Homotopy theory of curved algebras and derived complex geometry

In a recent joint work with Joan Bellier-Millès, we set up a homotopy theory of curved algebras over curved operads in which one can implement (with the appropriate modifications) usual constructions such as the bar-cobar adjunction and André-Quillen cohomology. The occurrence of curved structures in various research topics (symplectic topology, deformation theory, derived geometry, mathematical physics...) motivate the development of their homotopy theory, cohomology theory and deformation theory. Two important examples fitting in our framework are the homotopy unital curved A-infinity algebras encoding for instance the algebraic structure of Fukaya categories (as in the work of Fukaya-Oh-Ota-Ono) in symplectic topology, and formal integrable almost complex structures in complex geometry. The motivation behind the second example is to get an analogue of the Newlander-Nirenberg integrability theorem in derived complex analytic geometry. At the formal neighbourhood of a point, an integrable almost complex manifold can be described as an algebra over a certain curved operad. Our idea is to consider homotopy curved algebras over this curved operad and to glue these local data up to homotopy. Our results generalize to homotopy sheaves of curved algebras over a curved operad, so in particular they provide a framework to perform such a construction. During this talk, I will explain the various key notions and ideas of our work. If time permits, I will say a few words about the comparison between our model of derived complex analytic spaces and those developped by Lurie, Porta and Pridham.