# What is a derived Kähler manifold?

From what I understand, there exists a notion of derived $\mathbb{C}$-analytic space.

Let $T_{an}$ be the pregeometry in the sense of Lurie whose underlying $\infty$-category is the category of open sets in $\mathbb{C}^n$ and whose admissible morphisms are open immersions. Then a derived $\mathbb{C}$-analytic space is a $T_{an}$-structured $\infty$-topos $(X, \mathcal{O}_X)$ satisfying some requirements (see the works of M. Porta).

Moreover, there exists a notion of symplectic structure on derived stack (for example, see the survey of D. Calaque).

My naive question is: has anybody tried to figure out whether it's possible to define compatibility relations between complex and symplectic structures in the derived setting analogous to the classical $\omega(Ju, Jv)=\omega(u, v)$, $\omega(u, Ju)>0$? In other words, has anybody tried to define derived Kähler manifolds?

• A derived symplectic scheme (as opposed to stack) is just a smooth symplectic scheme, because the cotangent complex has to be concentrated in degree $0$. Thus a derived Kaehler manifold would just be a Kaehler manifold. – Jon Pridham Jun 11 '18 at 8:01
• @JonPridham oh I am ignorant about those. Do you think anything differing from the classical case can be constructed for stacks? – Aknazar Kazhymurat Jun 11 '18 at 13:36