Quadratically parametrized smooth maps from one complex projective space to another are constructed as projections of the Segre map of the complexification. A classification theorem relates equivalence classes of projections to congruence classes of matrix pencils. Maps from the 2-sphere to the complex projective plane, which generalize stereographic projection, and immersions of the complex pr...

The main theorem of this paper asserts that the inclusion of the space of projective Lagrangian planes into the space of Lagrangian submanifolds of complex projective space induces an injective homomorphism of fundamental groups. We introduce three invariants of exact loops of Lagrangian submanifolds that are modelled on invariants introduced by Polterovich for loops of Hamiltonian symplectomor...

The moduli space of Higgs bundles can be defined as a quotient an infinite-dimensional space. Moreover, by the Kuranishi slice method, it is equipped with structure normal complex In this paper, we will use analytic methods to show that quasi-projective. fact, following Hausel's symplectic cut construct and projective compactification space, hence prove quasi-projectivity. main difference betwe...