04.12.14 · algebraic-geometry / tropical

Logarithmic structures and log smooth morphisms

shipped3 tiersLean: none

Anchor (Master): Kato 1989 *Logarithmic structures of Fontaine-Illusie* (originator: the log structure $\alpha:M_X\to\mathcal{O}_X$, charts, fs log schemes, log smooth/log étale via the infinitesimal-lifting criterion, log differentials $\Omega^1_{X/S}(\log)$); Fontaine-Illusie 1980s (unpublished originating idea, $p$-adic Hodge theory); Kato 1996 *Fontaine-Illusie* (Trieste lectures) and Kato-Nakayama 1999 *Kodai Math. J.* 22 (Betti realisation); Ogus 2018 *Lectures on Logarithmic Algebraic Geometry* (canonical modern textbook); Gross-Siebert 2006 *J. Differential Geom.* 72 §1-§2 (divisorial and central-fibre log structures on toric degenerations); Gross-Siebert 2010 *J. Algebraic Geom.* 19 (intrinsic log structures, Kato fans); F. Kato 2000 *Math. Z.* 233 (log smooth deformation theory); Abramovich-Chen-Gillam-Huang-Olsson-Satriano-Sun 2013 *Logarithmic geometry and moduli* (Handbook of Moduli) survey

Intuition Beginner

A degenerating family of shapes can look smooth almost everywhere and then break at one moment. Picture a smooth doughnut shrinking in a one-parameter family until, at the final instant, it pinches into a circle with two points glued. The family is well behaved up to that last moment, where ordinary smoothness fails. The question is how to keep treating the broken final shape as if it were still part of a smooth family, instead of throwing it away as a defect.

Logarithmic geometry answers this by remembering, as extra bookkeeping attached to the space, exactly where the space is allowed to be singular and how the breaking happened. The bookkeeping is a record of which functions are permitted to vanish along the broken locus. A function that vanishes there is no longer treated as zero by accident; it is treated as a marked, controlled vanishing carrying combinatorial information.

The slogan is that a space carries its boundary as memory. The broken final shape, equipped with this memory, is "log smooth" even though it is singular in the plain sense. This lets a degenerating family behave smoothly all the way to the end, which is precisely what a toric degeneration of a Calabi-Yau 04.12.07 needs.

Visual Beginner

A two-panel schematic. Left panel: a smooth fibre of a family, drawn as an unbroken curve, with no marked points and a plain coordinate grid. Right panel: the broken central fibre, two components meeting along a node, with the node circled and a small tag reading "boundary memory" pointing at it. Along each component, a dotted line marks the divisor where functions are allowed to vanish; a little monoid diagram (a corner of a lattice cone) sits beside the node to indicate the combinatorial data recorded there.

The picture captures the central move: the right-hand broken space looks defective, but the added bookkeeping at the node — a sheaf of monoids recording allowed vanishings — promotes it to a log smooth object. The combinatorial cone beside the node is the tropical shadow of this bookkeeping, the same combinatorial data that organises the dual intersection complex.

Worked example Beginner

Take the simplest broken family: a coordinate line crossing inside a plane, modelled by the two axes meeting at the origin. Plainly the cross is singular at the origin, where the two branches meet. We attach the log bookkeeping and watch the origin become acceptable.

Step 1. The functions. On the plane with coordinates $x$ and $y$ , the two functions $x$ and $y$ each vanish along one of the two axes. Away from the origin, on each axis separately, one of the two functions stays nonzero and so is a unit.

Step 2. The memory. The log bookkeeping records the monoid generated by $x$ and $y$ under multiplication, together with all the genuine units (the nowhere-vanishing functions). At a point on one axis away from the origin, only one generator is allowed to vanish, so the recorded combinatorial corner is one-dimensional.

Step 3. The corner. At the origin, both $x$ and $y$ are allowed to vanish, so the recorded corner is the full two-dimensional lattice quadrant: the monoid of pairs of whole numbers. This quadrant is the combinatorial cone that beginner pictures draw beside the node.

Step 4. The promotion. The plain cross is singular at the origin, but with this quadrant of memory attached, the origin carries the same combinatorial structure as a smooth coordinate corner of a model toric space. In the log sense the cross is now smooth: it is locally modelled on a standard toric corner, and the singular origin has been absorbed into bookkeeping.

What this tells us: remembering the lattice quadrant at the crossing converts a plain singularity into a controlled, toric-model corner. The same move, applied along the whole broken central fibre of a toric degeneration, makes that fibre log smooth.

Check your understanding Beginner

Exercise (easy, multiple choice).

A log structure on a space is best described as:

A. A choice of logarithm function on the structure sheaf.
B. Extra bookkeeping — a sheaf of monoids recording which functions are allowed to vanish along a chosen locus — together with a multiplicative map to the functions on the space.
C. A second copy of the space glued along its boundary.
D. A metric that blows up logarithmically near the singular locus.

Hint

The word "log" is historical; the content is a record of allowed vanishings attached as a sheaf of monoids, not a logarithm or a metric.

Answer

B. A log structure is a sheaf of monoids $M_{X}$ together with a multiplicative map to the functions on the space that is an isomorphism over the nowhere-vanishing functions. It records, as monoid data, exactly which functions are permitted to vanish along a marked locus. Option A confuses the name "log" with a logarithm function. Option C describes a doubling construction, not the monoid bookkeeping. Option D describes an analytic metric condition unrelated to the monoid data. The whole point is that the bookkeeping is combinatorial and is what lets a broken space count as smooth in the log sense.

Exercise (easy, true-false).

True or false: a log smooth space must be smooth in the ordinary sense.

Hint

Recall the broken central fibre of a degenerating family: singular in the plain sense, yet the entire reason for log geometry is to treat it as smooth.

Answer

False. A log smooth space need not be ordinarily smooth — that mismatch is the entire reason the theory exists. A normal-crossings central fibre of a degenerating family is singular where its components meet, but once equipped with its boundary-memory log structure it becomes log smooth, locally modelled on a standard toric corner. The plain singularity is absorbed into combinatorial bookkeeping. If log smooth always meant ordinarily smooth, the apparatus would carry no information and the toric degenerations of 04.12.07 could not be handled.

Formal definition Intermediate+

Throughout, a monoid means a commutative monoid written multiplicatively; $N$ denotes the additive monoid of nonnegative integers and, for a cone $σ$ of 04.11.04, $S_{σ} = σ^{\lor} \cap M$ denotes the associated monoid of lattice points. We work over a base scheme; for the mirror-symmetry application the reader may take everything over $C$ .

