Riemann's bilinear relations

Intuition [Beginner]

Take a compact Riemann surface $X$ — a closed surface dressed with the data needed to ask whether a function on it is holomorphic — of genus $g$. The first homology of $X$ is generated by $2g$ independent loops, and these loops can be paired off into $g$ symplectic pairs $(a_i,b_i)$ that interlock like the handles of a $g$-holed pretzel. Around each loop a holomorphic 1-form has a periodic value: a complex number that records the form's "twist" along that loop.

Riemann's bilinear relations are two structural laws on the table of these periodic values. After packaging them into a $g\times g$ matrix $\tau$ (built from the $b$-loop integrals against a basis normalised on the $a$-loops), the matrix is symmetric: the value across a $b_i$-loop measured by the $i$-th form equals the value across the $b_j$-loop measured by the $j$-th form. And the imaginary part of $\tau$ is positive definite: a structural strict positivity that promotes $\tau$ to a respectable point in a high-dimensional generalisation of the upper half plane.

Why bother? Because $\tau$ is the entire complex-analytic fingerprint of $X$. Riemann showed that the surface, the integrals of every algebraic function on it, and the theta function that organises its divisors are all controlled by this single matrix and the two laws it obeys.

Visual [Beginner]

A schematic of a compact Riemann surface of genus $g=2$ with two symplectic pairs of loops $(a_1,b_1)$ and $(a_2,b_2)$ drawn around the two handles. Beside the surface, a $2\times 2$ complex matrix $\tau$ is shown with its diagonal entries marked symmetric across the diagonal, and the lower-right region shaded to indicate "imaginary part positive definite". An arrow labels the embedding $\tau \mapsto {\mathfrak{H}}_2$ into the genus-2 Siegel upper half space.

Worked example [Beginner]

Take the elliptic curve $X=\mathbb{C}/(1,\tau )$ where the lattice is generated by $1$ and a complex number $\tau$ in the upper half plane. The genus is $g=1$, so the period matrix is a single complex number — the same $\tau$.

The first bilinear relation says $\tau$ equals its own transpose. A $1\times 1$ matrix is automatically equal to its transpose, so the symmetry has no content in this case. The second bilinear relation says the imaginary part of $\tau$ is positive: $\mathrm{Im}\tau >0$. This is the upper-half-plane condition that defines a valid lattice $(1,\tau )$ in the first place — pick $\tau =i$ and the lattice is the square lattice with $\mathrm{Im}(i)=1>0$.

For $\tau =0.4+0.7i$, the relations read: symmetry (automatic for the $1\times 1$ matrix $(\tau )$, since $\tau =\tau$), and imaginary part $0.7>0$. Both pass, and the elliptic curve $\mathbb{C}/(1,0.4+0.7i)$ is a valid compact Riemann surface.

What this tells us: in genus $1$ the bilinear relations collapse to the single condition $\mathrm{Im}\tau >0$ that classifies elliptic curves up to isomorphism. In higher genus the relations carry more content, and the symmetry condition becomes a genuine constraint on the $g\times g$ matrix.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a compact Riemann surface of genus $g\ge 1$. Choose a symplectic basis $a_1,\dots ,a_g,b_1,\dots ,b_g$ of $H_1(X,\mathbb{Z})$ — a basis with intersection numbers $a_i\cdot b_j={\delta }_{ij}$ and $a_i\cdot a_j=b_i\cdot b_j=0$. By the Hodge decomposition 06.04.03, $\dim H^0(X,{\Omega }_X^1)=g$. Choose a basis ${\omega }_1,\dots ,{\omega }_g$ of $H^0(X,{\Omega }_X^1)$ normalised so that

$${\int }_{a_j}{\omega }_i={\delta }_{ij}.$$

The period matrix is the $g\times g$ complex matrix

$${\tau }_{ij}:={\int }_{b_j}{\omega }_i.$$

In the non-normalised form, where the $a$-periods do not start out as the identity, write $A_{ij}={\int }_{a_j}{\omega }_i$ and $B_{ij}={\int }_{b_j}{\omega }_i$ for $g\times g$ matrices $A,B$; the full period matrix is the $g\times 2g$ block $(A\mid B)$ (cf. 06.06.02).

Theorem (Riemann's bilinear relations, Riemann 1857). The normalised period matrix $\tau$ satisfies:

• (RB1) Symmetry: ${\tau }_{ij}={\tau }_{ji}$, i.e. $\tau ={\tau }^T$.
• (RB2) Positive-definite imaginary part: $\mathrm{Im}\tau >0$ as a $g\times g$ real symmetric matrix.

Equivalently in non-normalised form: $A{B}^T=B{A}^T$ and $\mathrm{Im}(B{A}^{-1})>0$.

The Siegel upper half space is

$${\mathfrak{H}}_g:=\{\tau \in M_{g\times g}(\mathbb{C}):\tau ={\tau }^T,\mathrm{Im}\tau >0\}.$$

The bilinear relations assert that the period matrix of a compact Riemann surface lies in ${\mathfrak{H}}_g$. Conversely, every $\tau \in {\mathfrak{H}}_g$ defines a complex torus $A_{\tau }={\mathbb{C}}^g/({\mathbb{Z}}^g+\tau {\mathbb{Z}}^g)$ that is a principally polarised abelian variety, with the polarisation determined by the alternating form on the lattice and the positivity (RB2).

Counterexamples to common slips.

