Categorification of Donaldson-Thomas invariants via perverse ...

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 31is Z × S. Then for any convergent p n → p 0 ∈ V × S such that d V f(p n ) ≠ 0 andd V f(p n ) → 0, we havef(p n )limn→∞ |d V f(p n )| = 0.Proof. We prove the case S = pt. The general case is exactly the same. Letν : Ṽ → V be the resolution of the ideal sheaf of Z so that ν−1 (Z) is a normalcrossing divisor in Ṽ . Let ˜p n be the unique lifting of p n in Ṽ − ν−1 (Z). Suppose n kis a subsequence so that ˜p nk → q ∈ ν −1 (Z). We investigate the limiting behaviorof |f(p nk )|/|df(p nk )|.By our choice of resolution ν, locally near q ∈ Ṽ , the pullback ν∗ (df) of theideal (df) is a principal ideal sheaf generated by some monomial ϕ = z k11 · · · zkm mwith k i > 0 for holomorphic functions z 1 , · · · , z r on V such that restricting to thefiber Ṽ they give a local coordinates of Ṽ centered at q. Let w 1, · · · , w r be localcoordinates of V centered at p 0 . Then ϕ divides ν ∗ ∂fi∂w ifor all i.We claim that ν ∗ f is divisible by z k1+11 · · · zm km+1 . Indeed if we expand nearz 1 = 0, ν ∗ f = c m1 z m11 + c m1+1z m1+11 + · · · , where c j are holomorphic in z 2 , · · · , z r ,and c m1 ≠ 0. Then ϕ divides∑ ∂f ∂w i=∂ ν ∗ f = m 1 c m1 z m1−11 + (m 1 + 1)c m1+1z m11 + · · · .∂w i ∂z 1 ∂z 1Hence m 1 ≥ k 1 + 1 and z k1+11 divides ν ∗ f. Likewise z ki+1i divides ν ∗ f for eachi. Therefore ν ∗ (f) ⊂ ν ∗ (df) √ ν ∗ (df) where √ ν ∗ (df) is the radical of ν ∗ (df). Wewrite ν ∗ f = ∑ g i · ν ∗ ∂f∂w iwith g i | ν −1 (Z) = 0. Thereforeν ∗ |f(p n k)||df(p nk )| = |ν ∗ f(˜p nk )|(∑( ∑ |ν ∗ ∂f∂w i(˜p nk )| 2 ) ≤ |gi (˜p 1/2 nk )| 2) 1/2−→ 0.Because Ṽ → V is proper, and because this convergence holds for all convergentf(psubsequence ˜p nk , we find that lim n)n→∞ ||df(p = 0.□n)||Proof of Proposition 6.2. We use the inner product and the hermitian metric on V xvia embedding V x∼ = Vx ⊂ Ω 0,1 (adE x ) s . We let ∇ V f t be the relative gradient vectorfield of f t on D C x ×D C x ×V x as the metric dual of d V f t , the differential of f t in the V xdirection. Note that ∇ V f t is a vertical vector field with respect to the projectionpr 12 : D C x × D C x × V x → D C x × D C x which is differentiable on each fiber. We define atime dependent vector field on the complement of Z = Z t = (d V f t = 0) by(6.3) ξ t = f 0 − f 1|∇ V f t | 2 ∇ V f t .We claim that it extends to a well defined vector field on some neighborhood of(x, x, x) in D C x × D C x × V x . It suffices to show that|ξ t | = |f 0 − f 1 ||∇ V f t |= |f 0 − f 1 ||d V f t |approaches zero as a point approaches Z. Since Z t = Z 0 = Z 1 by Lemma 6.4, wehave an inclusion (d V f t ) ⊃ (d V f i ) of ideals for i = 0, 1. Hence we can express thevertical partial derivatives of f 0 and f 1 as linear combinations of the vertical partialderivatives of f t . Thus in a neighborhood of (x, x, x), we have|d V f 0 | ≤ C |d V f t | and |d V f 1 | ≤ C |d V f t |