Local Donaldson–Thomas invariants of blowups of surfaces

Abstract

Using the degeneration formula for Doanldson-Thomas in- variants, we proved a formula for the change of Donaldson-Thomas in- variants of local surfaces under blowing up along points. 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 stable 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 in (Thomas) showed 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 invariants of X which are deformation invariant. Donaldson-Thomas invariants provide a new vehicle to study the geometry and other aspects of higher-dimensional varieties. It is important to understand these invariants. It is well-known (MNOP1, MNOP2)that there is a correspondence be- tween Donaldson-Thomas invariants and Gromov-Witten invariants. Both invariants are deformation independent. On the side of Gromov-Witten in- variants, Li and Ruan in (L-R) first established the degeneration formula of Gromov-Witten invariants in symplectic geometry. J. Li proved an algebraic geometry version of this degeneration formula. In (Hu1, Hu2), the author studied the change of Gromov-Witten invariants under the blowup. The author (Hu3) also studies the change of local Gromov-Witten invariants of Fano surfaces under the blowup. In the birational geometry of 3-folds, we have blowups and flops which are semistable degenerations. In (HL) the au- thors studied how Donaldson-Thomas invariants change under the blowup at a point, some flops and extremal transitions. Local del Pezzo surface used to play an important role in physics. Local de Pezzo surfaces are usually associated to phase transitions in the Kahler mod- uli space of various string, M-theory, and F-theory compactifications.Non- toric del Pezzo surfaces seem to be related to exotic physics in four, five and