I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the picture clearer before complicating too much the setting and hope things will not be that different then.

Let me consider a smooth projective variety $X$ defined over $\mathbb{C}$ (these hypotheses can obviously be relaxed), and a surjection $A \to A_0$ of commutative algebras over $\mathbb{C}$ defined by a nilpotent ideal. Taking an $E\in \mathrm{Coh}(X\times \mathrm{Spec} A_0)$, I guess a deformation of $E$ should be some $\mathcal{E}\in \mathrm{Coh}(X\times \mathrm{Spec} A)$ such that $i^\ast \mathcal{E}\simeq E$ where $i\colon X\times \mathrm{Spec} A_0 \hookrightarrow X\times \mathrm{Spec} A$ is the inclusion.

My question is: one can associate to $E$ various gadgets like $\Gamma(X,E)$, $H^\bullet (X, E)$, $Ext^\bullet(E,E)$: which is the relation between these objects calculated for $E$ and for $\mathcal{E}$? this would be equivalent to asking whether the functions $dim\Gamma(X,-)$, etc.. are locally constant on a moduli space of such sheaves.

Looking forward to thank you for giving me some examples and references as well

  • $\begingroup$ I am a little confused about this question. Already the simplest nontrivial example, namely line bundles of degree 0 on an elliptic curve, shows that $h^0$ is not locally constant (only semicontinuous). Can you clarify what I am misreading? $\endgroup$ – potentially dense Aug 31 '15 at 21:43
  • $\begingroup$ You should also require flatness of $\mathcal{E}$ over $\mathrm{Spec} (A)$. I don't think any of the things you mentioned are locally constant. Something that is locally constant is the Hilbert polynomial of the sheaf and the ''numerical'' Chern classes (see for example mathoverflow.net/questions/52667/chern-classes-in-flat-families). $\endgroup$ – Mattia Talpo Sep 1 '15 at 21:43

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