Real and complex ADHM quotients at positive and regular zero FI

For $g\in GL(V)$ put $x^{\mathrm{\prime }}=g\cdot x$ and $M(g{)}^2={\sum }_i\mathrm{\parallel }{B}_i^{\mathrm{\prime }}{\mathrm{\parallel }}^2+\mathrm{\parallel }{I}^{\mathrm{\prime }}{\mathrm{\parallel }}^2+\mathrm{\parallel }{J}^{\mathrm{\prime }}{\mathrm{\parallel }}^2$. A fixed word frame $C(x)$ exists by [F2]. Each column of $gC(x)=C(x^{\mathrm{\prime }})$ is a product of at most $k-1$ matrices $B_i^{\mathrm{\prime }}$ followed by $I^{\mathrm{\prime }}$ on a unit framing vector. Submultiplicativity and operator norm bounded by Frobenius norm give $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le A_x(1+M(g){)}^k$ for $A_x=\sqrt{k}\mathrm{\parallel }C(x{)}^{-1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. If $x$ is costable, the rows consisting of coordinate rows of $J{B}_{i_1}\cdots B_{i_m}$ span $V^{\ast }$: their common annihilator is the largest common $B_i$-invariant subspace in $\ker J$, hence zero. The same finite dimension-growth argument as for columns gives a row basis $D(x)$ using words of length at most $k-1$. Since $D(x^{\mathrm{\prime }})=D(x)g^{-1}$, the identical norm estimate gives $\mathrm{\parallel }{g}^{-1}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le A_x^{\mathrm{\prime }}(1+M(g){)}^k$. This also proves costability is open, since it is covered by the nonvanishing determinants of finitely many such row frames.

Define $F_{\zeta }(g)=M(g{)}^2-\zeta \log \det (g^{†}g)$. It is continuous and invariant under left $U(V)$. By [F4], the positive eigenvalues of $g^{†}g$ are at most $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2$, so $\det (g^{†}g)\le \mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^{2k}$. The column bound in step 1.1 therefore gives $\log \det (g^{†}g)\le C_x+2k^2\log (1+M(g))$, with $C_x=2k\log \max \{1,A_x\}$. Since $\log (1+M)\le M$ for $M\ge 0$, a sublevel $F_{\zeta }\le c$ with $\zeta >0$ bounds $M$ by the quadratic inequality $M^2-2\zeta k^{2}M-\zeta C_x\le c$, and hence bounds $g$. The same sublevel gives $\det (g^{†}g)\ge e^{-c\mathrm{/}\zeta }>0$; bounded entries and the adjugate formula therefore bound $g^{-1}$. The sublevel is closed in the finite matrix space, since any matrix limit remains invertible by this determinant bound. It is compact by [F3]. The nonempty sublevel at $F_{\zeta }(I)$ has a minimum, which is a global minimum since all points outside have larger value. For $\zeta =0$ and regular $x$, $F_0=M^2$ directly bounds $M$, and the two word bounds in step 1.1 bound both $g$ and $g^{-1}$; the same closedness/compactness argument gives a minimum.

Let $H=H^{†}$ and vary $g$ by $e^{tH}g$. Differentiating the finite trace expressions gives $\frac{d}{dt}{F}_{\zeta }(e^{tH}g){\mathrm{\mid }}_{t=0}=2\mathrm{tr}(H({\mu }_{\mathbb{R}}(x^{\mathrm{\prime }})-\zeta I))$: a $B_i^{\mathrm{\prime }}$ term contributes $2\mathrm{tr}H[B_i^{\mathrm{\prime }},B_i^{\mathrm{\prime }†}]$, the $I^{\mathrm{\prime }}$ term contributes $2\mathrm{tr}H{I}^{\mathrm{\prime }}{I}^{\mathrm{\prime }†}$, the $J^{\mathrm{\prime }}$ term contributes $-2\mathrm{tr}H{J}^{\mathrm{\prime }†}{J}^{\mathrm{\prime }}$, and $\log \det (g^{†}{e}^{2tH}g)$ has derivative $2\mathrm{tr}H$. These derivatives of $e^{tH}$ follow by applying the real-matrix supplier to $\rho (H)$ and using [F7]. Thus each minimum from step 2.1 is balanced, because the trace pairing on Hermitian matrices is nondegenerate. Conversely a balanced point is critical: these Hermitian directions and the skew-Hermitian directions tangent to the left unitary orbit span every real tangent direction of $GL(V)$, and the latter derivatives vanish by unitary invariance.

