Variation of Hodge structure on the Jacobian

Intuition [Beginner]

Take a smooth family of compact Riemann surfaces, parameterised by some base space — one surface for each parameter value, varying smoothly with the parameter. Each surface in the family has its own Jacobian, a complex torus that records how holomorphic 1-forms integrate around closed loops. As the parameter moves, the Jacobian moves with it: not just a single torus but a whole family of complex tori, with the same underlying topology but different complex structures.

The variation of Hodge structure on the Jacobian is the rule that says how the complex structure of the Jacobian changes as the base parameter changes. The integer lattice (the periods of integer cycles) stays put, locally; what moves is the choice of holomorphic versus anti-holomorphic directions inside the cohomology — equivalently, the position of the Jacobian's holomorphic tangent space inside the fixed total cohomology of the fibre. Recording this moving subspace at every parameter value gives a holomorphic mapping from the base into a target space called the Siegel upper half-space, the parameter space of all principally polarised abelian varieties of the right dimension.

The variation captures, in a single compact picture, every transcendental fact about how the period matrix of a curve changes as the curve is deformed: the modular geometry of moduli of curves, the Torelli theorem (a curve is recoverable from its Jacobian), the Schottky problem (which tori arise as Jacobians of curves), and the connection to integrable systems through the KP hierarchy.

Visual [Beginner]

A schematic of a one-parameter family of Riemann surfaces $X_s$ for $s$ in a base disc $S$, with each fibre's Jacobian $J(X_s)={\mathbb{C}}^g/{\Lambda }_s$ depicted as a complex torus floating above. As $s$ varies the topological lattice ${\mathbb{Z}}^{2g}$ stays fixed, but the holomorphic subspace $H^{1,0}(X_s)\subset H^1(X_s;\mathbb{C})$ (depicted as a half-plane inside a flat 2-plane on each fibre) tilts smoothly. An arrow labelled "period mapping" tracks each tilting subspace as a point of the target Siegel upper half-space ${\mathfrak{H}}_g$, drawn as a curved domain on the right.

Worked example [Beginner]

Take the Legendre family of elliptic curves once more: for each $\lambda \in \mathbb{C}\setminus \{0,1\}$, the curve $X_{\lambda }$ has equation $y^2=x(x-1)(x-\lambda )$. Each $X_{\lambda }$ is a compact Riemann surface of genus $1$, so its Jacobian $J(X_{\lambda })=\mathbb{C}/{\Lambda }_{\lambda }$ is itself a complex torus of dimension $1$.

Pick the holomorphic 1-form ${\omega }_{\lambda }=dx/y$ and the symplectic homology basis given by an $a$-cycle around the segment $[0,1]$ and a $b$-cycle around $[1,\lambda ]$. After normalising the $a$-period to $1$, the period matrix has the form $\Pi (\lambda )=(1\mid \tau (\lambda ))$, where $\tau (\lambda )$ is the ratio of the $b$-period to the $a$-period — a single complex number with positive imaginary part — and $\tau (\lambda )$ is the upper half-plane coordinate on the Jacobian. As $\lambda$ varies in the base $S=\mathbb{C}\setminus \{0,1\}$, the period $\tau (\lambda )$ traces out a holomorphic multivalued function on $S$ with values in the upper half-plane ${\mathfrak{H}}_1=\{\tau \in \mathbb{C}:\mathrm{Im}\tau >0\}$.

Concretely, near $\lambda =1/2$ one finds $\tau (\lambda )$ to first order from the ratio of two complete elliptic integrals; the function $\tau (\lambda )$ extends to a triple cover of the upper half-plane realising the modular curve $Y(2)={\mathfrak{H}}_1/\Gamma (2)$. The period mapping $\mathcal{P}:\mathbb{C}\setminus \{0,1\}\to {\mathfrak{H}}_1/\Gamma (2)$ is holomorphic and (in this genus-$1$ case) an isomorphism onto its image.

What this tells us: the variation of Hodge structure on the Jacobian of an elliptic-curve family is a holomorphic mapping into the upper half-plane modulo the symplectic monodromy; the target is the moduli space of polarised abelian varieties of dimension $1$, and for genus $1$ this is exactly the modular curve familiar from number theory. The same phenomenon, in the higher-rank Siegel upper half-space, governs every family of higher-genus Riemann surfaces.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\pi :\mathcal{X}\to S$ be a smooth proper morphism of complex manifolds with each fibre $X_s={\pi }^{-1}(s)$ a compact Riemann surface of fixed genus $g\ge 1$, $S$ smooth and connected. The cohomology bundle ${\mathcal{H}}^1=R^1{\pi }_{\ast }{\mathbb{C}}_{\mathcal{X}}$ is a rank-$2g$ local system on $S$ with fibre $H^1(X_s;\mathbb{C})$, equipped with the Gauss-Manin connection $\nabla$ (06.08.01).

Hodge filtration. Each fibre cohomology decomposes as $H^1(X_s;\mathbb{C})=H^{1,0}(X_s)\oplus H^{0,1}(X_s)$ with $\overline{H^{1,0}(X_s)}=H^{0,1}(X_s)$ and ${\dim }_{\mathbb{C}}{H}^{1,0}(X_s)=g$ (06.04.03). The Hodge filtration on the cohomology bundle is the single proper step $$ F^1 \mathcal{H}^1 = \pi_* \Omega^1_{\mathcal{X}/S} \subset \mathcal{H}^1, \qquad F^0 \mathcal{H}^1 = \mathcal{H}^1, $$ a rank-$g$ holomorphic subbundle of the rank-$2g$ total bundle.

Polarisation. The cup product ${\mathcal{H}}^1\otimes {\mathcal{H}}^1\to R^2{\pi }_{\ast }{\mathbb{C}}_{\mathcal{X}}\cong {\mathbb{C}}_S$ produces a non-degenerate alternating bilinear form $Q:{\mathcal{H}}^1\otimes {\mathcal{H}}^1\to {\mathbb{C}}_S$, the symplectic intersection form. On each fibre $Q$ realises the Riemann bilinear relations: $Q(F^1,F^1)=0$ (isotropy) and $iQ(\omega ,\bar{\omega })>0$ for non-zero $\omega \in F^1$ (positivity).

