A principle governing deformation theory with cohomology constraints in characteristic zero, generalizing Deligne's deformation theory principle, was developed together with B. Wang, M. Rubio in terms of dg Lie modules, and, more generally, $\text{L}\infty$ modules. An application of this theory is that for a generic compact Riemann surface the theta function is at every point on the Jacobian equal to its first non-zero Taylor term, up to a holomorphic change of local coordinates and multiplication by a local holomorphic unit.[-]

We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]

We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]

We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]

We show how higher (simplicial or differential graded) techniques are naturally required when studying infinitesimal deformation theory. We then outline the strict relationship between derived moduli spaces and the combination of classical moduli spaces plus infinitesimal derived/extended/higher deformation functions, as the former induce the latter, but the latter carry (in an appropriate sense) all the information of the former.
In the process, we consider explicit examples and highlight the role of differential graded Lie algebras and their generalisations.[-]