Definition (log structure; Kato 1989). Let $X$ be a scheme with structure sheaf $O_{X}$ , regarded as a sheaf of monoids under multiplication. A log structure on $X$ is a sheaf of commutative monoids $M_{X}$ on the étale site of $X$ together with a morphism of sheaves of monoids $$ \alpha : M_X \longrightarrow (\mathcal{O}_X, \cdot) $$ such that the restriction $α^{- 1} (O_{X}^{*}) \to O_{X}^{*}$ is an isomorphism of sheaves of monoids. A scheme with a log structure is a log scheme $(X, M_{X})$ . The isomorphism condition over $O_{X}^{*}$ identifies the units inside $M_{X}$ with the genuine units $O_{X}^{*}$ ; one writes $M_{X} \supseteq O_{X}^{*}$ accordingly.

Definition (ghost sheaf / characteristic monoid). The ghost sheaf of $(X, M_{X})$ is the quotient sheaf of monoids $$ \overline{M}_X := M_X / \mathcal{O}_X^*. $$ The ghost sheaf strips away the units and records the purely combinatorial vanishing data — the "tropical" content of the log structure. Its stalks are the monoids that appear in the dual intersection complex and the integral affine bookkeeping of 04.12.07.

Definition (pullback and associated log structure). Given a morphism of schemes $f : X \to Y$ and a log structure $β : N \to O_{Y}$ , the pre-log structure $f^{- 1} N \to f^{- 1} O_{Y} \to O_{X}$ generates a log structure, denoted $f^{*} N$ , by the universal pushout that turns a pre-log structure (a map $P \to O_{X}$ of sheaves of monoids with no condition on units) into one satisfying the isomorphism-over-units axiom: $f^{*} N = (f^{- 1} N \oplus_{(α \circ f)^{- 1} O_{X}^{*}} O_{X}^{*})$ . A morphism of log schemes $(X, M_{X}) \to (Y, N_{Y})$ is a pair $(f, f^{♭})$ with $f : X \to Y$ and $f^{♭} : f^{*} N_{Y} \to M_{X}$ a morphism of log structures over $O_{X}$ .

Example (divisorial log structure). Let $X$ be a regular scheme and $D \subseteq X$ a normal-crossings divisor. The divisorial log structure is $$ M_{(X,D)} := {, g \in \mathcal{O}_X : g\ \text{is invertible on}\ X \setminus D ,} ;\subseteq; \mathcal{O}_X, $$ with $α$ the inclusion. Its ghost stalk at a point lying on exactly $k$ branches of $D$ is $N^{k}$ : each branch contributes one generator recording the order of vanishing along it. This is the running example throughout: the central fibre $X_{0} \subseteq X$ of a toric degeneration carries the divisorial log structure of $X_{0}$ inside $X$ .

Example (toric log structure). Let $σ \subseteq N_{R}$ be a rational polyhedral cone with associated affine toric variety $X_{σ} = Spec C [S_{σ}]$ of 04.11.04, where $S_{σ} = σ^{\lor} \cap M$ . The monoid $S_{σ}$ maps into $C [S_{σ}]$ by $m \mapsto χ^{m}$ , and the associated log structure has ghost stalk $S_{σ} / S_{σ}^{*}$ at the torus-fixed point. The toric boundary divisor is normal-crossings exactly when $σ$ is simplicial-unimodular, in which case the toric and divisorial log structures agree.

Definition (chart). A chart for a log structure $M_{X}$ subordinate to a monoid $P$ is, on an étale neighbourhood, a morphism of sheaves of monoids $P \to M_{X}$ (with $P$ the constant sheaf) whose associated log structure is $M_{X}$ . A log structure is coherent if charts exist locally, integral if $M_{X}$ is a sheaf of integral monoids (cancellative: $ab = a c \Rightarrow b = c$ ), and fine if it is coherent and integral. A fine log structure is saturated if each stalk $\overline{M}_{X, \overset{x}{ˉ}}$ is a saturated monoid ( $n p \in \overline{M}$ with $n \geq 1$ implies $p \in \overline{M}$ inside the group it generates). A fine saturated (fs) log scheme is one with a fine saturated log structure; the fs condition is what aligns the monoids with the lattice cones $S_{σ}$ and is assumed throughout the Gross-Siebert programme.

Definition (standard log point). The standard log point is $Spec C$ equipped with the log structure $C^{*} \oplus N \to C$ , $(u, n) \mapsto u \cdot 0^{n}$ (so $α (u, 0) = u$ and $α (u, n) = 0$ for $n \geq 1$ ). Its ghost monoid is $N$ . A toric degeneration over $Spec C [[t]]$ restricts over the closed point to a log smooth morphism to the standard log point; this is the precise carrier of the central fibre's combinatorial data.

Definition (log smooth and log étale). A morphism of fine log schemes $f : (X, M_{X}) \to (S, M_{S})$ is log smooth if, étale locally, $f$ admits a factorisation through a chart by a map of monoids $Q \to P$ with $Q$ a chart of $S$ and $P$ a chart of $X$ , such that (i) the kernel and torsion of the cokernel of $Q^{gp} \to P^{gp}$ are finite groups of order invertible on $X$ , and (ii) the induced map $X \to S \times_{Spec Z [Q]} Spec Z [P]$ is (classically) smooth. It is log étale if moreover that classical map is étale. This is Kato's chart criterion: log smoothness is smoothness relative to a monomial model $Spec Z [Q] \to Spec Z [P]$ .

Definition (log differentials and log canonical sheaf). For $f : (X, M_{X}) \to (S, M_{S})$ the sheaf of log differentials $Ω_{X / S}^{1} (lo g)$ is the $O_{X}$ -module generated by ordinary differentials $d a$ for $a \in O_{X}$ and symbols $d lo g m$ for $m \in M_{X}$ , modulo the relations $d (α (m)) = α (m) d lo g m$ and $d lo g m = 0$ for $m \in O_{S}^{*}$ . When $X$ is regular with normal-crossings divisor $D$ and the log structure is divisorial, $Ω_{X / S}^{1} (lo g)$ is the classical sheaf of differentials with logarithmic poles along $D$ . The log canonical sheaf is its top exterior power $ω_{X / S}^{l o g} := det Ω_{X / S}^{1} (lo g)$ , which for the divisorial case is $ω_{X} (D)$ , the sheaf realising the adjunction divisor $K_{X} + D$ of 04.08.02.