Definition (polarised variation of Hodge structure of weight 1). The triple $(R^1 \pi_ \mathbb{Z}, F^1 \mathcal{H}^1, Q)$on$S$isapolarisedvariationofHodgestructureofweight$1$:anintegrallocalsystem,aholomorphicHodgesubbundleofthecomplexification,andapolarisationform,suchthatoneachfibretheHodgestructureispureofweight$1$ with the stated polarisation, the Hodge filtration varies holomorphically, and Griffiths transversality holds.*

Period domain $\mathcal{D}$ for weight-1 PHS. Fix a reference fibre $H=H^1(X_{s_0};\mathbb{C})$ of dimension $2g$ with its symplectic form $Q_0$. The period domain is $$ \mathcal{D} = \bigl{ F \subset H : \dim_{\mathbb{C}} F = g,\ Q_0(F, F) = 0,\ i,Q_0(v, \bar v) > 0\ \forall v \in F \setminus {0} \bigr}, $$ a complex submanifold of the Lagrangian Grassmannian $\mathrm{LG}(g,2g)$ cut out by the Hodge positivity condition. As a homogeneous space $\mathcal{D}=\mathrm{Sp}(2g,\mathbb{R})/U(g)$, a Hermitian symmetric domain of compact type. Choosing a symplectic basis of $H$ identifies $\mathcal{D}\cong {\mathfrak{H}}_g$ with the Siegel upper half-space $$ \mathfrak{H}_g = \bigl{ \tau \in M_g(\mathbb{C}) : \tau^T = \tau,\ \mathrm{Im},\tau > 0 \bigr} $$ of dimension $g(g+1)/2$. The arithmetic group $\mathrm{Sp}(2g,\mathbb{Z})$ acts on ${\mathfrak{H}}_g$ by $\tau \mapsto (A\tau +B)(C\tau +D{)}^{-1}$ for $\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in \mathrm{Sp}(2g,\mathbb{Z})$; the quotient ${\mathcal{A}}_g:={\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ is the moduli space of principally polarised abelian varieties of dimension $g$.

Period mapping. Let $\Gamma \subset \mathrm{Sp}(2g,\mathbb{Z})$ be the image of the monodromy representation $\rho :{\pi }_1(S,s_0)\to \mathrm{Aut}(H_1(X_{s_0};\mathbb{Z}),Q_0)$ of the integral local system $R^1{\pi }_{\ast }\mathbb{Z}$. The period mapping is $$ \mathcal{P} : S \to \mathcal{D} / \Gamma \cong \mathfrak{H}_g / \Gamma, \qquad s \mapsto [F^1(X_s)], $$ sending $s$ to the Hodge filtration on $H^1(X_s;\mathbb{C})$ identified with the reference $H$ via parallel transport along an arbitrary path. Different path choices produce $\Gamma$-equivalent points, so $\mathcal{P}$ is well-defined modulo monodromy. The lift $\tilde{\mathcal{P}}:\tilde{S}\to {\mathfrak{H}}_g$ to the universal cover is single-valued.

Griffiths transversality (weight-1 case). The constraint $$ \nabla F^1 \subseteq F^0 \otimes \Omega^1_S = \mathcal{H}^1 \otimes \Omega^1_S $$ is automatic since $F^0={\mathcal{H}}^1$ — the Hodge filtration has only one proper step. The substantive content of transversality at weight 1 is the Cauchy-Riemann condition that the period matrix $\Pi (s)$ depends holomorphically on $s$: differentiating the period ${\Pi }_{ij}(s)={\int }_{{\gamma }_j(s)}{\omega }_i(s)$ in the antiholomorphic direction picks up the antiholomorphic part of the integrand, and this part vanishes because ${\omega }_i$ is a holomorphic 1-form on each fibre.

Counterexamples to common slips.

• The integral lattice $R^1{\pi }_{\ast }\mathbb{Z}$ is locally constant, but the holomorphic subbundle $F^1{\mathcal{H}}^1\subset {\mathcal{H}}^1$ is not locally constant — it varies as a holomorphic subbundle, and the variation is exactly the period mapping into ${\mathfrak{H}}_g$.
• The Siegel upper half-space ${\mathfrak{H}}_g$ is not the upper half-plane raised to the $g$-th power: ${\mathfrak{H}}_g$ has dimension $g(g+1)/2$, while ${\mathfrak{H}}_1^g$ has dimension $g$. The two coincide only at $g=1$.
• Griffiths transversality at weight 1 is automatic because the Hodge filtration has one proper step. The substantive content is holomorphy of the period mapping, not a multi-step transversality constraint.

Key theorem with proof [Intermediate+]

Theorem (Griffiths 1968, holomorphy of the period mapping for weight-1 VHS). Let $\pi :\mathcal{X}\to S$ be a smooth proper family of compact Riemann surfaces of genus $g$ over a smooth connected complex base $S$. The period mapping $$ \mathcal{P} : S \to \mathfrak{H}_g / \mathrm{Sp}(2g, \mathbb{Z}), \qquad s \mapsto [\Pi(s)], $$ is holomorphic and equivariant under the monodromy representation $\rho :{\pi }_1(S,s_0)\to \mathrm{Sp}(2g,\mathbb{Z})$.

Proof. The argument has three steps: realise the period mapping as the Hodge filtration of a polarised VHS, differentiate the period integrals along base directions, and verify the Cauchy-Riemann condition from the holomorphy of the integrand 1-forms.

Step 1 — period mapping as Hodge filtration. By the formal-definition section, $(R^1{\pi }_{\ast }\mathbb{Z},F^1{\mathcal{H}}^1,Q)$ is a polarised pure variation of Hodge structure of weight $1$ on $S$. Choose a symplectic homology basis ${\gamma }_1,\dots ,{\gamma }_g,{\delta }_1,\dots ,{\delta }_g$ of $H_1(X_{s_0};\mathbb{Z})$ at the reference fibre, and a basis ${\omega }_1(s),\dots ,{\omega }_g(s)$ of $H^0(X_s,{\Omega }_{X_s}^1)=F^1{\mathcal{H}}_s^1$ varying holomorphically with $s$ on a simply-connected open $U\subset S$. The cycles parallel-transport to ${\gamma }_j(s),{\delta }_j(s)\in H_1(X_s;\mathbb{Z})$ via the Gauss-Manin local system. The period matrix is $$ \Pi(s) = \bigl(A(s) \mid B(s)\bigr) \in M_{g \times 2g}(\mathbb{C}), \qquad A_{ij}(s) = \int_{\gamma_j(s)} \omega_i(s),\quad B_{ij}(s) = \int_{\delta_j(s)} \omega_i(s). $$ After normalisation $A(s)=I_g$ (achievable on a dense open by Riemann's bilinear relations from 06.04.03, which assert that $A$ has full rank), the matrix $\tau (s):=A(s{)}^{-1}B(s)\in M_g(\mathbb{C})$ is symmetric with positive-definite imaginary part, hence $\tau (s)\in {\mathfrak{H}}_g$. The period mapping $\mathcal{P}{|}_U:U\to {\mathfrak{H}}_g$ is $s\mapsto \tau (s)$.

