Projective structures on a Riemann surface. III
For part I and II see Int. Math. Res. Not. 1996, 753-768 (1996; Zbl 0888.14012) and ibid. 1999, 685-716 (1999; Zbl 0939.14014). A projective structure on a Riemann surface \(X\) is a covering of \(X\) by holomorphic coordinate charts whose transition functions are Möbius transformations. Let \(\Delta\) denote the diagonal divisor of \(X\times X\) and let \(p_i: X\times X\to X\), \(i=1,2\), be the projection onto the \(i\)-th factor. Let \(K_X\) be the holomorphic cotangnet bundle of \(X\), \({\mathcal M}=p_1^*K_X\otimes p^*_2 K_X\otimes {\mathcal O}_{X\times X} (2\Delta)\), and denote by \(\Delta_n\) the \(n\)-th order infinitesimal neighborhood of \(\Delta\) in \(X\times X\). The authors introduce the notion of generalized projective structure on \(X\) as the choice of an extension of the trivialization of the line bundle \({\mathcal M}\) on \(\Delta_1\) to a trivialization on \(\Delta_n\) for \(n\geq 2\), and they provide several canonically equivalent descriptions of the space \({\mathcal S}_n(X)\) of generalized projective structures on \(X\). Moreover, they construct a canonical involution on \({\mathcal S}_n(X)\), and interpret it in the isomorphic spaces introduced above. The paper under review represents a rather interesting progress of the previous work (part I and II, loc. cit.) by both authors.
scientific article; zbMATH DE number 1780854