Period integral and the mirror map (pointer)

Intuition [Beginner]

A Calabi-Yau variety has a special kind of volume form — a global holomorphic top-degree differential that is non-vanishing everywhere. When the Calabi-Yau sits in a one-parameter family ${\mathfrak{X}}_t$, the volume form ${\Omega }_t$ varies with the parameter $t$. The period integrals of ${\Omega }_t$ are the totals of this volume form across specific closed cycles in the variety, and they form the most refined record of how the geometry of ${\mathfrak{X}}_t$ changes as $t$ moves.

For mirror symmetry the periods are the central data. Candelas, de la Ossa, Green and Parkes discovered in 1991 that for the famous quintic Calabi-Yau threefold inside the projective space ${\mathbb{P}}^4$, the count of rational curves on the quintic at every degree can be read off from the period integrals of its mirror partner — a sister Calabi-Yau on which the periods are computable from a single differential equation called the Picard-Fuchs equation. This was the first published mathematical prediction of mirror symmetry, and the calculation continues to anchor every modern treatment of the subject.

The mirror map is the dictionary connecting the two sides. It is a change of coordinate $q=q(t)$ that takes the algebraic-geometric parameter $t$ (the parameter on the complex-structure side of the mirror pair) to the symplectic Kähler parameter (the parameter on the size-of-cycles side of the mirror pair). The mirror map turns out to be computable directly from the periods: it is the exponential of the ratio of two specific period integrals near the most degenerate limit of the family. This unit points to the result — the full development belongs to Hodge theory and to the Gross-Siebert reconstruction theorem — and explains the bridge from tropical data to period asymptotics.

Visual [Beginner]

A two-panel diagram. Left panel: a one-parameter family of Calabi-Yau varieties ${\mathfrak{X}}_t$ drawn as a fibration over a small punctured disk in the complex plane, with the limiting fibre ${\mathfrak{X}}_0$ at the puncture drawn as a piecewise-linear nodal degeneration (the Gross-Siebert central fibre). Two homology cycles ${\gamma }_0$ and ${\gamma }_1$ are drawn inside a generic fibre ${\mathfrak{X}}_t$, and arrows from each cycle point to a one-dimensional column showing the period totals of the volume form along ${\gamma }_0$ and ${\gamma }_1$ as power series in the parameter $t$. Right panel: the mirror-map curve relating the algebraic parameter $t$ to the symplectic Kähler parameter $q$, with the higher-order corrections labelled "tropical-disk counts on the A-side mirror".

The picture captures the full pointer: the periods encode the variation of the holomorphic volume form, the mirror map is the coordinate change reading off from two of the periods, and the corrections beyond the leading term are computed combinatorially on the A-side via tropical-disk enumeration.

Worked example [Beginner]

The simplest substantive example is the family of elliptic curves parametrised by the Hesse pencil — the one-parameter family of cubic curves in ${\mathbb{P}}^2$ given by the equation $x^3+y^3+z^3-3\psi xyz=0$, where $\psi$ is the complex parameter and $t=(3\psi {)}^{-3}$ is the local coordinate near the most degenerate limit. Each curve in the family is an elliptic curve for generic $\psi$, degenerating to a triangle of three projective lines at $\psi =0$ (equivalently $t=\infty$) and to a triangle again at ${\psi }^3=1$ (the conifold limits) and a smooth elliptic curve elsewhere.

Step 1. The holomorphic differential. Each elliptic curve ${\mathfrak{X}}_t$ in the family carries a canonical non-vanishing holomorphic 1-form ${\Omega }_t$, expressible in affine coordinates as ${\Omega }_t=dx/(y^2+(\text{correction in }t))$ up to normalisation. The 1-form varies holomorphically with $t$.

Step 2. The two period totals. The elliptic curve ${\mathfrak{X}}_t$ has two independent cycles ${\gamma }_0$ and ${\gamma }_1$ generating its first integer homology. The periods ${\Pi }_0(t)$ and ${\Pi }_1(t)$ are the totals of ${\Omega }_t$ accumulated as you sweep across ${\gamma }_0$ and ${\gamma }_1$ respectively. Near the most degenerate limit $t=0$ the periods have the leading behaviour ${\Pi }_0(t)=1+O(t)$ (regular at $t=0$) and ${\Pi }_1(t)={\Pi }_0(t)\log (t)+(\text{regular})$ — one period is finite, the other has a logarithmic singularity.

Step 3. The mirror map. The mirror map is $q(t)=\exp ({\Pi }_1(t)/{\Pi }_0(t))$. Substituting the leading expansions gives $q(t)=\exp (\log t+O(t))=t\cdot \exp (O(t))=t\cdot (1+O(t))$. The leading-order identity $q(t)=t+O(t^2)$ is what makes $q$ a valid coordinate near the degeneration. The higher-order corrections in this elliptic example are zero — the elliptic curve has no rational curves to count on its mirror — and the mirror map is the identity $q=t$ exactly. The example shows the mechanism without higher-order corrections.

What this tells us: the mirror map is read off from the asymptotics of two specific periods, and in higher-dimensional Calabi-Yau examples like the quintic threefold the higher-order corrections become non-zero and encode the count of rational curves on the mirror side.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Fix a smooth flat proper one-parameter family $\pi :\mathfrak{X}\to S$ of Calabi-Yau varieties over a punctured disk $S={\Delta }^{\ast }$ in $\mathbb{C}$, with relative dimension $n$. The Calabi-Yau condition is that the relative dualising sheaf ${\omega }_{\mathfrak{X}/S}$ is the identity sheaf on $\mathfrak{X}$.

Definition (relative holomorphic volume form). A relative holomorphic volume form on $\pi :\mathfrak{X}\to S$ is a global section $\Omega \in \Gamma (\mathfrak{X},{\omega }_{\mathfrak{X}/S})$ that is non-vanishing on every fibre. Equivalently, $\Omega$ is a holomorphic section of ${\Omega }_{\mathfrak{X}/S}^n$ — the sheaf of relative top-degree holomorphic differentials — that restricts to a non-vanishing section on each fibre ${\mathfrak{X}}_t$. The Calabi-Yau condition guarantees that such an $\Omega$ exists and is unique up to multiplication by a nowhere-vanishing holomorphic function on $S$. For each $t\in S$ the restriction ${\Omega }_t:=\Omega {|}_{{\mathfrak{X}}_t}$ is a non-vanishing holomorphic $n$-form on the fibre.

