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

In 2020, Oh and Thomas constructed a virtual cycle [ X ] v i r ∈ A ∗ ( X ) [X]^{\mathrm {vir}} \in A_*(X) for a quasi-projective moduli space X 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 ( σ ) X(\sigma ) of an isotropic cosection σ \sigma of the obstruction sheaf O b X Ob_X of X X and construct a localized virtual cycle [ X ] l o c v i r ∈ A ∗ ( X ( σ ) ) [X]^{\mathrm {vir}} _\mathrm {loc}\in A_*(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 σ \sigma is surjective so that the virtual cycle vanishes, we construct a reduced virtual cycle [ X ] r e d v i r [X]^{\mathrm {vir}} _{\mathrm {red}} . As an application, we prove DT4 vanishing results for hyperkähler 4-folds. All these results hold for virtual structure sheaves and K-theoretic DT4 invariants.