Stable ADHM partition fixed points and tangent character

The infinitesimal $GL(V)$ action and the derivative of the complex ADHM equation form the complex $C^0=\mathrm{End}(V)\stackrel{d_0}{\to }{C}^1=\mathrm{End}(V)t_1\oplus \mathrm{End}(V)t_2\oplus \mathrm{Hom}(W,V)\oplus \mathrm{Hom}(V,W)t_1{t}_2\stackrel{d_1}{\to }{C}^2=\mathrm{End}(V)t_1{t}_2,$ where $d_0(\xi )=([\xi ,B_1],[\xi ,B_2],\xi I,-J\xi )$ and $d_1({\dot{B}}_1,{\dot{B}}_2,\dot{I},\dot{J})=[{\dot{B}}_1,B_2]+[B_1,{\dot{B}}_2]+\dot{I}J+I\dot{J}$. Expanding gives $d_1{d}_0(\xi )=[\xi ,[B_1,B_2]+IJ]=0$, so this is a complex.

If $d_0(\xi )=0$, then $\xi$ commutes with both $B_i$ and $\xi I=0$. Therefore $\xi B_{i_1}\cdots B_{i_m}I(w)=B_{i_1}\cdots B_{i_m}\xi I(w)=0$ for every word and $w$. Stability says those vectors span $V$, hence $\xi =0$ and $d_0$ is injective.

To show $d_1$ is onto, use the nondegenerate trace pairing on $\mathrm{End}(V)$. If $\mathrm{\Theta }$ annihilates its image, variation of $\dot{J}$ gives $\mathrm{\Theta }I=0$, and variations of ${\dot{B}}_1,{\dot{B}}_2$ give $[\mathrm{\Theta },B_1]=[\mathrm{\Theta },B_2]=0$. The same word calculation as in step 2.1 then gives $\mathrm{\Theta }=0$. Thus the annihilator of $\mathrm{im}{d}_1$ is zero, so $d_1$ is surjective.

Apply [F3] to the displayed equation and stability condition. Its proof uses the declared holomorphic implicit-function supplier, proves properness by word-frame bounds on both the gauge transformation and its inverse, and constructs holomorphic quotient charts. Thus the quotient here is a Hausdorff second-countable complex manifold, with tangent $\ker d_1\mathrm{/}\mathrm{im}{d}_0$ and complex dimension $2kN$. For $k=0$ the quotient is a point. This is the stable complex quotient; no real moment-map or sheaf/ASD identification is inferred.

At a $T$-fixed quotient point represented by $x$, [F3] gives the unique gauge lift $g(t)=C(x)\mathrm{diag}(e_{{\alpha }_{\mathrm{\ell }}}{t}_1^{a_{\mathrm{\ell }}}{t}_2^{b_{\mathrm{\ell }}})C(x{)}^{-1}$ from a standard-framing word basis. This is a smooth representation of the compact torus itself. The fixed-orbit equations give $g(t)I(w_{\alpha })=e_{\alpha }I(w_{\alpha })$ and $g(t)B_i=t_i{B}_{i}g(t)$, so all weights generated by stability have the form $e_{\alpha }{t}_1^i{t}_2^j$ with $i,j\ge 0$. Under $t_1{t}_{2}hJ=Jg(t{)}^{-1}$, a nonzero component from a $V$-weight $\chi$ to the $\beta$th framing line requires $\chi =e_{\beta }{t}_1^{-1}{t}_2^{-1}$. Distinct integer characters on the compact torus cannot satisfy this equality with any generated weight, so $J=0$. The moment equation then gives $[B_1,B_2]=0$. No Zariski-density or algebraic lifting assertion is used.

For each framing line, the commuting pair $B_1,B_2$ makes its cyclic span a quotient of $\mathbb{C}[x,y]$, with $x,y$ acting by $B_1,B_2$. The torus weights $e_{\alpha }{t}_1^i{t}_2^j$ are distinct between colors and between monomials, so the kernel is a monomial ideal. Its standard monomials form a finite downward-closed subset of ${\mathbb{Z}}_{\ge 0}^2$, hence a Young diagram $Y_{\alpha }$. Stability says the cyclic spans together exhaust $V$, so ${\sum }_{\alpha }\mathrm{\mid }{Y}_{\alpha }\mathrm{\mid }=\dim V=k$. Conversely, for any such tuple, take one basis vector per box, let each $B_i$ move a box one unit in direction $i$ and send it to zero when the next box is absent, let $I$ send the $\alpha$th framing vector to the origin box of $Y_{\alpha }$ when nonempty, and set $J=0$. Downward closure makes the shifts commute, the moment equation holds, and every box is generated from its origin. These data are stable and recover the given diagrams. The construction and recovery are inverse on gauge orbits, proving the fixed-point bijection.

