# Complex nearby cycles as normal and conormal sections A complex normal line has two useful sections. A normal vector on which the defining function has derivative one computes nearby cycles. The conormal covector given by that derivative computes vanishing cycles. The second assertion depends on the Fourier convention and on including the endpoint of a closed ray. We prove both comparisons, then deduce constructibility and the support bound for an arbitrary holomorphic function. Let \(k\) be a commutative ring of finite global dimension. Manifolds are complex analytic, Hausdorff and countable at infinity. An object of \(D^b_{w\text{-}\mathbb C\text{-}c}(k_X)\) has locally constant cohomology on a locally finite complex analytic stratification. Its coefficient modules may be infinite. The perfect constructible subcategory additionally requires perfect stalk complexes. No Noetherian hypothesis is imposed on \(k\). We retain the cycle normalization from Nearby cycles and the two monodromy triangles: \[ \phi_f(F)=\operatorname{Cone}\bigl(i^{-1}F\longrightarrow\psi_f(F)\bigr)[-1]. \tag{1} \] Thus the vanishing object here already contains the source's shift by \(-1\). The real covector associated to a complex covector is its real part. On a normal complex line the pairing is \(\operatorname{Re}(v\xi)\), without a conjugate. The Fourier transform uses the closed kernel \(\operatorname{Re}(v\xi)\leq0\). The normal-deformation comparison for weak real constructibility is proved in Nearby cycles through the normal deformation. Here the additional complex geometry is essential. The proof uses the bounded specialization estimate, Fourier test and whole-complex descent theorems in the precise forms stated below. *Original lesson text and solutions: CC0 1.0 Universal. Human mathematical sources are credited below.* ## Complex constructibility survives specialization Let \(Y\subset X\) be a closed complex submanifold. Put \(E=T_YX\), \(L=T_Y^*X\subset T^*X\), and \(\Lambda=\operatorname{SS}(F)\). The complex constructibility criterion proved in Complex microlocal stratifications and constructibility makes \(\Lambda\) closed complex analytic, complex-conic and real-isotropic. The bounded specialization estimate `SH02-CHE-001` gives \[ \operatorname{SS}(\nu_YF)\subset C_L(\Lambda), \tag{2} \] under the canonical normal/cotangent identification. In adapted holomorphic coordinates \((u,y)\), with \(Y=\{u=0\}\), write \((\alpha,\beta)\) for their complex covectors. The coordinates and signs of that identification are \[ (v,y;\alpha,\beta)\in T^*E \longleftrightarrow(\alpha,y;-v,\beta)\in T^*L \longleftrightarrow \bigl((0,y;\alpha,0);v,\beta\bigr)\in N_L(T^*X). \tag{3} \] Each map is holomorphic. The second map uses the symplectic normal identification \(K([w])(a)=\operatorname{Re}\Omega(w,a)\); its inverse is induced by \(-H\) with the Hamiltonian convention used earlier. The first is the Fourier cotangent map. These signs agree with the selected specialization contract. The analytic normal-cone argument in Analytic normal cones through complex deformation applies to the analytic set \(\Lambda\) and the analytic submanifold \(L\). Its accessible central fibre is analytic and invariant under complex normal scaling. The Lagrangian normal-cone theorem proved in Boundary forms and Lagrangian normal cones makes its image in \(T^*L\) real-isotropic. The holomorphic symplectic Fourier identification in (3) preserves this property on \(T^*E\). We must also check cotangent conicity in \(T^*E\), since normal scaling alone is a different action. A normal-cone sequence in these coordinates has \[ (t_jv_j,y_j;\alpha_j,t_j\beta_j)\in\Lambda, \qquad t_j>0,\quad t_j\to0, \tag{4} \] with all four displayed limiting coordinates finite. Multiplying the input covector by any fixed \(\lambda\in\mathbb C^*\) gives the same witnesses with \(\alpha_j,\beta_j\) replaced by \(\lambda\alpha_j,\lambda\beta_j\). Under (3) this is exactly complex dilation of the output cotangent covector at the unchanged base \((v,y)\). Therefore the bound in (2) is closed complex analytic, complex-conic and real-isotropic. The same constructibility criterion, applied to this bound, proves \[ \nu_YF\in D^b_{w\text{-}\mathbb C\text{-}c}(k_E), \qquad \mu_YF=(\nu_YF)^\wedge\in D^b_{w\text{-}\mathbb C\text{-}c}(k_{E^\vee}). \tag{5} \] For the second assertion use the complex Fourier theorem proved in Holomorphic operations and complex Fourier symmetries. Both objects are bounded and positively conic along their bundle fibres. The boundedness and conicity are part of the selected specialization/Fourier contracts, rather than consequences of finite-dimensional coefficient modules. For perfect constructible \(F\), Perfect operations and finite microlocal coefficients proves perfect stalks for both \(\nu_YF\) and \(\mu_YF\). That proof keeps the real deformation chamber and treats it using weak real constructibility and perfect internal Hom. We have not treated a positive real chamber as a holomorphic open subset. Combining the real perfection theorem with (5) proves both perfect complex constructibility statements. ## Fibre constancy after lifting the punctured normal line For now suppose \(Y=f^{-1}(0)\) is a regular analytic fibre: \(df_y\ne0\) for every \(y\in Y\). Its normal line is canonically identified with \(\mathbb C_v\times Y\) by \[ \ell_y([a])=df_y(a),\qquad s(y)=\ell_y^{-1}(1),\qquad s'(y)=df_y\in T_Y^*X. \tag{6} \] In the dual coordinate \(\xi\), \(s'\) is the section \(\xi=1\). Let \(G=\nu_YF\), and let \(e:Y\hookrightarrow E\) be the zero section. A reduced smooth zero set for a ramified defining function does not suffice for (6); \(z^m\), \(m>1\), has zero derivative at its reduced zero set. On \(E\setminus e(Y)\), the cohomology of \(G\) is locally constant on every \(\mathbb C^*\) fibre. Here is the relevant microsupport check. Positive conicity annihilates the real radial vector field. The full complex Euler calculation in the preceding Fourier lesson, using the complex-conic actual microsupport, annihilates the imaginary radial vector field as well. In local coordinates a vertical covector \(\alpha\) consequently satisfies \[ \operatorname{Re}(v\alpha)=0, \qquad \operatorname{Im}(v\alpha)=0. \tag{7} \] When \(v\ne0\), it has \(\alpha=0\). The submersion descent criterion for microsupport therefore makes \(G\) locally a whole derived pullback in these fibre coordinates, and in particular makes its cohomology locally constant on each punctured fibre. The same reasoning applies to \(\mu_YF\) away from its dual zero section, by (5). Use the cover \[ p:\mathbb C_w\times Y\longrightarrow\mathbb C_v^*\times Y, \qquad p(w,y)=(e^{2\pi iw},y), \qquad q(w,y)=y. \tag{8} \] The pullback \(p^{-1}(G|_{v\ne0})\) has locally constant cohomology on the contractible \(\mathbb C_w\) fibres. The exact contract `SH02-CON-CYLINDER` identifies its derived direct image with evaluation at \(w=0\), through the actual evaluation morphism. It applies twice, to the two real coordinates of \(w\), and retains the full extension data and arbitrary coefficient modules. Thus \[ Rq_*p^{-1}(G|_{v\ne0})\simeq s^{-1}G. \tag{9} \] This contract proves descent on a product with \(\mathbb R\) using closed-strip exhaustions and their evaluation