Abstract:
According to Gromov's theory,
smooth symplectic 2-surfaces in $\mathbb C\mathrm P^2$ share many properties with complex algebraic curves. The same phenomenon takes place in the real case. Namely, smooth symplectic surfaces invariant under the complex conjugation (we call them real pseudoholomorphic curves) have many common properties with plane projective real algebraic curves. An open question (Symplectic Isotopy Problem): does each connected component of the space of symplectic surfaces contain an algebraic curve? The same question can be asked in the real case and a negative answer will be given in the talk. We shall prove certain inequalities for the complex orientations of plane real algebraic curves which are not satisfied by an infinite series of real pseudoholomorphic curves.