Déterminant des opérateurs de Cauchy-Riemann réels et application à l'orientabilité d'espaces de modules de courbes réelles

The paper is devoted to the study of the orientability of the moduli spaces of real pseudoholomorphic curves in real symplectic manifolds. We begin by extending the results we obtained in \cite{article1}. Namely, we consider a complex vector bundle $(N,c_N)$ equipped with a real structure over a real curve $(\Sigma_g,c_{\Sigma})$ of genus $g \in \NN$, and we give a description of the action of an automorphism of $(N,c_N)$ on the orientations of the determinant bundle over the space of all Cauchy-Riemann operators on $(N,c_N)$. In the second part, we use the previous decription to compute the first Stiefel-Whitney class of the moduli space of real pseudoholomorphic curves in many cases. We show for example that the real pseudoholomorphic curves with non-empty real part in the complex projective space of dimension three form an orientable moduli space for a generic almost complex structure.