Step 2 — differentiation of the period integrals. Let $X_s$ be a one-parameter local deformation in a base direction $\partial /\partial s$. The cycles ${\gamma }_j(s),{\delta }_j(s)$ parallel-transport along the Gauss-Manin connection of 06.08.01, so they vary $C^{\infty }$ but locally constantly in the integral local system: their classes in $H_1(X_s;\mathbb{Z})$ are constant up to monodromy. Differentiating the period matrix, $$ \frac{\partial \Pi_{ij}}{\partial s}(s) = \int_{\gamma_j(s)} \frac{\partial \omega_i}{\partial s}(s) $$ because the cycle-derivative terms vanish for parallel-transported cycles (the Lie derivative of a closed form along a horizontal section is exact, hence has zero period). The integrand $\partial {\omega }_i/\partial s$ is not in general a holomorphic 1-form — it is a closed $(1,0)$-form on $X_s$ whose cohomology class lies in ${\mathcal{H}}_s^1$ but not necessarily in $F^1{\mathcal{H}}_s^1$. This is the Kodaira-Spencer derivative, the differential of the period mapping.

Step 3 — Cauchy-Riemann condition. Decompose the base direction $\partial /\partial s$ into its $(1,0)$ and $(0,1)$ parts. The antiholomorphic derivative satisfies $$ \frac{\partial \Pi_{ij}}{\partial \bar s}(s) = \int_{\gamma_j(s)} \frac{\partial \omega_i}{\partial \bar s}(s) = 0, $$ since ${\omega }_i(s)$ is holomorphic in $s$ as a section of the holomorphic bundle $F^1{\mathcal{H}}^1$ — the variation $\partial {\omega }_i/\partial \bar{s}$ vanishes. (Equivalently: pick a holomorphic local trivialisation of $F^1{\mathcal{H}}^1$ on $U$; its sections are by construction holomorphic in $s$.) Hence the period matrix $\Pi (s)$ is holomorphic in $s$, and so is the normalised period $\tau (s)=A(s{)}^{-1}B(s)\in {\mathfrak{H}}_g$. The local lift $\mathcal{P}{|}_U$ is holomorphic, and $\mathcal{P}:S\to {\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ is holomorphic globally because holomorphy is a local property and the action is by biholomorphisms. Equivariance under monodromy follows from the fact that path-deforming a cycle ${\gamma }_j$ around a loop in ${\pi }_1(S,s_0)$ produces a $\mathrm{Sp}(2g,\mathbb{Z})$-conjugate cycle. $\square$

The three-step structure follows Griffiths 1968 Periods of integrals on algebraic manifolds I §3 ^{[Griffiths 1968]} (combined with the Riemann bilinear relations from Voisin §10 ^{[Voisin Hodge Theory I]}). Voisin's textbook treatment in Hodge Theory and Complex Algebraic Geometry I §10 reframes the same argument via the Kodaira-Spencer map; Donaldson §13 specialises to the curve case with an explicit period-matrix differentiation ^{[Donaldson Riemann Surfaces]}.

Bridge. The period mapping proven holomorphic here is the transcendental shadow of the family $\pi :\mathcal{X}\to S$ — a single holomorphic record of every period integral against every locally-constant cycle. The construction builds toward the Jacobian-of-the-family: each fibre Jacobian $J(X_s)={\mathbb{C}}^g/{\Lambda }_s$ is a principally polarised abelian variety, the relative Jacobian $\mathrm{Jac}(\mathcal{X}/S)$ is a smooth proper family of PPAV over $S$, and $\mathcal{P}$ records the moduli class of $J(X_s)$ at each $s$. The Gauss-Manin connection $\nabla$ appears again in 06.08.01 as the differential operator that produces $\mathcal{P}$ via the Hodge filtration step $F^1{\mathcal{H}}^1\subset {\mathcal{H}}^1$; Jacobi inversion 06.06.06 gives the geometric content of the period, with every Jacobian generated by effective divisors of degree $g$. The Riemann bilinear relations from 06.04.03 are the positivity input landing $\mathcal{P}$ in ${\mathfrak{H}}_g$ rather than the larger Lagrangian Grassmannian. This is the same as the Jacobian map ${\mathcal{M}}_g\to {\mathcal{A}}_g$, whose injectivity is Torelli's theorem and whose image is the Schottky locus. This generalises to higher weight VHS where Griffiths transversality acquires substantive content. Putting these together, the variation of the Hodge filtration on the cohomology of a family of curves is the moduli of the family of Jacobians, and the bridge from this transcendental fact to computable algebra is the explicit description of ${\mathfrak{H}}_g$ as a homogeneous space combined with the Schottky problem cutting out which points of ${\mathcal{A}}_g$ come from curves.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib does not currently formalise the polarised variation of Hodge structure on the relative cohomology of a smooth proper family, the Siegel upper half-space ${\mathfrak{H}}_g$ as a Hermitian symmetric domain with $\mathrm{Sp}(2g,\mathbb{R})$-action, the period mapping $\mathcal{P}:S\to {\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$, the moduli space ${\mathcal{A}}_g$ of principally polarised abelian varieties, Torelli's theorem, or the Schottky locus. A proposed signature, in Lean 4 / Mathlib syntax, sketching the target statement:

[object Promise]

The proof depends on names that do not currently exist in Mathlib (the Siegel upper half-space as a Hermitian symmetric domain with explicit $\mathrm{Sp}(2g,\mathbb{R})$-action, the polarised variation of Hodge structure on the Gauss-Manin local system, the period mapping into ${\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$, the moduli space ${\mathcal{A}}_g$ of PPAV, Torelli's theorem, the Schottky locus, and Shiota's KP characterisation). Each is a candidate Mathlib contribution; until then this unit ships with lean_status: none.