• The bilinear relations need both a symplectic homology basis and the normalisation ${\int }_{a_j}{\omega }_i={\delta }_{ij}$. Dropping the normalisation gives the equivalent non-normalised relations on $(A\mid B)$, but mixing the two forms leads to wrong matrix shapes.
• The matrix $\tau$ is symmetric, not Hermitian: $\tau ={\tau }^T$ uses the transpose, not the conjugate transpose. The conjugate transpose would force $\tau$ real, which already fails for elliptic curves with $\mathrm{Im}\tau \ne 0$.
• "Positive definite" for $\mathrm{Im}\tau$ refers to the real symmetric matrix obtained by taking imaginary parts entrywise, viewed as a quadratic form on ${\mathbb{R}}^g$. The complex matrix $\tau$ itself need not be positive in any sense.
• The dimension of ${\mathfrak{H}}_g$ is $g(g+1)/2$ (count of independent entries in a symmetric $g\times g$ matrix), while the moduli space ${\mathcal{M}}_g$ of compact Riemann surfaces has dimension $3g-3$ for $g\ge 2$. The image of the period map is far from dense in ${\mathfrak{H}}_g$ once $g\ge 4$ — this is the Schottky problem.

Key theorem with proof [Intermediate+]

Theorem (Riemann 1857, bilinear relations). Let $X$ be a compact Riemann surface of genus $g$, $a_1,\dots ,a_g,b_1,\dots ,b_g$ a symplectic basis of $H_1(X,\mathbb{Z})$, and ${\omega }_1,\dots ,{\omega }_g$ a basis of $H^0(X,{\Omega }_X^1)$ normalised so that ${\int }_{a_j}{\omega }_i={\delta }_{ij}$. Then ${\tau }_{ij}={\int }_{b_j}{\omega }_i$ satisfies $\tau ={\tau }^T$ and $\mathrm{Im}\tau >0$.

Proof. The argument is in four steps: cut $X$ along the symplectic basis into a $4g$-gon, derive Riemann's bilinear identity by Stokes, deduce symmetry from a bidegree-counting vanishing, and deduce positivity from the Kähler positivity of $\sqrt{-1}\omega \wedge \bar{\omega }$.

Step 1 — bilinear identity for closed 1-forms. Cut $X$ along the loops $a_1,\dots ,a_g,b_1,\dots ,b_g$ to obtain a $4g$-gon $P$ with sides labelled $a_1,b_1,a_1^{-1},b_1^{-1},\dots ,a_g,b_g,a_g^{-1},b_g^{-1}$ in cyclic order. The interior of $P$ is simply connected, so any closed 1-form $\omega$ on $X$ has a primitive $f$ with $df=\omega$ on the interior of $P$. For two closed 1-forms $\omega ,\eta$ on $X$ with primitive $f$ for $\omega$ on $P$, Stokes' theorem on $P$ gives

$${\int }_X\omega \wedge \eta ={\int }_{P}d(f\eta )={\int }_{\partial P}f\eta .$$

Compute the boundary integral by pairing the sides $a_k,a_k^{-1}$ and $b_k,b_k^{-1}$ in $\partial P$. On $a_k^{-1}$ (the same path as $a_k$ but traversed backwards on the surface), the primitive $f$ jumps by the period $-{\int }_{b_k}\omega$ relative to its value on $a_k$ (the path from a point of $a_k^{-1}$ to the corresponding point of $a_k$ in $P$ goes around the loop $b_k$). Symmetrically, on $b_k^{-1}$ the primitive $f$ jumps by $+{\int }_{a_k}\omega$. Pairing the sides and summing:

$${\int }_X\omega \wedge \eta =\sum _{k=1}^g\left[{\int }_{a_k}\omega \cdot {\int }_{b_k}\eta -{\int }_{b_k}\omega \cdot {\int }_{a_k}\eta \right].$$

This is Riemann's bilinear identity. It holds for any pair of closed 1-forms.

Step 2 — symmetry (RB1) from holomorphic-holomorphic vanishing. Apply the identity with $\omega ={\omega }_i,\eta ={\omega }_j$ (both holomorphic, hence closed). On a Riemann surface there are no $(2,0)$-forms (real dimension is $2$, complex dimension is $1$, so bidegree $(p,q)$ requires $p,q\le 1$), so ${\omega }_i\wedge {\omega }_j\in {\mathcal{A}}^{2,0}(X)=0$. The left side of the bilinear identity vanishes:

$$0={\int }_X{\omega }_i\wedge {\omega }_j=\sum _{k=1}^g\left[{\int }_{a_k}{\omega }_i\cdot {\int }_{b_k}{\omega }_j-{\int }_{b_k}{\omega }_i\cdot {\int }_{a_k}{\omega }_j\right].$$

Substitute the normalisation ${\int }_{a_k}{\omega }_i={\delta }_{ik}$ and ${\int }_{a_k}{\omega }_j={\delta }_{jk}$:

$$0=\sum _{k=1}^g\left[{\delta }_{ik}{\tau }_{jk}-{\tau }_{ik}{\delta }_{jk}\right]={\tau }_{ji}-{\tau }_{ij}.$$

Hence ${\tau }_{ij}={\tau }_{ji}$, i.e. $\tau ={\tau }^T$. ✓ (RB1)