Counterexamples to common slips

The map $α$ is not assumed injective or surjective. Only its restriction over units is an isomorphism. The divisorial log structure has $α$ injective; the standard log point has $α$ neither injective nor surjective (the element $(1, 1)$ maps to $0$ ). The defining axiom constrains $α$ solely over $O_{X}^{*}$ .
Log smooth does not mean smooth. The toric degeneration $X \to Spec C [[t]]$ is log smooth with respect to the divisorial log structure of the central fibre, yet it is not smooth: the central fibre is singular. Confusing the two collapses the entire subject. The classical smoothness in Kato's criterion is smoothness of the map to the monomial model, not of $f$ itself.
The ghost sheaf is not a constant sheaf in general. On the central fibre of a degeneration $\overline{M}_{X}$ jumps in rank along the stratification: rank $0$ on the smooth open part, rank $1$ along the smooth locus of $D$ , rank $k$ where $k$ branches meet. The non-constancy of $\overline{M}_{X}$ is exactly the combinatorial data of the dual intersection complex.

Key theorem with proof Intermediate+

Theorem (Kato's structure theorem for log smooth morphisms; Kato 1989 Theorem 3.5). Let $f : (X, M_{X}) \to (S, M_{S})$ be a morphism of fine log schemes. Then $f$ is log smooth if and only if, étale locally on $X$ , there is a chart $(P_{X}, Q_{S}, θ : Q \to P)$ for $f$ such that:

(a) the kernel of $θ^{gp} : Q^{gp} \to P^{gp}$ and the torsion subgroup of its cokernel are finite of order invertible on $X$ ; and

(b) the induced morphism $$ X ;\longrightarrow; S \times_{\operatorname{Spec}\mathbb{Z}[Q]} \operatorname{Spec}\mathbb{Z}[P] $$ is classically smooth in the usual sense.

Moreover, when $f$ is log smooth, $Ω_{X / S}^{1} (lo g)$ is locally free of rank $dim X - dim S + rk (P^{gp} / Q^{gp})$ , and the symbols $d lo g p$ for $p$ a basis of $P^{gp} / Q^{gp}$ together with a transcendence basis of the classical fibre give a local frame.

Proof. We give the argument in three moves; full details occupy Kato 1989 §3.

Step 1 (lifting criterion to chart criterion). Kato defines log smoothness by the infinitesimal lifting property: for every strict square-zero closed immersion of fine log schemes $T_{0} ↪ T$ over $S$ , the map $Hom_{S} (T, X) \to Hom_{S} (T_{0}, X)$ is surjective (here "strict" means the log structure on $T_{0}$ is pulled back from $T$ ). The first move shows this lifting property is equivalent to the existence of a chart with property (b). Given a chart $θ : Q \to P$ , a lift of $T_{0} \to X$ amounts to lifting the monoid map $P \to Γ (T_{0}, M_{T_{0}})$ and the ring map compatibly; the obstruction to lifting the monoid part is absorbed by passing to the group completion $P^{gp}$ , where it is governed by $Ext^{1}$ in abelian groups and vanishes precisely when condition (a) holds. The residual obstruction is the classical lifting obstruction for the map in (b), which vanishes for all square-zero extensions exactly when that map is classically smooth.

Step 2 (the monomial model). The fibre product $Spec Z [P] \to Spec Z [Q]$ is the universal monomial model: it carries the canonical log structure associated to $P$ and is log smooth over the base with chart $Q$ by direct verification of the lifting property (lifting a map to $Spec Z [P]$ is lifting a monoid homomorphism $P \to Γ (T, M_{T})$ , always possible after the finite-order adjustment of (a) because $P$ is finitely generated and the target monoid is integral). Log smoothness is preserved under strict base change and composition, so $f$ is log smooth whenever it factors through this model by a classically smooth map. This proves the "if" direction.

Step 3 (log differentials and the rank count). For the monomial model the universal log derivation sends $χ^{p} \mapsto χ^{p} d lo g p$ , and $d lo g : P^{gp} \to Ω^{1}$ identifies $Ω_{Z [P] / Z [Q]}^{1} (lo g) ≅ O \otimes_{Z} (P^{gp} / Q^{gp})$ , free of rank $rk (P^{gp} / Q^{gp})$ . The classically smooth map in (b) contributes the ordinary relative differentials of its (constant-rank) cotangent sheaf. The log conormal sequence for the factorisation through the monomial model is exact and splits because the second map is classically smooth, so $Ω_{X / S}^{1} (lo g)$ is the direct sum and is locally free of the stated rank. The "only if" direction reverses Step 1: log smoothness forces the lifting property, which produces a chart with (a) and (b) by choosing charts that realise the cotangent data. $□$

Bridge. This structure theorem is the foundational reason the broken central fibre of a toric degeneration of 04.12.07 can be treated as a fibre of a smooth family: log smoothness is exactly classical smoothness relative to the monomial model $Spec Z [P] \to Spec Z [Q]$ , and the normal-crossings degeneration $X \to Spec C [[t]]$ realises the model $N \to N^{k}$ , $1 \mapsto (1, \dots, 1)$ , whose classical map is smooth even though $f$ is not. The chart criterion generalises ordinary smoothness — which is the empty-log case $P = Q$ — to the boundary-aware setting, and the log differential $d lo g t$ that the theorem produces is dual to the combinatorial direction in the ghost sheaf $\overline{M}_{X}$ that the dual intersection complex records. This builds toward the central-fibre log structure of Gross-Siebert, where the rank-jump of $\overline{M}_{X}$ along the stratification is the integral affine bookkeeping, and it appears again in the log Gromov-Witten theory of 04.12.15, whose stable log maps are maps log smooth over the standard log point. Putting these together, the chart criterion is the single technical bridge from plain singular geometry to the smooth-family behaviour the whole programme presupposes.

Exercises Intermediate+

A graded set covering the defining axiom, the ghost sheaf, charts and the fs condition, log smoothness via the chart criterion, and log differentials.

Exercise 1 (easy, multiple choice).

The defining axiom of a log structure $α : M_{X} \to O_{X}$ requires that:

A. $α$ is injective.
B. $α$ is surjective.
C. $α$ restricts to an isomorphism $α^{- 1} (O_{X}^{*}) \sim O_{X}^{*}$ .
D. $α$ is the zero map.

Hint

The only constraint is over the units of $O_{X}$ ; everything else about $α$ is unconstrained.

Answer

C. A log structure is a sheaf of monoids $M_{X}$ with a multiplicative map $α$ whose restriction over the units is an isomorphism. This identifies $O_{X}^{*}$ as a submonoid of $M_{X}$ and is what makes the quotient $\overline{M}_{X} = M_{X} / O_{X}^{*}$ the meaningful combinatorial invariant. Options A, B, D impose unwanted conditions: the divisorial log structure has $α$ injective but not surjective, and the standard log point has $α$ neither, so any of A, B, D would exclude basic examples.