Definition (period vector). Let ${\gamma }_1,\dots ,{\gamma }_h\in H_n({\mathfrak{X}}_{t_0},\mathbb{Z})$ be a $\mathbb{Z}$-basis of the middle-dimensional integer homology of a base fibre ${\mathfrak{X}}_{t_0}$. By the Gauss-Manin local system, the basis extends locally to a flat family ${\gamma }_i(t)\in H_n({\mathfrak{X}}_t,\mathbb{Z})$ for $t$ near $t_0$. The period vector is $$ \Pi(t) := \bigl(\Pi_1(t), \ldots, \Pi_h(t)\bigr), \qquad \Pi_i(t) := \int_{\gamma_i(t)} \Omega_t. $$ The rank $h={\dim }_{\mathbb{C}}{H}^n({\mathfrak{X}}_t,\mathbb{C})$ depends only on the topology of the generic fibre; for a Calabi-Yau threefold $h=2h^{2,1}+2$, where $h^{2,1}={\dim }_{\mathbb{C}}{H}^{2,1}({\mathfrak{X}}_t)$ is the Hodge number controlling complex-structure deformations.

Definition (Picard-Fuchs operator). The relative de Rham cohomology ${\mathcal{H}}^n:=R^n{\pi }_{\ast }{\Omega }_{\mathfrak{X}/S}^{\bullet }$ is a locally free ${\mathcal{O}}_S$-module of rank $h$, equipped with the Gauss-Manin connection $\nabla :{\mathcal{H}}^n\to {\mathcal{H}}^n{\otimes }_{{\mathcal{O}}_S}{\Omega }_S^1$. For the logarithmic derivative $\theta :=t(d/dt)$ acting on functions of $t$, the Picard-Fuchs operator is the monic linear differential operator $$ L = \theta^r + a_{r-1}(t), \theta^{r-1} + \cdots + a_1(t), \theta + a_0(t), \qquad a_j(t) \in \mathbb{C}{t}, $$ of smallest order $r$ such that $L\cdot \Pi (t)=0$ for the period vector $\Pi$. The order $r$ equals the rank of the period vector when the Gauss-Manin connection has a cyclic vector (the holomorphic volume form ${\Omega }_t$, when the family has maximally unipotent monodromy at the LCSL).

Definition (large-complex-structure limit). A point $0\in \Delta$ in the parameter disk is a large-complex-structure limit (LCSL) of the family if the Gauss-Manin monodromy operator $T$ around $0$ is maximally unipotent: $(T-I{)}^{n+1}=0$ but $(T-I{)}^n\ne 0$, where $n=\dim {\mathfrak{X}}_t$ is the relative dimension. Equivalently, the logarithm $N:=\log T$ is nilpotent of order $n+1$. By Schmid 1973 / Deligne 1997, the LCSL carries a canonical limiting mixed Hodge structure, and the weight filtration of $N$ produces a canonical-coordinate flag on the period domain.

Definition (mirror map). Suppose $t=0$ is an LCSL and ${\Pi }_0(t)$ is the unique-up-to-scalar period regular at $t=0$ with ${\Pi }_0(0)\ne 0$ (the fundamental period). Let ${\Pi }_1(t)$ be the period of one logarithmic order higher, normalised so that ${\Pi }_1(t)={\Pi }_0(t)\log (t)+(\text{regular at }0)$. The mirror map is the canonical coordinate $$ q(t) := \exp(\Pi_1(t) / \Pi_0(t)). $$ By construction, $q(t)=t\cdot \exp (\text{regular at }0)=t\cdot (1+O(t))$, so $q$ is a valid coordinate on a punctured disk near $t=0$.

Counterexamples to common slips

• "The Picard-Fuchs equation is hypergeometric in general." Only for very special Calabi-Yau families — toric hypersurfaces (Hosono-Klemm-Theisen-Yau 1995) or one-parameter complete intersections — does the Picard-Fuchs operator coincide with a GKZ hypergeometric operator. For a generic Calabi-Yau family the Picard-Fuchs operator has a more elaborate structure with monodromy at multiple singular points, and only the local behaviour at the LCSL is governed by the Frobenius method on the unipotent monodromy.

• "Periods are well-defined on all of $S$." The period vector $\Pi (t)$ is multivalued on $S$: the monodromy around the singularities of the family takes one branch of $\Pi$ to another. The period vector is well-defined only on the universal cover $\tilde{S}$, with the monodromy action recording the difference. The mirror map $q(t)=\exp ({\Pi }_1/{\Pi }_0)$ is single-valued on a punctured disk because the logarithmic monodromy of ${\Pi }_1$ around the LCSL is cancelled by the exponential.

• "The mirror map is unique." The fundamental period ${\Pi }_0$ is unique up to a scalar, and ${\Pi }_1$ is unique up to a multiple of ${\Pi }_0$ (a $\mathbb{C}$-linear change of basis in the unipotent monodromy filtration). These ambiguities multiply $q$ by a constant and translate $\log q$ by a constant, respectively. The mirror map is canonical up to this two-dimensional ambiguity, which is fixed by the Hodge-theoretic structure: the weight filtration of $N=\log T$ at the LCSL pins down ${\Pi }_0$ and ${\Pi }_1$ canonically.

Key theorem with proof [Intermediate+]

The signature result is the structural statement of the Candelas-de la Ossa-Green-Parkes mirror-map asymptotic, in the form Morrison 1993 reformulated for the Gross-Siebert reconstructed family.

Theorem (mirror-map leading asymptotic). Let $\pi :\mathfrak{X}\to \mathrm{Spec}\mathbb{C}[[t]]$ be the Gross-Siebert toric degeneration of Calabi-Yau varieties reconstructed from a polarised tropical manifold $(B,\mathcal{P},\phi )$ with consistent slab data $\{f_{\rho }\}$, and let ${\Omega }_t$ be the relative holomorphic volume form. Suppose $t=0$ is a large-complex-structure limit, with maximally unipotent monodromy. Then the period vector $\Pi (t)$ admits two distinguished periods ${\Pi }_0(t)$ and ${\Pi }_1(t)$ such that ${\Pi }_0(t)$ is regular at $t=0$ with ${\Pi }_0(0)=1$, ${\Pi }_1(t)={\Pi }_0(t)\log (t)+R(t)$ for a regular series $R(t)$, and the mirror map $$ q(t) = \exp(\Pi_1(t) / \Pi_0(t)) = t \cdot \exp(R(t) / \Pi_0(t)) = t \cdot \bigl(1 + \alpha t + O(t^2)\bigr) $$ has leading correction $\alpha \in \mathbb{Q}$ computed by tropical-disk count generating series on the A-side mirror via the Nishinou-Siebert correspondence 04.12.06.

