= Solution
Let $X$ be a connected <compact Riemann surface> of <geometric genus> $g$, equivalently a <smooth projective curve> over $\mathbb C$, and put $V=H^0(X,K_X)$. Choose a <symplectic basis> $a_1,\ldots,a_g,b_1,\ldots,b_g$ of $H_1(X,\mathbb Z)$ with $a_i\cdot b_j=\delta_{ij}$. For a closed <differential form> $\alpha$, write $A_i(\alpha)=\int_{a_i}\alpha$ and $B_i(\alpha)=\int_{b_i}\alpha$. The underlying bilinear identity is
$$
\int_X\alpha\wedge\beta=\sum_{i=1}^g\bigl(A_i(\alpha)B_i(\beta)-B_i(\alpha)A_i(\beta)\bigr).
$$
It holds for any two smooth closed one-forms. In particular, the <Riemann bilinear relations> for <holomorphic differential forms> are
$$
\sum_i\bigl(A_i(\omega)B_i(\eta)-B_i(\omega)A_i(\eta)\bigr)=0,
\qquad
i\sum_i\bigl(A_i(\omega)\overline{B_i(\omega)}-B_i(\omega)\overline{A_i(\omega)}\bigr)>0
$$
for $\omega,\eta\in V$ and $\omega\ne0$ in the inequality. The inequality fixes the orientation and the sign convention.
To prove the identity, cut $X$ along the <symplectic basis> to a fundamental polygon $P$. On $P$, a closed one-form $\alpha$ has a primitive $F$. The <Stokes theorem> gives
$$
\int_P\alpha\wedge\beta=\int_{\partial P}F\beta.
$$
Opposite copies of each cut have opposite orientations, while their values of $F$ differ by the corresponding period of $\alpha$. Pairing those edges therefore leaves $A_i(\alpha)\int_{b_i}\beta-B_i(\alpha)\int_{a_i}\beta$ for the $i$th handle. Summing proves the identity. The same calculation can be made with a small disk about the polygon's vertices removed; its extra boundary terms tend to zero, so the vertices cause no additional term.
For <holomorphic differential forms> $\omega,\eta$, their <wedge product of differential forms> vanishes: locally both are multiples of $dz$. This proves the first relation. If $\omega=f(z)\,dz$ and $z=x+iy$, then
$$
i\omega\wedge\overline\omega=2|f(z)|^2\,dx\wedge dy.
$$
A nonzero <holomorphic differential form> is nonzero on an open set, so the integral is strictly positive. Applying the identity to $\omega,\overline\omega$ proves the second relation.
These relations first prove that the map $V\to\mathbb C^g$ taking $a$-periods is injective: if every $A_i(\omega)$ vanishes, the positive integral just computed would vanish. Since $\dim V=g$ by the <Riemann-Roch theorem>, it is an isomorphism. Consequently there is a unique <basis> $\omega_1,\ldots,\omega_g$ with $\int_{a_j}\omega_i=\delta_{ij}$. Set $\tau_{ij}=\int_{b_j}\omega_i$. The first relation gives $\tau_{ij}=\tau_{ji}$, and the second gives, for any nonzero column $c$,
$$
i\int_X\left(\sum_i c_i\omega_i\right)\wedge\overline{\left(\sum_i c_i\omega_i\right)}=2c^t(\operatorname{Im}\tau)\overline c>0.
$$
Thus \b[the normalized <period matrix of a complex torus> is symmetric and has positive-definite imaginary part]:
$$
\boxed{\tau^t=\tau,\qquad Y=\operatorname{Im}\tau>0.}
$$
The integration map sends a cycle $\gamma$ to the functional $\omega\mapsto\int_\gamma\omega$ in $V^*$. In the normalized <basis> its image is the <period lattice>
$$
\Lambda=\mathbb Z^g+\tau\mathbb Z^g.
$$
Indeed, the real-linear map $(x,y)\mapsto x+\tau y$ from $\mathbb R^{2g}$ to $\mathbb C^g$ is an isomorphism: its imaginary part is $Yy$, and $Y$ is invertible. Its image of $\mathbb Z^{2g}$ is therefore discrete and has compact quotient. This proves that
$$
\boxed{\operatorname{Jac}(X)=V^*/\Lambda\cong\mathbb C^g/(\mathbb Z^g+\tau\mathbb Z^g)}
$$
is a well-defined $g$-dimensional <complex torus>, rather than merely a quotient by an arbitrary subgroup.
The relations also provide its canonical principal <polarization of a complex torus>. With the convention that a <Hermitian form> is conjugate-linear in its first argument, put $H(z,w)=\overline z^{,t}Y^{-1}w$ and $E=\operatorname{Im}H$. Symmetry of $\tau$ gives
$$
E(m+\tau n,m'+\tau n')=m^tn'-n^tm',\qquad E(v,iv)=H(v,v)>0.
$$
In particular $E$ is integral and unimodular on $\Lambda$, and it is compatible with the complex structure. These are exactly the positivity and integrality conditions for a <Riemann form on a complex torus>; they make the <complex torus> a <Jacobian variety> with a principal <polarization of a complex torus>. If instead one uses a first-argument-linear <Hermitian form>, the same alternating form is $-\operatorname{Im}H$. Changing the <symplectic basis> changes the coordinates but not this polarized quotient. For $g=0$, $V=0$ and the <Jacobian variety> is a point.
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