Quadratic vector classes on Riemann surfaces
The author defines the quadratic period classes of a closed, marked Riemann surface of genus \(g\geq 3\), following \textit{R. C. Gunning} [Probl. Analysis, Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 239- 247 (1970; Zbl 0211.105)] by associating to a pair of Abelian differentials du and dv, the integrals of udv along curves connecting preimages of a given base point in the universal covering surface of the given surface. The object of the paper is to study the action of \textit{D. Johnson}'s representation of the Torelli group [Math. Ann. 249, 225-242 (1980; Zbl 0409.57009)] on the quadratic periods. The results are rather technical. The quadratic periods are asserted to give more information about the surface than do the abelian periods, and lead to a moduli space intermediate between the Torelli space and Teichmüller space. An indication of the results obtainable in the theorem: Off a countable collection of codimension at least 1 subvarieties of \(T_{3,1}\) two surfaces have identical abelian and quadratic periods iff they are conformally equivalent and one differs from the other by an element of the kernel of Johnson's Torelli group. Furthermore, at least one quadratic period is not a polynomial in the abelian periods with Gaussian integral coefficients.
scientific article; zbMATH DE number 3981523
scientific article