Advanced results [Master]

The polarised variation of Hodge structure on the Jacobian of a family of compact Riemann surfaces is the foundational example of a polarised pure VHS of weight $1$, and the theory developed in the curve case generalises in three directions: to higher weight (with multi-step Hodge filtration and substantive Griffiths transversality), to non-compact / singular bases (via Deligne's mixed Hodge theory and Schmid's nilpotent-orbit theorem at boundary divisors), and to other coefficient categories ($p$-adic, motivic, automorphic).

Period domain as Hermitian symmetric space. The period domain $\mathcal{D}=\mathrm{Sp}(2g,\mathbb{R})/U(g)\cong {\mathfrak{H}}_g$ is a Hermitian symmetric space of compact type — one of the four classical families (Cartan types $A,B,C,D$, with ${\mathfrak{H}}_g$ corresponding to type $C_g$). The horizontal tangent distribution coincides with the full tangent bundle in this weight-1 case; integrability is automatic, and the period mapping is unobstructed. For weight $\ge 2$ the period domain is generally non-Hermitian-symmetric and the horizontal distribution is non-integrable except on the Hermitian symmetric sub-domains, which classifies Shimura variety VHS and makes them the special case where the period mapping is unobstructed.

Torelli theorem (Andreotti 1958). For $g\ge 1$, the Jacobian map ${\mathcal{T}}_g:{\mathcal{M}}_g\to {\mathcal{A}}_g$ is injective on isomorphism classes — a smooth projective curve $X$ is determined up to isomorphism by its principally polarised Jacobian $(\mathrm{Jac}(X),{\Theta }_X)$. Andreotti's 1958 proof ^{[Andreotti 1958]} reconstructs the canonical model of $X$ from the singular locus of the theta divisor for $g\ge 4$; for $g=2,3$ a separate hyperelliptic-aware argument applies (Torelli 1913 for the original statement, refined by Andreotti-Mayer 1967, Mumford 1971, Green 1984). Higher-dimensional Torelli (for hyperkähler manifolds, Calabi-Yau threefolds) is open in general except in special cases (Verbitsky 2013 for hyperkähler).

Schottky problem. The image ${\mathcal{J}}_g:={\mathcal{T}}_g({\mathcal{M}}_g)\subset {\mathcal{A}}_g$ is the Schottky locus of dimension $3g-3$ for $g\ge 2$, of codimension $g(g+1)/2-(3g-3)=g(g-5)/2+3$ in the $g(g+1)/2$-dimensional Siegel space. Coverage by $g$:

• $g=1,2,3$: codimension $0$; ${\mathcal{J}}_g={\mathcal{A}}_g$ (every PPAV is a Jacobian).
• $g=4$: codimension $1$; cut out by an explicit modular relation (Schottky 1888 ^{[Schottky 1888]}, a quartic theta-constant relation $r_4(\tau )=0$ on ${\mathfrak{H}}_4$).
• $g\ge 5$: codimension $\ge 3$; characterised by Welters's trisecant identity (Welters 1984, proved Krichever 2006), the Andreotti-Mayer locus of PPAV with theta divisor of unexpectedly large singular locus, or equivalently by Shiota's KP characterisation (Shiota 1986 ^{[Shiota 1986]}: the Novikov conjecture).

Shiota's theorem (Novikov conjecture). For a PPAV $(A,\Theta )$ with $A={\mathbb{C}}^g/\Lambda$ and theta function $\theta$, the function $$ u(x, y, t) = 2 , \partial_x^2 \log \theta\bigl(x V + y W + t U + z_0;, \tau\bigr) $$ on ${\mathbb{C}}^3$ for generic vectors $V,W,U\in {\mathbb{C}}^g$ and base point $z_0\in {\mathbb{C}}^g$ is a solution of the Kadomtsev-Petviashvili (KP) equation $$ \partial_x \bigl(4 u_t - 6 u u_x - u_{xxx}\bigr) = 3 u_{yy} $$ if and only if $(A,\Theta )\cong (\mathrm{Jac}(X),{\Theta }_X)$ for some smooth projective curve $X$. Shiota's 1986 theorem ^{[Shiota 1986]} (Invent. Math. 83) settles the Novikov conjecture posed in the 1970s following Krichever's discovery that Jacobian theta functions produce KP solutions. The KP characterisation is the most precise known general-genus answer to the Schottky problem.

Degenerations and limit mixed Hodge structure. For a family $\pi :\mathcal{X}\to S$ extending across a normal-crossings boundary divisor $D=\overline{S}\setminus S$, the Gauss-Manin connection has regular singularities along $D$ with quasi-unipotent local monodromy (Borel's monodromy theorem; Cattani-Kaplan-Schmid 1986). The limit mixed Hodge structure on the nearby fibre at a boundary point — controlled by Schmid's nilpotent-orbit theorem 1973 and the SL${}_2$-orbit theorem of Cattani-Kaplan-Schmid 1986 — encodes the asymptotic behaviour of the period $\Pi (s)\sim \log (s{)}^k\cdot (\text{holomorphic})$ via the unipotent monodromy and the limit Hodge filtration. For curve degenerations (genus-$g$ curve degenerating to a nodal curve), the limit MHS is computable explicitly and the limit Jacobian is a generalised Jacobian — an extension of an abelian variety by an algebraic torus.

Mixed Hodge structures (Deligne 1971-74). Pierre Deligne's Théorie de Hodge II-III ^{[Deligne 1971]} generalised pure VHS to mixed Hodge structures on the cohomology of singular or non-compact algebraic varieties. The cohomology of an open or singular algebraic variety carries a weight filtration $W_{\bullet }$ on the integral lattice and a Hodge filtration $F^{\bullet }$ on the complexification, with ${\mathrm{gr}}_k^W$ pure of weight $k$. Variation of mixed Hodge structures generalises VHS to families with non-compact / singular fibres; the Gauss-Manin connection extends and the period mapping target becomes a mixed period domain.

Quantum cohomology and mirror symmetry. For Calabi-Yau threefolds, the variation of Hodge structure on the middle cohomology $H^3$ is the B-model side of mirror symmetry. The mirror conjecture (Givental 1996, Lian-Liu-Yau 1997) identifies the Gauss-Manin connection on a family of Calabi-Yau quintics with the quantum cohomology connection of the mirror, recovering Gromov-Witten invariants from period integrals. The curve-case VHS treated here is the genus-zero foundational example.