Exercise 2 (easy, short-answer).

Compute the ghost stalk $\overline{M}_{X, \overset{x}{ˉ}}$ of the divisorial log structure at a point $\overset{x}{ˉ}$ lying on exactly three smooth branches of a normal-crossings divisor $D$ .

Hint

Each branch contributes one generator recording the order of vanishing along it, and the branches are independent.

Answer

The ghost stalk is $N^{3}$ . Locally $D = {x_{1} x_{2} x_{3} = 0}$ in regular coordinates, and a section of $M_{(X, D)}$ near $\overset{x}{ˉ}$ has the form $u \cdot x_{1}^{a_{1}} x_{2}^{a_{2}} x_{3}^{a_{3}}$ with $u$ a unit and $a_{i} \in N$ the orders of vanishing along the three branches. Quotienting by units $O_{X}^{*}$ removes $u$ and leaves the exponent vector $(a_{1}, a_{2}, a_{3}) \in N^{3}$ . The rank of the ghost stalk equals the number of branches through $\overset{x}{ˉ}$ , which is the codimension of the stratum, recording the combinatorial position of $\overset{x}{ˉ}$ in the stratification.

Exercise 3 (medium, short-answer).

Show that the standard node-smoothing family $X = {x y = t} \subseteq A^{2} \times A_{t}^{1} \to A_{t}^{1}$ , with $X$ given the divisorial log structure of the central fibre ${t = 0}$ and the base $A_{t}^{1}$ the standard log structure at the origin, is log smooth, by exhibiting the monoid chart.

Hint

Use the chart $Q = N \to P = N^{2}$ sending the base generator to $(1, 1)$ ; the monomial model is $Spec Z [N^{2}] \to Spec Z [N]$ , which is $x y = t$ .

Answer

Take the base chart $Q = N$ , generator mapping to $t$ , and the total chart $P = N^{2}$ , generators mapping to $x$ and $y$ . The monoid map $θ : Q \to P$ sends $1 \mapsto (1, 1)$ , encoding $t = x y$ . The group map $θ^{gp} : Z \to Z^{2}$ , $1 \mapsto (1, 1)$ , has zero kernel and cokernel $Z^{2} / ⟨(1, 1)⟩ ≅ Z$ , which is torsion-free, so condition (a) of Kato's criterion holds. The monomial model is $Spec Z [N^{2}] = A^{2} \to Spec Z [N] = A^{1}$ given by $(x, y) \mapsto x y$ , and $X = {x y = t}$ maps to the fibre product $A_{t}^{1} \times_{A^{1}} A^{2}$ by an isomorphism, which is classically smooth, so condition (b) holds. Hence $f$ is log smooth, with $Ω_{X / S}^{1} (lo g)$ free of rank $rk (Z^{2} / Z) = 1$ , generated by $d lo g x = - d lo g y$ . The total space is singular at the origin of the central fibre, yet the morphism is log smooth — the model phenomenon of the subject.

Exercise 4 (medium, short-answer).

Explain why the fine saturated (fs) condition is imposed, and give a coherent integral log structure whose ghost monoid fails to be saturated.

Hint

Saturation means: if $n p$ lies in the monoid for some $n \geq 1$ , then $p$ already lies in the monoid (inside the ambient group). A cusp-type monoid violates this.

Answer

The fs condition aligns the ghost monoids with the lattice-point monoids $S_{σ} = σ^{\lor} \cap M$ of genuine rational polyhedral cones, which are always saturated; saturation is precisely the condition that the monoid equals the lattice points of the cone it generates, with no missing interior lattice points. Without it the geometry of the log structure decouples from the toric/tropical combinatorics. A non-saturated example: the monoid $P = ⟨ 2, 3 ⟩ \subseteq N$ generated by $2$ and $3$ . Its group completion is $Z$ , and $1 = 3 - 2 \in / P$ while $2 \cdot 1 = 2 \in P$ , so $P$ is integral and finitely generated (hence gives a coherent integral log structure) but not saturated: $1$ is a missing lattice point of the cone $R_{\geq 0}$ . Its saturation is all of $N$ . Imposing fs excludes such cusp monoids and guarantees the dual-cone dictionary of 04.11.04 applies verbatim.

Exercise 5 (medium, short-answer).

For a regular variety $X$ with smooth divisor $D$ , identify $Ω_{X}^{1} (lo g D)$ and the log canonical sheaf, and relate the latter to the adjunction divisor $K_{X} + D$ of 04.08.02.

Hint

In local coordinates with $D = {z_{1} = 0}$ , the log differentials are spanned by $d z_{1} / z_{1}, d z_{2}, \dots, d z_{n}$ ; take the top exterior power.

Answer

In local coordinates $z_{1}, \dots, z_{n}$ with $D = {z_{1} = 0}$ , the sheaf $Ω_{X}^{1} (lo g D)$ is the locally free $O_{X}$ -module with frame $\frac{d z _{1}}{z _{1}}, d z_{2}, \dots, d z_{n}$ — ordinary differentials except a logarithmic pole allowed along $D$ . It sits in the residue exact sequence $0 \to Ω_{X}^{1} \to Ω_{X}^{1} (lo g D) res O_{D} \to 0$ , where the residue map sends $\frac{d z _{1}}{z _{1}} \mapsto 1$ . Its top exterior power is the log canonical sheaf $ω_{X}^{l o g} = det Ω_{X}^{1} (lo g D) = ω_{X} (D)$ , the canonical sheaf of 04.08.02 twisted by $O_{X} (D)$ . As a divisor class this is $K_{X} + D$ , the adjunction divisor: restricting $ω_{X} (D)$ to $D$ and applying the residue recovers the adjunction formula $ω_{D} ≅ (ω_{X} (D)) ∣_{D}$ . The log canonical sheaf is therefore the geometric meaning of $K_{X} + D$ , and a log Calabi-Yau pair is one with $K_{X} + D \sim 0$ .

Exercise 6 (hard, short-answer).

Define a log Calabi-Yau pair $(X, D)$ and explain why the central fibre of a toric degeneration of 04.12.07, with its divisorial log structure, is the motivating example.

Hint

Log Calabi-Yau means the log canonical class $K_{X} + D$ is linearly equivalent to zero, so a nowhere-vanishing log volume form exists.

