# Vanishing cycles as positive real support For a holomorphic function and a weakly complex constructible sheaf, vanishing cycles can be computed by local cohomology with support in its closed positive real halfspace. The comparison is an actual coefficient-triangle map. Its proof compares the lifted punctured neighborhood with one negative sector, using the local pushforward theorem over a complex curve. Let \(k\) be a commutative ring of finite global dimension, \(X\) a complex manifold that is Hausdorff and countable at infinity, and \(f:X\to\mathbb C\) holomorphic. Put \[ Y=f^{-1}(0),\quad i:Y\hookrightarrow X,\qquad A=\{x:\operatorname{Re}f(x)<0\},\qquad H=\{x:\operatorname{Re}f(x)\geq0\}. \tag{1} \] The input \(F\in D^b_{w\text{-}\mathbb C\text{-}c}(k_X)\) may have infinite coefficient modules. We will prove a natural isomorphism \[ i^{-1}R\Gamma_HF\simeq\phi_f(F), \tag{2} \] with the source convention \(\phi_f(F)=\operatorname{Cone}(i^{-1}F\to\psi_f(F))[-1]\). The fibre \(Y\) may be singular and \(df\) may vanish. Properness of \(f\) is not assumed. The local complex-curve pushforward theorem applies on a ball intersected with a sufficiently small inverse-image target neighborhood. Contractible-fibre descent uses the whole-complex theorem `SH02-CON-CYLINDER`. These inputs retain their analytic, conic, sheaf-operation and boundedness hypotheses. *Original lesson text and solutions: CC0 1.0 Universal. Human mathematical sources are credited below.* ## The normalized negative sector supplies a coefficient map Use the fixed cover from the monodromy lesson, \[ p:\mathbb C_w\longrightarrow\mathbb C, \qquad p(w)=e^{2\pi iw},\qquad \widetilde U=X\times_{\mathbb C}\mathbb C_w, \quad q:\widetilde U\longrightarrow X. \tag{3} \] Its image is \(X\setminus Y\). Write \(L_f=q_!k_{\widetilde U}\). The proper-support trace \(\operatorname{tr}:L_f\to k_X\) sums the finitely supported sheet coefficients. It gives the coefficient complex \[ K_f=[L_f\xrightarrow{\operatorname{tr}}k_X], \quad\text{in degrees }-1,0, \qquad \phi_f(F)=i^{-1}R\mathcal Hom(K_f,F). \tag{4} \] Let \(N=\{\lambda:\operatorname{Re}\lambda<0\}\). It has the normalized argument in \((\pi/2,3\pi/2)\), so the strip \[ S=\{w:1/4<\operatorname{Re}w<3/4\} \tag{5} \] maps homeomorphically onto \(N\). This fixes the lift of the negative sector, with \(-1\) lifted to \(1/2\), in the cover convention (3). Its pullback to \(X\) is an open subset \(\widetilde A\subset\widetilde U\) mapped homeomorphically by \(q\) onto \(A\). Extend its constant coefficient by zero into \(\widetilde U\) and apply \(q_!\). The resulting map is \[ \alpha_f:k_A\longrightarrow L_f, \qquad \operatorname{tr}\circ\alpha_f:k_A\longrightarrow k_X. \tag{6} \] The composite is the ordinary open-extension inclusion. Let \(C_f=[k_A\to k_X]\), again in degrees \(-1,0\). Open–closed localization gives \(C_f\simeq k_H\), in degree zero. The coefficient morphism \[ C_f\longrightarrow K_f=(\alpha_f,\mathrm{id}) \tag{7} \] induces, by contravariant internal Hom and restriction to \(Y\), \[ \phi_f(F)\longrightarrow i^{-1}R\Gamma_HF. \tag{8} \] It is a natural map of the two defining fibre triangles. Their middle term is \(i^{-1}F\), with the identity map. On their other terms it is \[ \beta_F: \psi_f(F)=i^{-1}Rq_*q^{-1}F \longrightarrow i^{-1}Rj_*j^{-1}F, \qquad j:A\hookrightarrow X. \tag{9} \] The cover adjunction and the open-set Hom interpretation identify this map with ordinary restriction from the entire lifted punctured neighborhood to \(\widetilde A\). The trace square in (6) makes it commute with the central unit. Thus proving (9) to be an isomorphism proves (8) to be an isomorphism, whose natural inverse is (2). No additional shift can enter this fibre-triangle comparison. ## Local curve pushforwards give a cofinal family of balls and target discs Fix \(x\in Y\), choose a relatively compact holomorphic coordinate chart around \(x\), and translate \(x\) to zero. The centered-ball germ theorem of the complex-curve lesson supplies arbitrarily small source radii \(r\) and target neighborhoods \(D\ni0\) for which \[ V=B_r(x)\cap f^{-1}(D),\qquad G_V=R(f|_V)_*(F|_V) \tag{10} \] is bounded weakly complex constructible on \(D\). This includes critical functions and arbitrary weak coefficients. The theorem's proof chooses radii below the first positive selected central critical value and a compact cutoff band; it works at arbitrarily small radii. Its repaired reciprocal exhaustion verifies every finite closed-level properness condition. We need the ordinary direct image in (10). On a complex curve, weak complex constructibility makes the cohomology locally constant away from a locally finite set of points. Shrink \(D\) to a small centered disc so that the only possible exceptional point of \(G_V\) in that disc is zero. Restricting the target disc also restricts \(V\); ordinary open-base restriction commutes with direct image, so (10) and its statement remain valid. We obtain nested cofinal neighborhoods \[ V_a=B_{r_a}(x)\cap f^{-1}(D_{\delta_a}), \quad r_a\downarrow0,\quad\delta_a\downarrow0, \qquad