Modular interpretation. For $g=1$ the Siegel upper half-space ${\mathfrak{H}}_1$ is the upper half-plane, the symplectic group $\mathrm{Sp}(2,\mathbb{Z})=\mathrm{SL}(2,\mathbb{Z})$ is the modular group, and modular forms are sections of line bundles on ${\mathfrak{H}}_1/\mathrm{SL}(2,\mathbb{Z})$. The variation of Hodge structure on a family of elliptic curves is precisely the geometric input for modular-form theory: periods of $dz/y$ are quasi-modular forms in the parameter $\tau$, and the Eisenstein series $E_4,E_6$ generating the ring of modular forms arise as Hodge-theoretic invariants of the universal elliptic curve. For $g\ge 2$ the analogous story is Siegel modular forms, sections of automorphic line bundles on ${\mathcal{A}}_g$, and the bridge from the Gauss-Manin geometry of curves to the automorphic theory of $\mathrm{Sp}(2g)$.

Generalisations. $p$-adic VHS via Faltings's $p$-adic Hodge theory (1987-88) and Berthelot rigid cohomology produces a $p$-adic analogue with Frobenius isocrystal structure; the Jacobian-of-a-curve case provides the de Rham realisation of the corresponding $p$-adic Galois representation. Real $\mathbb{R}$-VHS on a homogeneous period domain $G(\mathbb{R})/V$ is the natural setting for non-classical period domains where $G$ is not Hermitian symmetric; here the period mapping is constrained to the horizontal distribution and only special VHS arise from algebraic geometry (the integrality and quasi-projectivity problem of Griffiths). Motivic VHS (Saito 1988-90) realises VHS as the de Rham realisation of mixed motives, the categorical setting where Tannakian formalism and conjectures of Beilinson and Voevodsky apply.

Synthesis. The variation of Hodge structure on the Jacobian of a family of compact Riemann surfaces is the canonical polarised pure VHS of weight $1$, producing a holomorphic period mapping into a Hermitian symmetric domain whose target is the moduli of principally polarised abelian varieties. Read in the curve case, $\mathcal{P}:{\mathcal{M}}_g\to {\mathcal{A}}_g$ is the Jacobian map, injective by Torelli, with image the Schottky locus ${\mathcal{J}}_g$ of codimension $g(g-5)/2+3$; the explicit description is Schottky's quartic relation at $g=4$ and Shiota's KP characterisation for general $g$. Read in the Hodge-theoretic case, this is dual to a smooth lift of the Gauss-Manin local system to ${\mathfrak{H}}_g$, with the Cauchy-Riemann condition realising the substantive content of weight-1 Griffiths transversality. Read in the modular case, this generalises the elliptic-curve story: ${\mathfrak{H}}_1/\mathrm{SL}(2,\mathbb{Z})$ is the modular curve and the variation is the bridge to modular forms; for $g\ge 2$, this is the same as the geometry of Siegel modular forms on ${\mathcal{A}}_g$. Read in the integrable-systems case, this generalises to KP: the Riemann theta function on the Jacobian satisfies KP, and exactly the Jacobian points of ${\mathcal{A}}_g$ have this property. Putting these together, every smooth proper family of curves carries a polarised pure VHS of weight $1$, lifting to a holomorphic period mapping whose image is a constructible subvariety injecting from ${\mathcal{M}}_g$, and the resulting transcendental geometry encodes simultaneously modular-form theory, the Schottky problem, and the integrable-system characterisation of Jacobians.

Full proof set [Master]

Lemma (Hodge filtration as a holomorphic subbundle). Let $\pi :\mathcal{X}\to S$ be a smooth proper family of compact Riemann surfaces of genus $g$. Then $F^1 \mathcal{H}^1 := \pi_ \Omega^1_{\mathcal{X}/S}$isarank-$g$holomorphicsubbundleoftherank-$2g$cohomologybundle$\mathcal{H}^1 = R^1 \pi_* \mathbb{C}{\mathcal{X}}$,withquotient$\mathcal{H}^1 / F^1 \mathcal{H}^1 = R^1 \pi* \mathcal{O}_{\mathcal{X}}$ofrank$g$.*

Proof. The relative dualising sheaf ${\omega }_{\mathcal{X}/S}={\Omega }_{\mathcal{X}/S}^1$ on a smooth proper morphism of relative dimension $1$ is a line bundle on $\mathcal{X}$, and ${\pi }_{\ast }{\omega }_{\mathcal{X}/S}$ is locally free of rank $g$ on $S$ by the relative Hodge theorem (Grauert's coherence + Hodge-to-de-Rham degeneration in characteristic zero), with fibre $H^0(X_s,{\Omega }_{X_s}^1)$ of dimension $g$ by 06.04.03. The relative de Rham complex $0\to {\mathcal{O}}_{\mathcal{X}}\to {\Omega }_{\mathcal{X}/S}^1\to 0$ in degree $0,1$ produces the stupid filtration $F^p{\Omega }_{\mathcal{X}/S}^{\bullet }={\sigma }^{\ge p}{\Omega }_{\mathcal{X}/S}^{\bullet }$, and applying $R^1{\pi }_{\ast }$ to this filtration gives the Hodge filtration on relative cohomology: $F^1{\mathcal{H}}^1={\pi }_{\ast }{\Omega }_{\mathcal{X}/S}^1$ and $F^0{\mathcal{H}}^1={\mathcal{H}}^1$. The short exact sequence $0\to F^1{\mathcal{H}}^1\to {\mathcal{H}}^1\to {\mathcal{H}}^1/F^1\to 0$ identifies the quotient with $R^1{\pi }_{\ast }{\mathcal{O}}_{\mathcal{X}}$ via the spectral sequence of the stupid filtration on ${\Omega }_{\mathcal{X}/S}^{\bullet }$, of rank $g$ by Serre duality 06.04.04 applied fibrewise. $\square$

Lemma (Riemann bilinear relations on the family). For the same family, the symplectic intersection form $Q$ on ${\mathcal{H}}^1$ satisfies $Q(F^1,F^1)=0$ and $iQ(\omega ,\bar{\omega })>0$ for non-zero $\omega \in F^1$, fibrewise. Equivalently, in a normalised symplectic basis of $H_1(X_s;\mathbb{Z})$, the period matrix $\tau (s)=A(s{)}^{-1}B(s)$ satisfies ${\tau }^T=\tau$ and $\mathrm{Im}\tau >0$.

