# nLab Theta characteristic

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

cohomology

# Contents

## Idea

For $X$ a space equipped with a notion of dimension $dim X \in \mathbb{N}$ and a notion of Kähler differential forms, a $\Theta$-characteristic of $X$ is a choice of square root of the canonical characteristic class of $X$. See there for more details.

In complex analytic geometry and at least if the Theta characteristic is principally polarizing then its holomorphic sections are called theta functions.

## Examples

### Over Riemann surfaces

###### Proposition

For $\Sigma$ a Riemann surface, the choices of square roots of the canonical bundle correspond to the choice of spin structures.

For $X$ of genus $g$, there are $2^{2g}$ many choices of square roots of the canonical bundle.

###### Remark

The first statement remains true in higher dimensions over Kähler manifolds, see at Spin structure – On Kähler manifolds.

Revised on June 21, 2014 02:31:15 by Ingo Blechschmidt (46.244.242.152)