Answer

A log Calabi-Yau pair is a pair $(X, D)$ with $X$ a normal variety and $D$ a reduced divisor such that the log canonical divisor $K_{X} + D$ is linearly equivalent to zero, equivalently $ω_{X}^{l o g} = ω_{X} (D) ≅ O_{X}$ . This means there is a nowhere-vanishing global section of the log canonical sheaf — a holomorphic volume form with at worst log poles along $D$ , nonvanishing elsewhere — the log analogue of the Calabi-Yau holomorphic volume form. For a toric degeneration $X \to Spec C [[t]]$ the relative log canonical $ω_{X / S}^{l o g}$ is the structure sheaf (the Calabi-Yau condition (v) of 04.12.07), and the central fibre $X_{0}$ with the divisorial log structure of $X_{0} \subseteq X$ inherits a log Calabi-Yau structure: the nowhere-vanishing relative log volume form $Ω \in Γ (X, ω_{X / S}^{l o g})$ restricts to a log volume form on the singular fibre. Ordinary Calabi-Yau geometry breaks down on $X_{0}$ because $X_{0}$ is singular and its plain canonical sheaf is not invertible; the log canonical sheaf is invertible and isomorphic to the structure sheaf, so the log structure repairs exactly the property needed. This is why the Gross-Siebert programme is phrased in log geometry: the central fibre is a log Calabi-Yau space even though it is not a Calabi-Yau variety.

Exercise 7 (hard, short-answer).

Describe the Kato-Nakayama space $X^{l o g}$ of a log scheme over $C$ and explain how it makes the ghost sheaf visible as monodromy, connecting to the SYZ torus fibration of the Gross-Siebert picture.

Hint

The Kato-Nakayama construction is a real oriented blow-up replacing each point of the boundary stratum by a torus whose rank is the rank of the ghost monoid there.

Answer

For a fine saturated log scheme $(X, M_{X})$ over $C$ , the Kato-Nakayama space $X^{l o g}$ (Kato-Nakayama 1999) is a topological space with a proper continuous map $τ : X^{l o g} \to X^{an}$ to the underlying analytic space, defined as the set of pairs (a point $x$ , a homomorphism $M_{X, x}^{gp} \to S^{1}$ lifting the argument $ar g α$ on units). Over a point where the ghost monoid has rank $k$ , the fibre of $τ$ is the real torus $(S^{1})^{k}$ : the construction is an oriented real blow-up that replaces the boundary stratum by a torus bundle whose rank reads off the ghost-sheaf rank. The monodromy of this torus bundle around the boundary is governed by the integral affine transition data recorded in $\overline{M}_{X}$ . For the central fibre of a toric degeneration, $X^{l o g}$ realises the smooth nearby fibre topologically: the rank- $n$ torus appearing over the deepest stratum is exactly the SYZ torus, and the dual intersection complex of 04.12.07 is recovered as the base of this torus fibration with its singular integral affine structure. The Kato-Nakayama space is therefore the topological bridge from the algebraic log structure to the SYZ T-duality picture, turning combinatorial ghost-sheaf data into a genuine torus fibration whose monodromy encodes the mirror geometry.

Advanced results Master

We collect the deeper structural results: the log smooth deformation theory that controls smoothing of the central fibre, the intrinsic log structure of Gross-Siebert, and the role of log differentials in the period computation.

Log smooth deformation theory. F. Kato 1996 developed the deformation theory of log smooth morphisms in parallel with the classical Kodaira-Spencer theory. For a log smooth morphism $f : (X, M_{X}) \to (S, M_{S})$ the first-order deformations over a square-zero log extension are classified by $H^{1} (X, Θ_{X / S}^{l o g})$ and the obstructions by $H^{2} (X, Θ_{X / S}^{l o g})$ , where $Θ_{X / S}^{l o g} = H o m (Ω_{X / S}^{1} (lo g), O_{X})$ is the sheaf of log derivations. The decisive feature is that a normal-crossings central fibre, which is obstructed in the ordinary deformation theory (its plain tangent sheaf is not locally free), becomes unobstructed in the log category whenever the log structure is log smooth over the standard log point: the log tangent sheaf is locally free of the correct rank. This is the rigorous mechanism by which the broken central fibre smooths to the nearby Calabi-Yau fibres.

Friedman d-semistability as a log condition. The d-semistability obstruction of Friedman — the condition that the central fibre of a normal-crossings degeneration admit a smoothing — is exactly the condition that the divisorial log structure on the central fibre extends to a log smooth log structure over the standard log point. The combinatorial avatar on the dual intersection complex of 04.12.07 is the existence of a global integral affine structure with the correct monodromy; the log-geometric avatar is the local-to-global gluing of the standard-log-point structures, controlled by the $E x t^{1}$ that Friedman computes. The two viewpoints agree because the ghost sheaf $\overline{M}_{X}$ is the sheaf-theoretic record of the integral affine fan structure.

Intrinsic log structures and Kato fans. Gross-Siebert 2010 constructed the log structure on the central fibre intrinsically, without reference to an ambient smoothing, by gluing local toric models indexed by the cells of the polyhedral decomposition $P$ . The combinatorial datum is a Kato fan, a monoidal space locally modelled on $Spec$ of a monoid, and the intrinsic log structure is the pullback of the canonical log structure on this fan. The unobstructed-smoothing criterion becomes a consistency condition on the gluing — the same consistency that the scattering diagram of 04.12.07's successors enforces. This intrinsic construction is what frees the Gross-Siebert programme from needing the smoothing in advance: the central fibre, with its intrinsic log structure, determines the smoothing rather than presupposing it.

Log differentials and the period computation. The log de Rham complex $(Ω_{X / S}^{∙} (lo g), d)$ computes the cohomology of the nearby fibre with its limit mixed Hodge structure: the Steenbrink spectral sequence is built from $Ω_{X / S}^{∙} (lo g)$ restricted to the central fibre, and the monodromy weight filtration is read off the log structure. The nowhere-vanishing log volume form $Ω \in Γ (ω_{X / S}^{l o g})$ of a log Calabi-Yau is the class whose periods, integrated against vanishing cycles, give the period integrals and the mirror map. Thus the log canonical sheaf $K_{X} + D$ is not bookkeeping decoration but the carrier of the Hodge-theoretic output of the whole programme.

Synthesis. Putting these together, the log structure is the foundational reason the singular central fibre behaves like a smooth fibre: log smooth deformation theory makes the obstructed plain deformation problem unobstructed, which is exactly the statement that the log tangent sheaf is locally free where the ordinary one is not. The intrinsic Kato-fan construction generalises the divisorial log structure to a self-contained combinatorial object, so that the central fibre determines its own smoothing — the central insight that lets reconstruction run without the smoothing given in advance. The Friedman d-semistability obstruction is dual to the integral affine monodromy on the dual intersection complex, and the log de Rham complex builds toward the period and mirror-map output, where the log canonical class $K_{X} + D$ appears again in the Hodge-theoretic readout. The bridge is always the ghost sheaf $\overline{M}_{X}$ : it is simultaneously the combinatorial fan datum, the obstruction-theoretic gluing record, and the tropical shadow that the rest of the chapter manipulates.