Step 3 — positivity (RB2) from Kähler positivity. Apply the identity with $\omega ={\omega }_i,\eta =\overline{{\omega }_j}$. The form $\overline{{\omega }_j}$ is anti-holomorphic, hence closed (its complex conjugate ${\omega }_j$ is holomorphic, hence closed), so the bilinear identity applies. The left side is

$${\int }_X{\omega }_i\wedge \overline{{\omega }_j}.$$

The wedge ${\omega }_i\wedge \overline{{\omega }_j}$ is a $(1,1)$-form on $X$ with values in $\mathbb{C}$. Locally, write ${\omega }_i=f_i(z)dz$ and ${\omega }_j=f_j(z)dz$ in a coordinate $z$; then ${\omega }_i\wedge \overline{{\omega }_j}=f_i{\bar{f}}_{j}dz\wedge d\bar{z}$, and the form $\frac{i}{2}dz\wedge d\bar{z}=dx\wedge dy$ is the standard volume form on the chart. Hence

$$\frac{i}{2}{\int }_X{\omega }_i\wedge \overline{{\omega }_j}$$

is a Hermitian inner product on $H^0(X,{\Omega }_X^1)$, and is positive definite since $\frac{i}{2}{\omega }_i\wedge \overline{{\omega }_i}=|f_i{|}^{2}dx\wedge dy$ is a positive volume form whenever ${\omega }_i\ne 0$.

Compute the right side of the bilinear identity. Since $\overline{{\omega }_j}$ is anti-holomorphic, ${\int }_{a_k}\overline{{\omega }_j}=\overline{{\int }_{a_k}{\omega }_j}=\overline{{\delta }_{jk}}={\delta }_{jk}$ and ${\int }_{b_k}\overline{{\omega }_j}=\overline{{\tau }_{jk}}$. The bilinear identity gives

$${\int }_X{\omega }_i\wedge \overline{{\omega }_j}=\sum _{k=1}^g\left[{\delta }_{ik}\overline{{\tau }_{jk}}-{\tau }_{ik}{\delta }_{jk}\right]=\overline{{\tau }_{ji}}-{\tau }_{ij}.$$

By the symmetry (RB1) just established, ${\tau }_{ji}={\tau }_{ij}$, so $\overline{{\tau }_{ji}}=\overline{{\tau }_{ij}}$, and the right side equals $\overline{{\tau }_{ij}}-{\tau }_{ij}=-2i\mathrm{Im}{\tau }_{ij}$. Multiplying by $\frac{i}{2}$:

$$\frac{i}{2}{\int }_X{\omega }_i\wedge \overline{{\omega }_j}=\frac{i}{2}\cdot (-2i\mathrm{Im}{\tau }_{ij})=\mathrm{Im}{\tau }_{ij}.$$

The left side is the matrix of a positive-definite Hermitian form on $H^0(X,{\Omega }_X^1)$. The right side is the matrix $\mathrm{Im}\tau$. Hence $\mathrm{Im}\tau >0$ as a real symmetric matrix. ✓ (RB2)

Step 4 — equivalence of the non-normalised form. In a non-normalised basis where $A_{ij}={\int }_{a_j}{\omega }_i$ may be any invertible $g\times g$ complex matrix, change basis ${\omega }_i\mapsto (A^{-1}{)}_{ki}{\omega }_k$ to obtain a new basis ${\tilde{\omega }}_1,\dots ,{\tilde{\omega }}_g$ with ${\int }_{a_j}{\tilde{\omega }}_i={\delta }_{ij}$. The new $b$-period matrix is $\tilde{\tau }=B{A}^{-1}$ (matrix product). Applying (RB1) to $\tilde{\tau }$ gives $B{A}^{-1}=(B{A}^{-1}{)}^T=(A^{-1}{)}^T{B}^T$, equivalently $A^{T}B=B^{T}A$, equivalently $A{B}^T=B{A}^T$ after transposition. Applying (RB2) gives $\mathrm{Im}(B{A}^{-1})>0$. $\square$

The four-step structure follows Donaldson §12, with the bilinear identity derived as a Stokes-on-the-$4g$-gon computation; Forster §21 reorganises the same content around the residue calculus on the cut surface; Griffiths-Harris §2.7 develops the bilinear relations as a special case of the Hodge-Riemann relations on a polarised Hodge structure of weight one. The underlying content — bilinear identity, holomorphic-holomorphic vanishing, Kähler positivity — is identical.

Bridge. The bilinear relations proven here close the period theory begun in 06.04.03 Hodge decomposition and built up in 06.06.02 period matrix; the present unit builds toward the principal polarisation that promotes the Jacobian variety 06.06.03 from a complex torus to a projective abelian variety, and the bilinear identity appears again in the proof of 06.06.06 Jacobi inversion (where the differential of the Abel-Jacobi map is a Brill-Noether matrix whose Gram matrix against the bilinear pairing extracts the Riemann-Roch dimension count). The engine that converts the Hodge data into the Siegel upper half space is exactly the Stokes-on-the-$4g$-gon argument of Step 1: the topological symplectic structure of the homology, refined by the holomorphic structure of the differentials, descends through the bilinear identity to the matrix-level relations on $\tau$.

Read in the opposite direction, the Siegel upper half space is the moduli space of all data that could be the period matrix of a curve, and the image of the period map is the Jacobi locus — a $(3g-3)$-dimensional subvariety of the $g(g+1)/2$-dimensional space ${\mathfrak{H}}_g$. The geometric content of the Schottky problem is exactly the question of identifying this image, and the bilinear relations are its first cut: they reduce the matrix data from ${\mathbb{C}}^{g\times g}\times {\mathbb{C}}^{g\times g}$ (the space of $(A\mid B)$) down to ${\mathfrak{H}}_g$. The Riemann theta function $\theta (z;\tau )$ on ${\mathbb{C}}^g\times {\mathfrak{H}}_g$ (cf. 06.06.05) uses the bilinear relations directly in its quasi-periodicity, and through Riemann's vanishing theorem links the bilinear-relation data to the divisor theory of the curve. Putting these together, the foundational insight is that the topological symplectic intersection form on $H_1(X,\mathbb{Z})$, paired with the Kähler positivity of the Hodge decomposition, fully determines the matrix-level constraints on the period matrix; and the bridge is Riemann's bilinear identity, which threads the topology of the cut surface through the bidegree calculus of holomorphic forms.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib does not currently formalise the Siegel upper half space ${\mathfrak{H}}_g$, the period matrix of a smooth projective complex curve as a morphism into ${\mathfrak{H}}_g$, or Riemann's bilinear relations as a theorem. 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 symplectic basis of integral homology of a smooth projective curve, the integral pairing of holomorphic 1-forms against integral 1-cycles, Riemann's bilinear identity from cutting along the symplectic basis to a $4g$-gon, and the Kähler positivity of $\sqrt{-1}\omega \wedge \bar{\omega }$ on a compact Riemann surface). Each is a candidate Mathlib contribution; until then this unit ships with lean_status: none.

Advanced results [Master]

Riemann's bilinear relations are the analytical foundation of the modern theta-function theory of compact Riemann surfaces. The statement and proof presented above is the curve case; both relations generalise, and this section spells out the higher-dimensional generalisation, the connection to integrable systems, and the modular geometry of ${\mathfrak{H}}_g$.

Hodge-Riemann bilinear relations on a polarised Hodge structure. For a compact Kähler manifold $X$ of complex dimension $n$ with Kähler form $\omega \in H^{1,1}(X,\mathbb{R})$ and primitive cohomology $P^k(X):=\ker (L^{n-k+1}:H^k(X,\mathbb{C})\to H^{2n-k+2}(X,\mathbb{C}))$ (where $L$ denotes wedge with $\omega$), the Hodge-Riemann bilinear relations assert that the form $Q(\alpha ,\beta ):=(-1{)}^{k(k-1)/2}{\int }_X\alpha \wedge \beta \wedge {\omega }^{n-k}$ pairs $H^{p,q}(X)\cap P^k$ with $H^{n-p,n-q}(X)\cap P^k$ non-degenerately, vanishes on $H^{p^{\prime },q^{\prime }}(X)\cap P^k$ for $(p^{\prime },q^{\prime })\ne (n-p,n-q)$, and the form $i^{p-q}(-1{)}^{k(k-1)/2}Q(\alpha ,\bar{\alpha })>0$ for $\alpha \in H^{p,q}(X)\cap P^k$ non-zero. For $n=1$ and $k=1$, the primitive cohomology is all of $H^1$ (the Lefschetz operator on $H^1$ is into $H^3=0$), the bilinear form $Q$ becomes the cup product $H^1\otimes H^1\to H^2=\mathbb{C}$, and the Hodge-Riemann relations reduce to (RB1) and (RB2) on the period matrix.

Riemann theta function and quasi-periodicity. Define $\theta (z;\tau ):={\sum }_{n\in {\mathbb{Z}}^g}\exp (\pi i{n}^T\tau n+2\pi i{n}^{T}z)$ on ${\mathbb{C}}^g\times {\mathfrak{H}}_g$. The series converges absolutely and uniformly on compact subsets by Exercise 7. The quasi-periodicity reads

$$\theta (z+n;\tau )=\theta (z;\tau ),\theta (z+\tau m;\tau )=\exp (-\pi i{m}^T\tau m-2\pi i{m}^{T}z)\theta (z;\tau )$$

for $n,m\in {\mathbb{Z}}^g$. The first periodicity uses the integrality of the lattice; the second uses the symmetry (RB1) directly in the manipulation of the quadratic form $n^T\tau n$ under the shift $n\mapsto n+m$. Hence $\theta (\cdot ;\tau )$ descends to a holomorphic section of a holomorphic line bundle $L_{\tau }$ on the principally polarised abelian variety $A_{\tau }={\mathbb{C}}^g/{\Lambda }_{\tau }$. The line bundle $L_{\tau }$ is the theta line bundle, and its first Chern class is the principal polarisation. The vanishing locus ${\Theta }_{\tau }:=\{z:\theta (z;\tau )=0\}/{\Lambda }_{\tau }\subset A_{\tau }$ is the theta divisor, a divisor in the abelian variety whose class generates $\mathrm{Pic}(A_{\tau })$ modulo translations.

Schottky problem. The period map $j:{\mathcal{M}}_g\to {\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ sends a curve to its period matrix modulo the action of the symplectic group on ${\mathfrak{H}}_g$. The image $j({\mathcal{M}}_g)$ is the Jacobi locus, of complex dimension $3g-3$ (for $g\ge 2$), inside the moduli space ${\mathcal{A}}_g:={\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ of principally polarised abelian varieties of complex dimension $g(g+1)/2$. For $g\le 3$ the dimensions match ($3\cdot 1-3=0$ and $1$; $3\cdot 2-3=3$ and $3$; $3\cdot 3-3=6$ and $6$), so the period map is a birational equivalence. For $g\ge 4$ the codimension of the Jacobi locus is positive ($g(g+1)/2-(3g-3)=(g-2)(g-3)/2$ for $g\ge 3$), and the Schottky problem asks for explicit equations cutting out $j({\mathcal{M}}_g)$ inside ${\mathcal{A}}_g$. Several solutions are known:

