Categorification of Donaldson-Thomas invariants via perverse ...

38 YOUNG-HOON KIEM AND JUN LISince ∂F k∂w j(z, 0) involve the partial derivatives of the cs via the chain rule, we concludethat ∂F k∂w j(z, 0) ∈ I. This proves that F k (z, δ 1 (z), · · · , δ n (z)) − F k (z, 0) ∈ I 2 ,which is (7.4). This proves the Lemma.□Since f 0 −f 1 ∈ I 2 and I = (d V f t ), we can always find a time dependent holomorphicvertical vector field ξ t which satisfies (7.1) and vanishes along X (cf. [5]). Thisgives us the desired biholomorphic map Φ. This completes our proof of Theorem7.1. Note that we don’t need Lemma 6.5 for this choice of ξ t .Remark 7.4. (Gluing of polarization) To use the hard Lefschetz property of [30],we also need the gluing of polarization and monodromy for M • and the graduationgr W M • . Note that our gluing isomorphisms arise from continuous families of CScharts V . Since a continuous variation of rational numbers is constant, the obviouspolarizations for the constant sheaves Q V on V is constant in a continuous familyof CS charts. The induced polarizations on the perverse sheaves A • f [r] = φ f Q[r−1],defined in [30, §5.2], are functorial. Therefore the polarizations on A • f[r] glue togive us a polarization on P • . By the same argument, we have a polarization ongr W M • and the monodromy operators also glue .8. Gopakumar-Vafa invariantsIn this section, we provide a mathematical theory of the Gopakumar-Vafa invariantas an application of Theorem 3.16.8.1. Intersection cohomology sheaf. As before, let Y be a smooth projectiveCalabi-Yau 3-fold over C. From string theory ([9, 15]), it is expected that(1) there are integers n h (β), called the Gopakumar-Vafa invariants (GV forshort) which contain all the information about the Gromov-Witten invariantsN g (β) of Y in the sense that∑(8.1)N g (β)q β λ 2g−2 = ∑n h (β) 1 (2 sin( kλ ) 2h−2k 2 ) q kβg,βk,h,βwhere β ∈ H 2 (Y, Z), q β = exp(−2π ∫ β c 1(O Y (1)));(2) n h (β) come from an sl 2 × sl 2 action on some cohomology theory of themoduli space X of one dimensional stable sheaves on Y ;(3) n 0 (β) should be the Donaldson-Thomas invariant of the moduli space X.By using the global perverse sheaf P • constructed above and the method of [10],we can give a geometric theory for GV invariants.We recall the following facts from [30, 31].Theorem 8.1. (1) (Hard Lefschetz theorem) If f : X → Y is a projectivemorphism and P • is a perverse sheaf on X which underlies a pure (polarizable)MHM M • , then the cup product induces an isomorphismω k : p H −k Rf ∗ P • −→ p H k Rf ∗ P •where ω is the first Chern class of a relative ample line bundle.