maps. Iteration on \(\mathbb R^2\) is legitimate. It does not assert unrestricted nonproper base change or descent from a punctured fibre with nontrivial fundamental group. Ordinary positive-conic contraction identifies the left side of (9) with \(\psi_\ell(G)\). Explicitly, the cover direct image in the nearby definition has a lifted positive action: multiplying \(v\) by \(r>0\) translates \(w\) by \(\log r/(2\pi i)\). Contracting its ordinary direct image to the zero section and composing the two direct images gives \(Rq_*\) in (9). The normal comparison already proved therefore yields \[ \psi_f(F)\simeq\psi_\ell(G)\simeq s^{-1}\nu_YF. \tag{10} \] Fixing \(w=0\) fixes the lift of the normal section. Deck translation \(w\mapsto w+1\) remains the monodromy automorphism. Fibrewise local constancy on \(\mathbb C^*\) does not force this automorphism to be the identity. ## A dual halfspace has a closed-ray polar Let \(\tau:E\to Y\) and \(\tau^\vee:E^\vee\to Y\) be the bundle projections. Set \[ U=\{\xi\in\mathbb C:\operatorname{Re}\xi>0\}, \qquad R=\{v\in\mathbb C:\operatorname{Im}v=0, \ \operatorname{Re}v\geq0\}. \tag{11} \] The cohomology of \(\mu_YF\) is locally constant along the \(U\) fibres. A smooth product identification \(U\simeq\mathbb R^2\), chosen to take \(1\) to the origin, and the whole-complex cylinder descent give \[ s'^{-1}\mu_YF\simeq R\tau^\vee_*R\mathcal Hom(k_{U\times Y},\mu_YF). \tag{12} \] The internal Hom for this open coefficient means ordinary sections over the open halfspace, pushed into the ambient bundle. It does not mean compactly supported cohomology. In the real pairing \(\operatorname{Re}(v\xi)\), the positive polar of \(U\) is \(R\). Indeed, write \(v=u+iw\) and \(\xi=a+ib\). Then \[ \operatorname{Re}(v\xi)=ua-wb, \qquad a>0,\quad b\in\mathbb R. \tag{13} \] Nonnegativity for every such \(a,b\) forces \(w=0\) and \(u\geq0\), and these conditions suffice. The zero vector passes every inequality and must be included. The open-convex-cone test `SH02-FS-SECTIONS`, formula FS13, now gives \[ R\tau^\vee_*R\mathcal Hom(k_{U\times Y},G^\wedge) \simeq R\tau_*R\mathcal Hom(k_{R\times Y},G). \tag{14} \] To obtain an isomorphism of sheaves on \(Y\), apply the natural Fourier test to the restriction over every open base subset and its restriction maps. Its proof applies the inverse Fourier equivalence to both Hom arguments; the inverse image of the open-cone coefficient is its closed-polar coefficient. The inverse Fourier shifts and dual orientation lines cancel in this test. The real rank of the complex line remains two; (14) introduces no further shift or orientation factor. ## The slit and the cover have the same section complex Let \(O=\mathbb C\setminus R\). This slit plane is contained in \(\mathbb C^*\), and its inverse image has the distinguished open strip \[ S=\{w\in\mathbb C:0<\operatorname{Re}w<1\}. \tag{15} \] The restriction \(p|_S:S\to O\) is a homeomorphism. Write \(E^*=\mathbb C^*\times Y\) and \(j:E^*\hookrightarrow E\); the cover \(p\) has target \(E^*\). Extension by zero from \(S\) to the cover, proper-support direct image along \(p\), and then \(j_!\) give a coefficient morphism on \(E\) \[ \alpha:k_{O\times Y}\longrightarrow L=j_!p_!k_{\mathbb C_w\times Y}, \qquad \operatorname{tr}\circ\alpha: k_{O\times Y}\longrightarrow k_E. \tag{16} \] The trace is the composite of the punctured-target counit, extended by \(j_!\), with the open-inclusion counit \(j_!k_{E^*}\to k_E\). On \(E^*\) it sums finitely supported sheet coefficients. Consequently \(\operatorname{tr}\circ\alpha\) is the ordinary open-extension inclusion. Both two-term complexes below are objects on \(E\), with terms in degrees \(-1,0\): \[ B=[k_{O\times Y}\longrightarrow k_E], \qquad K=[L\xrightarrow{\operatorname{tr}}k_E], \qquad B\longrightarrow K=(\alpha,\mathrm{id}). \tag{17} \] Open–closed localization identifies \(B\simeq k_{R\times Y}\), in degree zero. Applying contravariant internal Hom to (17) gives \[ R\tau_*R\mathcal Hom(K,G) \longrightarrow R\tau_*R\mathcal Hom(k_{R\times Y},G). \tag{18} \] The left side equals \(\phi_\ell(G)\). This follows from the coefficient definition in the monodromy lesson and ordinary positive-conic contraction to \(e\). The coefficient \(K\), the target \(G\), and their internal Hom have the needed positive conicity; the lifted positive action preserves \(S\), so it also preserves the coefficient map in (17). We show that (18) is an isomorphism by comparing the Hom triangles of the two complexes in (17). Their \(k_E\) terms have the identity map. On the other terms, ordinary composition and the cover adjunction identify the map with restriction \[ Rq_*p^{-1}(G|_{v\ne0}) \longrightarrow R(\tau|_{O\times Y})_*(G|_{O\times Y}). \tag{19} \] It is restriction from the entire cover to \(S\). The whole cover and the open strip are each a product with a contractible real two-dimensional fibre. Their cohomology is vertically locally constant. Apply the cylinder descent contract, identifying \(S\) with \(\mathbb R^2\), and evaluate both sides at \(w=1/2\). The two evaluation maps commute with restriction. Both are isomorphisms, so (19) is an isomorphism. The evaluation point lies above \(v=-1\), which belongs to \(O\). In contrast, \(v=1\) lies on the removed ray. Evaluation at \(w=0\) from (9) is related to evaluation at \(w=1/2\) by transport in the contractible cover, rather than by pretending that the normal section \(1\) lies in the slit. This distinction allows arbitrary monodromy on the punctured normal line. The map between the two Hom triangles is now an isomorphism on both their other terms. Their fibre term (18) is an isomorphism too. Combining (12), (14), (18), and the normal vanishing comparison gives \[ \phi_f(F)\simeq\phi_\ell(\nu_YF) \simeq R\tau_*R\mathcal Hom(k_{R\times Y},\nu_YF) \simeq s'^{-1}\mu_YF. \tag{20} \] The coefficient map (17) fixes the slit branch used in this comparison. Formula (20) retains exactly the normalization (1). It does not append a further Fourier, real-rank or complex-orientation shift. Changing the lift is governed by deck transport; no trivialization of monodromy on the entire punctured line has been assumed. ## Constructibility and support for a critical function For a regular fibre, (5), (10), and (20), followed by holomorphic ordinary inverse image along \(s,s'\), prove weak complex constructibility of both cycles. If \(F\) has perfect stalks, both bundle objects have perfect stalks and ordinary inverse image retains them. This proves perfect complex constructibility as well. Now allow any holomorphic \(f:X\to\mathbb C\). Its zero set \(Y\) can be singular. Use the closed graph and the coordinate projection \[ g:X\hookrightarrow\mathbb C_t\times X, \quad g(x)=(f(x),x),\qquad H=g_*F, \quad Z=\{t=0\}\simeq X. \tag{21} \] The graph is proper and holomorphic. The operation theorem makes \(H\) weakly, or perfectly, complex constructible as appropriate. The target coordinate \(t\) has a regular fibre \(Z\), so the results just proved apply to \(\psi_t(H)\) and \(\phi_t(H)\). With \(a:Y\hookrightarrow Z\) the closed inclusion, the actual proper-on-support comparisons from Proper pushforwards of nearby and vanishing cycles give \[ \psi_t(H)\simeq a_*\psi_f(F), \qquad \phi_t(H)\simeq