Proof. Fibrewise this is the Riemann bilinear relations from 06.04.03: for $\omega ,{\omega }^{\prime }\in H^0(X_s,{\Omega }_{X_s}^1)$, the wedge product $\omega \wedge {\omega }^{\prime }$ is a section of ${\Omega }^{2,0}(X_s)=0$ on the curve $X_s$ (which has complex dimension $1$), so $Q(\omega ,{\omega }^{\prime })={\int }_{X_s}\omega \wedge {\omega }^{\prime }=0$, the isotropy condition. Positivity: $iQ(\omega ,\bar{\omega })=i{\int }_{X_s}\omega \wedge \bar{\omega }$ equals ${\int }_{X_s}|f{|}^2\frac{i}{2}dz\wedge d\bar{z}$ in a local coordinate where $\omega =fdz$, that is, ${\int }_{X_s}|f{|}^{2}dV>0$ for $\omega \ne 0$. The normalised period-matrix form follows from Riemann's bilinear identity $A{B}^T=B{A}^T$ and $\mathrm{Im}(B{A}^{-1})>0$ from 06.04.03, which yield ${\tau }^T=\tau$ and $\mathrm{Im}\tau >0$ for $\tau =A^{-1}B$ on the locus where $A$ is invertible (a dense open). $\square$

Theorem (holomorphy and equivariance of the period mapping, full statement). Statement and proof as in the Intermediate-tier Key theorem section.

Proof. The Intermediate-tier proof goes through using the two lemmas above as packaged inputs: Lemma 1 produces the Hodge filtration $F^1{\mathcal{H}}^1\subset {\mathcal{H}}^1$ as a rank-$g$ holomorphic subbundle of the rank-$2g$ cohomology bundle; Lemma 2 lands the period matrix in ${\mathfrak{H}}_g$ via Riemann's bilinear relations. Holomorphy of the period mapping reduces to the antiholomorphic-derivative computation $\partial \Pi /\partial \bar{s}=0$ of Step 3. Monodromy equivariance follows from the locally-constant lifting of cycles by the Gauss-Manin connection. $\square$

Corollary (Torelli's theorem for curves, Andreotti 1958). The Jacobian map ${\mathcal{T}}_g:{\mathcal{M}}_g\to {\mathcal{A}}_g$ is injective on isomorphism classes for $g\ge 1$.

Proof. The genus-$1$ case is classical: an elliptic curve is determined up to isomorphism by its $j$-invariant, which is a function of the period $\tau \in {\mathfrak{H}}_1$, recovering $X$ from $\mathrm{Jac}(X)=X$.

For $g\ge 2$, Andreotti's 1958 proof ^{[Andreotti 1958]} reconstructs $X$ from $(\mathrm{Jac}(X),{\Theta }_X)$ via the geometry of the theta divisor: the canonical embedding $X\hookrightarrow {\mathbb{P}}^{g-1}=\mathbb{P}(H^0(X,K_X{)}^{\ast })$ for non-hyperelliptic $X$ realises $X$ as the Gauss image of the smooth locus ${\Theta }_{\mathrm{sm}}$ of the theta divisor — the map ${\Theta }_{\mathrm{sm}}\to \mathbb{P}(T_0\mathrm{Jac}(X))=\mathbb{P}(H^0(X,K_X{)}^{\ast })$ sending a smooth point $p\in \Theta$ to the projective normal direction at $p$ has image equal to the canonical curve. Reconstructing the canonical model recovers $X$ up to isomorphism. The hyperelliptic case ($g=2,3$ or arbitrary $g$ with $X$ hyperelliptic) is handled separately: a hyperelliptic curve is determined by its $2g+2$ Weierstrass points, and these can be recovered from the singular locus of $\Theta$ by Andreotti-Mayer 1967. $\square$

Corollary (dimension and codimension of the Schottky locus). For $g\ge 2$, the Schottky locus ${\mathcal{J}}_g:={\mathcal{T}}_g({\mathcal{M}}_g)\subset {\mathcal{A}}_g$ has dimension $3g-3$ and codimension $$ \mathrm{codim}_{\mathcal{A}_g} \mathcal{J}_g = \frac{g(g+1)}{2} - (3g - 3) = \frac{g(g - 5) + 6}{2} = \frac{(g - 1)(g - 6)}{2} + 3. $$

Proof. Riemann's count $\dim {\mathcal{M}}_g=3g-3$ for $g\ge 2$ is classical (Riemann 1857; rigorous via Mumford's GIT construction 1965). Torelli's theorem (Corollary above) is injective on closed points; combined with the fact that $\mathcal{P}$ is generically immersive (the differential is generically of full rank by a Kodaira-Spencer argument), $\dim {\mathcal{J}}_g=\dim {\mathcal{M}}_g=3g-3$. The dimension of ${\mathcal{A}}_g={\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ is $g(g+1)/2$. Subtracting gives the codimension. $\square$

Corollary (Schottky's relation for $g=4$). For $g=4$, the Schottky locus ${\mathcal{J}}_4\subset {\mathcal{A}}_4$ has codimension $1$ and is cut out by a single explicit modular relation, the Schottky relation $r_4(\tau )=0$, a polynomial in the second-order theta constants $\theta \left[\begin{array}{c}ϵ\\ {ϵ}^{\prime }\end{array}\right](0;\tau )$ for half-integer characteristics $ϵ,{ϵ}^{\prime }\in (\frac{1}{2}\mathbb{Z}/\mathbb{Z}{)}^4$.

Proof. Codimension $1$ from the previous corollary at $g=4$. The explicit relation $r_4$ was discovered by Schottky 1888 ^{[Schottky 1888]} using the Schottky-Jung relations among theta constants induced by an étale double cover of the curve; Igusa 1981 proved that $r_4$ alone cuts out ${\mathcal{J}}_4$ scheme-theoretically. The relation has degree $16$ in the theta constants and is the unique (up to scalar) $\mathrm{Sp}(8,\mathbb{Z})$-modular form of its weight vanishing on ${\mathcal{J}}_4$. $\square$

Corollary (Shiota 1986, KP characterisation of Jacobians). Let $(A,\Theta )=({\mathbb{C}}^g/\Lambda ,\Theta )$ be an indecomposable principally polarised abelian variety with theta function $\theta$. Suppose there exist generic vectors $V,W,U\in {\mathbb{C}}^g$ and $z_0\in {\mathbb{C}}^g$ such that $$ u(x, y, t) := 2 , \partial_x^2 \log \theta(xV + yW + tU + z_0;, \tau) $$ satisfies the KP equation ${\partial }_x(4u_t-6u{u}_x-u_{xxx})=3u_{yy}$ on ${\mathbb{C}}^3$. Then $(A,\Theta )$ is the Jacobian of a smooth projective curve $X$.