Bring each torus-transformed representative back to the fixed representative by the inverse unique gauge lift from step 5.1. The variations then transform as $t_i\mathrm{End}(V)$ for ${\dot{B}}_i$, as $V^{\vee }{\mathbf{W}}^{\vee }$ for $\dot{I}$, and as $t_1{t}_2\mathbf{W}\mathbf{V}$ for $\dot{J}$; the complex-equation target transforms as $t_1{t}_2\mathrm{End}(V)$. Thus the torus representations in the complex of step 1.1 are $C^0={\mathbf{V}}_{\vec{Y}}^{\vee }{\mathbf{V}}_{\vec{Y}}$, $C^1=(t_1+t_2){\mathbf{V}}_{\vec{Y}}^{\vee }{\mathbf{V}}_{\vec{Y}}+{\mathbf{V}}_{\vec{Y}}^{\vee }{\mathbf{W}}^{\vee }+t_1{t}_2\mathbf{W}{\mathbf{V}}_{\vec{Y}}$, and $C^2=t_1{t}_2{\mathbf{V}}_{\vec{Y}}^{\vee }{\mathbf{V}}_{\vec{Y}}$. Exactness at $C^0$ and $C^2$ follows from steps 2.1 and 3.1, so the tangent character is $[C^1]-[C^0]-[C^2]$, which is precisely the expression in the Statement.

The label set in step 6.1 is finite: for $k>0$ every box of a size-at-most-$k$ downward-closed diagram has both coordinates less than $k$, so the possible tuples lie in a finite collection of subsets; $k=0$ has only the empty tuple. Hence the fixed locus is a finite set in the Hausdorff quotient from [F3], and each point is isolated. By [F3] the quotient has a finite affine atlas of word-frame slices $S_C$, and on each slice the normalized action is $s\mapsto D_C(t{)}^{-1}\cdot (t\cdot s)$ with $D_C(t)=\mathrm{diag}(e_{{\alpha }_{\mathrm{\ell }}}{t}_1^{a_{\mathrm{\ell }}}{t}_2^{b_{\mathrm{\ell }}})$. Each ambient matrix-entry coordinate restricts to a weight vector and these coordinates generate the affine slice coordinate ring, so the action extends algebraically to $T_{\mathbb{C}}$ and the slices are invariant. A point fixed by the compact torus has every nonzero-weight coordinate equal to zero, since each nontrivial integer character is nontrivial on that compact torus; conversely these equations make the point fixed by $T_{\mathbb{C}}$. Thus the two fixed loci have the same point set. The $T_{\mathbb{C}}$-fixed subscheme is smooth by [F4], because $T_{\mathbb{C}}$ is diagonalizable and hence linearly reductive. In an invariant affine slice its fixed ideal is generated by the images of the nonzero-weight matrix coordinates; linearizing that ideal and applying the Jacobian-kernel result in [F4] identifies its tangent with the invariant tangent subspace. A smooth fixed scheme with a finite point set has dimension zero at every point, so that subspace is zero. Therefore every complex tangent weight is a nonzero character. Its associated $u_{\chi }$ is a nonzero integer linear polynomial in the algebraically independent variables and is invertible in $K$. The tangent has complex dimension $2kN$, so the Euler product has exactly $2kN$ factors counted with multiplicity.

For each fixed $k$, there are finitely many partitions of $k$, hence finitely many $N$-tuples of total size $k$; every summand $E_{\vec{Y}}^{-1}$ is in $K$ by step 7.1. Therefore the coefficient is a finite sum and the sequence of coefficients defines the formal power series $Z_N^{\mathrm{f}\mathrm{p}}(q)\in K⟨\text{\hspace{0.17em}⁣}⟨q⟩\text{\hspace{0.17em}⁣}⟩$. At $k=0$, $V=0$, the empty tuple is the unique fixed point, the Euler product is empty and equals $1$, so the constant coefficient is $1$.