Full proof set Master

Proposition 1 (ghost sheaf of a divisorial log structure). Let $X$ be regular and $D = ⋃_{i} D_{i} \subseteq X$ a simple normal-crossings divisor with smooth components $D_{i}$ . Then the ghost sheaf of the divisorial log structure $M_{(X, D)}$ has stalk at a geometric point $\overset{x}{ˉ}$ equal to $N^{I (\overset{x}{ˉ})}$ , where $I (\overset{x}{ˉ}) = {i : \overset{x}{ˉ} \in D_{i}}$ indexes the branches through $\overset{x}{ˉ}$ .

Proof. Choose étale-local coordinates $z_{1}, \dots, z_{n}$ at $\overset{x}{ˉ}$ so that the branches through $\overset{x}{ˉ}$ are $D_{i} = {z_{i} = 0}$ for $i \in I (\overset{x}{ˉ})$ , with $∣ I (\overset{x}{ˉ}) ∣ = k$ after reindexing. A section of $M_{(X, D)}$ near $\overset{x}{ˉ}$ is a regular function invertible on the complement of $D$ , hence of the form $g = u \cdot z_{1}^{a_{1}} \dots z_{k}^{a_{k}}$ with $u \in O_{X, \overset{x}{ˉ}}^{*}$ and $a_{i} \in N$ (negative exponents are excluded because $g$ is regular, not merely rational; exponents along branches not through $\overset{x}{ˉ}$ are zero because $g$ is invertible there). The map sending $g$ to its exponent vector $(a_{1}, \dots, a_{k})$ is a homomorphism $M_{(X, D), \overset{x}{ˉ}} \to N^{k}$ that is surjective (the monomials $z_{i}$ realise the generators) with kernel exactly the units $u$ , since $g$ is a unit iff all $a_{i} = 0$ . Therefore $\overline{M}_{(X, D), \overset{x}{ˉ}} = M_{(X, D), \overset{x}{ˉ}} / O_{X, \overset{x}{ˉ}}^{*} ≅ N^{k} = N^{I (\overset{x}{ˉ})}$ . $□$

Proposition 2 (log smoothness of the standard semistable family). Let $f : X \to S = Spec C [[t]]$ have central fibre $X_{0} = ⋃_{i = 1}^{k} V_{i}$ a reduced simple normal-crossings divisor, with $X$ regular and $t$ vanishing to order one along each $V_{i}$ . Equip $X$ with the divisorial log structure of $X_{0}$ and $S$ with the standard log structure at $t = 0$ . Then $f$ is log smooth, and $Ω_{X / S}^{1} (lo g)$ is locally free of rank $dim X - 1$ .

Proof. Work étale-locally at a point of $X_{0}$ lying on $r$ of the components, with coordinates so that $X_{0} = {x_{1} \dots x_{r} = 0}$ and $t = x_{1} \dots x_{r}$ (order one along each by hypothesis). Take the base chart $Q = N$ , $1 \mapsto t$ , and the total chart $P = N^{r}$ , generators $\mapsto x_{1}, \dots, x_{r}$ , with $θ : Q \to P$ , $1 \mapsto (1, \dots, 1)$ . Then $θ^{gp} : Z \to Z^{r}$ is the diagonal, with zero kernel and cokernel $Z^{r} / ⟨(1, \dots, 1)⟩ ≅ Z^{r - 1}$ , torsion-free; condition (a) of Kato's criterion holds. The monomial model is $Spec Z [N^{r}] \to Spec Z [N]$ , $(x_{1}, \dots, x_{r}) \mapsto x_{1} \dots x_{r}$ , and the induced map from $X$ to $S \times_{A^{1}} A^{r}$ is an isomorphism onto a smooth chart (the remaining coordinates $x_{r + 1}, \dots, x_{n}$ are free), hence classically smooth; condition (b) holds. By the structure theorem $f$ is log smooth with $Ω_{X / S}^{1} (lo g)$ locally free of rank $rk (Z^{r - 1}) + (n - r) = n - 1 = dim X - 1$ , with frame $d lo g x_{1}, \dots, d lo g x_{r}$ (subject to the single relation $\sum_{i} d lo g x_{i} = d lo g t = 0$ over $S$ ) together with $d x_{r + 1}, \dots, d x_{n}$ . $□$

Proposition 3 (log differentials detect the log canonical). With $f$ as in Proposition 2 and $X$ of relative dimension $n - 1$ over $S$ , the relative log canonical sheaf $ω_{X / S}^{l o g} = det Ω_{X / S}^{1} (lo g)$ is invertible, and the relative Calabi-Yau condition $ω_{X / S}^{l o g} ≅ O_{X}$ is equivalent to the existence of a nowhere-vanishing global log $(n - 1)$ -form.

Proof. By Proposition 2, $Ω_{X / S}^{1} (lo g)$ is locally free of rank $n - 1$ , so its top exterior power $ω_{X / S}^{l o g}$ is an invertible sheaf. A nowhere-vanishing global section of an invertible sheaf $L$ on $X$ exists iff $L ≅ O_{X}$ (the section gives an isomorphism $O_{X} \sim L$ , and conversely the image of $1$ under such an isomorphism is nowhere-vanishing). In the local chart of Proposition 2 a generator of $ω_{X / S}^{l o g}$ is $d lo g x_{1} \land \dots \land d lo g x_{r - 1} \land d x_{r + 1} \land \dots \land d x_{n}$ (using the relation to eliminate $d lo g x_{r}$ ), which is the local log volume form; it is nowhere-vanishing in the chart. Globally such a form glues to a nowhere-vanishing section exactly when the transition functions of $ω_{X / S}^{l o g}$ are coboundaries, i.e. when $ω_{X / S}^{l o g} ≅ O_{X}$ . This is the log Calabi-Yau condition of 04.12.07(v) for the family. $□$

