nLab cohomology jump loci

Idea

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Definition

Let $\mathcal{M}$ be a moduli space parametrizing objects $P$ that come with $\mathbb{C}$ vector spaces $H^{i}(P)$ for $i\in \mathbb{Z}$ (e.g. the objects can be studied through a cohomology theory).

The cohomology jump loci of $\mathcal{M}$ are defined as

$$\mathcal{V}^{i}_{k} = \{P\in \mathcal{M} | \dim H^{i}(P) \geq k\} \,.$$

And for every $i$ there is a chain of inclusions

$$\mathcal{M}=\mathcal{V}^{i}_{0}\subset\mathcal{V}^{i}_{1}\subset\mathcal{V}^{i}_{2}\subset \,...$$

References

Introductions and surveys include

• Nero Budur and Botong Wang, Recent Results on Cohomology Jump Loci (arXiv:1507.06714)