Products of pairs of Dehn twists and maximal real Lefschetz fibrations (Q2840045)

This paper addresses the problem of existence and uniqueness of a factorization of a given element of the modular group PSL(2, \(\mathbb Z\)) into a product of two Dehn twists. A geometric application is the realization of a given real elliptic Lefschetz fibration by an algebraic one. It is a classical result of Moishezon that any elliptic Lefschetz fibration is algebraic. For a real Lefschetz fibration, it means a Lefschetz fibration \(p: X\rightarrow B\) equipped with a pair of real structures (i.e., an orientation preserving involutive autodiffeomorphism whose real part is either empty or of highest possible dimension) on \(X\) and \(B\) commuting with \(p\). In general, a real Lefschetz fibration does not admit an algebraic structure (\textit{i.e.} \(p\) is holomorphic and the involutions are antiholomorphic). In this paper, it is shown that when there is a section of \(p\) (called Jacobian) and when the total Betti number of the real part \(X_{\mathbb R}\) is same as that of \(X\) in \(\mathbb Z_2\) coefficients (called maximal), the real Lefschetz fibration admits an algebraic structure. Moreover, the section could be chosen as a holomorphic one.