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A positive-FI point with a non-locally-free framed sheaf

Example

Take $V = W = C$ with their standard Hermitian products and let $ζ > 0$ . For scalar maps $B_{1}, B_{2} \in C$ and $I, J \in C$ , the complex and real moment maps are $μ_{C} = [B_{1}, B_{2}] + I J = I J$ and $μ_{R} = ∣ I ∣^{2} - ∣ J ∣^{2}$ ; positive-FI solutions satisfy $μ_{C} = 0$ and $μ_{R} = ζ$ . Stability means $I (W)$ spans $V$ , and costability means no nonzero common invariant subspace is contained in $\ker J$ . The datum $B_{1} = B_{2} = 0$ , $I = \sqrt{ζ}$ , $J = 0$ is stable and positive-FI, but not costable, so it is outside the zero-FI regular locus. Under the complex $G L (1, C)$ action its stable orbit has representative $I = 1$ , whose framed $P^{2}$ monad on homogeneous coordinates $[z_{0} : z_{1} : z_{2}]$ is

$O (- 1) \overset{(- z_{1}, - z_{2}, 0)^{T}}{\to} O^{\oplus 3} \overset{(z_{2}, - z_{1}, z_{0})}{\to} O (1),$

whose middle cohomology is the ideal sheaf of $p = [1 : 0 : 0]$ ; it is torsion-free and not locally free at $p$ . Thus this positive-FI point lies on the non-locally-free boundary excluded from the ordinary zero-FI framed ASD correspondence.

Example references

ADHM linear data, stability, costability and FI levels

Real and complex ADHM quotients at positive and regular zero FI

Framed P² monads and stable ADHM data

Donaldson, §§2 and 6, printed pp. 3 and 10–11

Nakajima–Yoshioka, §3.1, printed pp. 17–19

Facts & Assumptions

Given: $N = k = 1$ , a positive real level $ζ > 0$ , and the ADHM, quotient and framed monad conventions in the cited items.

[F1]

The complex moment is $[B_{1}, B_{2}] + I J$ , the real moment is $[B_{1}, B_{1}^{†}] + [B_{2}, B_{2}^{†}] + I I^{†} - J^{†} J$ , stability means the $B_{i}$ -words applied to $I (W)$ span $V$ , and costability forbids a nonzero common invariant subspace in $\ker J$ . ADHM linear data, stability, costability and FI levels

[F2]

The positive-FI real quotient represents the stable complex orbit, while at zero FI the real quotient represents exactly the stable-and-costable locus. Real and complex ADHM quotients at positive and regular zero FI

[F3]

For a stable complex datum the framed $P^{2}$ monad has maps $α = (z_{0} B_{1} - z_{1}, z_{0} B_{2} - z_{2}, z_{0} J)^{T}$ and $β = (- z_{0} B_{2} + z_{2}, z_{0} B_{1} - z_{1}, z_{0} I)$ , and its middle cohomology is a framed torsion-free sheaf; local freeness is equivalent to costability. Framed P² monads and stable ADHM data

Proof

Given: The datum and hypotheses in the Example.

1.1F1F2algebra

For $V = W = C$ , both commutators vanish. Thus $μ_{C} = I J = 0$ and $μ_{R} = ∣ I ∣^{2} - ∣ J ∣^{2} = ζ$ . Since $I \neq 0$ , the words of length zero applied to $I (W)$ span $V$ , so the datum is stable. Since $J = 0$ , the nonzero subspace $V$ is invariant and contained in $\ker J$ , so the datum is not costable; in particular it is not a zero-FI regular datum. By [F2], its positive-FI complex orbit has a representative with $I = 1$ after the $G L (1, C)$ change of basis $g = ζ^{- 1 / 2}$ .

2.1F3step 1.1algebra

Substituting that representative in [F3] gives $α = (- z_{1}, - z_{2}, 0)^{T}$ and $β = (z_{2}, - z_{1}, z_{0})$ . Their product is $- z_{2} z_{1} + z_{1} z_{2} = 0$ . The row $β$ is surjective at every projective point because its three entries have no common zero, and $α$ is injective as a sheaf map because it is nonzero away from $p$ . Let $K = \ker β$ . Projection to the third coordinate restricts to a morphism $γ : K \to O$ . On the affine chart $z_{0} = 1$ , put $x = z_{1}$ and $y = z_{2}$ ; a vector in $K$ has the form $(u, v, - y u + x v)$ , so $γ (K) = (x, y)$ and its kernel is generated by $(- x, - y, 0)$ , the image of $α$ . On any chart away from $p$ where $z_{1}$ or $z_{2}$ is a unit, the same equations show that $γ$ is onto and its kernel is generated by $α$ . Hence $K / im α ≅ I_{p}$ globally, where $I_{p}$ is the ideal sheaf of $p$ .

3.1F2F3step 2.1algebra∎

At $p$ , the local ideal $(x, y)$ is not principal: if it were generated by one element in the regular local ring $C [x, y]_{(x, y)}$ , its quotient by the maximal ideal would have dimension at most one, whereas $(x, y) / (x, y)^{2}$ has dimension two. Therefore $I_{p}$ is not locally free at $p$ . Its framing on $L_{\infty}$ is the canonical trivialization because $p \notin L_{\infty}$ . This exhibits a positive-FI stable point outside the costable, locally-free zero-FI locus; the correspondence with ordinary ASD moduli is restricted to that zero-FI locus and does not absorb this boundary point.

A positive-FI point with a non-locally-free framed sheaf

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources