Super Atiyah classes and obstructions to splitting of supermoduli space

Understand

The first obstruction to splitting a supermanifold S is one of the three components of its super Atiyah class, the two other components being the ordinary Atiyah classes on the reduced space M of the even and odd tangent bundles of S.

• We evaluate these classes explicitly for the moduli space of super Riemann surfaces ("super moduli space") and its reduced space, the moduli space of spin curves.
• These classes are interpreted in terms of certain extensions arising from line bundles on the square of the varying (super) Riemann surface.
• These results are used to give a new proof of the non-projectedness of ${\mathfrak{M}}_{g,1}$, the moduli space of super Riemann surfaces with one puncture.