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

In 2020, Oh and Thomas constructed a virtual cycle [ X stretchy="false">] v mathvariant="normal">i mathvariant="normal">r ∈ A ∗ encoding="application/x-tex">X(\sigma ) an isotropic cosection alttext="sigma"> encoding="application/x-tex">\sigma obstruction sheaf O b O b encoding="application/x-tex">Ob_X construct localized l o c right-parenthesis mathvariant="normal">l mathvariant="normal">o mathvariant="normal">c _\mathrm {loc}\in A_*(X(\sigma )) . This is achieved by further localizing Oh-Thomas class Edidin-Graham’s square root Euler special orthogonal bundle. When surjective so vanishes, reduced e d r"> mathvariant="normal">e mathvariant="normal">d _{\mathrm {red}} As application, vanishing results hyperkähler 4-folds. All these hold structure K-theoretic invariants.