In the paper under review, the author studies certain moduli spaces of sheaves on Enriques surfaces which lead to new examples of Calabi-Yau manifolds. In order to present the main result, let us recall some crucial definitions. Let \((X,H)\) be a smooth projective polarized surface. For a pure \(1\)-dimensional sheaf \(F\) on \(X\) we define Fitting support. Let us recall that such a pure dimension \(1\) sheaf has homological dimension \(1\), i.e., there exists a length one locally free resolution of \(F\), \[ 0 \to L_{1} \stackrel{a}{\rightarrow} L_{0} \rightarrow F \rightarrow 0. \] The Fitting support of \(F\) is the subscheme of \(X\) defined by the equation \(\text{det} \,a = 0\). For the main result, we consider only general Enriques surfaces \(T\).
Main Result. Let \(|C|\) be a genus \(g \geq 2\) linear system on \(T\), \(d= g - 1\) an integer, \(H\) a generic polarization, and let \(N \rightarrow |C|\) be the component of the moduli space of \(H\)-semistable sheaves on \(T\) with Fitting support in \(|C|\) and Euler characteristic equal to \(\chi = d - g + 1\) that contains sheaves supported on irreducible curves. Suppose the divisibility of \(C\) in the Néron-Severi space \(\mathrm{NS}(T)\) is coprime with \(2(d - g + 1)\). Then we have the following:
i) \(N\) is a smooth \((2g - 1)\)-dimensional Calabi-Yau variety.
ii) For \(g = 2\), one gets Calabi-Yau \(3\)-folds with the following Hodge diamond: \[ \begin{gathered} 1\\ 0 \quad 0\\ 0 \quad 10 \quad 0\\ 1 \quad 10 \quad 10 \quad 1. \end{gathered} \]
iii) There is a surjection \(\mathbb{Z}/(2) \rightarrow \pi_{1}(N)\) which, under some natural assumption (please consult Assumption 2.17 therein), turns out to be an isomorphism. Under the same assumption, one can additionally show that for \(g \geq 3\) we have \(h^{2}(N) = 11\).