Proof. Shiota 1986 ^{[Shiota 1986]} (Invent. Math. 83). The proof produces a curve $X$ inside $A$ as the orbit closure of the KP flow at the origin: the KP flow is generated by the vector field $V$ together with a hierarchy of higher commuting flows, and the trajectory of the origin in $A$ under the full hierarchy traces out an algebraic curve whose Jacobian is $A$. The reverse direction (Krichever's construction 1977) realises every smooth projective curve's Jacobian as a KP solution via the Krichever-Baker-Akhiezer function on the curve, producing a soliton hierarchy from the $\theta$-function pulled back to the Jacobian. Together these give a bidirectional characterisation: $A$ is a Jacobian iff its theta function generates a KP solution. $\square$

Connections [Master]

• Gauss-Manin connection 06.08.01. The Gauss-Manin connection on the cohomology bundle ${\mathcal{H}}^1=R^1{\pi }_{\ast }{\mathbb{C}}_{\mathcal{X}}$ is the canonical flat structure that organises the variation of Hodge structure: the integral lattice $R^1{\pi }_{\ast }\mathbb{Z}$ is the horizontal data, and the Hodge subbundle $F^1{\mathcal{H}}^1\subset {\mathcal{H}}^1$ varies with the parameter against this fixed integral background. The period mapping in this unit is the Hodge filtration of the Gauss-Manin local system, and its holomorphy is a direct consequence of the differential calculus of $\nabla$ together with the holomorphy of the basis 1-forms. Without the Gauss-Manin connection there is no notion of "fixing the integral lattice while moving the holomorphic subspace," and the variation of Hodge structure has no setting.

• Jacobi inversion theorem 06.06.06. Jacobi inversion identifies the Jacobian $\mathrm{Jac}(X)$ with the symmetric power ${\mathrm{Sym}}^g(X)$ modulo linear equivalence, giving the geometric interpretation of points of the Jacobian as effective divisors of degree $g$ on the curve. In a family of curves, the Abel-Jacobi map gives a family-version identification of the relative Jacobian $\mathrm{Jac}(\mathcal{X}/S)$ with the relative symmetric power ${\mathrm{Sym}}_S^g(\mathcal{X})$ modulo linear equivalence, and the variation of Hodge structure on ${\mathcal{H}}^1$ encodes how this geometric data deforms with the parameter $s\in S$. The principal polarisation realised by the theta divisor through Riemann's vanishing theorem is the polarisation form $Q$ on ${\mathcal{H}}^1$ at the level of the variation of Hodge structure.

• Hodge decomposition on a compact Riemann surface 06.04.03. The fibrewise Hodge decomposition $H^1(X_s;\mathbb{C})=H^{1,0}(X_s)\oplus H^{0,1}(X_s)$ is the pointwise data of the variation; the Riemann bilinear relations $A{B}^T=B{A}^T$ and $\mathrm{Im}(B{A}^{-1})>0$ are the polarisation conditions that promote the pointwise decomposition to a polarised structure, and the fact that $F^1{\mathcal{H}}^1={\pi }_{\ast }{\Omega }_{\mathcal{X}/S}^1$ has rank $g$ across the family is the family-version of the curve Hodge identifications.

• Period matrix 06.06.02. The period matrix $\Pi (s)=(A(s)\mid B(s))$ of a fibre $X_s$ is the local coordinate description of the period mapping at $s$, and the normalised form $\tau (s)=A(s{)}^{-1}B(s)\in {\mathfrak{H}}_g$ is the explicit lift to the Siegel upper half-space. The Riemann bilinear relations are the pointwise constraints that hold uniformly along the period mapping, and the variation of Hodge structure organises the period matrices of the fibres into a single holomorphic mapping into the moduli of PPAV.

• Jacobian variety 06.06.03. The fibre Jacobian $J(X_s)=(F^1{\mathcal{H}}_s^1{)}^{\ast }/R_1{\pi }_{\ast }{\mathbb{Z}}_s$ is constructed from the variation data — the Hodge subbundle and the integral lattice — at each $s$. The relative Jacobian $\mathrm{Jac}(\mathcal{X}/S):=(F^1{\mathcal{H}}^1{)}^{\ast }/R_1{\pi }_{\ast }\mathbb{Z}$ is a smooth proper family of principally polarised abelian varieties over $S$, and the period mapping records the moduli class of $J(X_s)$ at each $s$.

• Theta function 06.06.05. Riemann's theta function $\theta (z;\tau )$ on ${\mathbb{C}}^g\times {\mathfrak{H}}_g$ is the universal section of the principal polarisation line bundle on the family of PPAV over ${\mathfrak{H}}_g$; it pulls back along the period mapping to give the relative theta function on the family of Jacobians of the curve family. The Schottky problem and Shiota's KP characterisation are stated directly in terms of the theta function and its zero divisor — the theta divisor — making $\theta$ the load-bearing transcendental object of the variation.

• Holomorphic 1-form 06.06.01. The space of holomorphic 1-forms $H^0(X_s,{\Omega }_{X_s}^1)$ is the fibre of the Hodge subbundle $F^1{\mathcal{H}}^1$ at $s$; sections of $F^1{\mathcal{H}}^1$ over $S$ are families of holomorphic 1-forms on the fibres varying holomorphically with $s$, and the period mapping is computed by integrating these 1-forms against parallel-transported integer cycles.

• Riemann-Roch theorem for compact Riemann surfaces 06.04.01. Riemann-Roch on each fibre supplies the rank computation $\dim H^0(X_s,{\Omega }_{X_s}^1)=g$ and the duality $H^1(X_s,{\mathcal{O}}_{X_s})\cong H^0(X_s,{\Omega }_{X_s}^1{)}^{\ast }$ that promotes the pair $(F^1{\mathcal{H}}^1,{\mathcal{H}}^1/F^1{\mathcal{H}}^1)$ to a perfect-pairing Hodge filtration; without Riemann-Roch the rank counts of the variation are not justified.

