# Obtaining a deformation from its cohomology class

Classical (infinitesimal) deformations of a complex manifold $X$ are parametrized by $H^1(X,TX)$, where the coefficients are taken to be the sheaf of sections of the tangent bundle $TX$.

Obstructions to obtaining an actual differentiable family of complex manifolds "live in" $H^2$. When this space is trivial, $X$ can be deformed along any tangent direction and one obtains a one-parameter family of complex manifolds.

The above is what I understand for example from Kodaira's book Complex manifolds and deformation of complex structures.

I am wondering whether there is an explicit description of obtaining such a differentiable family. Given a complex manifold $X$ explicitly, i.e. by a good cover $\{U_i\}$ and transition functions $\{\phi_{ij}\}$, and given a cohomology class $\sigma$, how do I obtain an explicit description of a deformation of $X$ in terms of the transition functions $\{\phi_{ij}\}$ and $\sigma$?

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See Hodge theory 1 by Claire Voisin, chapter 9. –  Qfwfq Oct 15 at 23:35
It might be worth looking at Chapter 6 of Huybrechts' Complex Geometry. –  Michael Albanese Oct 16 at 3:59
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