nLab Theta characteristic

Context

Differential geometry

differential geometry

synthetic differential geometry

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)