• Serre duality on a curve 06.04.04. The fibrewise Serre duality $H^1(X_s,{\mathcal{O}}_{X_s})\cong H^0(X_s,{\Omega }_{X_s}^1{)}^{\ast }$ is the pointwise content of the polarisation form $Q$ on ${\mathcal{H}}^1$; the family-version of the duality is the perfect pairing $F^1{\mathcal{H}}^1\otimes ({\mathcal{H}}^1/F^1{\mathcal{H}}^1)\to {\mathcal{O}}_S$ that pairs the Hodge subbundle with its quotient and is the structural input for the principal polarisation of the family of Jacobians.

• Sheaf cohomology of holomorphic line bundles 06.04.02. The cohomology $R^1{\pi }_{\ast }{\mathcal{O}}_{\mathcal{X}}$ and ${\pi }_{\ast }{\Omega }_{\mathcal{X}/S}^1$ on the base $S$ are computed via the Čech / sheaf cohomology of the relative de Rham complex; the variation of Hodge structure on the Jacobian is built directly on top of the sheaf-cohomology machinery applied to the relative differential operators.

Historical & philosophical context [Master]

Phillip Griffiths developed the theory of variation of Hodge structure in a four-paper series Periods of integrals on algebraic manifolds I-III in 1968-70 ^{[Griffiths 1968]} (Amer. J. Math. 90, 568-626 and 805-865; Publ. Math. IHÉS 38, 125-180), with antecedents in Phillip Griffiths and James Carlson's earlier Inventiones work on infinitesimal variations. Griffiths introduced the period mapping into the period domain $\mathcal{D}$ as a Hermitian symmetric (or more generally, complex homogeneous) space, proved the infinitesimal period relations (Griffiths transversality $\nabla F^p\subseteq F^{p-1}\otimes {\Omega }_S^1$), and developed the systematic theory of polarised pure variation of Hodge structure as the Hodge-theoretic content of a smooth proper family of compact Kähler manifolds. The weight-1 case treated in this unit is the curve-theoretic foundational example; Griffiths emphasised it precisely because the period domain is Hermitian symmetric and the period mapping is unobstructed, so that the resulting transcendental geometry can be compared to the algebraic moduli of curves.

The Torelli theorem for curves traces to Ruggiero Torelli's 1913 Sulle varietà di Jacobi (Rend. R. Acc. Lincei 22, 98-103). The modern proof reconstructing the canonical model from the singular locus of the theta divisor is due to Aldo Andreotti's 1958 On a theorem of Torelli ^{[Andreotti 1958]} (Amer. J. Math. 80, 801-828); Andreotti and Alan Mayer's 1967 On period relations for abelian integrals on algebraic curves extended the theorem to handle the hyperelliptic cases via a separate analysis of the singular locus of $\Theta$. Generic Torelli for higher-genus curves and refinements of the Schottky-related geometry are due to Mark Green 1984 and Phillip Griffiths-Joe Harris 1980. Higher-dimensional Torelli theorems (for $K3$ surfaces, hyperkähler manifolds, certain Calabi-Yau threefolds) are open in general except in special cases; Misha Verbitsky proved Torelli for hyperkähler manifolds in 2013 (Duke Math. J. 162).

The Schottky problem was posed and partially solved by Friedrich Schottky in 1888 Zur Theorie der Abelschen Functionen von vier Variablen ^{[Schottky 1888]} (J. Reine Angew. Math. 102, 304-352), giving the explicit modular relation $r_4(\tau )=0$ that cuts out the Schottky locus ${\mathcal{J}}_4\subset {\mathcal{A}}_4$ in genus $4$. Schottky and Heinrich Jung extended the relation system in 1909, leading to the Schottky-Jung relations — additional theta-constant identities holding on Jacobians but not on general PPAV. The general-genus Schottky problem was reformulated by Sergei Novikov in the mid-1970s as a question about the Kadomtsev-Petviashvili (KP) equation: Novikov conjectured that the Riemann theta function on a PPAV solves KP iff the PPAV is a Jacobian. The conjecture was proved by Takahiro Shiota in 1986 Characterization of Jacobian varieties in terms of soliton equations ^{[Shiota 1986]} (Invent. Math. 83, 333-382). Shiota's proof builds on Igor Krichever's 1977 construction (Funct. Anal. Appl. 11) of KP solutions from data on a Riemann surface and the Baker-Akhiezer function; Krichever's 2006 proof of the Welters trisecant conjecture (Russian Math. Surveys 61) gave a parallel characterisation via secant varieties of the Kummer image. Other approaches: the Andreotti-Mayer locus (PPAV whose theta divisor has unexpectedly large singular locus, 1967) and Beauville-Debarre's Prym variety analysis (1980s).

The deeper Hodge-theoretic foundations were laid by Pierre Deligne in Théorie de Hodge II-III ^{[Deligne 1971]} (Publ. Math. IHÉS 40, 5-57 and 44, 5-77, 1971-74) generalising pure VHS to mixed Hodge structures on the cohomology of singular or non-compact algebraic varieties. The degeneration theory of pure VHS at boundary divisors was completed by Wilfried Schmid in 1973 Variation of Hodge structure: the singularities of the period mapping (Invent. Math. 22, 211-319) with the nilpotent-orbit theorem, extended by Eduardo Cattani, Aroldo Kaplan, and Wilfried Schmid in 1986 Degeneration of Hodge structures (Ann. of Math. (2) 123, 457-535) with the SL${}_2$-orbit theorem and the limit mixed Hodge structure. Morihiko Saito's Modules de Hodge polarisables (1988) and Mixed Hodge modules (1990) provide the categorical / sheaf-theoretic framework that subsumes pure and mixed VHS into a single formalism with six-functor calculus.

The textbook expositions are Claire Voisin's Hodge Theory and Complex Algebraic Geometry I+II (Cambridge Studies 76 and 77, 2002-03) ^{[Voisin Hodge Theory I]} §10–§11, which develops the theory with the curve case as an extended worked example, and Donaldson's Riemann Surfaces (Oxford GTM 22, 2011) §13 ^{[Donaldson Riemann Surfaces]}, which presents the variation on the Jacobian via the period mapping into the Siegel upper half-space at the level of the explicit period matrix. Mumford's Tata Lectures on Theta I-III (Birkhäuser PM 28, 43, 97, 1983-91) ^{[Mumford Tata Lectures]} is the standard reference for the theta-function side and the Schottky problem; Cox-Katz Mirror Symmetry and Algebraic Geometry (AMS 1999) ^{[Cox-Katz Mirror Symmetry]} develops the Calabi-Yau VHS application to mirror symmetry.

Bibliography [Master]

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