Hall HN factorization, regular logarithms and potential Euler integration

Internal part 1 — HN filtrations from finite dimension, with the King sign fixed. Let $\delta (d)=\sum d_i$ and $Z_{\theta }(d)=-\theta (d)+i\delta (d)$. Its phase increases with the slope ${\mu }_{\theta }(d)=\theta (d)\mathrm{/}\delta (d)$. In particular phase 1/2 and ordinary Z-semistability are exactly King θ(E)=0, θ(A)≤0 for subobjects A.

A fixed nonzero representation E has only finitely many possible dimension vectors of subobjects, so their slopes have a maximum μ. Among subobjects of slope μ choose A of largest δ. If A and B both have maximal slope, then their intersection has slope at most μ. Additivity of dimension and θ gives slope(A+B)≥μ. It equals μ by maximality, so maximality of δ forces B⊂A. Thus A is the unique largest maximal-slope subobject. All its subobjects have slope≤μ, so it is semistable. Every nonzero subobject of E/A has slope strictly smaller: its inverse image would otherwise have slope larger than μ or have slope μ with larger dimension. Repeat on E/A. Dimension decreases, giving a finite semistable filtration with strictly decreasing slopes. The same largest-subobject characterization at every quotient proves uniqueness. This is an actual finite-dimension HN argument, not an assumed existence theorem.

For semistable A,B with μ(A)>μ(B), Hom(A,B)=0. A nonzero image is a quotient of A, hence has slope≥μ(A), and a subobject of B, hence slope≤μ(B), a contradiction. This also gives the usual alternative uniqueness proof by comparing the first factors.

Internal part 2 — Families, Hall product and the HN identity. For a family of representations of dimension d, its subrepresentations of dimension e are the closed incidence locus in the product of vertex Grassmannians: each arrow must preserve the chosen vertex subbundles. This locus is projective over the family base. Its proper image is closed. Because there are finitely many e≤d, excluding forbidden subobject dimensions gives the open semistable locus, and excluding forbidden quotient dimensions gives the corresponding phase-interval opens.

Hall convolution uses first factor as SUBOBJECT and second as QUOTIENT. The stack over two representations E,F is the groupoid of extensions 0→E→G→F→0, with identified outer terms. Associativity is the actual equivalence between either iterated extension groupoid and the stack of length-three filtrations; inverse constructions use subobjects and their inverse images. This works for locally free vertex bundles on all test schemes and respects the middle-object map. The zero representation supplies the unit.

Fix a total-dimension truncation k. There are finitely many possible nonzero dimension vectors, hence finitely many possible phases for this fixed θ. Let $1_{\phi }$ be the stack function of semistable objects of phase φ. For an interval I write $1_I$ for objects whose HN factors have phases in I, including zero. Then

$1_I={\prod }_{\phi \in I}^{descending}{1}_{\phi }$.

Indeed the convolution is the stack of filtrations with semistable factors of the given descending phases, inserting zero factors for missing phases. Forgetting this filtration induces an equivalence on geometric objects and their automorphisms by HN uniqueness. Its representable finite-type fibres are single points. The Hall stack-function equality follows by the following explicit characteristic-zero stratification, rather than claiming that all such morphisms are isomorphisms: pass to reduced finite-type atlas strata; a quasi-finite geometrically bijective morphism has equal function fields in characteristic zero and is an isomorphism over a dense open after clearing finitely many denominators; repeat on the smaller closed complement. Nilpotent thickenings have the same stack-function class by the closed/open relation with empty complement. The isomorphisms and stabilizer identifications descend on these strata. Thus the classes agree. This proves the identity in every finite truncation and hence in the inverse-limit Hall algebra.

Internal part 3 — Triangular factorization and scattering consistency. For a positive-graded nilpotent Lie algebra truncated at total degree k and a weight θ, split its graded pieces into θ-positive, θ-zero and θ-negative subalgebras. Multiplication $G_{+}(\theta )\times G_0(\theta )\times G_{-}(\theta )\to G$ is bijective. To prove this without a finite-dimensionality assumption on the graded pieces, use the BCH polynomial in the nilpotent algebra and solve degree by degree: at each degree the unknown new components enter by their sum, and the three sign summands are a direct sum; all other terms were determined in lower degree. This gives existence and uniqueness in finitely many steps.

Apply this to the fixed element $1_A$ of all representations. The HN identity gives

$1_A=1_{(1\mathrm{/}2,1)}(\theta )$ · $1_{ss}(\theta )$ · $1_{(0,1\mathrm{/}2)}(\theta )$.

Consequently its unique middle factor is exactly $1_{ss}(\theta )$. Subdivide weight space by the finitely many hyperplanes θ(d)=0, δ(d)≤k. On each relative chamber all relevant signs are fixed. The positive factor $F(\theta )={\mathrm{\Pi }}_{+}^{\theta }(1_A)$ is constant on those chambers. At a generic wall point, zero degrees are the positive multiples of its primitive normal n; the middle factor lies in precisely that wall subgroup. Crossing in the positive n direction changes F from $F_0$ to $F_0\cdot 1_{ss}$. With the source convention ε=−sign(d/dt θ(n)) and reverse crossing order, the path-ordered product is therefore

$\mathrm{\Phi }(\gamma )=F(\gamma (1){)}^{(-1)}F(\gamma (0))$.

This follows one crossing at a time and telescopes in the stated noncommutative order. Every closed generic path has product 1. Increasing k gives compatible finite diagrams, so defines the consistent Hall diagram in the degree completion.

Its support is exactly the set of weights with a nonzero semistable representation. A semistable representation has a Jordan–Hölder filtration in its fixed-slope abelian category (dimension again bounds its length); a stable factor satisfies strict inequalities for all proper subobject dimension vectors. Its semistability cone therefore contains a relatively open region in its dimension hyperplane around the chosen weight. This supplies a genuine codimension-one wall through every point of the semistable support, including points on joints by closure. Conversely the finitely many subobject inequalities defining a chosen representation's semistability are closed; if the wall's generic semistable locus is nonempty, one such representation remains semistable throughout that wall cone and its faces. The Hall label is nonzero without an Euler noncancellation assumption: pull its degree-d class back along a geometric semistable object Spec $C\to M_d$. Since $M_d^{ss}\to M_d$ is an open immersion, this pullback is the Hall coefficient 1. Thus deleting trivial labels leaves exactly this support, not just a larger hyperplane arrangement.

Internal part 4 — Extension stacks prove regularity and commutator divisibility. Work on varieties X,Y mapping to the representation stacks, and stratify so their vertex bundles are trivial and the linear maps below have constant ranks. For outer terms E,F, put

$C0(F,E)={\oplus }_i\mathrm{Hom}(F_i,E_i),C1(F,E)={\oplus }_{a:i\to j}\mathrm{Hom}(F_i,E_j)$.

An extension in a vertex splitting has upper-right arrow blocks $z_a$. Every relation imposes a LINEAR equation in these blocks, since a product of upper-triangular matrices has exactly one upper-right factor. Let Z1(F,E) be their kernel. Change of splitting acts by the vertex vector group C0 via its coboundary d0. The actual extension groupoid is [Z1/C0]. Rank stratification makes ker d0=Hom(F,E) and Z1/im d0=Ext1(F,E) vector bundles. Choose local splittings of these vector-bundle sequences. The C0-torsor Z1→[Z1/C0] is Zariski locally trivial after every scheme pullback, since its affine-translation cocycle descends to an extension of O by a vector bundle, which splits on affine opens (O is a projective module there); hence its class is $L^{(-\dim C0)}[Z1\to M]$. After removing the coboundary affine directions this is $L^{(-h_{FE})}[\mathrm{Ext}1(F,E)\to M]$, with $L=t^2$ and $h_{FE}=\dim \mathrm{Hom}(F,E)$. It belongs to $H_{reg}$. This proves regular Hall closure for the generators and hence their regular coefficient combinations.

Decompose Ext1 into zero and nonzero parts. Scaling a nonzero extension leaves its middle object isomorphic, and its projective quotient carries the universal extension with its subobject twisted by O(1); thus the middle-object map descends to P Ext1. This is an actual family construction, and the complement of the zero section is its $G_m$-torsor. On the rank stratum,

$[X]\ast [Y]=L^{(-h_{FE})}([X\times Y\to M_{\text{direct-sum}}]+(L-1)[P\mathrm{Ext}1(F,E)\to M])$.

Interchanging E,F gives the corresponding expression. The two direct-sum maps are canonically isomorphic. Their coefficient difference $L^{(-h_{FE})}-L^{(-h_{EF})}$ is divisible by L−1 in the permitted Laurent coefficient ring. The remaining terms already have this factor. Hence every regular commutator is divisible by L−1, proving the regular Poisson bracket. All these are stack-function equalities over the middle-object stack, not only identities after forgetting the map.

Internal part 5 — The potential Hessian gives the exact skew form. Let r1(F,E):C1(F,E)→C2(F,E) be the linearized relation map; $C2(F,E)={\oplus }_{a:i\to j}\mathrm{Hom}(F_j,E_i)$. The trace pairings identify C2(F,E) with $C1(E,F{)}^{\vee }$. The two off-diagonal blocks of the polynomial tr W on representations of dimension d(E)+d(F) give the mixed Hessian. Equality of mixed derivatives and cyclic invariance of trace show that r1(F,E) is the adjoint of r1(E,F), with the same rank. This is a finite polynomial identity, valid for every finite potential; no Ginzburg-CY theorem is needed.

Since $h_{FE}=\dim C0(F,E)-\mathrm{rank}d0$ and $e_{FE}=\dim$ ker r1(F,E)−rank d0,

$h_{FE}-e_{FE}=\dim C0(F,E)-\dim C1(F,E)+\mathrm{rank}r1(F,E)$.

Subtract the reversed expression. The vertex terms are symmetric and the Hessian ranks equal, so

$(h_{EF}-e_{EF})-(h_{FE}-e_{FE})={\sum }_{a:i\to j}(d(E{)}_{j}d(F{)}_i-d(E{)}_{i}d(F{)}_j)=⟨d(E),d(F)⟩$.

This matches the source skew form and the SUBOBJECT-first Hall product. It proves the needed Hom/Ext1 skew identity directly in the Jacobian representation category.

Internal part 6 — Euler integration on regular classes. Use compactly supported Euler characteristic, equal to ordinary Euler for complex finite-type varieties, and the source virtual-Poincaré coefficient normalization. With the printed convention $\mathrm{{\rm Y}}(Y)=\sum b_i(Y)t^i$ for smooth projective Y, Euler specialization is $t=-1$; in particular $L=t^2$ maps to 1. A sentence in the source saying t maps to 1 cannot be copied together with that positive-Betti convention (an elliptic curve has Υ=1+2t+t² and Euler zero). Nothing in the stated Hall Poisson formula changes, since its denominator is L−1. This fixes the normalization explicitly instead of identifying signed and unsigned Betti sums.

Define $I([X\to M_d])=e_c(X)x^d$ and specialize regular coefficients accordingly. Additivity and Zariski-fibration multiplicativity are the coefficient/scissor normalization of this Euler specialization. The extension-stack formula proves multiplicativity: affine extension fibres contribute 1 and projective fibres contribute e, so the product before division has Euler e(X)e(Y). For the divided commutator, the direct-sum coefficient contributes $h_{EF}-h_{FE}$, while projective extensions contribute $e_{FE}-e_{EF}$. Their sum is

$(h_{EF}-e_{EF})-(h_{FE}-e_{FE})=⟨d(E),d(F)⟩$.

Stratify X×Y by the constant dimensions; the right side depends only on dimension vectors, so summing gives

$I((L-1{)}^{(-1)}(a\ast b-b\ast a))=\{I(a),I(b)\}$.

Thus I is the required graded Poisson homomorphism on regular Hall classes. Its graded completion and exponential group homomorphism follow term by term in each finite nilpotent quotient. No analytic convergence is asserted.

Internal part 7 — Framed objects supply the regular logarithm. We now supply the no-poles assertion, rather than making Joyce's theorem an input. Fix a dimension truncation k and use the abelian category $C_{\theta }$ of θ-semistable objects of weight zero, including zero. It is abelian: for a morphism E→F its image is a quotient of E and a subobject of F, hence has θ both ≥0 and ≤0; its kernel and cokernel have θ zero and inherit semistability. The same subobject inequality proves extension closure. Finite representation dimension makes it a length category for the argument below.

Frame E by m vectors at every vertex, for a fixed m≥1. Call (E,s) framed-stable when E lies in $C_{\theta }$ and NO proper $C_{\theta }$-subobject, including the zero subobject, contains all the frame vectors. Zero has its unique framed object and supplies the unit. For each fixed nonzero d the framed-stable locus is an OPEN of the finite-type representation/frame scheme: exclude zero frames and the images of the finitely many projective subrepresentation incidence schemes of dimensions $e<d$ with θ(e)=0 and containing the frames. There are only finitely many e. No Hilbert–Mumford criterion or general GIT theorem is needed.

It has no automorphisms. If g fixes the frame, ker(g−1) is a $C_{\theta }$-subobject containing it; framed stability forces $g=1$. The same kernel argument for an infinitesimal endomorphism shows Lie $Aut=0$. The quotient by the vertex GL group is a finite-type algebraic space $F_d$, with its actual universal framed module. Here is a direct chart construction. At x choose dim G regular functions whose differentials on the orbit tangent space are independent (Lie $Aut=0$ makes that tangent space dim G). Their common level S is a transverse local slice. In the incidence scheme $Z=\{(g,u):f_i(gu)=f_i(x)\}$, the Jacobian in the smooth G variables is invertible at (1,x), so Z→U is étale there by the explicit Jacobian criterion. The change (g,u)↦(g,gu) identifies this with an open of G×S→U. Thus S→[U/G] is étale. These slices cover, and their relation is the represented equality relation inside G×S×S. Trivial inertia makes it an étale equivalence relation, yielding the algebraic space by its sheaf definition. Triviality also holds over arbitrary test schemes: relative inertia is unramified by fibre Lie vanishing; the identity section is open, covers every geometric fibre and hence is the whole inertia. This constructs the needed chart instead of assuming a free GIT quotient.

For applicability here $F_d$ is separated. A generic frame-preserving isomorphism of two R-families has vertex matrices over Frac R. Multiply all by the smallest power ${\pi }^a$ making every matrix integral and some matrix nonzero modulo π. If $a>0$, its special-fibre module map is nonzero and kills every frame, contradicting the kernel argument in $C_{\theta }$. Thus the original matrices are integral. Apply the same argument to the inverse to get an R-isomorphism. The Isom functor is an affine finite-type transporter, so this is its proper valuative criterion and proves separatedness. This argument only needs semistability and framed stability of the two special fibres; no projectivity of $F_d$ is assumed. All normalization and finite/quasi-finite facts used in the preceding dense-open construction are the actual finite-type-field and affine localization inputs, without assuming that arbitrary source-local finite algebras possess compatible descent data.

The class $[F_d\to M_d]$ belongs to $H_{reg}$. It suffices to stratify a separated finite-type algebraic space X over C by schemes. Choose a surjective etale atlas U→X with U a finite disjoint union of affine finite-type schemes. The geometric fibre cardinalities have a finite bound. This can be checked on a finite affine etale atlas of X: the resulting scheme morphisms are quasi-finite of finite type; source-local finite factorizations in F3 give a finite cover of each quasi-compact source by opens inside finite schemes. A finite algebra on a Noetherian affine base has a fixed finite number of module generators, bounding every geometric fibre length; summing over that finite cover gives a fibre-cardinality bound. Let n be the largest attained cardinality and let $X_n$ be its locus. This locus is open, and $U_n=U{\times }_X{X}_n\to X_n$ is finite etale. Here is a local verification of both assertions, which supplies the missing compatibility instead of gluing the finite factorizations. At a geometric point with n distinct lifts, take the fibre product of pointed etale neighborhoods of those lifts. After shrinking this etale base, the pulled-back atlas has n sections which are pairwise disjoint: their coincidence loci are closed because U→X is separated, and miss the chosen point. Each section is open since the morphism is etale. Every geometric fibre on this neighborhood contains these n points, so by maximality it contains no others. The n open sections therefore cover the pulled-back atlas and identify it with n disjoint copies of that base. Their base images form open neighborhoods in X; these prove that $X_n$ is open and the atlas over it is etale-locally a finite disjoint union, hence finite etale. The descent data here are the actual atlas pullbacks and their canonical cocycles. Form the ordered-full-fibre cover $T\subset (U_n{)}_{X_n}^n$ by deleting all coincidence diagonals. Those diagonals are open and closed, since $U_n\to X_n$ is finite etale. Thus T is finite over $U_n$, is a scheme, and is quasi-affine: $U_n$ is open in the affine disjoint union U, and a finite scheme over a quasi-affine scheme is quasi-affine, by extending its finite algebra generators to an affine closure after clearing denominators. Permuting the n entries gives a free $S_n$-action, and $T\mathrm{/}{S}_n=X_n$ as etale sheaves: an ordering of a finite etale fibre exists etale-locally, and two orderings differ by exactly one permutation. A finite group on a quasi-affine scheme has invariant affine neighborhoods of each finite orbit. Indeed take the joint graph of a quasi-affine embedding and all its translates, an invariant quasi-affine embedding into an affine scheme; choose a function vanishing on the boundary and nonzero on the orbit, and multiply its translates. Its nonvanishing locus is invariant affine. The affine quotients Spec $B^{S_n}$ glue to a scheme representing $T\mathrm{/}{S}_n$. Thus X contains the nonempty open scheme $X_n$. Apply this construction to each reduced irreducible closed remainder; Noetherian induction and a finite affine stratification give a finite sum of varieties. Nilpotent thickenings do not alter stack-function classes by the scissor relation. This proves the asserted regularity over $M_d$.