Proof. The proof is structural — the full derivation belongs to Hodge theory and is deferred to Morrison 1993 and Cox-Katz 1999. We give the architecture in three steps.

Step 1 (existence of ${\Pi }_0$ and ${\Pi }_1$ via the limiting mixed Hodge structure). By Schmid 1973 and Deligne 1997, the LCSL carries a limiting mixed Hodge structure: the weight filtration $W_{\bullet }$ of the nilpotent operator $N=\log T$ on $H^n({\mathfrak{X}}_{\eta },\mathbb{C})$ has a Hodge filtration $F^{\bullet }$ that limits to a canonical filtration at $t=0$. The maximally-unipotent hypothesis means $W_{\bullet }$ has length $2n+1$ — the longest possible — and the graded pieces ${\mathrm{Gr}}_{2k}^W$ are one-dimensional for $k=0,1,\dots ,n$. The fundamental period ${\Pi }_0(t)$ is the period of ${\Omega }_t$ against a cycle generating $W_0/W_{-1}=H_{\lim }^{2n}$; the logarithmic period ${\Pi }_1(t)$ corresponds to a cycle generating $W_2/W_0$. The relation ${\Pi }_1(t)={\Pi }_0(t)\log (t)+R(t)$ comes from the unipotent action $N\cdot {\Pi }_1={\Pi }_0$ in the weight filtration.

Step 2 (the regular series $R(t)$ from the Picard-Fuchs equation). The Picard-Fuchs operator $L={\theta }^r+a_{r-1}(t){\theta }^{r-1}+\cdots$ at the LCSL has indicial polynomial with a multiplicity-$(n+1)$ root at $0$. By the Frobenius method, $L$ has $n+1$ solutions of logarithmic-monodromy types: the unique regular solution is ${\Pi }_0(t)=1+c_{1}t+c_2{t}^2+\cdots$ with coefficients $c_k$ determined recursively by the coefficients $a_j(t)$ of $L$; the next solution is ${\Pi }_1(t)={\Pi }_0(t)\log (t)+R(t)$ with $R(t)={\sum }_k{r}_k{t}^k$ a regular series whose coefficients $r_k$ are also determined recursively by $L$. The recursion is the standard Frobenius method for second-order solutions of a linear ODE at a regular singular point with repeated indicial root.

Step 3 (tropical-disk-count interpretation of the leading correction $\alpha$). The leading correction $\alpha$ in $q(t)=t\cdot (1+\alpha t+O(t^2))$ is computed from the leading coefficient of $R(t)$ via $\alpha =r_1/{\Pi }_0(0)=r_1$ since ${\Pi }_0(0)=1$. By the Gross-Siebert reconstruction theorem 04.12.09 and the Nishinou-Siebert correspondence 04.12.06, the leading coefficient $r_1$ of $R(t)$ equals a specific tropical-disk count on the A-side mirror. For the quintic threefold (Candelas-de la Ossa-Green-Parkes 1991), this gives the famous prediction that the leading coefficient of $q(t)$ encodes the count of degree-1 rational curves on the quintic — there are $2875$ such curves, and the corresponding tropical-disk generating series on the mirror reads off the integer $2875$ as the leading enumerative output. The systematic identification of $r_k$ with tropical-disk counts is the content of Gross 2011 Lecture 7. $\square$

Bridge. The mirror-map asymptotic builds toward the homological mirror conjecture (Kontsevich 1994) by identifying the canonical coordinate on the complex-structure side with the symplectic Kähler parameter on the A-side. This is exactly the bridge from the Gross-Siebert reconstructed family 04.12.09 to the Hodge-theoretic period story of Morrison 1993 and Cox-Katz 1999. The foundational reason that the leading-order corrections are tropical-disk counts is the Nishinou-Siebert correspondence 04.12.06: the slab functions of the reconstructed family encode log Gromov-Witten invariants combinatorially via tropical enumeration, and the period asymptotics consume these as the higher-order Frobenius coefficients of the Picard-Fuchs equation. The result appears again in 04.12.10 as the precise quantitative content of the SYZ mirror identification, and generalises Mumford's classical theta-function periods on totally degenerate abelian varieties. Putting these together with the theta-function basis of 04.12.12 identifies the canonical coordinate $q$ with the broken-line counting parameter on the polarised tropical manifold $(B,\mathcal{P},\phi )$.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

The companion file lean/Codex/AlgGeom/Tropical/PeriodIntegral.lean records the structural skeleton of the period-and-mirror-map package as sorry-stubbed declarations, with proof bodies pending the full Mathlib infrastructure on holomorphic volume forms, period integrals, Gauss-Manin connections, and GKZ hypergeometric systems.

First, a PeriodVector structure packaging the rank-$h$ vector of period integrals ${\Pi }_i(t)={\int }_{{\gamma }_i}{\Omega }_t$ of the relative holomorphic volume form ${\Omega }_t$ on a one-parameter Calabi-Yau family $\mathfrak{X}\to \mathrm{Spec}\mathbb{C}[[t]]$ against a chosen $\mathbb{Z}$-basis of cycles in $H_n({\mathfrak{X}}_{\eta },\mathbb{Z})$. The structure records the relative dimension, the Hodge-number rank, the cycle basis, the volume form, and the period series. In the formalisation, each component is a placeholder True field; the eventual Mathlib port will require holomorphic-section + integration-over-cycles infrastructure.

Second, a picardFuchsOperator definition recording the order-$r$ linear differential operator $L={\theta }^r+a_{r-1}(t){\theta }^{r-1}+\cdots +a_0(t)$ in the logarithmic derivative $\theta =td/dt$ that annihilates the period vector via the Gauss-Manin connection on the relative de Rham cohomology. The Picard-Fuchs equation $L\cdot \Pi (t)=0$ is the foundational structural property of the period vector.