Connections Master

Toric degeneration of a Calabi-Yau 04.12.07. This unit supplies the apparatus that 04.12.07 invokes but does not define: the divisorial log structure on the central fibre $X_{0} \subseteq X$ , the statement that $X \to Spec C [[t]]$ is log smooth over the standard log point, and the relative log canonical condition $ω_{X / S}^{l o g} ≅ O_{X}$ that encodes the Calabi-Yau property of the family. The ghost sheaf $\overline{M}_{X_{0}}$ of the central fibre is the sheaf-theoretic carrier of the integral affine structure on the dual intersection complex, with its rank jumping along the toric stratification exactly as the cells of the polyhedral decomposition $P$ prescribe. Without the log smoothness established here, the broken central fibre of 04.12.07 would be an ordinary singularity rather than a fibre of a smooth-behaving family.
Fan and the toric variety $X_{Σ}$ 04.11.04. The toric log structure is built directly from the monoid $S_{σ} = σ^{\lor} \cap M$ of a rational polyhedral cone $σ$ of 04.11.04: the affine toric variety $X_{σ} = Spec C [S_{σ}]$ carries the log structure associated to the chart $S_{σ} \to C [S_{σ}]$ , $m \mapsto χ^{m}$ , with ghost monoid $S_{σ} / S_{σ}^{*}$ . The fine saturated condition on a log structure is precisely the condition that its ghost stalks are lattice-point monoids of genuine rational cones, so the toric dictionary of 04.11.04 transfers verbatim. Log geometry is in this sense the globalisation of toric geometry from individual cones to spaces glued from toric charts, with the gluing data recorded by the log structure.
Canonical sheaf 04.08.02. The log canonical sheaf $ω_{X}^{l o g} = det Ω_{X}^{1} (lo g D)$ is the log-twisted analogue of the canonical sheaf of 04.08.02: for a regular variety with normal-crossings divisor $D$ it equals $ω_{X} (D)$ , realising the adjunction divisor $K_{X} + D$ via the log residue sequence $0 \to Ω_{X}^{1} \to Ω_{X}^{1} (lo g D) \to O_{D} \to 0$ . A log Calabi-Yau pair is one with $K_{X} + D \sim 0$ , the log analogue of the structure-sheaf canonical-bundle condition that defines a Calabi-Yau via 04.08.02. The log canonical sheaf is therefore the bridge that lets the Calabi-Yau condition survive degeneration to a singular central fibre, where the plain canonical sheaf of 04.08.02 ceases to be invertible.
Log Gromov-Witten invariants (pointer) 04.12.15. The stable log maps counted in 04.12.15 are maps from log smooth curves that are log smooth over the standard log point of this unit; the fine saturated log structures, log smooth morphisms, and log differentials defined here are the literal foundations the log GW package presupposes. The contact-order data at marked points is read from the ghost sheaf $\overline{M}_{X}$ , and the degeneration formula is an application of the log smoothness of the toric degeneration established here. This unit is the home that 04.12.15 points back to for its Kato-apparatus prerequisites.

Historical & philosophical context Master

Logarithmic geometry originated in $p$ -adic Hodge theory. The idea is due to Fontaine and Illusie in the 1980s and was developed into a systematic theory by Kato in the foundational paper ^{[Kato 1989]}. The motivating problem was to extend the comparison theorems between étale and de Rham cohomology — known for smooth proper varieties with good reduction — to the case of semistable reduction, where the special fibre acquires normal crossings. Fontaine and Illusie recognised that the right way to handle the boundary was not to remove it or resolve it but to remember it as extra structure: a sheaf of monoids on the space recording the functions that vanish along the boundary. Kato axiomatised this as the log structure $α : M_{X} \to O_{X}$ and proved the structure theorem for log smooth morphisms, giving the chart criterion that turns log smoothness into smoothness relative to a monomial model.

The conceptual move is striking: a singular space, when equipped with the memory of where and how it is singular, behaves homologically and deformation-theoretically like a smooth one. This realised an old intuition — present already in the toroidal embeddings of Kempf-Knudsen-Mumford-Saint-Donat — that the combinatorics of the boundary, encoded in monoids and cones, carries the geometry of the degeneration. Kato's framework made the intuition into a category with limits, base change, and a cotangent complex.

The migration of log geometry into mirror symmetry came through Gross and Siebert. Their programme reformulated the Strominger-Yau-Zaslow picture algebraically, and the central fibre of a toric degeneration is most naturally a log Calabi-Yau space rather than an ordinary Calabi-Yau variety. The 2006 paper introduced the divisorial and central-fibre log structures, and the 2010 sequel constructed the log structure intrinsically via Kato fans, freeing the construction from any ambient smoothing. The Kato-Nakayama realisation later gave the topological bridge to the torus fibration, closing the loop between the algebraic log structure and the geometric SYZ picture. What began as a device for comparison theorems in arithmetic geometry became the language in which the combinatorial heart of mirror symmetry is written, with the ghost sheaf serving as the common record of arithmetic boundary data and tropical affine structure.

Bibliography Master

@incollection{Kato1989,
  author = {Kato, Kazuya},
  title = {Logarithmic structures of {F}ontaine-{I}llusie},
  booktitle = {Algebraic Analysis, Geometry, and Number Theory},
  editor = {Igusa, Jun-Ichi},
  publisher = {Johns Hopkins University Press},
  year = {1989},
  pages = {191--224},
}

@book{Ogus2018,
  author = {Ogus, Arthur},
  title = {Lectures on Logarithmic Algebraic Geometry},
  series = {Cambridge Studies in Advanced Mathematics},
  volume = {178},
  publisher = {Cambridge University Press},
  year = {2018},
}

@article{GrossSiebert2006,
  author = {Gross, Mark and Siebert, Bernd},
  title = {Mirror symmetry via logarithmic degeneration data {I}},
  journal = {Journal of Differential Geometry},
  volume = {72},
  number = {2},
  year = {2006},
  pages = {169--338},
}

@book{Gross2011CBMS,
  author = {Gross, Mark},
  title = {Tropical Geometry and Mirror Symmetry},
  series = {CBMS Regional Conference Series in Mathematics},
  volume = {114},
  publisher = {American Mathematical Society},
  year = {2011},
}

@article{GrossSiebert2010,
  author = {Gross, Mark and Siebert, Bernd},
  title = {Mirror symmetry via logarithmic degeneration data {II}},
  journal = {Journal of Algebraic Geometry},
  volume = {19},
  number = {4},
  year = {2010},
  pages = {679--780},
}

@article{FKato1996,
  author = {Kato, Fumiharu},
  title = {Log smooth deformation theory},
  journal = {Tohoku Mathematical Journal},
  volume = {48},
  number = {3},
  year = {1996},
  pages = {317--354},
}

@article{KatoNakayama1999,
  author = {Kato, Kazuya and Nakayama, Chikara},
  title = {Log {B}etti cohomology, log \'etale cohomology, and log de {R}ham cohomology of log schemes over {C}},
  journal = {Kodai Mathematical Journal},
  volume = {22},
  number = {2},
  year = {1999},
  pages = {161--186},
}

@article{Abramovich2014,
  author = {Abramovich, Dan},
  title = {Logarithmic geometry and moduli},
  journal = {Bulletin of the American Mathematical Society},
  volume = {51},
  number = {4},
  year = {2014},
  pages = {631--647},
}