Every framed E in $C_{\theta }$ has a unique smallest $C_{\theta }$-subobject A containing its frame: take finite intersections until dimension stops decreasing. Intersections remain in $C_{\theta }$ by its abelian property. The framed A is framed-stable and E/A lies in $C_{\theta }$. Conversely this is precisely the filtration selected by a framed-stable subobject. Family incidence and the same finite-type, geometric-point-equivalence stratification used for the HN identity therefore give the ACTUAL Hall equality

${\mathrm{\Theta }}_m(g)=F\ast g,g=1_{ss}(\theta ),F=1+{\sum }_{d\mathrm{\ne }0}[F_d\to M_d]$,

where ${\mathrm{\Theta }}_m$ multiplies degree d by $L^{m\delta (d)}$. The left side parametrizes all frames, a vector bundle of rank mδ(d) over the semistable stack. The right side parametrizes their unique generated $C_{\theta }$-subobject and quotient, with the first Hall factor as subobject. Thus $F={\mathrm{\Theta }}_m(g)g^{(-1)}$ lies in $1+H_{reg}$, and this equality respects the maps to the middle-object stack.

Set $a=(L-1)\log g$, initially an element of the Hall algebra over C(t). We prove $a_d\in H_{reg}$ by induction on total degree. Suppose all lower-degree coefficients are regular, put $u=a_{<d}\mathrm{/}(L-1)$, and $h=\exp (-u)g$. Then h has no nonzero coefficients below degree d and its degree-d term is $a_d\mathrm{/}(L-1)$. The element $F_0={\mathrm{\Theta }}_m(\exp u)\exp (-u)$ is regular: write ${\mathrm{\Theta }}_{m}u=u+v$, where $v=({\mathrm{\Theta }}_m{a}_{<d}-a_{<d})\mathrm{/}(L-1)$ is regular. The BCH polynomial log(exp(u+v)exp(−u)) has in each Lie monomial at least one v. Commuting u with a regular element preserves regularity, since [a,b] is divisible by L−1 by internal part 4. Every other bracket of regular elements is regular too. Thus $\log F_0$ and its exponential lie in $H_{reg}$ in each nilpotent truncation. This is a finite polynomial calculation, not a convergence assertion.

At degree d the equation $F={\mathrm{\Theta }}_m(g)g^{(-1)}$ now gives

$F_d-(F_0{)}_d=((L^{m\delta (d)}-1)\mathrm{/}(L-1))a_d$.

Conjugation by exp u changes no degree-d term, because u has positive degree; this proves the displayed coefficient identity. Its scalar is $1+L+\cdots +L^{m\delta (d)-1},a$ UNIT in the specified regular coefficient ring (it is exactly a projective-space polynomial). Both terms on the left are regular, so $a_d$ is regular. Start with the smallest positive degree and continue to k. As k was arbitrary this proves

$(L-1)\log 1_{ss}(\theta )\in {\widehat{H}}_{reg}$.

This is the required regular logarithmic factorization, reconstructed by actual finite-type framed moduli and the BCH induction, without taking a no-poles theorem as a premise. It applies at joints as well as general wall points; no generic θ hypothesis was used.

Internal part 8 — Integrated scattering and exact output. Write ε(d,θ) for the degree-d coefficient of the regular logarithm just proved. The completed Poisson homomorphism in internal part 6 sends $(L-1{)}^{(-1)}\epsilon$ to $e(\epsilon )x^d$ in the target Poisson Lie algebra. All coefficients here are rational: the framed classes have integer Euler characteristics, and the induction uses only rational BCH coefficients and inverses of nonzero integer specializations of the projective polynomials. It therefore transports the consistent Hall path products of internal part 3 to consistent path products, with wall element

$\exp ({\sum }_{d:\theta (d)=0}J(d,\theta )x^d),J(d,\theta )=e(\epsilon (d,\theta ))$.

This is exactly the specified potential-case Joyce normalization, not an arbitrary weighted DT invariant. The Hall diagram has the exact support proved in internal part 3; the integrated diagram may of course have zero numerical wall labels, which does not alter that Hall-support assertion. The construction is formal in degree and makes no analytic-growth claim.