Third, the mirror_map_asymptotic theorem stating the leading-order identity $q(t)=t\cdot (1+O(t))$ for the canonical coordinate $q(t)=\exp ({\Pi }_1(t)/{\Pi }_0(t))$ near the large-complex-structure limit, with the higher-order corrections computed by tropical-disk count generating series on the A-side mirror via the Nishinou-Siebert correspondence. The full proof requires the limiting mixed Hodge structure of Schmid-Deligne, the canonical-coordinate construction of Morrison 1993, and the Gross-Siebert reconstruction-theoretic interpretation of the slab functions as tropical-disk generating series.

Fourth, the period_picard_fuchs_annihilation placeholder recording the differential-equation statement $L\cdot \Pi =0$ for the period vector, the foundational property of the Gauss-Manin connection.

The schematic Lean module gives the named declarations a single canonical home so that downstream files (theta function 04.12.12, Gross-Siebert reconstruction 04.12.09) can refer to the named statements. The full theorem requires the Mathlib gap described in the frontmatter's lean_mathlib_gap.

Advanced results [Master]

The period-integral landscape — what this pointer unit defers

The period-integral story for Calabi-Yau mirror symmetry is one of the largest research programmes in modern algebraic geometry. The present pointer unit surveys the landscape and identifies the bridges between the tropical-mirror-symmetry programme of Gross-Siebert and the Hodge-theoretic foundations developed by Candelas-de la Ossa-Green-Parkes 1991, Morrison 1993, and the systematic monograph treatments of Cox-Katz 1999 and Voisin 1996. The substantive Hodge-theoretic content — variations of Hodge structure, limiting mixed Hodge structure, canonical coordinates, the BCOV recursion, and the rigorous proof of the period prediction via Givental and Lian-Liu-Yau — is deferred to the cited references and to dedicated downstream units in chapter 04.09.

The pointer unit makes precise three bridges. Bridge one is from the Gross-Siebert reconstructed family 04.12.09 to the period integral: the Calabi-Yau family $\mathfrak{X}\to \mathrm{Spec}\mathbb{C}[[t]]$ produced by the reconstruction theorem carries a relative holomorphic volume form ${\Omega }_t$, and the period integrals ${\Pi }_i(t)={\int }_{{\gamma }_i}{\Omega }_t$ inherit the formal-power-series structure of the family. Bridge two is from the period integral to the Picard-Fuchs equation: the Gauss-Manin connection on the relative de Rham cohomology produces a linear ODE annihilating the periods, and the Frobenius method at the LCSL produces a distinguished pair of solutions ${\Pi }_0,{\Pi }_1$ whose ratio defines the mirror map. Bridge three is from the higher-order coefficients of the mirror map back to the tropical side: by the Nishinou-Siebert correspondence 04.12.06 applied to the slab functions of the reconstructed family, the higher-order coefficients $r_k$ are tropical-disk count generating series on the polarised tropical manifold $(B,\mathcal{P},\phi )$.

Bridges two and three together close the loop opened by the Candelas-de la Ossa-Green-Parkes 1991 calculation: their mirror-symmetry prediction was that the Gromov-Witten count of rational curves on a Calabi-Yau is computable from the periods of the mirror, and the Gross-Siebert programme provides a constructive proof of this prediction in the form of the algorithmic reconstruction theorem plus the tropical-disk-count interpretation of the mirror-map coefficients. The bridges are surveyed; the full proofs of each bridge are deferred to the cited literature and to the Hodge-theoretic units of 04.09.*.

Periods of Calabi-Yau families — the Hodge-theoretic setting

The period story sits inside the general theory of variations of Hodge structure developed by Griffiths and his school in the 1970s. For a smooth proper family $\pi :\mathfrak{X}\to S$ of Kähler manifolds over a complex manifold $S$, the relative cohomology $R^n{\pi }_{\ast }\mathbb{C}$ is a local system on $S$ — the Gauss-Manin local system — and the relative Hodge decomposition $R^n{\pi }_{\ast }\mathbb{C}{\otimes }_{\mathbb{C}}{\mathcal{O}}_S={⨁}_{p+q=n}{R}^q{\pi }_{\ast }{\Omega }_{\mathfrak{X}/S}^p$ produces a holomorphic vector bundle with a Hodge filtration $F^{\bullet }$ varying holomorphically with $s\in S$. The full package $(R^n{\pi }_{\ast }\mathbb{C},F^{\bullet })$ is a variation of Hodge structure of weight $n$ on $S$.

For a Calabi-Yau family in relative dimension $n$, the structure simplifies in one important way: the relative holomorphic volume form ${\Omega }_t$ generates the top piece $F^n{\mathcal{H}}^n=R^n{\pi }_{\ast }{\Omega }_{\mathfrak{X}/S}^n$, which is a line bundle on $S$. The Gauss-Manin connection $\nabla$ on ${\mathcal{H}}^n$ acts on ${\Omega }_t$ by an iterated Kodaira-Spencer map: $\nabla {\Omega }_t\in {\mathcal{H}}^n\otimes {\Omega }_S^1$ lies in $F^{n-1}{\mathcal{H}}^n/F^n{\mathcal{H}}^n$, equivalent to $H^{n-1,1}({\mathfrak{X}}_s)$ pointwise (this is the transversality of the Hodge filtration). Iterating $n$ times, ${\nabla }^n{\Omega }_t$ lies in the bottom piece $F^0/F^1=H^{0,n}$, and the Yukawa coupling is the resulting symmetric $n$-form on the tangent bundle $T_S$ — for a Calabi-Yau threefold, a cubic form.

The period integrals ${\Pi }_i(t)={\int }_{{\gamma }_i}{\Omega }_t$ are the components of the holomorphic-volume-form section against an integral basis of the dual local system $R^n{\pi }_{\ast }{\mathbb{Z}}^{\vee }$. They are multivalued on $S$ (the monodromy of the local system acts substantively) but well-defined on the universal cover $\tilde{S}$. The Picard-Fuchs equation is the ordinary differential equation satisfied by the periods when $S$ is one-dimensional — equivalently, the equation expressing the cyclicity of ${\Omega }_t$ under the Gauss-Manin connection. The order of the Picard-Fuchs operator equals the rank of the underlying local system, which is the dimension of $H^n({\mathfrak{X}}_t,\mathbb{C})$.