Diagonalize $H$ by [F4], with real eigenvalues ${\lambda }_i$. Along the preceding path the squared $B_i^{\mathrm{\prime }}$ entries have factors $e^{2t({\lambda }_a-{\lambda }_b)}$, the squared $I^{\mathrm{\prime }}$ entries have $e^{2t{\lambda }_a}$ and the squared $J^{\mathrm{\prime }}$ entries have $e^{-2t{\lambda }_a}$. The log determinant is affine in $t$. Hence $F_{\zeta }^{\mathrm{\prime }\mathrm{\prime }}$ is a sum of $4w^2\mathrm{\mid }z{\mathrm{\mid }}^2{e}^{2tw}$ over those entries and their listed weights $w$, and is nonnegative. It can vanish only when $[H,B_i^{\mathrm{\prime }}]=0$, $H{I}^{\mathrm{\prime }}=0$ and $J^{\mathrm{\prime }}H=0$. Stability of $x^{\mathrm{\prime }}$ then forces $H=0$, by applying $H$ to all spanning word vectors. Thus the path is strictly convex for every nonzero $H$. Given two critical matrices $g_1,g_2$, diagonalize the positive matrix $(g_2{g}_1^{-1}{)}^{†}(g_2{g}_1^{-1})$ and take the positive logarithms of its eigenvalues to write $g_2{g}_1^{-1}=u{e}^H$ with $u$ unitary and $H$ Hermitian. The endpoints of $e^{tH}{g}_1$ are both critical, after removing $u$ by invariance. Strict convexity forces $H=0$, so $g_2=u{g}_1$. This proves uniqueness of the unitary orbit at either level.

Suppose now that $x$ is stable and ${\mu }_{\mathbb{R}}(x)=0$. For a common invariant subspace $S\subseteq \ker J$, write $V=S\oplus S^{\perp }$ and $B_i=\left(\begin{array}{cc}{A}_i& C_i\\ 0& D_i\end{array}\right)$, $I=(I_S,I_{S^{\perp }}{)}^T$, $J=(0,J_{S^{\perp }})$. The trace on $S$ of the real equation is $0={\sum }_i\mathrm{\parallel }{C}_i{\mathrm{\parallel }}^2+\mathrm{\parallel }{I}_S{\mathrm{\parallel }}^2$. Thus all $C_i$ and $I_S$ vanish. Then $S^{\perp }$ is invariant under both $B_i$ and contains $I(W)$, so stability gives $S^{\perp }=V$, hence $S=0$. This proves automatic costability at zero real level. Combining with existence and uniqueness from steps 2.1–4.1 proves both asserted bijections of orbit sets.

At a balanced datum $x$, let $L_x:\mathrm{Herm}(V)\to \mathrm{Herm}(V)$ be $H\mapsto \frac{d}{dt}{\mu }_{\mathbb{R}}(e^{tH}\cdot x){\mathrm{\mid }}_{t=0}$. Step 4.1 gives $2\mathrm{tr}(H{L}_{x}H)=F_{\zeta }^{\mathrm{\prime }\mathrm{\prime }}(e^{tH}){\mathrm{\mid }}_{t=0}>0$ for nonzero $H$, so $L_x$ is injective and hence invertible in finite equal dimensions. The map $(H,y)\mapsto {\mu }_{\mathbb{R}}(e^H\cdot y)-\zeta I$ is smooth by [F7] and the polynomial formula for ${\mu }_{\mathbb{R}}$. By [F5] the balancing $H=H(y)$ is locally $C^r$ for every finite $r$, and uniqueness glues these branches to the same smooth map. The same invertible derivative along the Hermitian gauge directions in $T_x{X}^s$ proves that the real-level locus is a smooth submanifold. The unitary group is compact (closed bounded matrices), its action there is free by [F2] and proper (inverse images are closed subsets of $U(V)$ times a compact coordinate projection); therefore its quotient is smooth by [F6]. The map to the complex quotient is smooth using local submersion sections in [F6]. On any word-frame chart [F2], the local balancing map just constructed, followed by the unitary quotient map, is its smooth inverse; step 4.1 proves independence of all choices. Thus the bijections are diffeomorphisms. The compact torus commutes with gauge actions and preserves both the complex equation and ${\mu }_{\mathbb{R}}$, so the bijections are equivariant. For $k=0$ all matrices and equations are empty and all quotients are points, proving every boundary case stated. No sheaf or ASD identification was used.