The sheaf of 1-forms of a versal deformation

This is a report about the PhD Thesis of B. Fr\ohlich (Bochum). It is well known and usually not difficult to describe the tangent space of a semiuniversal deformation at a point.

For instance, if $f:{\mathcal X}\to S$ is the semiuniversal deformation of a compact complex manifold $X=f^{-1}(0)$, $0\in S$, then $T_0S\cong H^1(X,\Theta_X)$.
However, it is usually much harder to give a description of the sheaf of 1-forms $\Omega^1_S$ of the basis of the semiuniversal deformation. There is one case where the answer to this problem is known.
Let $Q$ be the Quot scheme of a coherent sheaf $\mathcal F$ over a smooth projective variety $X$ of dimension $d$ as constructed by Grothendieck. There is a universal subsheaf $\mathcal K\subseteq p^*(\mathcal F)$,
where $p:X\times Q\to X$ is the projection, and a universal quotient $\mathcal G=p^*(\mathcal F)/\mathcal K$. By a result of M.~Lehn, the sheaf of 1-forms of $Q$ is then isomorphic to the relative Ext-sheaf $\mathcal Ext^d_p(\mathcal K, \mathcal G\otimes p^*(\omega_X))$.

In this talk we show how to deduce such formulas for the sheaf of 1-forms in general as soon as there is a cotangent complex controlling deformations. This result will in particular apply to deformations of complex spaces, sheafs or the Quot space of singular varieties $X$.