Localizing virtual cycles for Donaldson-Thomas invariants of Calabi-Yau 4-folds

Recently Oh-Thomas constructed a virtual cycle $[X{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}\in A_{\ast }(X)$ for a quasi-projective moduli space $X$ of stable sheaves or complexes over a Calabi-Yau 4-fold against which DT4 invariants may be defined as integrals of cohomology classes. In this paper, we prove that the virtual cycle localizes to the zero locus $X(\sigma )$ of an isotropic cosection $\sigma$ of the obstruction sheaf $O{b}_X$ of $X$ and construct a localized virtual cycle $[X{]}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{v}\mathrm{i}\mathrm{r}}\in A_{\ast }(X(\sigma ))$. This is achieved by further localizing the Oh-Thomas class which localizes Edidin-Graham’s square root Euler class of a special orthogonal bundle. When the cosection $\sigma$ is surjective so that the virtual cycle vanishes, we construct a reduced virtual cycle $[X{]}_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{v}\mathrm{i}\mathrm{r}}$. As an application, we prove DT4 vanishing results for hyperk"ahler 4-folds. All these results hold for virtual structure sheaves and K-theoretic DT4 invariants.