Betti numbers of real elliptic surfaces (Q2473584)

This article deals with real regular elliptic surfaces. An elliptic surface is a morphism \(\pi:X\to \mathbb{P}^1\) where \(X\) is a complex, compact analytic surface and such that all but a finite number of fibers \(X_t=\pi^{-1}(t)\) are nonsingular curves of genus 1. Moreover, \(X\) is said to be regular if \(H^1(X, {\mathcal O}_X)=\{0\}\). A real elliptic surface is a pair \((X, \sigma)\), where \(\sigma\) is an antiholomorphic involution of the elliptic surface \(X\) such that \(\tau\circ\pi=\pi\circ \sigma\) and \(\tau:\mathbb{P}^1\to\mathbb{P}^1\) is complex conjugation. The real part of \((X, \sigma)\) is the set \(X(\mathbb R)\) of points of \(X\) fixed by \(\sigma\). The authors prove the inequality \(\pi_0(X(\mathbb R))\leq 5\chi({\mathcal O}_X)\), where \(\pi_0(X(\mathbb R))\) is the number of connected components of the real part of a real regular elliptic surface \((X, \sigma)\) admitting a real section, and \(\chi({\mathcal O}_X)\) is its holomorphic Euler characteristic. Moreover, this bound is attained for all values of \(\chi({\mathcal O}_X)\). The proof, that is clever and well written, is not at all elementary. Among its the main ingredients it is worthwhile to mention the following. First it uses a nice result which has its own interest: Let \(\pi:X\to \mathbb{P}^1\) be a real, regular, elliptic surface relatively minimal and admitting a real section. Then, there exists a real, regular, elliptic surface \(\pi':Y\to \mathbb{P}^1\) admitting a real section such that \(\chi({\mathcal O}_X)= \chi({\mathcal O}_Y)\) and \[ h_1(Y(\mathbb R))=2\pi_0(X(\mathbb R)) \quad \text{and}\quad h_1(X(\mathbb R))=2\pi_0(Y(\mathbb R)) \] Secondly, the authors provide a proof of an unpublished result by Kharlamov that states the inequality \(h_1(X(\mathbb R))\leq h^{1,1}(X)\), where \(X\) is a real, regular, elliptic surface without multiple fibers.