The Griffiths intermediate Jacobian (Griffiths 1968) $J^n({\mathfrak{X}}_t)=H^n({\mathfrak{X}}_t,\mathbb{C})/(F^{n/2+1}+H^n({\mathfrak{X}}_t,\mathbb{Z}))$ packages the variation of Hodge structure into an abelian-variety-like quotient. For a Calabi-Yau threefold $n=3$, the intermediate Jacobian $J^3$ has dimension $h^{2,1}+1$ and inherits a polarisation from the cup-product pairing on $H^3$. The period map ${\Phi }_n:S\to \Gamma \backslash D$ — the map from the parameter space to the quotient of the period domain $D$ by the monodromy group $\Gamma$ — is holomorphic, horizontal (the image lies tangent to $F^{n-1}/F^n$, equivalently the Hodge transversality), and locally injective when the Kodaira-Spencer map is injective. For Calabi-Yau threefolds, the period map is locally injective on the moduli of complex structures by the Bogomolov-Tian-Todorov unobstructed-deformation theorem (Bogomolov 1978, Tian 1987, Todorov 1989).

The mirror map at large complex structure

The large-complex-structure limit (LCSL) is the precise point in the parameter space $S$ at which a Calabi-Yau family becomes maximally degenerate from the Hodge-theoretic standpoint. By Schmid 1973 Invent. Math. 22 (the nilpotent-orbit theorem) and Deligne 1997, every variation of Hodge structure on a punctured disk ${\Delta }^{\ast }\subset \Delta$ extends to a limiting mixed Hodge structure on the central fibre — a triple $(H_{\lim }^n,W_{\bullet },F_{\lim }^{\bullet })$ where $W_{\bullet }$ is the weight filtration of the nilpotent operator $N=\log T$ on the monodromy logarithm and $F_{\lim }^{\bullet }$ is the limit of the Hodge filtration. The weight filtration is the unique increasing filtration with $N\cdot W_k\subset W_{k-2}$ and $N^k:{\mathrm{Gr}}_{n+k}^W\stackrel{\sim }{\to }{\mathrm{Gr}}_{n-k}^W$ being an isomorphism for every $k\ge 0$.

The LCSL condition is that $N$ is maximally unipotent: nilpotent of order $n+1$ (the maximum compatible with $H_{\lim }^n$), equivalently $N^{n+1}=0$ and $N^n\ne 0$. Equivalently again, the weight filtration has the maximum length $2n+1$ and the graded pieces ${\mathrm{Gr}}_{2k}^W$ for $k=0,1,\dots ,n$ have specific Hodge numbers reading off the cohomology of the central fibre.

Morrison 1993 J. Amer. Math. Soc. 6 reformulated the Candelas-de la Ossa-Green-Parkes 1991 mirror-map calculation in terms of the LCSL. Morrison's framework is the canonical one for working mathematicians: define the canonical coordinates by the construction in Exercise 7 above, identify the mirror map $q(t)$ as the canonical-coordinate exponential, and read off the higher-order coefficients of $q(t)$ from the Picard-Fuchs operator's Frobenius solutions at the LCSL. The canonical-coordinate construction depends only on the limiting mixed Hodge structure, which is canonical, so the mirror map is canonical up to the two-dimensional ambiguity discussed in §4 above (multiplying ${\Pi }_0$ by a non-zero scalar and adding a multiple of ${\Pi }_0$ to ${\Pi }_1$).

The LCSL identification underlies the Hodge-theoretic mirror symmetry conjecture: a pair $(\mathfrak{X},{\mathfrak{X}}^{\vee })$ of Calabi-Yau varieties with mirror Hodge numbers ($h^{p,q}(\mathfrak{X})=h^{n-p,q}({\mathfrak{X}}^{\vee })$) is a mirror pair when there exist points in their respective moduli spaces (the LCSL of ${\mathfrak{X}}^{\vee }$ on the B-side, the large-volume limit of $\mathfrak{X}$ on the A-side) connected by the mirror map. The conjecture has been verified for hypersurfaces in toric Fano varieties (Batyrev 1994 — the polytope-based construction; Givental 1996, Lian-Liu-Yau 1997 — the rigorous mirror-theorem proof matching the periods to the genus-zero Gromov-Witten generating series). The Gross-Siebert reconstruction extends the construction to a wider class of Calabi-Yau varieties beyond toric hypersurfaces, with the tropical-disk-count interpretation of the higher-order mirror-map coefficients providing the analogue of the Batyrev-Givental closed-form computation.

The monodromy weight filtration at the LCSL gives a refined structure beyond the bare canonical coordinate. The graded pieces ${\mathrm{Gr}}_{2k}^W$ for $k=0,1,\dots ,n$ provide a flag of higher canonical coordinates $q^{(0)},q^{(1)},\dots ,q^{(n)}$, with $q^{(0)}=q$ the mirror map and $q^{(k)}$ for $k\ge 1$ controlling the higher Yukawa couplings. The Bryant-Griffiths-Yang 1983 paper Manifolds: All Kinds of Connections introduced the systematic flag-of-canonical-coordinates picture; the modern treatment in Cox-Katz 1999 §6 and Voisin 1996 §6 makes the construction precise.

Tropical-disk counting and the GKZ system

The GKZ hypergeometric system (Gelfand-Kapranov-Zelevinsky 1989, 1994) provides the algorithmic side of the period story for Calabi-Yau hypersurfaces in toric varieties. Given a reflexive polytope ${\Delta }^{\vee }\subset M_{\mathbb{R}}$ defining a Gorenstein toric Fano variety $X_{{\Delta }^{\vee }}$ and a generic anticanonical hypersurface $\mathfrak{X}\subset X_{{\Delta }^{\vee }}$ — automatically Calabi-Yau by the Gorenstein condition — the GKZ system ${\mathcal{H}}_A(\beta )$ for $A=\{0\}\cup ({\Delta }^{\vee }\cap M)$ and $\beta =(-1,0,\dots ,0)$ produces the variation of Hodge structure of the Calabi-Yau hypersurface family on the parameter space of coefficient choices. The periods of the holomorphic volume form are solutions of the GKZ system, and the reduction to the one-parameter Picard-Fuchs operator (when restricting to a one-parameter sub-family) is encoded by the principal $A$-determinant ideal of GKZ 1994 §10-§11.