G_a=R(f|_{V_a})_*(F|_{V_a}), \tag{11} \] with \(G_a\) cohomologically locally constant on \(D_{\delta_a}^*\). Choose each next radius and disc inside the previous ones; the curve theorem and open target restriction permit this. The \(V_a\) are open neighborhoods of \(x\), are contained in shrinking coordinate balls, and are therefore cofinal in the ordinary neighborhood system. It is not required that \(f(B_{r_a})\subset D_{\delta_a}\). The actual domain in (11) is the intersection. The curve theorem applies to this intersection, and these intersections supply the cofinal source neighborhoods used below. ## The actual cover-to-sector restriction is an isomorphism Let \[ P_a=\{w:|e^{2\pi iw}|<\delta_a\}, \qquad S_a=P_a\cap S. \tag{12} \] The cover \(P_a\to D_{\delta_a}^*\) is a local homeomorphism, \(P_a\) is an open halfplane, and \(S_a\) is an open half-strip. Each is diffeomorphic to \(\mathbb R^2\). The latter maps homeomorphically onto \(D_{\delta_a}\cap N\). Ordinary base change along a local homeomorphism is valid for any map in the other direction. To check its actual morphism, restrict to an evenly covered target open set and one sheet, where the base map is a homeomorphism. The two preimages and their restriction maps identify, so open-base restriction of derived direct image gives the isomorphism there. These local identifications prove the global base-change morphism; they use no properness assertion. Apply this to \(f|_{V_a}\) over the punctured disc. Direct-image composition and open restriction give natural identifications \[ \begin{aligned} R\Gamma(q^{-1}V_a;q^{-1}F) &\simeq R\Gamma(P_a;p^{-1}(G_a|_{D_{\delta_a}^*})),\\ R\Gamma(V_a\cap A;F) &\simeq R\Gamma(S_a;p^{-1}(G_a|_{D_{\delta_a}^*})|_{S_a}). \end{aligned} \tag{13} \] The identifications carry the restriction in (9) to restriction from \(P_a\) to \(S_a\), by their open-set and base-change naturality. The coefficient \(p^{-1}(G_a|_{D_{\delta_a}^*})\) has locally constant cohomology on the simply connected \(P_a\). Choose any \(w_a\in S_a\). The cylinder descent contract, iterated on two real coordinates after product identifications of \(P_a,S_a\), gives isomorphisms from both section complexes to evaluation at \(w_a\). They commute with the actual restriction map. Consequently \[ R\Gamma(q^{-1}V_a;q^{-1}F) \longrightarrow R\Gamma(V_a\cap A;F) \quad\text{is an isomorphism.} \tag{14} \] This uses the whole bounded complex, not only its separate cohomology modules, and does not impose finite generation. Nontrivial monodromy on the punctured target disc remains allowed; both evaluation arguments take place on the contractible cover and its selected strip. Take the filtered stalk system over the cofinal \(V_a\). The two systems in (14) compute the stalks of \(Rq_*q^{-1}F\) and \(Rj_*j^{-1}F\) at \(x\). Their maps are the restrictions induced by the globally defined coefficient branch (6), so they commute with all smaller-neighborhood restriction maps. Exact filtered colimits on cohomology show that \((\beta_F)_x\) is an isomorphism. Since this holds at every \(x\in Y\), (9) is an isomorphism on \(Y\). The fibre-triangle map (8) is then an isomorphism, proving (2). The auxiliary coordinate balls, discs and evaluation points prove that a previously defined natural map is invertible. They are not part of its definition. The branch normalization is part of the fixed covering convention: translating the strip by an integer gives the corresponding deck-translated comparison. With (3)–(5) fixed, (2) is natural in \(F\). ## Endpoints, coefficients and critical functions The support in (2) is the closed halfspace, with \(\operatorname{Re}f=0\) included. Its complementary sector is the strictly negative halfplane. It is local cohomology \(R\mathcal Hom(k_H,F)\), not ordinary restriction of \(F\) to \(H\) extended by zero, and not compactly supported cohomology of \(H\). If \(F\) has perfect stalks, its cycles are perfect complex constructible by the preceding section theorem. The comparison therefore also proves that the restricted support object in (2) has perfect stalks. The proof of invertibility itself retains arbitrary weak coefficients throughout. For \(f=0\), the two negative-sector and punctured-cover objects are zero, so the comparison is the identity on \(F\). No regular-fibre hypothesis was used. ## Exercises with complete solutions ### The ball must be read over a smaller target germ *Difficulty: Intermediate.* For \(f(z_1,z_2)=z_1\) and \(F=k_{\{z_2=z_1\}}\), compute the image of the support inside a ball of radius \(r\). Explain why the proof uses \(B_r\cap f^{-1}(D_\delta)\), and why neighborhoods of this form with \(r\to0\) are cofinal at zero. **Solution.** On the support, the squared norm is \(2|z_1|^2\), so the support in the open ball maps to \(D_{r/\sqrt2}\). The entire ball maps to \(D_r\). Ordinary pushforward of the support's open disc, viewed on a target containing \(D_r\), acquires a real circle boundary at radius \(r/\sqrt2\); it is not weakly complex constructible across that boundary. On a smaller disc \(D_\delta\) with \(\delta