• Schottky 1888. For $g=4$, a single explicit polynomial relation among theta-null values cuts out the Jacobi locus inside ${\mathcal{A}}_4$. The relation is the Schottky-Jung relation, refined by Igusa.
• Andreotti-Mayer 1967. The Jacobi locus is contained in the locus $N_{g-4}$ of principally polarised abelian varieties whose theta divisor has at least a $(g-4)$-dimensional singular locus.
• Novikov-Shiota. Novikov conjectured that the Jacobi locus inside ${\mathcal{A}}_g$ is characterised by the property that the function $u(x,y,t):=-2{\partial }_x^2\log \theta (xV+yU+tW+z_0;\tau )$ solves the KP equation $\frac{3}{4}{u}_{yy}=(u_t-\frac{3}{2}u{u}_x-\frac{1}{4}{u}_{xxx}{)}_x$ for some constant vectors $V,U,W\in {\mathbb{C}}^g$ and a starting point $z_0$. Shiota 1986 proved Novikov's conjecture by constructing the curve from the KP-flow data on the Jacobian.
• Welters trisecant conjecture, Krichever 2006. The Jacobi locus is characterised by the existence of a single trisecant line of the Kummer variety embedding $A_{\tau }\to {\mathbb{P}}^{2^g-1}$. Krichever's proof uses the algebraic-integrability data of the KP hierarchy.

Modular geometry of ${\mathfrak{H}}_g$. The symplectic group $\mathrm{Sp}(2g,\mathbb{Z})$ acts on ${\mathfrak{H}}_g$ by

$$\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\cdot \tau :=(A\tau +B)(C\tau +D{)}^{-1},$$

generalising the $\mathrm{SL}(2,\mathbb{Z})$ action on the upper half plane. The quotient ${\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ is the Siegel modular variety ${\mathcal{A}}_g$, the moduli space of principally polarised abelian varieties. Modular forms on ${\mathfrak{H}}_g$ — holomorphic functions transforming by an automorphy factor under $\mathrm{Sp}(2g,\mathbb{Z})$ — generalise the classical modular forms on the upper half plane (the case $g=1$). The graded ring of Siegel modular forms is finitely generated (Igusa 1967 for $g=2$; later refinements for higher genus), and its Proj is ${\mathcal{A}}_g$.

Synthesis. The bilinear relations are the engine that converts the topological data of a compact Riemann surface — the symplectic intersection form on $H_1(X,\mathbb{Z})$ — into the analytical data of a point in the Siegel upper half space, and through that point produces every transcendental object on the Jacobian. The four-step proof — bilinear identity from Stokes on the $4g$-gon, holomorphic-holomorphic vanishing, Kähler positivity, change of basis — is the curve case of the general Hodge-Riemann bilinear relations on a polarised Hodge structure of weight one. Read in the opposite direction, every $\tau \in {\mathfrak{H}}_g$ defines a principally polarised abelian variety $A_{\tau }$ via the lattice ${\mathbb{Z}}^g+\tau {\mathbb{Z}}^g$, and the question of which $\tau$ come from curves — the Schottky problem — is the question of identifying the Jacobi locus $j({\mathcal{M}}_g)$ inside the Siegel modular variety ${\mathcal{A}}_g={\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$.

The geometric content of the Riemann theta function is the explicit holomorphic section of the principal-polarisation line bundle on $A_{\tau }$ whose vanishing locus is the theta divisor; quasi-periodicity of $\theta$ under lattice translations uses (RB1) in an essential way. The dimension count ${\dim }_{\mathbb{C}}{\mathfrak{H}}_g=g(g+1)/2$ versus ${\dim }_{\mathbb{C}}{\mathcal{M}}_g=3g-3$ exhibits the period map as a codimension-$(g-2)(g-3)/2$ inclusion for $g\ge 3$ — a strict inclusion that opens the Schottky problem and the modern theory of Jacobi loci. Putting these together, the period matrix, the principal polarisation, the theta function, the theta divisor, the Schottky problem, the KP-hierarchy characterisation, and the modular geometry of ${\mathfrak{H}}_g$ together form a single connected structure that the bilinear relations organise; and the curve case of the Hodge-Riemann pairing is the foundational identification on which the entire Jacobian-side analytic theory of compact Riemann surfaces rests.

Full proof set [Master]

Lemma (Stokes on a $4g$-gon). Let $X$ be a closed Riemann surface of genus $g$ with symplectic basis $a_1,\dots ,a_g,b_1,\dots ,b_g$ of $H_1(X,\mathbb{Z})$. Cutting $X$ along the symplectic basis produces a $4g$-gon $P$ with sides $a_1,b_1,a_1^{-1},b_1^{-1},\dots ,a_g,b_g,a_g^{-1},b_g^{-1}$ in cyclic order, and the interior of $P$ is simply connected.

