Minimal varieties with trivial canonical classes, I.
A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold – which is viewed as the obstruction to the existence of a Q-invariant Berezin volume – is not well know. We review the basic ideas and then apply this technology to various examples, including -algebroids and higher Poisson manifolds.
We define and make an initial study of (even) Riemannian supermanifolds equipped with a homological vector field that is also a Killing vector field. We refer to such supermanifolds as Riemannian Q-manifolds. We show that such Q-manifolds are unimodular, i.e., come equipped with a Q-invariant Berezin volume.
We describe explicitly the moduli spaces of polystable holomorphic structures with on a rank two vector bundle with and for all minimal class VII surfaces with and with respect to all possible Gauduchon metrics . These surfaces are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When is a half or parabolic Inoue surface, is always a compact one-dimensional complex disc. When is an Enoki surface, one obtains a complex disc with finitely...