a_*\phi_f(F). \tag{22} \] These are objects supported on \(a(Y)\). Refine their analytic stratifications in \(Z\) compatibly with the analytic subset \(Y\). Their restrictions are locally constant on the resulting strata of \(Y\); ordinary stalk restriction retains the perfect condition. Thus both cycles are weakly complex constructible on \(Y\), and are perfect constructible when \(F\) is. This argument does not assign a smooth normal line to a singular fibre. The selected contract `SH02-MO-MICROLOCAL-SUPPORT` gives \[ \operatorname{supp}(\mu_YF) \subset T_Y^*X\cap\operatorname{SS}(F) \tag{23} \] for a smooth \(Y\). Here and below support is the closed support of the cohomology sheaves. For a regular fibre, pulling (23) back along \(s'\) and using (20) gives the closed bound \[ \operatorname{supp}(\phi_f(F)) \subset\{y\in f^{-1}(0):(y;df_y)\in\operatorname{SS}(F)\}. \tag{24} \] For a critical function use (21)–(22). At \((0,x)\), the covector \(dt\) restricts to the graph as \(df_x\). More explicitly, graph transpose pullback sends \((c,\xi)\) to \(c\,df_x+\xi\). The proper direct-image estimate `SH02-MO-PROPER-PUSH`, formula MO8, implies \[ ((0,x);dt)\in\operatorname{SS}(g_*F) \ \Longrightarrow\ (x;df_x)\in\operatorname{SS}(F). \tag{25} \] Apply the regular support bound to \(t,H\) and use (22). It gives (24) for arbitrary \(f\), including \(df_x=0\). Closedness matters: the preimage of the closed set \(\operatorname{SS}(F)\) under \(y\mapsto(y;df_y)\) is closed, so the assertion bounds closed support and not merely individual nonzero stalks. Finally, if \(p\notin\operatorname{SS}(F)\), choose an open cotangent neighborhood \(V\) disjoint from microsupport. For every point \(x\) and every local holomorphic function \(h\) with \(h(x)=0\) and \((x;dh_x)\in V\), restriction of microsupport to the domain of \(h\) and (24) give \[ \phi_h(F)_x=0. \tag{26} \] This proves the uniform forward holomorphic test. Quadratic cycles and the holomorphic microsupport test proves the reverse implication using generic microlocal models, transfer of the test to those models, and the quadratic calculation. ## Exercises with complete solutions ### The endpoint and two Fourier calibrations *Difficulty: Introductory.* Compute the polar of \(U=\{\operatorname{Re}\xi>0\}\) using the pairing in (13). Over a field, check (20) for \(G=k_{\mathbb C}\) and \(G=k_{\{0\}}\), with \(f(v)=v\). Explain why the ray's endpoint changes the answer. **Solution.** Allowing every real \(b\) forces \(w=0\), and then every \(a>0\) forces \(u\geq0\). Thus the polar is the closed ray \(R\), including zero. For \(k_{\mathbb C}\), nearby cycles are \(k\) and the central-to-nearby map is the identity, so \(\phi=0\). The Fourier transform is \(k_{\{0\}}[-2]\), with the canonical real rank-two orientation, and restriction at \(\xi=1\) is zero. For \(k_{\{0\}}\), the nearby object is zero and (1) gives \(\phi=k\); its Fourier transform is the constant \(k\) in degree zero, agreeing at \(1\). Equivalently, \(R\operatorname{Hom}(k_R,k_{\{0\}})=k\), since closed support at zero belongs to \(R\). Removing the endpoint makes that Hom zero: the coefficient then has zero stalk at zero and closed-embedding adjunction computes it there. The second calibration would fail. ### A slit cannot be evaluated at the removed normal section *Difficulty: Intermediate.* For (15), locate lifts of \(v=1\) and \(v=-1\). Prove directly that restriction from the cover to \(S\) induces an isomorphism of ordinary derived section complexes for a cohomologically locally constant complex on the cover. Describe the effect of choosing the strip \(m<\operatorname{Re}w