Moduli spaces of complex affine and dilation surfaces

Abstract We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech [W. A. Veech, Flat surfaces, Amer. J. Math. 115 1993, 3, 589–689], we show that the moduli space 𝒜 g , n ⁢ ( 𝒎 ) {{\mathcal{A}}_{g,n}(\boldsymbol{m})} of genus g affine surfaces with cone points of complex order 𝒎 = ( m 1 ⁢ … , m n ) {\boldsymbol{m}=(m_{1}\ldots,m_{n})} is a holomorphic affine bundle over ℳ g , n {\mathcal{M}_{g,n}} , and the moduli space 𝒟 g , n ⁢ ( 𝒎 ) {{\mathcal{D}}_{g,n}(\boldsymbol{m})} of dilation surfaces is a covering space of ℳ g , n {\mathcal{M}_{g,n}} . We then classify the connected components of 𝒟 g , n ⁢ ( 𝒎 ) {{\mathcal{D}}_{g,n}(\boldsymbol{m})} and show that it is an orbifold- K ⁢ ( G , 1 ) {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].