Moduli spaces of complex affine and dilation surfaces

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech [W. A. Veech, Flat surfaces, Amer. J. Math. 115 (1993), no. 3, 589-689], we show that the moduli space A(g,n)(m/ of genus g affine surfaces with cone points of complex order m = (m(1) ... , m(n)/ is a holomorphic affine bundle over M-g,M-n, and the moduli space Dg,n(m/ of dilation surfaces is a covering space of M-g,M-n. We then classify the connected components of D-g,D-n(m/ and show that it is an orbifold-K(G, 1/, where G is the framed mapping class group of [A. Calderon and N. Salter, Framed mapping class groups and the monodromy of strata of Abelian differentials, preprint 2020].