# Isolated phases and local characteristic-cycle indices

A differential graph can meet a characteristic cycle at a singular or degenerate point. The local number still has a cohomological meaning: it is the Euler characteristic of the sheaf's closed half-space test. We will prove this without replacing the isolated intersection by a transverse one. The proof converts that local test into a compact global calculation while controlling every covector created at the edge of a coordinate ball.

[Differential sections and proper-below Euler indices](differential-sections-and-proper-below-euler-indices.md#compact-cohomology-uses-the-opposite-section) supplies the compact-cohomology formula with its opposite differential section. The [supported section-intersection comparison](continuous-sections-and-supported-cycle-intersections.md#the-complete-supported-section-intersection-comparison) fixes its ordered cup and trace. For the geometric cutoffs we use [full limiting-sum isotropy](../../sheaf-proof-readings/src/SH03/limiting-cotangent-sums-and-characteristic-inverse-images.md#limiting-sum-isotropy), including unbounded cancellation, and the [direct critical-set proof](../../sheaf-proof-readings/src/SH03/isotropic-cotangent-transport-and-discrete-critical-values.md#direct-critical-set-proof) of locally finite selected values under properness on the base support.

The sheaf maps are the actual restriction in [the small-ball theorem](../../sheaf-proof-readings/src/SH03/small-balls-central-fibres-and-supported-cohomology.md#the-three-comparisons), its [closed cutoff in a coordinate chart](../../sheaf-proof-readings/src/SH03/small-balls-central-fibres-and-supported-cohomology.md#local-chart-cutoff), the [ordinary open-boundary estimate](../../sheaf-proof-readings/src/SH02/microsupport-operations.md#sh02-mo-boundary--four-ways-to-impose-a-boundary), and the [supported finite-band comparison (MO32)](../../sheaf-proof-readings/src/SH02/microsupport-operations.md#sh02-mo-morse--cohomology-between-two-levels). [Perfect internal Hom](../../sheaf-proof-readings/src/SH03/perfect-operations-and-finite-microlocal-coefficients.md#perfect-inverse-images-tensors-and-internal-hom) gives the bounded constructible cutoff objects. The geometric inputs include analytic curve selection and subanalytic closure, inverse-image and proper-image calculus; the [singular one-form argument](../../sheaf-proof-readings/src/SH03/subanalytic-sets-and-limiting-tangent-directions.md#one-forms-on-a-singular-set) and the full normal-cone proof state their use of these inputs. Below we prove the two radial exclusions, establish the uniform boundary control, and follow the particular support maps from the local test to the compact index.

Throughout, \(X\) is a finite-dimensional real analytic manifold, Hausdorff and countable at infinity, \(k\) is a field of characteristic zero, and \(F\in D^b_{\mathbb R\text{-}c}(k_X)\) has perfect stalks. Complexes are globally bounded. The closed support is \(\pi\operatorname{SS}(F)\), which can contain a point where the ordinary stalk vanishes. All cotangent conicity uses positive scalars.

*Original lesson text and solutions: CC0 1.0 Universal. Human mathematical sources retain their named credit and their own terms.*

## The local number and the closed test

Let \(x_0\in X\), let \(\varphi\) be real analytic near \(x_0\), and put

\[
\sigma_\varphi(x)=(x;d\varphi_x),\qquad
L_\varphi=\sigma_\varphi(X),\qquad
p_0=\sigma_\varphi(x_0).
\qquad\text{(1)}
\]

After restricting to a neighborhood of \(x_0\), assume

\[
L_\varphi\cap\operatorname{SS}(F)\subset\{p_0\}.
\qquad\text{(2)}
\]

The intersection can be empty. Neither \(p_0\neq0\), a smooth microsupport, a nondegenerate Hessian nor transversality is required. Define the local test complex

\[
J_{\varphi,x_0}(F)=
\bigl(R\Gamma_{\{\varphi\geq\varphi(x_0)\}}F\bigr)_{x_0}.
\qquad\text{(3)}
\]

This is sheaf cohomology with a closed support condition, followed by the ordinary stalk. It is not the ordinary stalk of \(F\), and it is not in general the ambient point costalk.

Recall how the right-hand side below is normalized. With \(\pi:T^*X\to X\), the graph section has its canonical supported unit with coefficient \(P=\pi^!k_X\), and the cycle has coefficient \(E=\pi^{-1}\omega_X\). Cup the section first with the cycle second, using the specified exceptional tensor map \(P\otimes E\to\omega_{T^*X}\). Its support is contained in \(L_\varphi\cap\operatorname{SS}(F)\). Under (2), this is a class supported at \(p_0\). The section comparison transports it properly to the base support \(\{x_0\}\), and the composed closed-point and global traces give its scalar. Denote this number by
\(\#([\sigma_\varphi]\cap CC(F))_{p_0}\). An empty support gives zero. The subscript records localization of the supported class, rather than a signed count assumed to exist only for transverse smooth manifolds.

**Local index theorem.** Under (2), \(J_{\varphi,x_0}(F)\) is a bounded finite-dimensional complex and

\[
\chi\bigl(J_{\varphi,x_0}(F)\bigr)
=\#\bigl([\sigma_\varphi]\cap CC(F)\bigr)_{p_0}.
\qquad\text{(4)}
\]

Finiteness is already justified independently of the equality: the closed analytic inequality defines a subanalytic set \(Z\), and
\(R\Gamma_ZF=R\mathcal Hom(k_Z,F)\) is bounded constructible with perfect stalks by the internal-Hom theorem. If \(p_0\notin\operatorname{SS}(F)\), the defining microsupport test makes (3) zero, while the intersection support is empty. In dimension zero the assertion is the normalized point trace of the finite coefficient complex. We therefore prove the remaining case in a positive-dimensional coordinate chart.

Subtract \(\varphi(x_0)\), choose coordinates with \(x_0=0\), and write

\[
\varphi(0)=0,\qquad
\rho(x)=|x|^2,\qquad
A=\operatorname{SS}(F).
\qquad\text{(5)}
\]

For the rest of the local proof, \(X\) denotes this open coordinate chart and \(F\) its restriction; choose it so that \(\varphi\) is analytic on all of it and (2) holds there. Take \(\overline B_R\subset X\). All geometric carriers used in a critical-value argument will first be restricted over this closed ball, while the sheaf and its differentials remain defined on an open neighborhood of it. This proves the needed base properness without extending the analytic phase outside its original domain.

## Two radial exclusions on a punctured ball

We need a carrier containing both signs of the conormal to the zero phase, even if that level is singular at zero. Put

\[
Z_0=\{\varphi=0\},\qquad
B=\operatorname{SS}(k_{Z_0})\cup
\operatorname{SS}(k_{Z_0})^a.
\qquad\text{(6)}
\]

Here \(a(x;\xi)=(x;-\xi)\). The sheaf \(k_{Z_0}\) is constructible with perfect stalks. Thus \(B\) is closed, positive-conic, subanalytic and isotropic. Its base lies in \(Z_0\). At a regular point of the level set, the closed-submanifold microsupport calculation includes every \(\lambda\,d\varphi_x\), \(\lambda\in\mathbb R\). At a critical point these covectors are zero, and zero belongs to the microsupport over \(Z_0\). Therefore

\[
(x;\lambda\,d\varphi_x)\in B
\quad\text{if }\varphi(x)=0,\quad \lambda\in\mathbb R.
\qquad\text{(7)}
\]

The full limiting-sum theorem makes \(A\widehat+B\) a closed conic subanalytic isotropic set. This is available even if the same-base sum is not closed.

Apply the discrete-critical-value theorem to \(\rho\), first on \(A\cap\pi^{-1}\overline B_R\), then on \((A\widehat+B)\cap\pi^{-1}\overline B_R\). Restriction to a subanalytic base subset preserves the canonical-form condition, and both sets remain closed and conic. Their base projections are compact: scaling any fibre covector to zero shows that the projection equals the inverse image of the carrier under the zero section, hence is closed in \(\overline B_R\). Thus \(\rho\) is proper on each projection. Each selected radial-value set is closed and locally finite in \(\mathbb R\). Choose \(0<r_0<R\) so that neither contains a value in \((0,r_0^2]\). This one radius gives both exclusions, at every base point and not just along a chosen sequence:

\[
\begin{aligned}
A\cap L_\rho\cap\{0<|x|\leq r_0\}&=\varnothing,\\
(A\widehat+B)\cap L_\rho
\cap\{0<|x|\leq r_0\}&=\varnothing.
\end{aligned}
\qquad\text{(8)}
\]

We have not declared an unrestricted cotangent projection proper. The compact base restriction is what permits the theorem. The second exclusion will be used precisely at zero phase on the boundary of the eventual ball.

## Positive phase excludes every nonnegative radial multiplier

There is \(r_1>0\) such that

\[
0<|x|\leq r_1,\quad \varphi(x)>0,\quad c\geq0
\quad\Longrightarrow\quad
(x;c\,d\rho_x+d\varphi_x)\notin A.
\qquad\text{(9)}
\]

The multiplier \(c\) has no prescribed upper bound. A proof that treated only a bounded interval of multipliers would not justify the later ordinary-boundary estimate.

**Proof.** Suppose (9) fails at arbitrarily small radii. Normalize each offending pair of coefficients:

\[
u=\frac{c}{\sqrt{1+c^2}},\qquad
v=\frac1{\sqrt{1+c^2}}.
\qquad\text{(10)}
\]

Positive conicity gives
\((x;u\,d\rho_x+v\,d\varphi_x)\in A\).
Consider the subanalytic set

\[
\mathcal E=
\left\{(x,u,v):
\begin{array}{l}
x\neq0,\ \varphi(x)>0,\ u\geq0,\ v>0,\\
u^2+v^2=1,\ (x;u\,d\rho_x+v\,d\varphi_x)\in A
\end{array}\right\}.
\qquad\text{(11)}
\]

Membership is an analytic inverse-image condition on \(A\). The coefficient circle is compact. A subsequence of offending points therefore approaches a point \((0,u_0,v_0)\) in the closure of \(\mathcal E\). Its limit may have \(v_0=0\); that is how unbounded original multipliers are retained.

Curve selection supplies an analytic arc \((x(t),u(t),v(t))\) through that limit, lying in \(\mathcal E\) for small \(t>0\). The cotangent arc
\[
q(t)=\bigl(x(t);u(t)d\rho_{x(t)}+v(t)d\varphi_{x(t)}\bigr)
\]
lies in \(A\). Isotropy and the singular one-form pullback rule make its canonical one-form zero along the arc, including an arc contained in the singular locus. Consequently

\[
u(t)\frac{d}{dt}\rho(x(t))
+v(t)\frac{d}{dt}\varphi(x(t))=0.
\qquad\text{(12)}
\]

Both scalar functions are analytic, vanish at zero, and are strictly positive for \(t>0\). Their first nonzero Taylor terms have positive coefficients and positive integer orders. Both derivatives are therefore strictly positive for all sufficiently small positive \(t\). Since \(u(t)\geq0\) and \(v(t)>0\), the left side of (12) is strictly positive. This contradiction proves (9). \(\square\)

The normalization proved the required compactness before selecting an arc. It did not discard sequences with \(c\to+\infty\), nor assume that the selected covector lies in the regular part of \(A\).

## Extend an actual small-ball test by ordinary direct image

Set \(H=R\Gamma_{\{\varphi\geq0\}}F\), a bounded constructible complex by perfect internal Hom. Choose \(R'<R\) and form \(H'=H\otimes k_{\overline B_{R'}}\). Closed restriction followed by closed direct image identifies this with the closed cutoff; its stalks are those of \(H\) on the ball and zero elsewhere, so constructibility and the original cohomological bounds are retained. Its closed support is compact. Apply the small-ball theorem to the coordinate map \(x:X\to\mathbb R^n\) and \(H'\). It gives \(r_2>0\) such that for every \(0<r<\min(r_2,R')\), the actual restriction \(R\Gamma(B_r;H')\to H'_0\) is an isomorphism. Since \(H'=H\) on these smaller balls, choose once and for all
\(0<r<\min(r_0,r_1,r_2,R')\). The resulting map is

\[
R\Gamma(B_r;H)\xrightarrow{\sim}H_0
=J_{\varphi,0}(F).
\qquad\text{(13)}
\]

The closed cutoff was used only to justify (13); it agrees with \(H\) on a whole neighborhood of \(\overline B_r\). The next cutoff is instead an ordinary open direct image of \(F\). Keeping these two constructions distinct preserves both the actual restriction map in (13) and the boundary sign needed below.

For \(j:B_r\hookrightarrow X\), use the ordinary extension

\[
F_r=Rj_*j^{-1}F
\simeq R\mathcal Hom(k_{B_r},F).
\qquad\text{(14)}
\]

It is bounded constructible with perfect stalks. Its closed support \(D_r\) lies in \(\overline B_r\), hence is compact. Open internal-Hom adjunction, followed by direct-image composition, identifies

\[
\begin{aligned}
R\Gamma_{\{\varphi\geq0\}}(X;F_r)
&=R\Gamma\bigl(X;R\mathcal Hom(k_{\{\varphi\geq0\}},Rj_*j^{-1}F)\bigr)\\
&\simeq R\Gamma\bigl(B_r;
 R\mathcal Hom(k_{\{\varphi\geq0\}\cap B_r},j^{-1}F)\bigr)\\
&=R\Gamma(B_r;H).
\end{aligned}
\qquad\text{(15)}
\]

For a closed \(Z\) in the chart, open internal-Hom adjunction is the natural comparison
\(R\mathcal Hom(k_Z,Rj_*G)\simeq Rj_*R\mathcal Hom(j^{-1}k_Z,G)\).
Here \(j^{-1}k_Z=k_{Z\cap B_r}\), and restriction of the supported sheaf \(H\) to \(B_r\) is the same internal Hom. Composition of ordinary direct images therefore proves (15) for the specified maps. It is compatible with the map from sections to the stalk in (13). No nonproper closed-fibre base-change assertion is being used for \(j\).

At a boundary point \(|x|=r\), \(d\rho_x\neq0\). The first exclusion in (8) and positive conicity exclude every positive multiple of \(d\rho_x\) from \(A_x\). This exclusion has the neighbourhood control required by the ordinary boundary theorem. Choose a compact annulus \(Q\) about the sphere, contained in \(\{0<|x|<r_0\}\). There is \(\eta>0\) such that for all \(x\in Q\), \(p\in A_x\), and \(c\geq0\),

\[
|p-c\,d\rho_x|\geq\eta\bigl(|p|+c|d\rho_x|\bigr).
\]

Indeed, failure would give a sequence with the ratio tending to zero. Divide \(p\) and \(c\,d\rho\) by the positive denominator. Compactness of \(Q\), the continuous nonvanishing radial differential, and closed positive conicity of \(A\) give equal nonzero limiting vectors: the first lies in \(A\), and the second is a positive radial covector. This contradicts (8). The same normalized compactness separates the positive radial directions from the unit covectors of \(A\) in one neighbourhood of the sphere. Thus the boundary hypothesis holds on a collar, and unbounded cancellation in the ordinary sum cannot occur there.

For \(B_r=\{\rho<r^2\}\), the strict-normal polar on its boundary is \(\mathbb R_{\leq0}d\rho\). The ordinary open-boundary theorem requires avoidance of its opposite positive ray, exactly the exclusion just verified, and gives

\[
\operatorname{SS}(F_r)_x
\subset A_x+\mathbb R_{\leq0}\,d\rho_x
\quad (|x|=r).
\qquad\text{(16)}
\]

On the interior \(F_r=F\), and outside the closed ball it is zero. The sign in (16) belongs to \(Rj_*\); changing the extension changes that sign.

Let \(C=T^*_{\partial B_r}X\), the full conormal of the analytic sphere. It is closed, conic, subanalytic and isotropic. Define

\[
\Gamma=
\bigl(A\cup(A\widehat+C)\bigr)
\cap\pi^{-1}\overline B_r.
\qquad\text{(17)}
\]

The finite union and subanalytic subset rules preserve isotropy; the limiting-sum theorem proves the needed closedness and subanalyticity. The ordinary sums in (16) lie in \(A\widehat+C\), so
\(\operatorname{SS}(F_r)\subset\Gamma\).
The base projection of \(\Gamma\) is compact, by the zero-section argument used for (8), so \(\varphi\) is proper there. Apply the discrete-critical-value theorem to this fixed carrier after \(r\) has been chosen. Its selected values have no nonzero accumulation at zero. Fix \(\delta>0\). The nonzero selected values in \([-\delta,\delta]\) form a finite set; choose \(0<\epsilon<\delta\) smaller than the absolute value of every member of that set. If the set is empty, any \(0<\epsilon<\delta\) works. This single choice works for both signs of the phase and every later \(0<t<\epsilon\):

\[
\Gamma\cap L_\varphi\cap
\{0<|\varphi|\leq\epsilon\}=\varnothing.
\qquad\text{(18)}
\]

Zero is allowed as a selected value. The two-sided gap will control the negative phase band and the compact support comparison.

## A supported Morse map becomes compact cohomology

Put \(U=\{\varphi>-\epsilon\}\), and apply the supported finite-band Morse theorem to \(\psi=-\varphi\) and \(F_r\). Properness on support holds because \(D_r\) is compact. For every \(0<t<\epsilon\), the changing band is
\[
0<\psi\leq t,\qquad\text{equivalently }-t\leq\varphi<0.
\]
The required negative covector in the supported comparison (MO32) is \(-d\psi=d\varphi\). It is excluded from \(\operatorname{SS}(F_r)\) by (18) on the entire half-open band \(0<\psi\leq t\). The starting level \(\psi=0\), where the desired local test may be nonzero, is not among the levels that require exclusion. The terminal level is included and is covered by the closed endpoint in (18). The actual inclusion-of-supports map is therefore an isomorphism

\[
R\Gamma_{\{\varphi\geq0\}}(X;F_r)
\xrightarrow{\sim}
R\Gamma_{\{\varphi\geq-t\}}(X;F_r).
\qquad\text{(19)}
\]

To pass to compact support in \(U\), restrict the support family to \(D_r\). The sets
\[
K_t=D_r\cap\{\varphi\geq-t\},\qquad 0<t<\epsilon,
\]
are compact and lie in \(U\). They are cofinal for compact coefficient supports in \(U\): if \(K\subset D_r\cap U\) is compact, its minimum phase \(m\) is greater than \(-\epsilon\); choose \(t\in(0,\epsilon)\) with \(-t\leq m\). Then \(K\subset K_t\).

Here is the derived comparison with its maps. Write \(i:D_r\hookrightarrow X\). The restriction unit identifies \(F_r\) with \(i_*i^{-1}F_r\), since its cohomology vanishes off \(D_r\). Resolve \(i^{-1}F_r\) by injectives on \(D_r\) and apply the exact closed direct image, which preserves injectives. This gives a representative with every term supported on \(D_r\). Sections of it supported in \(\{\varphi\geq-t\}\) are precisely sections supported on \(K_t\). Closed-support excision computes these sections on \(U\) as well. Termwise compact sections on \(U\) are the filtered union over this cofinal family. Injectives restrict to injectives on \(U\) and are acyclic for compact sections on a locally compact space; exact filtered colimits then commute with cohomology. The compatible support inclusions (19) consequently give

\[
R\Gamma_{\{\varphi\geq0\}}(X;F_r)
\xrightarrow{\sim}R\Gamma_c(U;F_r|_U).
\qquad\text{(20)}
\]

The first object maps to the support-family colimit through (19); all those arrows are already isomorphisms. This explains the actual comparison in (20). No equality between ordinary cohomology and compact cohomology on an arbitrary noncompact open set was used.

Apply the compact Euler index theorem on \(U\) to the analytic function \(\psi=-\varphi:U\to(-\infty,\epsilon)\). The coefficient support is \(D_r\cap U\), closed in \(U\). For \(s<\epsilon\), its closed support sublevel is exactly \(D_r\cap\{\psi\leq s\}\): that inequality already puts it in \(U\). It is a closed subset of compact \(D_r\), and therefore compact in \(U\). Moreover
\[
L_\psi\cap\operatorname{SS}(F_r|_U)^a
\quad\text{corresponds under the antipode to}\quad
L_\varphi\cap\operatorname{SS}(F_r|_U).
\]
The latter is compact: by (18) it has no points with \(-\epsilon<\varphi<0\), so it is the closed graph intersection over the compact set \(D_r\cap\{\varphi\geq0\}\). Thus both hypotheses of that theorem are satisfied. It yields

\[
\chi_c(U;F_r|_U)
=\#\bigl([\sigma_\varphi]\cap CC(F_r|_U)\bigr).
\qquad\text{(21)}
\]

The positive section on the right follows from \(-\psi=\varphi\). No dimension sign is appended to the antipodal comparison.

## Exclude boundary intersections in all three phase regions

Inside \(B_r\), (2) leaves at most \(p_0\). We prove that there is no graph intersection over \(\partial B_r\cap U\).

If \(-\epsilon<\varphi(x)<0\), (18) excludes it directly.

If \(\varphi(x)>0\), membership of \(d\varphi_x\) in the right side of (16) would give

\[
d\varphi_x=p-c\,d\rho_x,\qquad
p\in A_x,\quad c\geq0.
\qquad\text{(22)}
\]

Hence \(p=d\varphi_x+c\,d\rho_x\), contradicting (9).

Finally suppose \(\varphi(x)=0\). In (22), \(c=0\) would contradict the isolated-intersection hypothesis, since \(x\neq0\). If \(c>0\), then (7) and positive conicity give

\[
d\rho_x=c^{-1}p-c^{-1}d\varphi_x
\in A_x+B_x
\subset (A\widehat+B)_x.
\qquad\text{(23)}
\]

This contradicts the second exclusion in (8). The zero-phase argument needs the negative phase multiple \(-c^{-1}d\varphi_x\). The symmetric carrier in (6) supplies that exact covector. A graph of \(d\varphi\) alone would not supply all those multiples.

We have proved

\[
L_\varphi\cap\operatorname{SS}(F_r|_U)\subset\{p_0\},
\qquad\text{(24)}
\]

with no new boundary contribution. Choose an open neighbourhood \(V\) of zero with \(V\subset B_r\cap U\). The canonical ordinary restriction identifies \((F_r|_U)|_V\) with \(F|_V\). Characteristic cycles restrict through their defining kernel unit and evaluated trace, and the section class restricts through its exceptional counit. Hence the two ordered cups agree on \(T^*V\). By (24), the global cup on \(T^*U\) is supported on the same single point as this local cup. Closed-support excision identifies their classes with that support retained. Open-extension trace composition, followed by the section's proper projection and the closed point trace, identifies their scalar numbers. Thus (21) is exactly the local number in (4), with no orientation change from shrinking the ambient chart. Taking Euler characteristics in the actual comparisons (13), (15), (20) and (21) proves (4). \(\square\)

The compact computation is a device for reading the original local supported class. The radial boundary has been checked before forgetting its support, so none of its possible contributions is hidden in an Euler cancellation.

## A transverse conormal intersection reads the negative Hessian

Let \(Y\subset X\) be a closed smooth submanifold for which \(k_Y\) is \(\mathbb R\)-constructible; a closed real analytic submanifold is one instance. Let \(x_0\in Y\), with \(d\varphi_{x_0}\in T_Y^*X\), and suppose the graph of \(d\varphi\) and the conormal intersect transversely there.

In coordinates tangent and normal to \(Y\), their common tangent vectors are exactly the vectors in
\(\ker\operatorname{Hess}_{x_0}(\varphi|_Y)\).
Transversality therefore means that this restricted Hessian is nondegenerate. Let \(q\) be its number of negative eigenvalues. The intersection is locally isolated.

Put \(d=\dim Y\). The bracket \([T_Y^*X]\) has the [fixed graph normalization](transverse-pullback-of-normalized-conormal-cycles.md#a-supported-point-unit-survives-every-tangential-rank), while the [constant coefficient formula](transporting-characteristic-cycles-through-a-graph.md#globally-constant-finite-coefficients-fix-conormal-normalization) is \(CC(k_Y)=(-1)^d[T_Y^*X]\). For the stated smooth constructible support this is the same local calculation as in the analytic case: straighten \(Y\) in a smooth chart, retain the positive tangent-fibre Thom map in the constant directions, and compose with the normal closed-embedding trace. The graph-normalized tangent zero cycle is \((-1)^d\) times that Thom map. Their orientation lines pair on changes of frame, so the local coefficient identity glues. The [closed-embedding microsupport formula](../../sheaf-proof-readings/src/SH03/small-balls-central-fibres-and-supported-cohomology.md#closed-embedding-microsupport-proof) gives \(SS(k_Y)=T_Y^*X\); constructibility makes this an allowed subanalytic carrier. This argument retains the smooth-submanifold scope and does not choose a new orientation for the bracket.

The actual conormal support calculation in [Pure and simple sheaves from directional tests](../../microlocal-composition-and-pure-sheaves/src/pure-and-simple-sheaves-from-directional-tests.md#conormal-sheaves-give-the-normalization-explicitly) uses Morse coordinates
\(\varphi|_Y-\varphi(x_0)=|u|^2-|v|^2\), with \(\dim v=q\).
The closed-test triangle compares constants on a small ball in \(Y\) with constants on its negative-phase part. For \(q=0\) that part is empty. For \(q>0\) it retracts to \(S^{q-1}\), and the restriction from constants is the augmentation. Its derived fibre is the negative-direction orientation line in degree \(q\), locally identified with \(k[-q]\). Thus (4) gives

\[
\begin{aligned}
\#\bigl([\sigma_\varphi]\cap CC(k_Y)\bigr)_{p_0}&=(-1)^q,\\
\#\bigl([\sigma_\varphi]\cap[T_Y^*X]\bigr)_{p_0}&=(-1)^{d+q}.
\end{aligned}
\qquad\text{(25)}
\]

The second equality follows from the first by the fixed coefficient \((-1)^d\). It depends on the dimension of the supporting submanifold, not on an independently chosen ambient sign. For example, a minimum of the constant sheaf on a line has local test \(k\) and index \(+1\), while its graph-normalized zero section has intersection number \(-1\); the characteristic cycle is the negative of that generator. Finite families of transverse pure intersections use the first equality for their test coefficients and the second when they are expressed in these conormal generators.

## Examples and exercises with solutions

### A degenerate phase can have local number zero, one or minus one

*Difficulty: Introductory.*

Take \(X=\mathbb R\), \(F=k_X\), \(x_0=0\), and \(\varphi(t)=a\,t^m\), with \(a\neq0\) and integer \(m\geq2\). Compute the test complex and local number. Explain why the calculation for \(m\geq3\) is not a transverse Hessian calculation.

**Solution.** The microsupport is the zero section. Its intersection with \(d\varphi=am\,t^{m-1}dt\) is exactly the zero covector at zero. The local test triangle is the fibre of the restriction from \(k\) on a small interval to cohomology of its negative-phase part. If \(m\) is odd, that part is a single half-interval. The restriction \(k\to k\) is the identity, so the fibre is zero and the local number is zero.

If \(m\) is even and \(a>0\), there is no negative-phase part. The test complex is \(k\), giving number \(1\). If \(m\) is even and \(a<0\), the negative-phase part has two components. The restriction is the diagonal \(k\to k\oplus k\); its fibre is \(k[-1]\), giving number \(-1\). In the last case the closed test is supported only at zero and agrees with the line's point costalk. For \(m\geq3\), the Hessian at zero vanishes. These conclusions follow from the actual closed test and the isolated local index theorem, including its degenerate and empty-number cases.

### Cubic tests distinguish open and closed rays

*Difficulty: Intermediate.*

Let \(A=k_{[0,\infty)}\) and \(B=k_{(0,\infty)}\) on \(\mathbb R\). At zero compute the local test complexes and numbers for \(\varphi=t^3\) and \(\varphi=-t^3\).

**Solution.** Both graphs meet the full microsupport only at the zero covector over zero: away from the endpoint, the only supported covectors are zero and \(d\varphi\neq0\). At the endpoint \(d\varphi=0\). The zero covector remains in the closed microsupport of \(B\) even though \(B_0=0\).

For \(t^3\), the closed test side is \(t\geq0\). The complementary negative interval has no coefficients for either sheaf. The support triangles therefore give \(J(A)=k\) and \(J(B)=0\). For \(-t^3\), the complementary positive interval has cohomology \(k\). The closed-ray restriction \(A_0=k\to k\) is the identity, giving \(J(A)=0\). The open ray has restriction \(B_0=0\to k\), whose derived fibre is \(k[-1]\). Hence the complete table is

| Sheaf | Cubic positive test | Local number | Cubic negative test | Local number |
|---|---|---:|---|---:|
| \(k_{[0,\infty)}\) | \(k\) | \(1\) | \(0\) | \(0\) |
| \(k_{(0,\infty)}\) | \(0\) | \(0\) | \(k[-1]\) | \(-1\) |

The table uses the full microsupport, the ordinary stalk and the actual complementary restriction. Replacing the closed support by the nonzero-stalk locus would remove the open-ray endpoint incorrectly.

### A singular crossing retains its vertex correction

*Difficulty: Advanced.*

In \(\mathbb R^2\), let \(C\) be the union of the coordinate axes, let \(F=k_C\), and take \(\varphi(x,y)=x^2-y^2\). Compute the local number at the origin directly from the supported test, and verify it through cycle additivity.

**Solution.** On the punctured horizontal axis, \(F\) is the constant sheaf and its microsupport is the horizontal-axis conormal. The differential \((2x,-2y)\) belongs to that conormal only when \(x=0\). On the punctured vertical axis, the analogous condition forces \(y=0\). Off \(C\) the sheaf vanishes locally. Thus the graph has just the isolated zero-covector intersection at the origin, where the microsupport need not be smooth.

The stalk \(F_0\) is \(k\). In the negative phase the support consists of the two punctured vertical arms. Their cohomology is \(k\oplus k\), and restriction sends a local constant \(v\) to \((v,v)\). The fibre is \(k[-1]\), so the local number is \(-1\).

Write \(L_x,L_y\) for the two axes. The sequence
\[
0\longrightarrow k_C\longrightarrow
k_{L_x}\oplus k_{L_y}
\xrightarrow{(a,b)\mapsto a(0)-b(0)}k_{\{0\}}
\longrightarrow0
\qquad\text{(26)}
\]
is exact on every stalk: away from the origin it is the identity on the relevant axis, and at the origin it is the diagonal kernel of the difference map. Additivity gives
\(CC(k_C)=CC(k_{L_x})+CC(k_{L_y})-CC(k_{\{0\}})\).
Each axis has dimension one, so in the graph-normalized generators the same equality is
\(CC(k_C)=-[T_{L_x}^*\mathbb R^2]-[T_{L_y}^*\mathbb R^2]-[T_{\{0\}}^*\mathbb R^2]\).
For the first axis the restricted Hessian has \(q=0\), so its normalized conormal intersection in (25) is \(-1\), multiplied by its cycle coefficient \(-1\) to give \(+1\). For the second axis \(q=1\), so the normalized conormal intersection is \(+1\), multiplied by \(-1\) to give \(-1\). These are exactly the two sheaf-test numbers. The point sheaf has test \(k\) for every phase taking value zero there, so its number is \(1\). The sum \(1-1-1=-1\) agrees with the supported calculation. Omitting the vertex term would give the wrong answer.

### A zero local Euler number need not be a zero local test

*Difficulty: Intermediate.*

Let \(P=k^2\oplus k^2[1]\), \(F=P_{\mathbb R}\), and \(\varphi=-t^4\). Compute the local test and characteristic-cycle number at zero.

**Solution.** The full microsupport is the zero section, and the graph of \(-4t^3dt\) has just the isolated intersection at zero. The negative-phase complement is the punctured interval. Its coefficient complex is \(P\oplus P\), and the map from the stalk \(P\) is the diagonal. The split difference sequence of whole complexes makes its derived fibre \(P[-1]\). It has \(k^2\) in cohomological degrees zero and one, and is nonzero.

Its Euler characteristic is \(2-2=0\). Likewise \(CC(F)=-\chi(P)[T^*_{\mathbb R}\mathbb R]=0\) by finite coefficient additivity, so its local intersection number is zero. The vanishing is finite graded cancellation. It gives no reason to remove the zero section from the full microsupport or to declare the supported test acyclic.

### Extension by zero creates two extra boundary tests

*Difficulty: Advanced.*

Let \(F=k_{\mathbb R}\), \(B=(-r,r)\), \(r>0\), and \(\varphi=t^2\). Compare \(Rj_*k_B\) with \(j_!k_B\), where \(j:B\hookrightarrow\mathbb R\). Compute all graph-test numbers and the compact Euler indices.

