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- Title
- LOCALIZING VIRTUAL CYCLES FOR DONALDSON-THOMAS INVARIANTS OF CALABI-YAU 4-FOLDS
- KIAS Author
- Park, Hyeonjun,Kiem, Young-Hoon,Park, Hyeonjun
- Journal
- JOURNAL OF ALGEBRAIC GEOMETRY, 2023
- Archive
-
- Abstract
- In 2020, Oh and Thomas constructed a virtual cycle [X](vir)? A(*)(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(s) of an isotropic cosection s of the obstruction sheaf Ob(X) of X and construct a localized virtual cycle [X](vir) (loc) ? A(*)(X(s)). 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 s is surjective so that the virtual cycle vanishes, we construct a reduced virtual cycle [X](vir)( red). As an application, we prove DT4 vanishing results for hyperka center dot hler 4-folds. All these results hold for virtual structure sheaves and K-theoretic DT4 invariants.