides of the above picture, completing the above diagram: