Behrend weighted Euler identity for proper symmetric-obstruction schemes

Let E be the symmetric POT, ℰ=h¹/h⁰(E∨), of virtual rank zero. We construct only the needed intersection operations for its additive bundle-stack charts, not a Chow theory for arbitrary Artin stacks.

Use integral closed substacks, with their stack dimension and generic scheme length, and principal divisors of rational functions on them. These notions are explicit on [F₁/F₀]: closed substacks are invariant closed subschemes of F₁ modulo F₀, and rational functions are their invariant rational functions. Divisor orders are computed on the smooth additive atlas, where they are ordinary scheme orders. A second chart is compared on its fibre product; flat pullback of cycles and of the divisor of the same rational function agrees there. Surjective smooth pullback detects equality of cycles, so this defines the cycle and principal-divisor notions intrinsically. Only equality of cycles, not equality of Chow classes, is being descended. Denote their cycle/rational-equivalence quotient by $A_i(\mathcal{E})$.

Localization for these groups has the direct scheme-style proof: extend a cycle from an open by closure; extend a rational function by the same rational function on its integral closure cycle, whose extra divisor is supported on the closed complement. Proper representable pushforward is defined by the generic field degree and image dimension, retaining the unchanged generic additive stabilizer; flat pullback has its usual generic-length definition. The identities of proper push with principal divisors and of proper push with Cartesian flat pullback are checked on the common smooth bundle charts as SCHEME CYCLE identities, using the same field-degree/divisor-of-norm calculation. Thus they descend to these groups. This is not the construction of operations on arbitrary Artin stacks; the additive bundle charts and their closed cone substacks are the whole class used here.

For a vector-bundle stack π:ℰ→X of virtual rank zero, define π*[Z]=[ℰ_Z]. Its local bundle charts are smooth and geometrically integral over an integral Z; consequently ℰ_Z is integral and reduced, of the required stack dimension dim Z. The divisor of a base rational function pulls to its ordinary flat divisor on those charts. Thus π* is a well-defined homomorphism on integral Chow groups and commutes with proper base pushforward by the preceding cycle calculation.

Suppose ℰ=[F₁/F₀] globally over a scheme X, with rankF₁=rankF₀. First $\pi \ast :A_i(X)\to A_i(\mathcal{E})$ is SURJECTIVE in the cycle/rational-equivalence convention just defined. Project an integral cycle of ℰ to the closure Z of its image in X. On a dense affine open of Z the vector bundles and a nonzero maximal differential minor trivialize and split the complex at its generic rank. There ℰ_Z ≅ $A_Z^a$ × $B(G_a{)}^b$, with $a=b$.

Over K(Z), a closed integral substack is just an integral subvariety of $A^a$ times $B(G_a{)}^b$, since that additive stabilizer acts trivially after the splitting. The ACTUAL scheme affine-space Chow homotopy says its cycle is rationally equivalent to zero unless its affine-space dimension is a, in which case it is a multiple of the whole affine space. The finitely many equations and rational functions in this relation spread over a smaller dense open of Z; appending the same trivial additive factor gives a genuine stack principal-divisor relation there. Localization leaves a cycle with lower-dimensional image in Z. Induction on the dimension of the image proves surjectivity. There is no finite field cover, residual finite gerbe, averaging or division in this argument.

Let a:F₁→ℰ be the smooth atlas of relative dimension rankF₀. Its flat cycle pullback is well-defined by the invariant divisor description in additive cycle construction. The composite aπ is the ordinary vector-bundle pullback F₁→X, an integral Chow isomorphism by the published scheme theorem. Therefore π* is also injective: πα=0 implies aπα=0 and then α=0. Together with surjectivity this proves the bundle-stack homotopy isomorphism in the global-resolution case, including agreement of these cycle groups with the usual bundle-presentation convention. Its inverse is exactly 0^!_{F₁}a, the given BF zero-section formula.

This also supplies independence of a global resolution without a hidden chain-comparison assertion. For another atlas H₁→ℰ, $P=F_1{\times }_{\mathcal{E}}$ H₁ is a vector-bundle torsor over each atlas. The two composites P→X coincide. Pullback of the SAME cone to P and the inverse of that composite pullback give the same class. The torsor pullback is an integral isomorphism by its projective compactification P(W)∖P(V), projective-bundle formula and localization. Thus the concrete global-resolution formula agrees with the intrinsic inverse just proved.

There exists a projective scheme Y and a proper p:Y→X such that every integral closed Z⊂X has an integral closed Z'⊂Y mapping birationally onto Z. To construct it, apply the actual proper Chow lemma to obtain projective Y₀→X which is an isomorphism over a dense open U. Every Z meeting U has its birational strict-transform closure in Y₀. On the lower-dimensional closed complement X∖U perform the same construction inductively and add its envelope by disjoint union. Dimension drops on every remaining component, so the process terminates with a finite disjoint union of projective schemes. This is stronger than merely assuming that one birational modification lifts every subvariety; that latter assertion need not hold. Nilpotents cause no problem because cycles and rational functions use integral reduced closed subschemes.

The fibre product $R=Y{\times }_X$ Y is projective over C: X is separated, so R is closed in Y×Y. Both Y and R therefore admit global two-term bundle presentations of the pulled perfect complex. Here is the exact existence argument used for that assertion. On a projective Noetherian scheme, large positive twists kill higher cohomology of the finitely many coherent cohomology sheaves of E. The resulting hypercohomology edge map makes their globally generating sections actual derived maps from negative ample powers to E. Kill the top cohomology by finitely many such maps and continue downward, giving a bounded-above vector-bundle resolution. The uniform Tor amplitude [−1,0] makes its cokernel at degree −1 flat: its Tor₁ with every module is the degree−2 cohomology of the derived tensor, which is zero. The cokernel is finitely presented, hence locally free, and truncation gives the global two-term resolution. The finite-free/flat and ample-generation inputs are the actual scheme suppliers, with AC/DC inherited. No regularity of Y or R is used. For such an envelope there is an exact coequalizer $A_i(R)$ --(p₁*−p₂*)→ $A_i(Y)$ --p*→ $A_i(X)$ →0.

Here is the full argument needed, which also works after base change to ℰ. Surjectivity follows by choosing the birational lift of every integral cycle. On CYCLE groups the kernel is generated by the fibre-product differences: for an integral W⊂Y with image Z, choose a birational lift Z'→Z. A component of W×_Z Z' maps birationally to W. If $\dim Z<\dim W$, its other projection has smaller-dimensional image, giving the relation [W]−0. If dimensions agree and W→Z has degree e, the two projections give [W]−e[Z']. Subtract these for all W in a cycle in the kernel; the remaining coefficient of each chosen [Z'] is exactly the vanishing pushforward coefficient. This proves the kernel assertion on cycles. If a cycle pushes to a sum of principal divisors downstairs, lift each integral divisor-carrying subvariety birationally and use the same rational function there. Its divisor pushes to that original divisor. Subtract these principal divisors upstairs; the adjusted cycle has zero cycle pushforward and the preceding argument applies. This proves the coequalizer on Chow groups, not merely on geometric fibres.

The base change ℰ_Y→ℰ is likewise an envelope for every integral cone or closed substack appearing here. If its integral image lies over Z, base change by the chosen birational Z'→Z gives a component birational to it, with the identical generic additive stabilizer. Thus the SAME cycle/field-degree/principal-divisor proof gives $A_i({\mathcal{E}}_R)$ → $A_i({\mathcal{E}}_Y)$ → $A_i(\mathcal{E})$ →0.

All maps are proper representable maps, so the push/divisor calculation of additive cycle construction applies. These identities compare cycles and their rational functions on the actual proper fibre product, rather than inferring Chow descent from affine or étale covers.

Pullback by ℰ→X gives a commutative diagram of the two coequalizers in proper-envelope cycle descent. The vertical maps on Y and R are isomorphisms by global-resolution homotopy calculation and the global presentations in projective-envelope construction. Therefore the induced map $\pi \ast :A_i(X)$ ≅ $A_i(\mathcal{E})$ is an isomorphism: identify both middle and relation groups through their isomorphisms and take the same cokernel. This proves the inverse bundle-stack homotopy operation required in the proper target. It can be computed by any chosen projective envelope and its global bundle formula. It is independent of the envelope because it is the inverse of the intrinsically defined pullback, and agrees with every global-resolution BF formula by global-resolution homotopy calculation. In particular the proper-scheme associated intrinsic cone class is well-defined by $[X,E{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=(\pi \ast {)}^{-1}[C_X]$. This construction uses only additive bundle-stack cycles and proper projective envelopes of the proper scheme; no general intersection theory for unrelated Artin stacks has been asserted. For a proper f:X'→X and the pulled bundle stack, flat cycle pullback commutes with the proper pushforward. Write any class of ℰ_{X'} uniquely as π_{X'}α, using the just-proved isomorphism. Then (π_X)^{-1} $f_{\mathcal{E}}\ast$ π_{X'}α = (π_X)^{-1}π_X* $f_{\ast }\alpha$ = $f_{\ast }\alpha$. This proves EXACTLY the required proper compatibility of the intrinsic zero-section inverse. One may check it on projective envelopes first; the cycle diagram proves it globally and entails no extra functoriality assumption.

Let Q be a rank-d vector bundle on a proper scheme Z, and let q:ℰ→Q be a linear morphism whose numerator F₁→Q is surjective in its local bundle presentations. Then $K=q^{-1}(0)$ is a representable regular codimension-d closed substack. For an integral base Z, the needed identity is $[K]=\pi \ast (c_d(Q)\cap [Z])$ in $A_{\ast }(\mathcal{E})$.

Choose a projective envelope Y→Z as in projective-envelope construction and its dominant component Y₀ birational to Z. On Y₀ the pulled perfect complex has a global resolution; its numerator F₁→Q is surjective, and its bundle kernel N has quotient Q. The actual scheme vector-subbundle/zero-section excess identity gives $[N]=p_{F_1}\ast (c_d(Q)\cap [Y_0])$ after the zero-section homotopy identification. Pullback along the additive atlas detects the resulting equality of Chow classes by the isomorphism proved in global-resolution homotopy calculation. Thus $[K_{Y_0}]={\pi }_{Y_0}\ast (c_d(Q)\cap [Y_0])$. Push it to ℰ. The left side is [K] because $K_{Y_0}\to K$ is generically an isomorphism with unchanged generic stabilizer. The right side is $\pi \ast (c_d(Q)\cap [Z])$ by proper/flat compatibility and the ordinary scheme Chern projection formula. This proves the displayed identity globally. Linearity handles a general base cycle. Applying (π*)^{-1} gives precisely the regular-section top Chern identity; no operational identity on an arbitrary Artin stack is being assumed.

The intrinsic inverse and regular quotient-section calculations supply the three formal identities and nothing beyond them. For the Nash application, Z is the proper Nash modification, Q its tautological cotangent quotient, and ℰ is the pulled rank-zero obstruction bundle stack. Its kernel is proper birational to the corresponding intrinsic cone component by the already completed local conormal calculation. The global weighted-Euler comparison can therefore use these proved identities without inserting a quasi-projectivity or global-resolution condition into the original proper-scheme claim.

We first make the virtual-class/Aluffi comparison GLOBAL in the original proper-scheme scope. Write ℰ=h¹/h⁰(E∨), a vector-bundle stack of virtual rank zero. Symmetry and the POT h⁰-isomorphism identify its coherent quotient ob=h¹(E∨) with Ω_X. Let Z be an integral base cycle occurring in the intrinsic cone, $d=\dim Z$, and let ν:Y→Z be its intrinsic Nash modification. Its rank-d tautological cotangent quotient Q is a quotient of νΩ_Z, hence also of νΩ_X. This gives a global linear morphism q:ℰ_Y→Q. In any local two-term bundle presentation its numerator $F_1\to Q$ is a surjective bundle map and its denominator $F_0$ acts in its kernel. Consequently $K_Z=q^{-1}(0)$ is a representable regular closed substack of codimension d in ℰ_Y. This regular-section assertion is smooth-local on the stack and follows from the displayed surjective bundle map; it is not a Chow descent assertion.

By [F2], locally the reduced components of the obstruction cone are the conormal closures of their bases, with generic multiplicities $m_Z$. On the dense smooth part of Z the kernel defining $K_Z$ is precisely that conormal kernel modulo the same tangent action. The proper map $K_Z\to \mathcal{E}$ induced by Nash modification therefore maps generically isomorphically onto that intrinsic cone component, retaining its generic stabilizer and introducing no degree factor. It factors through the reduced component: in a bundle chart the source is reduced and all defining functions of that component vanish at its points. Hence $[C_X]={\sum }_Z{m}_Z({\eta }_Z{)}_{\ast }[K_Z]$ as actual cone cycles in ℰ. This equality is checked at generic points and by descent of closed cone algebras, which is legitimate; no equality of Chow classes is glued from opens. The same local normal-cone multiplicity calculation gives $c_X={\sum }_Z(-1{)}^{\dim Z}{m}_Z[Z]$.

Let π:ℰ_Y→Y. The regular quotient-section identity proved above gives $[K_Z]=\pi \ast (c_d(Q)\cap [Y])$, with the virtual rank zero grading. Applying its inverse zero-section operation from the proved intrinsic inverse in [F1] yields $0_{\mathcal{E}}Y![K_Z]=c_d(Q)\cap [Y]=(-1{)}^d{c}_d(Q\vee )\cap [Y]$. Proper compatibility and linearity now give [X]^vir=Σ_Z(−1)^d $m_Z{\nu }_{\ast }(c_d(Q\vee )\cap [Y])=c_0^M(c_X)=({\alpha }_X{)}_0$. This is one global proper kernel/Nash computation. It needs no global vector-bundle resolution of E, global smooth embedding of X, or quasi-projectivity of X. If the associated class is supplied by a global resolution, the same equalities are the ordinary scheme kernel/excess calculation after its torsor pullback; thus the intrinsic and explicit conventions agree.

It remains to prove the Euler-index identity for an arbitrary proper integral base cycle V⊂X of dimension d; let N→V be its intrinsic Nash modification and T its rank-d Nash tangent bundle. The quasi-projective restriction in the algebraic topology wrappers is not silently imposed on V. A proper separated algebraic V has compact Hausdorff analytification and a finite affine open cover. Compactness can be reduced to the projective case by the actual Chow-lemma supplier lem-chow-lemma-proper-noetherian: its proper quasi-projective cover is projective over C, has compact analytification and maps surjectively to V; separatedness makes the analytic target Hausdorff. Each affine chart has the published local Whitney and control constructions. Finite compatible global strata are constructed by their same dimension-drop procedure: choose smooth dense strata, remove the Whitney-failure loci in each of the finitely many affine charts, and repeat on the lower-dimensional closed remainder. Those failure loci are locally algebraic proper subsets by the affine construction, so their finite union remains lower dimensional; the process terminates. Refine also for the finitely many Nash maps and quotient-bundle charts used here. The tube projections and radii are local to these strata; their controlled right inverses are glued with partitions of unity on compact stratum cores. The overlap equations are linear in those right inverses, so this patching preserves them. Thus the needed controlled collars and finite chain models are obtained on the compact V itself. No global projective compactification of V is needed or claimed.

More explicitly, no global algebraic Whitney theorem outside the suppliers' scope is required here. Start with a finite affine open cover $U_i$ of the reduced proper $V$. Prescribe the closed boundaries of these opens and the finitely many support sets in the general separated finite-type smooth-frontier stratification supplier carried by [F4]. On each affine piece, apply the quasi-projective Whitney construction to the Nash morphism and these prescribed sets. To pass from these finitely many local refinements to global strata, first refine by their algebraic closures and boundaries. On each remaining smooth global stratum remove the closures of the Whitney-failure sets for its finitely many incident pairs in each chart. The local Whitney construction supplies proper lower-dimensional algebraic failure closures, so their finite union is lower dimensional. Repeat on this closed remainder. Dimension strictly drops, yielding a finite global Whitney stratification; every Whitney condition is local in the affine charts. The same construction makes the Nash map stratumwise submersive, by first removing its differential rank-drop closures. Thus the actual finite Whitney and proper stratified-submersion hypotheses used below are met.

To put the controlled construction in the suppliers' precise ambient-manifold setting, choose finitely many local analytic embeddings $z_i:U_i\to C^{N_i}$ and nonnegative smooth cutoffs ψ_i supported inside these charts with Σψ_i>0, and normalize their sum. Such cutoffs are made from local ambient bump functions and glued; chart transition coordinates are restrictions of holomorphic ambient functions. The map $\mathrm{\Psi }=({\psi }_i,{\psi }_i{z}_i{)}_i$, with zero extension outside $U_i$, embeds compact V into a Euclidean space. It is injective because at any point some ψ_i is positive and $z_i$ can be recovered as $({\psi }_i{z}_i)\mathrm{/}{\psi }_i$. It is a homeomorphism onto its image by compactness. On that chart Ψ extends smoothly to the ambient chart with this same smooth left inverse, so it is locally an ambient smooth embedding and preserves the Whitney conditions. Thus V is the actual compact Whitney-stratified subset of a smooth manifold to which the cited controlled-field construction applies. No algebraic or global complex embedding of V has been asserted.

The intrinsic Nash modification also exists without a global smooth embedding or resolution property. The relative Grassmannian of rank-d locally free quotients of the coherent Ω_V is constructed on finite affine module presentations; quotient functors identify the constructions on overlaps. Its Plücker line is the quotient of Λ^dΩ_V and is relatively ample, so its morphism to V is projective. Close the smooth-locus cotangent quotient to obtain N. All local Nash tangent bundles agree by their quotient description and glue. Therefore N is proper and complex oriented in top dimension; it may be singular.

The stratified source of the Nash map has its own smooth ambient realization, as required by [F4]. On an affine chart $U\subset V$ choose a closed embedding $U\hookrightarrow {\mathbb{C}}^m$. The Nash space $N_U$ is the closure of the Gauss graph $x\mapsto T_{x}U$ from the smooth locus inside $U\times Gr(d,m)$, with its reduced structure. It is a closed algebraic subset of the smooth manifold ${\mathbb{C}}^m\times Gr(d,m)$; its Nash projection is the restriction of the first projection. This agrees with step 23.1: the surjection ${\mathcal{O}}_U^m\twoheadrightarrow {\mathrm{\Omega }}_U$ identifies its rank-$d$ quotient Grassmannian with the corresponding closed incidence locus, and both Nash spaces are closures of the identical smooth-locus quotient. Apply the finite dimension-drop Whitney refinement of step 22.1 also to $N$, its chart overlaps, the Nash map and the boundary sets. Over a small stratum disk in $U$, let ${\pi }_S$ and ${\rho }_S$ be the smooth ambient tube projection and radius, with disk coordinates on $S$. The maps $(z,P)\mapsto {\pi }_S(z)$ and $(z,P)\mapsto {\rho }_S(z)$ are smooth ambient extensions of their composites with the Nash projection. A holomorphic projection straightening $S$ extends by the same first projection when used in the product-index calculation below. These extensions exist on an ambient neighbourhood of the whole Nash inverse image of the tube, including all Grassmannian directions. Transverse radius boundaries, with their ambient collars, give locally closed Whitney subsets retaining the boundary strata. Over a smaller disk the closed normal pairs are proper: the Nash map is projective and the closed tube has compact normal fibres. The stratumwise submersions and their boundary restrictions are verified in the next two steps. Thus the proper controlled-trivialization result applies to these embedded source pairs with its ambient-map hypothesis verified, rather than to the embedding of $V$ alone.

Take the finite controlled Whitney stratification just constructed, refined so every proper Nash projection occurring is a stratified submersion over its target strata. Such a refinement is obtained on each local finite-presentation map by removing the rank-drop loci and iterating in smaller target dimension, and then taking the finite common Whitney refinement. This is the same finite affine refinement used by the controlled-submersion supplier. It allows the local product transport to retain the tautological bundle charts; a vector bundle over a contractible stratum disk times a compact fibre is pulled back from that fibre up to a homotopy preserving the fibre.

For the radial index and its constancy one needs normal pairs, rather than only the Nash inverse image of a stratum. Fix a small stratum disk and a closed normal tube over its compact closure, with radius boundary transverse to all incident strata. Its inverse image under the projective Nash map is compact. On every source stratum the composite of the Nash map and the tube projection is a submersion; the same holds on its radius boundary by the submersive projection/radius pair. The controlled-submersion supplier therefore gives a proper local product of this normal Nash pair and its boundary over a smaller disk. In the Nash-source ambient embedding of step 24.1 the composite tube map and radius are restrictions of the displayed smooth ambient maps, so the ambient-map hypothesis is satisfied; holomorphic target coordinate projections may be used locally to retain the complex horizontal tangent orientation. Contracting the disk in this product gives the bundle homotopy. Thus neither the normal pair nor its radial boundary data is inferred from a trivialization of the fibre alone.

A continuous stratified tangent field ξ on V lifts to the Nash bundle by the same vector at a point (x,P) of N. Indeed Whitney A puts $T_{x}S$ inside every limiting tangent plane P above x∈S, so ξ(x)∈P. This is continuous in each bundle chart. We construct fields with the following radial extension rule. On a tube around a lower stratum S, project onto its tangent field by the tube projection π_S, and add a vertical outward radial component for its radius ρ_S. Away from S the equations for a vertical field w are dπ_S(w)=0 and dρ_S(w)>0. Whitney B makes (π_S,ρ_S) a submersion there. Its minimum-norm right inverses on compact links, rescaled by ρ_S, yield a field of norm O(√ρ_S), hence tending to zero at S. Whitney A supplies bounded horizontal right inverses tending to the identity on $T_S$. Adding the horizontal lift of ξ_S therefore tends continuously to ξ_S at S. A finite controlled partition of unity preserves these linear equations and their strict radial inequality. Induction over adjacent strata gives a continuous tangent field on the union, with the specified radial extension rule.

This construction is not claiming arbitrary vector fields are controlled. It chooses the field from the explicit projection/radius equations. Their solution space for a prescribed tangent field is convex after shrinking the tube and fixing a positive radial bound. Two resulting lifts are consequently homotopic through nowhere-zero link fields. A regular-radius collar gives the same comparison between different sufficiently small balls. Changing local embedding can be compared in their product; the extra smooth normal directions give a contractible choice of lifts and preserve the tautological quotient. These facts prove the independence asserted in the definition of Eu and the compatibility of local definitions on chart overlaps.

We make explicit the comparison of the radial convention with the algebraic Segre convention; its input is the controlled radial field already constructed in steps 26.2–26.1. Work in a local complex embedding of a reduced integral $d$-dimensional $V$, with $p$ as a stratum, and let $\mu :N\to V$ be its Nash modification with tangent bundle $T$. Put $F={\mu }^{-1}(p)$ with its scheme-theoretic fibre ideal. For $d>0$ blow up that ideal, $b:B={\mathrm{Bl}}_{F}N\to N$, and denote its exceptional effective Cartier divisor by $D$. The coordinate differences $x-p$ resolve on $B$ to a direction line $L\subset {\mathbb{C}}^m\times B$. The vector $x-p$ is a section of this line with divisor $D$, so $L\simeq {\mathcal{O}}_B(D)$, with the positive sign. This sign is also checked on an ordinary blowup of a smooth point: $L{\mathrm{\mid }}_D={\mathcal{O}}_{{\mathbb{P}}^{d-1}}(-1)={\mathcal{O}}_B(D){\mathrm{\mid }}_D$.

Let p lie on a complex s-dimensional stratum S, and let v be a tangent vector field with an isolated zero at p. Radially extend v by the preceding rule, without new zeroes in the normal direction. The lift to T has zero set exactly over p in a sufficiently small closed product neighbourhood. Its local relative Euler obstruction equals ind_S(v,p)·Eu_V(p).

At every point of $D$ the limiting secant line lies in the limiting tangent plane by Whitney B for the point stratum $\{p\}$. Fix a Hermitian inner product in the local ambient chart. Orthogonal projection of $L$ into $b^{\ast }T$ is therefore injective along $D$, and remains injective in a neighbourhood of $D$ by compactness of $F$ and $D$. It defines a complex line subbundle with quotient $Q$ of rank $d-1$. The projected radial vector is the product of the divisor section of $L$ with this nowhere-zero line inclusion. On a small link its inner product with $x-p$ is the squared norm of the projected vector, hence strictly positive. The lifted controlled radial field also has positive radial component. Their straight-line homotopy in the Nash tangent plane has positive radial component throughout, so is nonzero on that link. Both sections therefore have the same relative Euler obstruction. This uses the actual Whitney B and controlled radial hypotheses; normalization of an arbitrary function on the smooth locus would not suffice.

To verify this coefficient rather than assume it, first straighten the complex stratum holomorphically in a local smooth ambient chart and use its holomorphic tangent-coordinate projection. The radial extension made with the controlled tube is homotopic on the link to this one, by the same positive-radius right-inverse interpolation; this does not change its relative Euler class. Use the refined local tube product for this proper local Nash projection, with its source embedding and smooth ambient extension from step 24.1. Its differential is complex linear, and its projection of a nearby limiting tangent plane onto $T_S$ is surjective by Whitney A; its minimum-norm right inverse splits the Nash bundle continuously into a horizontal oriented rank-s complex bundle and the vertical rank-(d−s) complement. The horizontal component of the constructed field is exactly v pulled back from S; the normal component is radial. The controlled product and a bundle homotopy identify this with a direct-product section on a stratum disk times the compact normal Nash data, fixed on the relevant boundary. The Thom class of a direct sum is the product of the two Thom classes, and the relative fundamental cycle is the product cycle with its complex orientation. Hence evaluation of the relative Euler obstruction multiplies the tangent local degree by the normal radial obstruction. Taking v to be the ordinary outward radial field on the complex stratum disk gives tangent degree +1 and gives exactly the full local radial field defining Eu_V(p). Thus the normal factor is Eu_V(p), proving the formula including its sign. This reasoning uses a product of relative pairs and Thom classes, not a claim that the Nash space itself is smooth.

The relative Thom product for $L\oplus Q$ now evaluates this local obstruction as the degree of $c_{d-1}(Q)\cap [D]$. The computation is valid on a singular $B$: take its top-dimensional complex fundamental cycle and cap the pulled-back Thom classes, with no smooth Poincaré–Hopf assertion on $B$. Properness and $b_{\ast }[B]=[N]$ identify this relative evaluation with the one on $N$. Since $c(Q)=b^{\ast }c(T)\mathrm{/}(1+D)$, the result is $\mathrm{deg}\{b^{\ast }c(T)\cap (D-D^2+D^3-\cdots )\cap [B]{\}}_0=\mathrm{deg}\{c(T{\mathrm{\mid }}_F)\cap s(F,N){\}}_0$. Here the final equality is the blowup definition of the Segre class and proper projection formula. The algebraic/topological Chern and divisor cycle comparisons used in this calculation are exactly the ones constructed; they are independent of the Euler-index result. For $d=0$ the bundle has rank zero and both definitions give the length-one reduced point. This proves the required agreement of the radial Nash convention with Behrend equation (3), once the preceding controlled local constructions apply. It supplies the missing local sign and fibre convention; it does not by itself verify the global tube construction or the bundle-stack Chow descent in steps 1.1–25.1.

The same proper local product transports the normal factor along S. It follows that Eu_V is constant on these finitely many strata and is constructible; no constructibility theorem for Eu is required in addition. On the smooth locus the Nash modification is the identity, and the radial field has local degree one, so Eu_V=1 there.

Construct ξ inductively over increasing stratum dimension. On the already treated lower strata use their controlled tubes and the radial extension rule. For a stratum S, remove small controlled tubes of its lower boundary and round the finitely many transverse corners. A compact smooth core $K_S$ with collared ends remains. On its boundary the radial extension points inward into this core: it points away from the deleted lower-stratum tubes. Its outward normal component has the opposite sign from that of the core boundary. Choose an extension to the tangent bundle of $K_S$ matching the prescribed collars, and perturb it generically in the interior to have finitely many isolated nondegenerate zeroes. A partition of unity extends the prescribed collar field. For the perturbation, take finitely many coordinate balls in the interior and bump multiples of their tangent-coordinate frames. The universal finite-parameter perturbation is a submersion in the fibre direction because one nonzero bump frame spans each tangent fibre. The actual manifold Morse–Sard theorem therefore gives a parameter whose section is transverse to zero. Compactness makes its zero set finite, and the bumps leave the collar fixed. On the ends retain the existing nonzero radial extension. This adds no zeroes outside the interior of $K_S$. Poincaré–Hopf on $K_S$ gives Σ_{p∈S, ξ(p)=0} ind_S(ξ,p)=χ_c(S).

Specifically the field is inward on the boundary, so negate it to use the published outward theorem. S has real dimension 2s; negating a field multiplies each local degree by (−1)^{2s}=1, so its index sum is $\chi (K_S)$. The stratum retracts onto a finite core, and Poincaré duality in even real dimension gives ${\chi }_c(S)=\chi (S)=\chi (K_S)$. The same identity follows from the finite relative chain models of the collars. Zero-dimensional strata are included: a point has tangent index one and Euler characteristic one. Continue the radial extension to higher strata as in the radial-field construction. There are finitely many strata, so ξ has finitely many zeroes on compact V and is nowhere zero elsewhere. Later extensions preserve its values on the lower strata.

Lift ξ to the Nash tangent bundle. Its zero set is the finite disjoint union of the compact fibres ν^{-1}(p) above the finitely many zeroes of ξ. The relative Euler class of T is supported in arbitrarily small disjoint neighbourhoods of these fibres. Excision in the Thom-pullback definition therefore makes its evaluation on [N] the sum of the local relative evaluations. By the local product-index calculation this gives $⟨e(T_R),[N]⟩$ =Σ_{ξ(p)=0} ${\mathrm{ind}}_{S_p}(\xi ,p)$ Eu_V(p) =Σ_S χ_c(S) Eu_V(S) =χ_c(V,Eu_V).

Here $T_R$ is the underlying real oriented rank-2d bundle. The first equality is valid on singular N: the Thom class of a bundle can be capped with its complex-oriented top-dimensional fundamental cycle, and excision is applied to that cycle and its relative finite chain models. There is no application of smooth-manifold Poincaré–Hopf to N.

Finally this Euler evaluation is the degree of the algebraic top Chern class used by the Mather definition. The compatibility can be checked from its construction. Complex-oriented algebraic cycles define the cycle-class map: a principal divisor is a boundary by taking the graph of its rational function and an oriented arc between zero and infinity, with the local argument principle giving exactly the divisor multiplicities. Thus rational equivalences map to zero; proper maps push the same oriented cycles. For a line bundle the algebraic first Chern class is intersection with its Cartier zero divisor, whose local complex orientation is the winding-number Thom class. Thus its cycle class is the topological first Chern class. Pull back a general bundle to its complete flag bundle; the successive quotient lines split its total Chern class. The projective-bundle formulas on Chow cycles and on the oriented topological bundle imply that pullback is injective and that the same line-bundle identity descends. Proper pushforward preserves the degree of zero cycles and their oriented cycle classes. Hence $c_d(T)$ has cycle class $e(T_R)$, also on the singular base N, since Chern classes act operationally on its fundamental cycle. This is a comparison of the line-bundle/flag/Thom constructions, rather than an invocation of MacPherson naturality. It proves the assertion for V and therefore, by linearity, for every cycle c on a proper complex scheme.

Apply the proved proper Euler-obstruction index linearly to the finite integral cycle $c_X$ on the underlying reduced proper scheme. It gives ${\chi }_c(X,{\nu }_X)={\chi }_c(X,Eu(c_X))={\int }_X{c}_0^M(c_X)$. The global intrinsic kernel/Nash calculation above identifies that class with [X]^vir. Therefore #^vir(X)=deg[X]^vir=χ_c(X,ν_X), exactly the original full proper-scheme assertion. Nonreduced structure is retained in the intrinsic cone multiplicities $m_Z$. The additional finite-quotient/gerbe/smooth-stack scope is only the source qualification in the Statement; no formula for arbitrary proper DM stacks is asserted.