Riemann vanishing theorem ($\{z:\theta (z,B)=0\}=W_{g-1}-\kappa$)

For the period matrix of a genus-$g$ compact Riemann surface, there is a vector $\kappa$ depending on the marking and base point such that the Riemann theta function vanishes exactly on this translate of the image of the Abelian sum map. Pairing sides of the cut surface proves that a nonzero pullback $\theta (u(p)-z)$ has a degree-$g$ zero divisor whose Abel sum is $z+\kappa$. At a theta zero that pullback includes the base point. Irreducibility and dimension of the effective-divisor locus give equality of supports; the primitive First Chern class of the theta line bundle makes the divisor reduced.

If $p\mapsto \theta (u(p)-z)$ is not identically zero, its zero divisor $D_z$ has these properties. The logarithmic derivative changes by $-2\pi i{\omega }_j$ across the paired $b_j$ translation, giving zero count $g$. For a variation of $z$, the logarithm changes there by $2\pi i\delta z_j$; integrating by parts in the first-moment contour integral gives $\delta \mathrm{AJ}(D_z)=\delta z$. Thus the difference is constant. Weighted zero divisors, rather than a distinct-zero assumption, make the statement valid at collisions.

The vector of Riemann constants is the parameter-independent class obtained from the zeros of $p\mapsto \theta (u(p)-z)$ using the displayed formula. Under this sign convention the theta divisor is $W_{g-1}-\kappa$. It depends on the base point and normalized symplectic marking; some accounts use its negative. Changing the marking or sign convention must also change the translation formula consistently.