The Donaldson-Thomas invariants under blowups and flops

Abstract

Using the degeneration formula for Doanldson-Thomas invariants, we proved formulae for blowing up a point and simple flops. Given a smooth projective Calabi-Yau 3-fold X, the moduli space of stable sheaves on X has virtual dimension zero. Donaldson and Thomas (D-T) defined the holomorphic Casson invariant of X which essentially counts the number of sta- ble bundles on X. However, the moduli space has positive dimension and is singular in general. Making use of virtual cycle technique (see (B-F) and (L-T)), Thomas showed in (Thomas) that one can define a virtual moduli cycle for some X including Calabi-Yau and Fano 3-folds. As a consequence, one can define Donaldson-type in- variants of X which are deformation invariant. Donaldson-Thomas invariants pro- vide a new vehicle to study the geometry and other aspects of higher-dimensional varieties. It is important to understand these invariants. Much studied Gromov-Witten invariants of X are the counting of stable maps from curves to X. In (MNOP1, MNOP2), Maulik, Nekrasov, Okounkov, and Pandharipande discovered relations between Gromov-Witten invarints of X and Donaldson-Thomas invariants constructed from moduli spaces of ideal sheaves of curves on X. They conjectured that these two invariants can be identified via the equations of partition functions of both theory. This suggests that many phe- nomena on Gromov-Witten theory have the counterparts in Donaldson-Thomas theory. Donaldson-Thomas invariants are deformation independent. In the birational geometry of 3-folds, we have blowups and flops. Donaldson-Thomas invariants couldn't be effective in studying birational geometry unless we understand how invariants change under birational operations. Li and Ruan in (L-R) studied how Gromov-Witten invariants change under a flop for Calabi-Yau 3-fold. They proved that one can identify the 3-point functions of X and the flop X f of X up to some transformation of the q variables. The same question was aslo studied by Liu and Yau in (L-Y) recently using the J. Li's degeneration formula from algebraic geometry. In (Hu1, Hu2), the first author studied the change of Gromov-Witten