Question

Suppose I have a (2,4) complete intersection 3-fold $X\subset\mathbb P_{\mathbb C}^5$ with 118 nodal ($A_1$) singularities (if it simplifies things then assume it's the intersection of the Grassmanian $G(2,4)$ with a quartic). It is known (for example see Theorem 4.3 and Corollary 4.4 in Dimca's "Singularities and Topology of Hypersurfaces") that $H_1(X,\mathbb Z)=H_1(\mathbb P^3,\mathbb Z)=0$ and that $H_4(X,\mathbb Z)$ is torsion free.

Does anyone know if taking a crepant resolution $\tilde{X}$ of these 118 nodes could introduce torsion into $H_1$ or $H_4$ of the resolution? Specifically, does it introduce 2-torsion?

Answer 1

If I am not mistaken, the mapping $\tilde X\to X$ blows down a disjoint union of 2-dimensional spheres, so it follows from the homology sequence of pair that it induces isomorphisms of both $H_1$'s and $H_4$'s (with integer coefficients).