Proof. The standard polygon model of a closed orientable surface of genus $g$ is the $4g$-gon with the sides identified in the pattern $a_1{b}_1{a}_1^{-1}{b}_1^{-1}\cdots a_g{b}_g{a}_g^{-1}{b}_g^{-1}$. The interior is homeomorphic to an open disc, hence simply connected. The boundary identification recovers the surface as a CW complex with a single $0$-cell (all corners of $P$ identify), $2g$ $1$-cells (the labelled sides), and a single $2$-cell (the interior of $P$). The Euler characteristic is $1-2g+1=2-2g$, matching the topological invariant of the genus-$g$ surface. $\square$

Lemma (Riemann's bilinear identity). For closed 1-forms $\omega ,\eta$ on a closed Riemann surface $X$ of genus $g$,

$${\int }_X\omega \wedge \eta =\sum _{k=1}^g\left[{\int }_{a_k}\omega \cdot {\int }_{b_k}\eta -{\int }_{b_k}\omega \cdot {\int }_{a_k}\eta \right].$$

Proof. By the previous lemma, the cut surface $P$ is simply connected, so $\omega =df$ on $P$ for a primitive $f$ (potentially multivalued on the original $X$). By Stokes:

$${\int }_X\omega \wedge \eta ={\int }_{P}df\wedge \eta ={\int }_{P}d(f\eta )={\int }_{\partial P}f\eta ,$$

where the second equality uses $d\eta =0$ (since $\eta$ is closed). The boundary $\partial P$ consists of the labelled sides $a_1,b_1,a_1^{-1},b_1^{-1},\dots$ in cyclic order. Pair each side with its inverse: on $a_k^{-1}$, the primitive $f$ satisfies $f({\sigma }_k(p))=f(p)-{\int }_{b_k}\omega$ where ${\sigma }_k$ is the identification of $a_k$ with $a_k^{-1}$ on $X$ (the path from a point on $a_k$ to the identified point on $a_k^{-1}$ traverses the loop $b_k$ once, and $\omega$ has period ${\int }_{b_k}\omega$ along $b_k$). Hence ${\int }_{a_k}f\eta +{\int }_{a_k^{-1}}f\eta =-{\int }_{b_k}\omega \cdot {\int }_{a_k}\eta$. Symmetrically the $b_k,b_k^{-1}$ pairing contributes ${\int }_{a_k}\omega \cdot {\int }_{b_k}\eta$. Summing over $k$ gives the stated identity. $\square$

Theorem (Riemann's bilinear relations, full statement). Statement and proof as in the Intermediate-tier Key theorem section.

Proof. The Intermediate-tier proof goes through using the bilinear-identity lemma as the packaged input. Step 2 (symmetry) applies the identity to $\omega ={\omega }_i,\eta ={\omega }_j$ and uses bidegree vanishing ${\omega }_i\wedge {\omega }_j\in {\Omega }^{2,0}(X)=0$ on the curve. Step 3 (positivity) applies the identity to $\omega ={\omega }_i,\eta =\overline{{\omega }_j}$ and uses Kähler positivity of $\frac{i}{2}\omega \wedge \bar{\omega }$ as a positive volume form. Step 4 (non-normalised form) is a change-of-basis computation. $\square$

Corollary (period matrix lies in ${\mathfrak{H}}_g$). For any compact Riemann surface $X$ of genus $g$ and any choice of symplectic homology basis, the normalised period matrix $\tau$ lies in the Siegel upper half space ${\mathfrak{H}}_g$.

Proof. (RB1) gives $\tau ={\tau }^T$, the symmetry condition; (RB2) gives $\mathrm{Im}\tau >0$, the positivity condition. Together these are the defining conditions of ${\mathfrak{H}}_g$. $\square$

Corollary (Jacobian as principally polarised abelian variety). Let $X$ be a compact Riemann surface of genus $g$ with normalised period matrix $\tau$. The Jacobian $\mathrm{Jac}(X)={\mathbb{C}}^g/({\mathbb{Z}}^g+\tau {\mathbb{Z}}^g)$ is a principally polarised abelian variety, with polarisation given by the symplectic form on the lattice and the positive Hermitian form $h(z,w)=z^T(\mathrm{Im}\tau {)}^{-1}\bar{w}$ on ${\mathbb{C}}^g$.

Proof. The lattice ${\Lambda }_{\tau }={\mathbb{Z}}^g+\tau {\mathbb{Z}}^g$ has full rank $2g$ in ${\mathbb{C}}^g$ because $\mathrm{Im}\tau$ is invertible by (RB2): the $2g$ vectors $e_1,\dots ,e_g,\tau e_1,\dots ,\tau e_g$ are $\mathbb{R}$-linearly independent iff the real $2g\times 2g$ matrix $\left(\begin{array}{cc}{I}_g& \mathrm{Re}\tau \\ 0& \mathrm{Im}\tau \end{array}\right)$ is invertible iff $\det \mathrm{Im}\tau \ne 0$. The complex torus $A_{\tau }={\mathbb{C}}^g/{\Lambda }_{\tau }$ inherits the structure of a complex Lie group from the additive structure of ${\mathbb{C}}^g$. The alternating form $E((m,n),(m^{\prime },n^{\prime }))=m^T{n}^{\prime }-n^T{m}^{\prime }$ on ${\Lambda }_{\tau }\cong {\mathbb{Z}}^{2g}$ corresponds to the symplectic intersection form on $H_1(X,\mathbb{Z})$ in the symplectic basis. The Hermitian form $h(z,w)=z^T(\mathrm{Im}\tau {)}^{-1}\bar{w}$ is positive-definite by (RB2). Compatibility (the Riemann form conditions) follows from $\tau ={\tau }^T$ (RB1): $E$ extended $\mathbb{R}$-linearly to ${\mathbb{C}}^g$ satisfies $E(iz,iw)=E(z,w)$ and $E(z,iz)=h(z,z)>0$, by direct computation using (RB1) and (RB2). Principality is the unimodularity of the symplectic basis, automatic by Poincaré duality on $H_1(X,\mathbb{Z})$. $\square$

Corollary (quasi-periodicity of theta). For $\tau \in {\mathfrak{H}}_g$, $z\in {\mathbb{C}}^g$, $n,m\in {\mathbb{Z}}^g$,

$$\theta (z+n;\tau )=\theta (z;\tau ),\theta (z+\tau m;\tau )=\exp (-\pi i{m}^T\tau m-2\pi i{m}^{T}z)\theta (z;\tau ).$$

Proof. For $\theta (z+n;\tau )$, substitute $z\mapsto z+n$ in the series and absorb the integer shift into the index: $\theta (z+n;\tau )={\sum }_{n^{\prime }\in {\mathbb{Z}}^g}\exp (\pi i{n}^{\prime T}\tau n^{\prime }+2\pi i{n}^{\prime T}z+2\pi i{n}^{\prime T}n)$. The factor $\exp (2\pi i{n}^{\prime T}n)$ equals $1$ for $n^{\prime },n\in {\mathbb{Z}}^g$, recovering $\theta (z;\tau )$.

For $\theta (z+\tau m;\tau )$, change index $n^{\prime }=n+m$:

$$\theta (z+\tau m;\tau )=\sum _{n\in {\mathbb{Z}}^g}\exp (\pi i{n}^T\tau n+2\pi i{n}^T(z+\tau m)).$$

Reindex with $n=n^{\prime }-m$:

$$=\sum _{n^{\prime }\in {\mathbb{Z}}^g}\exp (\pi i(n^{\prime }-m{)}^T\tau (n^{\prime }-m)+2\pi i(n^{\prime }-m{)}^T(z+\tau m)).$$

Expand the quadratic using (RB1) (the symmetry $\tau ={\tau }^T$ allows the cross-term simplification $m^T\tau n^{\prime }=n^{\prime T}\tau m$):

$$=\sum _{n^{\prime }\in {\mathbb{Z}}^g}\exp (\pi i{n}^{\prime T}\tau n^{\prime }-2\pi i{n}^{\prime T}\tau m+\pi i{m}^T\tau m+2\pi i{n}^{\prime T}z+2\pi i{n}^{\prime T}\tau m-2\pi i{m}^{T}z-2\pi i{m}^T\tau m).$$

The $-2\pi i{n}^{\prime T}\tau m+2\pi i{n}^{\prime T}\tau m$ terms cancel. The remaining $z$- and $\tau$-independent factor is $\exp (\pi i{m}^T\tau m-2\pi i{m}^{T}z-2\pi i{m}^T\tau m)=\exp (-\pi i{m}^T\tau m-2\pi i{m}^{T}z)$. The remaining sum is $\theta (z;\tau )$. $\square$

Corollary (period matrix is in ${\mathfrak{H}}_g$, dimension count). The Jacobi locus $j({\mathcal{M}}_g)\subset {\mathcal{A}}_g$ has complex dimension $3g-3$ for $g\ge 2$, while ${\dim }_{\mathbb{C}}{\mathcal{A}}_g=g(g+1)/2$.

Proof. The moduli space ${\mathcal{M}}_g$ of smooth projective curves of genus $g$ has complex dimension $3g-3$ for $g\ge 2$ (the Riemann count, deformation theory of curves). The period map $j:{\mathcal{M}}_g\to {\mathcal{A}}_g$ is injective by Torelli's theorem, so $\dim j({\mathcal{M}}_g)=3g-3$. The Siegel modular variety ${\mathcal{A}}_g={\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ has the same complex dimension as ${\mathfrak{H}}_g$, namely $g(g+1)/2$ (the dimension of the space of symmetric $g\times g$ matrices). For $g\ge 4$ the codimension is $(g-2)(g-3)/2\ge 1$, opening the Schottky problem. $\square$

Connections [Master]

• Period matrix 06.06.02. The period matrix $\tau \in M_{g\times g}(\mathbb{C})$ is the data on which the bilinear relations operate; the present unit promotes the period-matrix construction to a map into the Siegel upper half space ${\mathfrak{H}}_g$, and the bilinear relations are the structural constraints satisfied by every period matrix of a compact Riemann surface.

• Hodge decomposition on a compact Riemann surface 06.04.03. The Hodge decomposition supplies the basis ${\omega }_1,\dots ,{\omega }_g$ of $H^0(X,{\Omega }_X^1)=H^{1,0}(X)$ used in the period-matrix definition, and the Kähler positivity $\frac{i}{2}\omega \wedge \bar{\omega }\ge 0$ used in Step 3 of the proof of (RB2). The bilinear relations are the matrix-level shadow of the Hodge-Riemann bilinear relations on the polarised Hodge structure $H^1(X,\mathbb{Z})$ of weight one.

• Jacobi inversion theorem 06.06.06. The bilinear relations promote the Jacobian $\mathrm{Jac}(X)={\mathbb{C}}^g/{\Lambda }_{\tau }$ from a complex torus to a principally polarised abelian variety, supplying the polarisation that the Jacobi inversion theorem uses to identify the theta divisor with the image of the degree-$(g-1)$ Abel-Jacobi map.

• Jacobian variety 06.06.03. The Jacobian is a principally polarised abelian variety because its period matrix lies in ${\mathfrak{H}}_g$. Without (RB1) the Jacobian would not be polarised; without (RB2) it would not even be a complex torus (the lattice would fail to have full rank).

• Theta function 06.06.05. The Riemann theta function $\theta (z;\tau )$ on ${\mathbb{C}}^g\times {\mathfrak{H}}_g$ uses (RB1) in its quasi-periodicity (Corollary above); without the symmetry of $\tau$ the cross-terms in the lattice-translation computation would not cancel, and $\theta$ would fail to descend to a section of a line bundle on the Jacobian.

