Moduli of real algebraic surfaces, and their superanalogues. Differentials, spinors, and Jacobians of real curves

The survey is devoted to various aspects of the theory of real algebraic curves. The involution defined by complex conjugation induces an antiholomorphic involution $\tau\colon P\to P$ on the complexification $P$ of a real curve. This involution acts on all structures related to the Riemann surface $P$, namely, on vector bundles, Jacobians, Prymians, and so on. The greater part of the survey is devoted to finding topological invariants and studying the corresponding moduli spaces. Statements of these problems were inspired by applications of the theory of real curves to problems in mathematical physics (theory of solitons, string theory, and so on).