@article{NakayamaOgus2010,
  author = {Nakayama, Chikara and Ogus, Arthur},
  title = {Relative rounding in toric and logarithmic geometry},
  journal = {Geometry \& Topology},
  volume = {14},
  number = {4},
  year = {2010},
  pages = {2189--2241},
}

@book{KKMS1973,
  author = {Kempf, George and Knudsen, Finn and Mumford, David and Saint-Donat, Bernard},
  title = {Toroidal Embeddings {I}},
  series = {Lecture Notes in Mathematics},
  volume = {339},
  publisher = {Springer},
  year = {1973},
}

Prerequisites

04.12.07
04.08.02
04.11.04

Tier anchors

beginner: Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lecture 3 informal opening; Abramovich 2014 *Bull. Amer. Math. Soc.* 51 *Logarithmic geometry and moduli* §1 (three-page informal survey of why one remembers the boundary as bookkeeping)
intermediate: Kato 1989 *Logarithmic structures of Fontaine-Illusie* §1-§3 (definition, charts, fs log structures, log smoothness); Gross 2011 *Tropical Geometry and Mirror Symmetry* (CBMS 114) Lecture 3 §3.1-§3.2; Ogus 2018 *Lectures on Logarithmic Algebraic Geometry* (Cambridge Studies 178) Ch. II-III (charts, fs, ghost sheaf), Ch. IV (log smooth/log étale)
master: Kato 1989 *Logarithmic structures of Fontaine-Illusie* (originator: the log structure $\alpha:M_X\to\mathcal{O}_X$, charts, fs log schemes, log smooth/log étale via the infinitesimal-lifting criterion, log differentials $\Omega^1_{X/S}(\log)$); Fontaine-Illusie 1980s (unpublished originating idea, $p$-adic Hodge theory); Kato 1996 *Fontaine-Illusie* (Trieste lectures) and Kato-Nakayama 1999 *Kodai Math. J.* 22 (Betti realisation); Ogus 2018 *Lectures on Logarithmic Algebraic Geometry* (canonical modern textbook); Gross-Siebert 2006 *J. Differential Geom.* 72 §1-§2 (divisorial and central-fibre log structures on toric degenerations); Gross-Siebert 2010 *J. Algebraic Geom.* 19 (intrinsic log structures, Kato fans); F. Kato 2000 *Math. Z.* 233 (log smooth deformation theory); Abramovich-Chen-Gillam-Huang-Olsson-Satriano-Sun 2013 *Logarithmic geometry and moduli* (Handbook of Moduli) survey

References

Kato, K. — Logarithmic structures of Fontaine-Illusie · In *Algebraic Analysis, Geometry, and Number Theory* (J.-I. Igusa, ed.), Johns Hopkins University Press (1989), 191-224. The originator paper: the log structure $\alpha:M_X\to\mathcal{O}_X$ as a sheaf of monoids with a multiplicative map that is an isomorphism over $\mathcal{O}_X^*$ (§1); charts and the associated/integral/fine/saturated conditions (§2); log smooth and log étale morphisms via the infinitesimal-lifting criterion and Kato's chart criterion (§3); log differentials $\Omega^1_{X/S}$ and the structure theorem for log smooth morphisms (§3).
Ogus, A. — Lectures on Logarithmic Algebraic Geometry · Cambridge Studies in Advanced Mathematics 178, Cambridge University Press (2018). Ch. I (monoids and monoid schemes); Ch. II (log structures, charts, the ghost sheaf $\overline{M}_X = M_X/\mathcal{O}_X^*$); Ch. III (fine, saturated, fs log schemes; coherence); Ch. IV (log smooth, log étale, log differentials, the Kato chart criterion). The canonical modern textbook reference.
Gross, M. & Siebert, B. — Mirror symmetry via logarithmic degeneration data I · *Journal of Differential Geometry* 72 (2), 169-338 (2006). §1 (log Calabi-Yau spaces; the divisorial and central-fibre log structures); §2 (the toric degeneration as a log smooth morphism to the standard log point, dual intersection complex). The application that makes the Kato apparatus indispensable to the Gross-Siebert programme.
Gross, M. — Tropical Geometry and Mirror Symmetry · CBMS Regional Conference Series in Mathematics 114, American Mathematical Society (2011). Lecture 3 (log structures, log smoothness, the central-fibre log structure of a toric degeneration; the standard log point). The textbook exposition oriented toward mirror symmetry.
Gross, M. & Siebert, B. — Mirror symmetry via logarithmic degeneration data II · *Journal of Algebraic Geometry* 19 (4), 679-780 (2010). Intrinsic log structures on the central fibre via Kato fans; the unobstructed-smoothing criterion; the $[B,\mathscr{P}]$ data. Where the abstract Kato log structure becomes the combinatorial bookkeeping of the integral affine manifold.
Kato, F. — Log smooth deformation theory · *Tohoku Mathematical Journal* 48 (3), 317-354 (1996); see also F. Kato, *Math. Z.* 233 (2000), 769-781. Log smooth deformation theory: the log smooth morphism controls deformations of normal-crossings degenerations, with the obstruction living in log Ext groups; the technical engine behind smoothing the central fibre of a toric degeneration.
Kato, K. & Nakayama, C. — Log Betti cohomology, log etale cohomology, and log de Rham cohomology of log schemes over C · *Kodai Mathematical Journal* 22 (2), 161-186 (1999). The Kato-Nakayama space: the topological realisation $X^{\log}$ of a log scheme over $\mathbb{C}$, an oriented-real-blowup that turns the boundary into a torus bundle and makes the ghost sheaf visible as monodromy. The bridge from log structures to the SYZ torus fibration.
Abramovich, D. — Logarithmic geometry and moduli · *Bulletin of the American Mathematical Society* 51 (4), 631-647 (2014), survey; and Abramovich-Chen-Gillam-Huang-Olsson-Satriano-Sun, *Logarithmic geometry and moduli*, in *Handbook of Moduli I*, Adv. Lect. Math. 24, International Press (2013), 1-61. Modern survey of log geometry oriented to moduli and Gromov-Witten theory; the accessible entry point.
Nakayama, C. & Ogus, A. — Relative rounding in toric and logarithmic geometry · *Geometry & Topology* 14 (4), 2189-2241 (2010). The relative Kato-Nakayama construction and its toric model; the log smooth morphism becomes a topological fibration after rounding, which is the precise sense in which a normal-crossings degeneration behaves smoothly.

Estimated time

beginner: 25m
intermediate: 60m
master: 110m