geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The Kodaira vanishing theorem for complex geometry says that if $X$ is a Kähler manifold and $L$ a holomorphic line bundle on $X$ which is positive, then the abelian sheaf cohomology of $X$ with coefficients in the sheaf of sections of the tensor product
with the canonical line bundle $\Omega^{n,0}_X$ is concentrated in degree 0:
The statement is due to Kunihiko Kodaira.
Lecture notes include