**Solution.** Ordinary interval cohomology at the two endpoints gives
\(Rj_*k_B=k_{[-r,r]}\).
At \(r\), its microsupport normal is a nonpositive multiple of \(dt\); at \(-r\), it is a nonnegative multiple of \(dt\). The graph differential is respectively \(2r\,dt\) and \(-2r\,dt\), so neither endpoint contributes. The only graph intersection is the minimum at zero, with supported test \(k\) and number \(1\). Its compact Euler characteristic is also \(1\).

For \(j_!k_B=k_{(-r,r)}\), the endpoint normals reverse. Both graph covectors now occur. At either endpoint the ordinary stalk is zero, while the complementary lower-phase interval inside \(B\) has cohomology \(k\). The support triangle is the fibre of \(0\to k\), namely \(k[-1]\), giving number \(-1\) at each endpoint. The interior minimum still contributes \(1\). The sum is \(1-1-1=-1\), equal to \(\chi_c(B;k)\).

This is why (14) and the negative coefficient in (16) belong to the proof. The exceptional extension has its own valid compact index, with additional boundary contributions; substituting it would not identify that global number with the original local test.

### A negative phase gap removes an artificial endpoint contribution

*Difficulty: Advanced.*

Take \(F=k_{\mathbb R}\), \(\varphi=t\), \(x_0=0\), and \(F_r=k_{[-r,r]}\). Compare the compact index on \(U_\epsilon=\{t>-\epsilon\}\) when \(0<\epsilon<r\) and when \(\epsilon>r\). Relate the difference to the phase gap.

**Solution.** For the original sheaf the graph of \(dt\) misses the zero-section microsupport. The local support triangle at zero compares \(k\) with \(k\) on a negative interval by the identity, so the local test and number vanish.

For \(F_r\), the graph meets the positive conormal at the left endpoint \(-r\), and misses the negative conormal at the right endpoint \(r\). The left endpoint's test is \(k\), since the closed test side \(t\geq-r\) locally contains its coefficient support. Its phase value is \(-r\).

If \(0<\epsilon<r\), the coefficient support in \(U_\epsilon\) is \((-\epsilon,r]\). Its compact cohomology vanishes: in a slightly larger compact interval, the support triangle removes the left endpoint, and restriction of constants to that endpoint is the identity. The graph has no remaining intersection, and the compact index is zero as required.

If \(\epsilon>r\), the entire closed interval lies in \(U_\epsilon\). Compact cohomology is \(k\), and the artificial left-endpoint intersection contributes \(1\). The asserted phase gap now fails, because the selected value \(-r\) lies in \((-\epsilon,0)\). This larger choice does not satisfy the hypothesis of (19). The discrepancy verifies the role of the negative-band exclusion rather than contradicting the local theorem.

## References and the next step

M. Kashiwara, [*Index theorem for constructible sheaves*](https://www.numdam.org/item/AST_1985__130__193_0/), Astérisque 130 (1985), Proposition 5.1 and Lemma 5.2, printed pp. 200–201, identify the generic transverse local test and its negative-Hessian degree. Theorem 9.1, pp. 207–208, states the supported real-analytic test index on a relatively compact region with no graph intersection over its boundary; the paper indicates a deformation-to-generic-position proof. The proof here instead obtains the isolated, possibly degenerate formula by the radial cutoff and the actual support comparisons below. Its section-first cup and graph-normalized conormal convention give \((-1)^q\) for \(CC(k_Y)\) and \((-1)^{\dim Y+q}\) for its normalized conormal generator, as in (25).

The proof above obtains the isolated, possibly degenerate formula without a transverse perturbation. It proves both radial exclusions, keeps the compactified multiplier arc when the original multiplier is unbounded, and uses the symmetric zero-phase carrier to control the missing negative multiple. The signed ordinary-boundary estimate is applied with uniform collar separation. The small-ball restriction, support inclusions, support-family colimit and compact index are compared as actual maps. Each boundary phase region is excluded before the final point trace, so an artificial boundary contribution is not hidden in a numerical cancellation.

For related complex-analytic developments, David B. Massey's [*A Little Microlocal Morse Theory*](https://arxiv.org/abs/math/0006185v2), version 2, 18 January 2001, studies isolated vanishing-cycle support and cohomological decompositions with principal-ideal-domain coefficients; §5 includes perturbation and radial-boundary arguments. This scholarly reference concerns a different complex-analytic setting. It is not used here to prove the real analytic cutoff estimates or to change the field hypothesis.

The next step applies the local theorem to finitely many transverse pure intersections and their ordered inertia degrees, then uses the finite sheaf-theoretic Morse filtration to obtain inequalities. Human sources retain their named authorship and their own terms. The lesson text and complete solutions have the stated CC0 license.