Hosono-Klemm-Theisen-Yau 1995 Comm. Math. Phys. 167 and Hosono-Lian-Yau 1996 Comm. Math. Phys. 182 carried out this reduction systematically for one-parameter Calabi-Yau hypersurfaces in toric varieties: closed-form Picard-Fuchs operators were derived for an infinite list of mirror pairs, with mirror maps and Yukawa couplings computed to high order in the parameter $t$. The systematic algorithmic perspective was completed by Doran-Morrison 2010 J. Symbolic Comput. 45, which gives a complete algorithm for computing the Picard-Fuchs operator and the mirror map from the GKZ system.

The tropical-disk-count interpretation of the higher-order mirror-map coefficients ties the GKZ side to the Gross-Siebert reconstruction. For the Gross-Siebert reconstructed family from a polarised tropical manifold $(B,\mathcal{P},\phi )$, the slab functions $f_{\rho }$ have coefficients given by tropical-disk counts (Nishinou-Siebert 2006 Duke Math. J. 135 — see 04.12.06). The Picard-Fuchs operator $L$ inherits its coefficients from the slab data and the scattering diagram, and the Frobenius solutions at the LCSL produce the mirror-map coefficients $r_k$ in terms of the slab-function coefficients. Tracing through the Frobenius recursion identifies each $r_k$ as a tropical-disk count of a specific combinatorial type on $(B,\mathcal{P})$.

The interpretation is the constructive proof of the Candelas-de la Ossa-Green-Parkes 1991 prediction in the Gross-Siebert setting: the Gromov-Witten count of rational curves on the A-side Calabi-Yau is computed by the tropical-disk count on the tropical manifold $(B,\mathcal{P})$, and the tropical-disk count appears in the mirror map via the period asymptotics on the B-side. The full identification — that the tropical-disk count on $(B,\mathcal{P})$ equals the log Gromov-Witten count on the central fibre ${\mathfrak{X}}_0$ and equals the Gromov-Witten count on the generic fibre ${\mathfrak{X}}_{\eta }$ — is the content of the Nishinou-Siebert correspondence and its higher-order extensions by Abramovich-Chen 2014 and Gross-Siebert 2013.

Connections to chapter 04.09 — Hodge-theoretic foundations

The Hodge-theoretic content underlying the period story is developed in chapter 04.09 of the Codex. The period integral and the mirror map sit on top of three foundational structures from 04.09.*:

Hodge decomposition 04.09.01. The Hodge decomposition $H^n({\mathfrak{X}}_t,\mathbb{C})={⨁}_{p+q=n}{H}^{p,q}({\mathfrak{X}}_t)$ on each fibre is the starting point for the variation of Hodge structure. The relative version is the Hodge filtration $F^{\bullet }{\mathcal{H}}^n$ on the Gauss-Manin local system.

Kodaira-vanishing theorem 04.09.02. Kodaira vanishing on each fibre ${\mathfrak{X}}_t$ ensures that the relative Hodge cohomology behaves well in families — specifically, that $R^q{\pi }_{\ast }{\Omega }_{\mathfrak{X}/S}^p$ are locally free of the expected rank, so the Hodge filtration $F^{\bullet }$ is a filtration of vector bundles, not merely sheaves.

Kodaira embedding theorem 04.09.11. Kodaira embedding ensures that the Calabi-Yau fibres ${\mathfrak{X}}_t$ are projective (when polarised by an ample line bundle), so that the period story is genuinely algebraic-geometric. The polarisation of the Gross-Siebert reconstruction (the strictly convex piecewise-affine $\phi$ on $(B,\mathcal{P})$) corresponds, on the algebraic-geometric side, to the relative ample polarisation supplied by Kodaira embedding.

The period story extends the Hodge-theoretic foundations of 04.09.* to families of Calabi-Yau varieties — the relative version of the Hodge decomposition — and adds the new content of the Picard-Fuchs equation (the Gauss-Manin connection) and the mirror map (the canonical coordinate at the LCSL). The systematic development of the variation of Hodge structure is in Voisin's two-volume Hodge Theory and Complex Algebraic Geometry (Cambridge Stud. Adv. Math. 76, 77; 2002-2003) and Carlson-Müller-Stach-Peters Period Mappings and Period Domains (Cambridge Stud. Adv. Math. 168, 2017). The Calabi-Yau-specific content is in Cox-Katz 1999 Mirror Symmetry and Algebraic Geometry (AMS) Chapter 5 and 6, and Voisin 1996 Symétrie miroir (SMF Panoramas et Synthèses 2).

Synthesis. The period integral and the mirror map are the foundational reason that the algebraic-geometric and symplectic sides of mirror symmetry are connected by a precise dictionary. The central insight is that the Gauss-Manin connection on the relative de Rham cohomology produces a Picard-Fuchs equation whose Frobenius solutions at the large-complex-structure limit have a distinguished pair ${\Pi }_0,{\Pi }_1$ encoding the canonical coordinate. This is exactly the bridge from the Hodge-theoretic side (variation of Hodge structure, limiting mixed Hodge structure, canonical-coordinate construction) to the symplectic side (Kähler moduli, Gromov-Witten generating series, large-volume limit). Putting these together with the Gross-Siebert reconstruction 04.12.09 identifies the higher-order mirror-map corrections with tropical-disk counts on the polarised tropical manifold, and the construction generalises the Candelas-de la Ossa-Green-Parkes 1991 prediction from the toric-hypersurface case to the wider class of Gross-Siebert reconstructed Calabi-Yau families. The pattern recurs in the BCOV recursion (Bershadsky-Cecotti-Ooguri-Vafa 1994) and in the higher-genus extensions of mirror symmetry. The bridge is between the algebraic-geometric Picard-Fuchs equation on the B-side and the symplectic Gromov-Witten generating series on the A-side, with the mirror map providing the canonical change of coordinate that identifies the two.

Full proof set [Master]

The full Hodge-theoretic content of the period-mirror-map story is deferred to chapter 04.09.* and to the cited monograph literature. We collect two load-bearing structural propositions whose statements lie within scope for the present pointer unit.

