On derived Calabi-Yau varieties

The chromatic structure of stable homotopy theory is organized by the heights of one-dimensional formal groups. A Calabi-Yau variety gives rise to a one-dimensional formal group by its deformation cohomology. Specifically the formal Brauer groups of K3 surfaces have either finite heights between one and ten or the infinite height, and can be expected to give the stable homotopic information up to the tenth chromatic layer. Polarized K3 surfaces are classified by a Deligne-Mumford stack whose strata in terms of the heights of formal Brauer groups generate generalized cohomology theories. We ask: can these cohomology theories be represented by E_∞ -ring spectra? In order to solve this realization problem, we formulate the generalization of K3 surfaces in derived algebraic geometry and state the moduli problems, the solution of which solves the realization problem and give an analogue of the spectrum of topological modular forms in the case of elliptic curves.