Witt groups of curves and surfaces (Q463972)

Let \(C\) be a smooth curve over an algebraically closed field of characteristic not two. In the article under review, the author gives a complete computation of the Grothendieck-Witt and Witt groups of \(C\) in terms of the Picard group and étale cohomology groups with \(\mathbb{Z}/2\) -coefficients. Let \(X\) be a smooth complex curve or smooth complex surface. The author in [Doc. Math., J. DMV 16, 465--511 (2011; Zbl 1277.11031)] introduced two comparison homomorphisms \(gw^{i}:\mathrm{GW} ^{i}(X)\rightarrow \mathrm{KO}^{2i}(X(\mathbb{C}))\) and \(w^{i}:\mathrm{W} ^{i}(X)\rightarrow \frac{\mathrm{KO}^{2i}(X(\mathbb{C}))}{\mathrm{K}(X( \mathbb{C}))}.\) In the article under review, the author proves that for a smooth complex curve \(X\), the comparison homomorphisms \(gw^{i}\) are surjective and the comparison homomorphisms \(w^{i}\) are isomorphisms. When \(X\) is a smooth complex surface, the same claim holds if and only if every continuous complex line bundle over \(X\) is algebraic.