Open questions on fundamental groups of complements of curves

After the remarkable discovery of the Donaldson invariants and the Sieberg- Witten invariants (SW), there was hope that using them one could distinguish among difierent connected components of moduli space of surfaces. In our earlier work we indicated that we believe that these invariants are not flne enough for this difierentiation and a more geometrical approach is needed. Indeed in 1997, M. Manetti produced examples of surfaces which are difieomorphic but are not a deformation of each other, and thus it is clear that a more direct geometric approach is needed. We want to suggest the following distinguishing invariant: Let X be a complex algebraic surface of general type embedded inCP n . Take a generic projection of X to CP 2 and let SX be its branch curve. Clearly …1(CP 2 ¡ SX) is stable on a connected component of moduli space of surfaces. We believe that these groups can distinguish among difierent components. We base our belief on the structure of such groups which already have been computed. Let C 2 be a big a-ne piece of CP 2 s.t. SX is transversal to the line at inflnity. We denote: OPEN QUESTIONS ON FUNDAMENTAL GROUPS OF COMPLEMENTS OF CURVES. Available from: https://www.researchgate.net/publication/241490456_OPEN_QUESTIONS_ON_FUNDAMENTAL_GROUPS_OF_COMPLEMENTS_OF_CURVES [accessed Jan 3, 2016].