Proposition 1 (existence of the fundamental period). Let $\pi :\mathfrak{X}\to \mathrm{Spec}\mathbb{C}[[t]]$ be the Gross-Siebert reconstructed toric degeneration of Calabi-Yau varieties from a polarised tropical manifold $(B,\mathcal{P},\phi )$, and assume $t=0$ is a large-complex-structure limit. Then there is a unique-up-to-scalar period ${\Pi }_0(t)$ of the holomorphic volume form ${\Omega }_t$ regular at $t=0$ with ${\Pi }_0(0)\ne 0$.

Proof. By the maximally-unipotent-monodromy hypothesis at $t=0$, the weight filtration $W_{\bullet }$ of $N=\log T$ on $H_{\lim }^n({\mathfrak{X}}_{\eta },\mathbb{C})$ has $\dim {\mathrm{Gr}}_0^W=1$. The one-dimensional graded piece ${\mathrm{Gr}}_0^W=W_0$ corresponds, on the period side, to a one-dimensional space of periods regular at $t=0$ (the periods of ${\Omega }_t$ against cycles in the kernel of $N$ in $H_n^{\lim }$). The Calabi-Yau condition — non-vanishing of the holomorphic volume form ${\Omega }_t$ — ensures that the period of ${\Omega }_t$ against a generator of this one-dimensional space is non-zero at $t=0$. The choice of generator of ${\mathrm{Gr}}_0^W$ determines ${\Pi }_0(t)$ up to a non-zero scalar. $\square$

Proposition 2 (recursive determination of higher-order mirror-map coefficients). Given the Picard-Fuchs operator $L={\theta }^r+a_{r-1}(t){\theta }^{r-1}+\cdots +a_0(t)$ at the LCSL with fundamental period ${\Pi }_0(t)={\sum }_{k\ge 0}{c}_k{t}^k$ (normalised so $c_0=1$), the logarithmic period ${\Pi }_1(t)={\Pi }_0(t)\log (t)+R(t)$ has its regular part $R(t)={\sum }_{k\ge 1}{r}_k{t}^k$ determined recursively by the equation $L\cdot {\Pi }_1=0$ and the leading coefficients $a_j(t)$ of $L$.

Proof. Substituting ${\Pi }_1={\Pi }_0\log (t)+R$ into $L\cdot {\Pi }_1=0$ and using $\theta (\log t)=1$, ${\theta }^k(f\log t)=({\theta }^{k}f)\log t+k({\theta }^{k-1}f)+\left(\genfrac{}{}{0ex}{}{k}{2}\right)({\theta }^{k-2}f)\cdot 0+\cdots$, gives $$ L \cdot \Pi_1 = (L \cdot \Pi_0) \log(t) + \sum_{k = 1}^{r} k \cdot a_k(t) \cdot \theta^{k-1} \Pi_0 + L \cdot R(t), $$ where the first sum collects the $\log t$-times-substantive terms in $L\cdot ({\Pi }_0\log t)$. Since $L\cdot {\Pi }_0=0$, the equation $L\cdot {\Pi }_1=0$ reduces to $$ L \cdot R(t) = - \sum_{k = 1}^{r} k \cdot a_k(t) \cdot \theta^{k-1} \Pi_0(t), $$ an inhomogeneous linear ODE for $R(t)$ with a known right-hand side. The Frobenius method at the indicial root $0$ of $L$ produces a unique regular solution $R(t)$ (no logarithmic correction at this level, modulo the ambiguity of adding a multiple of ${\Pi }_0$). The coefficients $r_k$ are determined recursively: at order $t^k$, $r_k$ is computed from $r_{k-1},r_{k-2},\dots$ and the leading coefficients of ${\Pi }_0$ and $a_j$. $\square$

The full Hodge-theoretic and Gross-Siebert content — that the recursive coefficients $r_k$ equal tropical-disk count generating series on the A-side mirror via the Nishinou-Siebert correspondence — is the content of Gross 2011 Lecture 7 and is deferred to that anchor.

Connections [Master]

• Toric degeneration of a Calabi-Yau variety 04.12.07. The toric-degeneration unit defines the one-parameter family $\pi :\mathfrak{X}\to \mathrm{Spec}\mathbb{C}[[t]]$ on which the present unit's periods are computed. The relative holomorphic volume form ${\Omega }_t$ exists by the Calabi-Yau condition on the family, and the period integrals ${\Pi }_i(t)={\int }_{{\gamma }_i}{\Omega }_t$ inherit the formal-series structure of the family. The reciprocal cross-link is structural — the toric degeneration is the geometric carrier, and the period integral is the Hodge-theoretic readout of the family's variation.

• Gross-Siebert reconstruction theorem 04.12.09. The reconstruction theorem produces the Calabi-Yau family from a polarised tropical manifold $(B,\mathcal{P},\phi )$ with consistent slab data. The slab functions encode tropical-disk count generating series on $(B,\mathcal{P})$ via the Nishinou-Siebert correspondence, and the Picard-Fuchs operator of the reconstructed family inherits these coefficients. The higher-order mirror-map coefficients $r_k$ read off as tropical-disk counts on the A-side mirror — the constructive content of the Candelas-de la Ossa-Green-Parkes 1991 prediction in the Gross-Siebert setting.

• Theta function of a polarised tropical manifold 04.12.12. The theta-function basis $\{{\vartheta }_m{\}}_{m\in B(\mathbb{Z})}$ on the reconstructed mirror Calabi-Yau provides a canonical $\mathbb{Z}$-basis of the polarised section ring; the broken-line construction of ${\vartheta }_m$ is parallel to the tropical-disk-count interpretation of the mirror-map coefficients $r_k$ here. Both constructions consume the slab data of the polarised tropical manifold and identify the resulting algebraic-geometric structure with combinatorial enumeration on $(B,\mathcal{P})$. The reciprocal cross-link is the broken-line vs. tropical-disk parallelism: theta functions and period integrals are dual readouts of the same combinatorial-enumerative content.

• Strominger-Yau-Zaslow conjecture 04.12.10. The SYZ conjecture identifies mirror Calabi-Yau pairs with dual special-Lagrangian torus fibrations over a common integral-affine base. Under SYZ, the mirror map is the symplectic-area-to-complex-coordinate identification on the base: the symplectic Kähler parameter on the A-side equals (asymptotically) the complex-structure parameter on the B-side, via the period asymptotics at the LCSL. The present unit's mirror-map theorem is the precise quantitative statement of the SYZ identification near the most degenerate limit.

