Artin vanishing on complex affine varieties

At a critical point on a complex manifold of dimension $s$, a strongly plurisubharmonic function has tangent Morse index at most $s$. Indeed, if its real negative subspace had dimension larger than $s$, it would meet its image under multiplication by $\sqrt{-1}$ nontrivially, so it would contain a real two-plane spanned by $v,\sqrt{-1}v$; the sum of the Hessian on these two vectors is a positive multiple of the Levi form on $v$, positive for a strongly plurisubharmonic function, whereas both Hessian values would instead be negative, a contradiction.

Suppose a constructible complex $A$ has stalks satisfying ${\mathcal{H}}^q(A{)}_x=0$ for $q>d-s$ on each stratum of complex dimension $s$, and put $c=d-s$. For $c=0$ the normal Morse pair is the point pair and its cohomology has the asserted upper bound zero. Assume the upper normal-Morse bound for all smaller codimensions. The complex link at a codimension-$c$ stratum has dimension at most $c-1$, and we use $c-1$ as an allowed upper dimension even when the variety is nonpure; an empty link has zero cohomology. A critical point of its collar-compatible logarithmic strongly plurisubharmonic exhaustion lies on a link stratum of dimension $c-1-c^{\mathrm{\prime }}$, where $c^{\mathrm{\prime }}$ is its codimension relative to that allowed upper dimension and satisfies $0\le c^{\mathrm{\prime }}\le c-1<c$. Before refinement its stalk bound is the original codimension bound, and refinement lowers the stratum dimension and therefore weakens the required upper bound, so the same stalk hypothesis holds. By the induction hypothesis its normal Morse cohomology vanishes above $c^{\mathrm{\prime }}$, and by step 1.1 its tangential index is at most $c-1-c^{\mathrm{\prime }}$; thus its relative link cohomology vanishes above $c-1$. Starting with the empty sublevel and applying the long exact pair sequences at each critical value proves that the link cohomology vanishes above $c-1$, and the collared-link and regular-interval clauses of [F1] identify this calculation with the full complex link.

The normal Morse triangle has stalk cohomology in one term and link cohomology in the other; in degrees $q>c$ the stalk term vanishes by the hypothesis on $A$, and the link term in degree $q-1$ vanishes by step 2.1, so its long exact sequence makes normal Morse cohomology vanish above $c$. This completes the codimension induction. For $Z$ itself, at a critical point on an $s$-dimensional stratum the normal degree is at most $d-s$ and the tangential index is at most $s$, so [F1] makes each relative group vanish above $d$; the finitely many critical-value exact sequences, followed by the regular interval above the last value, give $H^q(Z,A)=0$ for $q>d$.

A degree-zero sheaf $F$ satisfies the stalk hypothesis of step 2.1 because every codimension $d-s$ is nonnegative and $H^q(F)=0$ for $q\mathrm{\ne }0$, which proves its dimension-$d$ bound. Replace $Z$ by the closed algebraic support closure of $F$, of dimension $e$: extension by zero along that closed embedding recovers $F$ and its global cohomology, so the same argument gives the sharpened bound $q>e$.

For $P\in {}^p{D}_c^{\le 0}(Z;k)$ use its finite hypercohomology spectral sequence $E_2^{a,b}=H^a(Z,{\mathcal{H}}^b(P))$. By [F2] the support dimension of ${\mathcal{H}}^b(P)$ is at most $-b$, so step 4.1 makes $E_2^{a,b}=0$ whenever $a+b>0$, and the bounded spectral sequence therefore gives $H^q(Z,P)=0$ for $q>0$. For $P\in {}^p{D}_c^{\ge 0}$, apply this result to $DP\in {}^p{D}_c^{\le 0}$ and the duality in [F2] to obtain $H_c^q(Z,P)=0$ for $q<0$. Empty spaces and zero sheaves are covered by the zero spectral sequence.