• Holomorphic 1-form 06.06.01. The basis ${\omega }_1,\dots ,{\omega }_g$ of holomorphic 1-forms is the input to the period matrix, and the bidegree-counting fact ${\omega }_i\wedge {\omega }_j\in {\Omega }^{2,0}(X)=0$ on a curve is the key identity that gives (RB1) from the bilinear identity.

• Abel-Jacobi map 06.06.04. The Abel-Jacobi map $A:{\mathrm{Div}}^0(X)\to \mathrm{Jac}(X)$ takes values in a principally polarised abelian variety because of the bilinear relations; Abel's theorem and Jacobi inversion combined identify ${\mathrm{Pic}}^0(X)\cong \mathrm{Jac}(X)$ as principally polarised abelian varieties.

• Riemann-Roch theorem for compact Riemann surfaces 06.04.01. The dimension count $\dim H^0(X,{\Omega }_X^1)=g$ (used in defining the size of the period matrix) is a Riemann-Roch consequence; the bilinear relations operate on the resulting $g$-dimensional space of holomorphic 1-forms.

• Serre duality on a curve 06.04.04. The Hodge pairing $H^{1,0}(X)\times H^{0,1}(X)\to \mathbb{C}$ implicit in the positivity argument of (RB2) is the Serre duality pairing for the line bundle $L={\mathcal{O}}_X$. The bilinear relations recover the matrix form of this pairing on the integer lattice of periods.

Historical & philosophical context [Master]

Bernhard Riemann established the bilinear relations and the Siegel-upper-half-space data of compact Riemann surfaces in his 1857 paper Theorie der Abel'schen Functionen ^{[Riemann 1857]} (Crelle 54, 115-155), the foundational paper of the modern theory of Riemann surfaces. Riemann's text introduced the Riemann surface as the natural domain of multivalued algebraic functions, defined the period matrix as the table of integrals of holomorphic differentials against integral 1-cycles, derived the bilinear identity from Stokes-type reasoning on the cut surface, and stated both relations (RB1) and (RB2). Riemann then built on the bilinear relations to construct the Riemann theta function and prove Riemann's vanishing theorem, identifying the zero locus of theta with the image of ${\mathrm{Sym}}^{g-1}(X)$ under the degree-$(g-1)$ Abel-Jacobi map shifted by the Riemann constant.

Riemann's proof of the bilinear identity used the (then-controversial) Dirichlet principle: the assertion that minimisers of the Dirichlet energy exist on a compact Riemann surface. Karl Weierstrass had pointed out a logical gap in the Dirichlet principle in 1869, leaving Riemann's bilinear-relations proof temporarily without rigorous foundation. David Hilbert closed the gap in 1900 by proving the Dirichlet principle via the direct method of the calculus of variations, restoring Riemann's argument. Hermann Weyl's 1913 Die Idee der Riemannschen Fläche (Teubner) gave the first modern textbook treatment, defining abstract Riemann surfaces and presenting the bilinear relations with rigorous foundations.

Friedrich Schottky 1888 Zur Theorie der Abelschen Functionen von vier Variabeln ^{[Schottky 1888]} (Crelle 102, 304-352) was the first paper to study the Schottky problem — characterising which points of ${\mathfrak{H}}_g$ come from curves — and gave a single explicit polynomial relation among theta-null values that cuts out the Jacobi locus $j({\mathcal{M}}_4)\subset {\mathcal{A}}_4$. The Schottky-Jung relations (Schottky-Jung 1909) extended the framework. Andreotti-Mayer 1967 On period relations for abelian integrals on algebraic curves (Ann. Sc. Norm. Sup. Pisa 21, 189-238) established the singularity-of-theta-divisor characterisation. The Novikov conjecture characterising Jacobians via the KP equation was proven by Takahiro Shiota 1986 Characterization of Jacobian varieties in terms of soliton equations ^{[Shiota 1986]} (Invent. Math. 83, 333-382); the Welters trisecant conjecture was proven by Igor Krichever 2006 Characterizing Jacobians via trisecants of the Kummer variety (Ann. Math. 172, 485-516).

The transition from Riemann's transcendental treatment to the modern Hodge-theoretic framework was constructed by André Weil, Pierre Deligne, and Phillip Griffiths through the 1950s-1970s. Weil 1948 Variétés abéliennes et courbes algébriques (Hermann) gave the algebraic-geometric construction of the Jacobian as a principally polarised abelian variety, replacing Riemann's analytic construction. Mumford's Tata Lectures on Theta I-III (1983-1991) ^{[Mumford Tata Lectures I; ref: TODO_REF Mumford Tata Lectures II]} presented the bilinear relations as the case $n=1,k=1$ of the Hodge-Riemann relations on a polarised Hodge structure, and developed the Schottky problem and the modern theta-divisor geometry. Donaldson's Riemann Surfaces (Oxford GTM 22, 2011) §12 ^{[Donaldson Riemann Surfaces]} presents the bilinear relations via the Stokes-on-the-$4g$-gon argument used here, integrated with the Hodge / Dirichlet-energy proof of $\dim H^0(X,{\Omega }_X^1)=g$ in §10. Griffiths-Harris Principles of Algebraic Geometry (1978) §2.7 ^{[Griffiths-Harris]} develops the bilinear relations as a special case of the Hodge-Riemann pairing on a Kähler manifold, and integrates them with the Brill-Noether stratification of the Picard variety.

Bibliography [Master]

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