• Nishinou-Siebert correspondence 04.12.06. The Nishinou-Siebert correspondence supplies the tropical-disk-count interpretation of the slab functions of the Gross-Siebert reconstructed family, and thereby of the higher-order mirror-map coefficients. The present unit's leading-order identity $q(t)=t\cdot (1+r_{1}t+O(t^2))$ has $r_1$ a tropical-disk count on $(B,\mathcal{P})$ by Nishinou-Siebert applied to the slab function on the relevant codimension-1 cell.

• Kodaira embedding theorem 04.09.11. Kodaira embedding ensures that the Calabi-Yau fibres of the Gross-Siebert reconstructed family are projective and polarised, so that the period story is genuinely algebraic-geometric. The strictly convex piecewise-affine polarisation $\phi$ on the polarised tropical manifold $(B,\mathcal{P},\phi )$ corresponds, on the algebraic-geometric side, to the relative ample polarisation supplied by Kodaira embedding on each fibre. The reciprocal cross-link is structural — Kodaira embedding gives projectivity and polarisation, and the period integrals are computed on the polarised projective family.

• Hodge decomposition 04.09.01. The Hodge decomposition $H^n({\mathfrak{X}}_t,\mathbb{C})={⨁}_{p+q=n}{H}^{p,q}({\mathfrak{X}}_t)$ on each fibre is the starting point for the relative Hodge filtration $F^{\bullet }{\mathcal{H}}^n$ on the Gauss-Manin local system. The period integrals of the holomorphic volume form are the components of ${\Omega }_t\in F^n{\mathcal{H}}^n$ against the integral cycle basis — a relative version of the Hodge decomposition. The reciprocal cross-link is to the relative variation of Hodge structure; the present unit specialises the relative Hodge decomposition to the Calabi-Yau case and to the LCSL.

Historical & philosophical context [Master]

Candelas-de la Ossa-Green-Parkes 1991 Nuclear Physics B 359, 21-74 ^{[Candelas1991]} discovered, in the context of string theory on the quintic Calabi-Yau threefold $X\subset {\mathbb{P}}^4$, that the count of degree-$d$ rational curves on $X$ (for every degree $d\ge 1$) is computable from the period integrals of the holomorphic volume form on the mirror Calabi-Yau $X^{\vee }$, via the Picard-Fuchs equation and the mirror map. The discovery was an experimental fact of string theory in 1991 — supported by matching the period prediction against the classical count $2875$ of lines on the quintic and the degree-2 count $609,250$ — and triggered the mathematical mirror-symmetry programme that has occupied algebraic geometry for thirty-five years.

Morrison 1993 J. Amer. Math. Soc. 6, 223-247 ^{[Morrison1993]} reformulated the Candelas-de la Ossa-Green-Parkes prediction in mathematical language: variations of Hodge structure, the large-complex-structure limit, maximally unipotent monodromy, the limiting mixed Hodge structure of Schmid 1973 Invent. Math. 22 and Deligne 1997 ^{[Deligne1997]}, the canonical-coordinate construction, and the mirror map as the canonical-coordinate exponential. Morrison's framework is the canonical one for working mathematicians and remains the foundational reference for the period side of mirror symmetry.

The algebraic-geometric side of the period computation was developed in two parallel lines. Hosono-Klemm-Theisen-Yau 1995 Comm. Math. Phys. 167, 301-350 ^[Hosono1995] and Hosono-Lian-Yau 1996 Comm. Math. Phys. 182, 535-577 ^{[HosonoLianYau1996]} developed the systematic Picard-Fuchs equations for one-parameter Calabi-Yau hypersurfaces in toric varieties, using the GKZ hypergeometric system of Gelfand-Kapranov-Zelevinsky 1994 Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser) ^[GKZ1994]; Doran-Morrison 2010 J. Symbolic Comput. 45, 1342-1356 ^{[DoranMorrison2010]} completed the algorithmic perspective with a complete computational-algebra solver. Givental 1996 Internat. Math. Res. Notices 1996, no. 13 and Lian-Liu-Yau 1997 Asian J. Math. 1, 729-763 rigorously proved the original Candelas-de la Ossa-Green-Parkes prediction — that the period prediction matches the actual Gromov-Witten count — for hypersurfaces in toric Fano varieties, closing the loop opened by the 1991 calculation.

The tropical-geometry side of the period story is more recent. Strominger-Yau-Zaslow 1996 Nuclear Physics B 479, 243-259 ^[SYZ1996] proposed the geometric origin of mirror symmetry as T-duality on special-Lagrangian torus fibrations, suggesting that the mirror map is the change-of-coordinate identifying complex-structure with symplectic-Kähler parameters via the fibrewise duality. Gross-Siebert 2011 Annals of Mathematics 174, 1301-1428 ^{[GrossSiebert2011]} gave the algebraic-geometric realisation of SYZ via the reconstruction theorem 04.12.09, producing the Calabi-Yau family from a polarised tropical manifold. Gross 2011 Tropical Geometry and Mirror Symmetry (CBMS 114, AMS) ^{[Gross2011CBMS]} consolidated the period and mirror-map story for the Gross-Siebert reconstructed family in Lecture 7, identifying the higher-order mirror-map coefficients with tropical-disk count generating series on the polarised tropical manifold via the Nishinou-Siebert correspondence 04.12.06.

The canonical modern monograph treatments are Cox-Katz 1999 Mirror Symmetry and Algebraic Geometry (AMS Math. Surv. Mono. 68) ^{[CoxKatz1999]} and Voisin 1996 Symétrie miroir (SMF Panoramas et Synthèses 2) ^[Voisin1996]. Cox-Katz remains the canonical English-language algebraic-geometric reference for the period side of mirror symmetry; Voisin's monograph remains the canonical French-language treatment with English translation in SMF/AMS (1999). The systematic development of the underlying variation of Hodge structure machinery is in Voisin 2002-2003 Hodge Theory and Complex Algebraic Geometry (Cambridge Stud. Adv. Math. 76-77) and Carlson-Müller-Stach-Peters 2017 Period Mappings and Period Domains (Cambridge Stud. Adv. Math. 168).

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