Categorification of Donaldson-Thomas invariants via perverse ...

40 YOUNG-HOON KIEM AND JUN LI8.2. GV invariants from perverse sheaves. In this subsection, we assume thatdet Ext • π(E, E) admits a square root so that we have a perverse sheaf P • and aMHM M • which are locally the perverse sheaf and MHM of vanishing cycles for alocal CS functional.Remark 8.2. In [11], it is proved that if the Calabi-Yau 3-fold Y is simply connectedand H ∗ (Y, Z) is torsion-free, then det Ext • π(E, E) admits a square root. Forinstance, when Y is a quintic threefold, we have the desired perverse sheaf andMHM.Since the semi-normalization γ : ˜X → X is bijective, the pullback ˜P • of P ••is a perverse sheaf and γ ∗ ˜P ∼ = P • . By Theorem 7.1, P • lifts to a MHM M •and its pullback ˜M • satisfies rat( ˜M • ) = ˜P • since rat preserves Grothendieck’s sixfunctors ([31]). Let ˆM • = gr W ˜M • be the graded object of ˜M • with respect to theweight filtration W . Then ˆM • is a direct sum of polarizable Hodge modules ([32]).Let ˆP • = rat( ˆM • ) which is the graduation gr W ˜P • by the weight filtration of ˜P •because rat is an exact functor ([31]).By [30, §5], the hard Lefschetz theorem and the decomposition theorem holdfor the semisimple polarizable MHM ˆM • . Hence by applying the functor rat, weobtain the hard Lefschetz theorem and the decomposition theorem for ˆP • . Therefore,we can apply the argument in §8.1 to obtain an action of sl 2 × sl 2 on thehypercohomology H ∗ ( ˜X, ˆP • ) to writeH ∗ ( ˜X, ˆP • ) ∼ = ⊕ (( 1 ) ⊗h2 ) L ⊕ 2(0) L ⊗ R h .hDefinition 8.3. We define the Gopakumar-Vafa invariant asn h (β) := T r Rh (−1) H R.The GV invariant n h (β) is integer valued and defined by an sl 2 × sl 2 representationspace H ∗ ( ˜X, ˆP • ) as expected from [9].Proposition 8.4. The number n 0 (β) is the Donaldson-Thomas invariant of X.Proof. Recall that the DT invariant is the Euler number of X weighted by theBehrend function ν X on X and that ν X (x) for x ∈ X is the Euler number of thestalk cohomology of P • at x. Therefore the DT invariant of X is the Euler numberof H ∗ (X, P • ).Since the semi-normalization γ : ˜X → X is a homeomorphism, γ∗ ˜P• ∼ = P • andH ∗ (X, P • ) ∼ = H ∗ (X, γ ∗ ˜P • ) ∼ = H ∗ ( ˜X, ˜P • ) so thatDT (X) = ∑ k(−1) k dim H k (X, P • ) = ∑ k(−1) k dim H k ( ˜X, ˜P • ).Since ˜P • has a filtration W with ˆP • = gr W ˜P • , we have the equality of alternatingsums ∑(−1) k dim H k ( ˜X, ˜P • ) = ∑ (−1) k dim H k ( ˜X, ˆP • ).kkSince the Euler number of the torus part ( ( 1 2 ) ) ⊗hL ⊕ 2(0) L is zero for h ≠ 0,∑(−1) k dim H k ( ˜X, ˆP • ) = T r R0 (−1) H R= n 0 (β).This proves the proposition.k□