Proof. Choose bases and write \(T(x,y,z)=y^tM(x)z\), where \(M(x)=x_0A+x_1B\) is a pencil of two-by-two matrices. Put \(D(x)=\det M(x)\). Equations (18) imply \[ M(x)z=0,\qquad y^tM(x)=0,\qquad y^tAz=y^tBz=0. \tag{19}\] Rank two for \(M(x)\) is impossible. If its rank is one, its adjugate is a nonzero multiple of \(zy^t\), so the determinant derivative formula and (19) give \(\partial_{x_0}D(x)=\partial_{x_1}D(x)=0\). If \(M(x)=0\), its adjugate is zero and the same conclusion holds. For a nonzero binary quadratic \(D\), the gradient is a nonzero linear map in characteristic zero and has at most one projective zero. All qualifying indices then have the same \(x\), and there are at most two.

It remains to consider \(D\equiv0\). Every nonzero matrix in the pencil has rank one. A one-dimensional pencil makes \(T\) a product of three linear forms. For a two-dimensional pencil write \(A=uv^t\) and \(B=u'v'^t\). The equality \[0=\det(A+B)=\det[u,u']\det[v,v']\] shows that the pencil has a common left or right factor. After permuting the variables, both cases have the form \(T=l(x)q(y,z)\). If the bilinear form \(q\) has rank two, the last two partial contractions force \(l(x)=0\), again allowing at most two indices. If \(q\) has rank one, write \(T=l(x)m(y)k(z)\). Equations (18) then hold precisely when at least two of \(l(x),m(y),k(z)\) vanish. Each of the three distinguished projective coordinate values occurs at most twice. If \(N\) indices qualify, counting these incidences gives \(2N\leq6\), hence \(N\leq3\). ◻

Proof. If axis \(i\) at \(g\) met axis \(j\) at \(h\ne g\), the coordinates of \(g\) and \(h\) outside \(\{i,j\}\) would agree. There is at least one such coordinate, contradicting injectivity there. This proves disjointness.

Put \(V_i=H^0(E_i,D_i)\). Formula (15) gives \(\dim V_i=2\) and \(h^0(D_i(-q))=1\) for every \(q\in E_i\). Evaluation is therefore basepoint-free and defines a degree-two map \(\varphi_i:E_i\to\mathbb P(V_i^*)\), each of whose fibers has at most two distinct points. The marks \[\bigl(\varphi_1(g_1),\varphi_2(g_2),\varphi_3(g_3)\bigr), \qquad g=(g_1,g_2,g_3)\in K,\] have at most two indices in each coordinate fiber, since \(K\to E_i\) is injective. The product formula for sections identifies a section of \(D_1\boxtimes D_2\boxtimes D_3\) with a tensor in \(V_1\otimes V_2\otimes V_3\). Its restriction to axis \(i\) at \(g\) is contraction by evaluation at the other two coordinates. Vanishing on \(Z(g)\) is exactly (18). Lemma 18 forbids this for four or more indices for a nonzero section. Finally, the ideal of a closed subscheme with this support is contained in the radical ideal of the reduced union, which proves the last assertion. ◻

Proof. Over an affine \(U=\operatorname{Spec}R\subset B\), choose a finite affine cover \(V_i\) of \(V_U\). All finite intersections are affine because \(V_U\) is separated. Its Čech complex has terms \[ C_R^q=\prod_{i_0<\cdots<i_q} \Gamma(V_{i_0}\cap\cdots\cap V_{i_q},\mathcal F). \tag{22}\] It is bounded. Each term is \(R\)-flat: for a module on an affine scheme, flatness over \(R\) is checked at the total-space stalks, where it is the assumed relative flatness. For any \(R\)-algebra \(R'\), \(C_R^\bullet\otimes_RR'\) is exactly the affine Čech complex of the ordinary pullback of \(\mathcal F\) to \(V_U\times_RR'\). A bounded complex of flat modules is \(K\)-flat, so this tensor product also computes the derived tensor product. Affine Čech cohomology therefore proves (21) for affine base changes, and those isomorphisms glue for an arbitrary \(T\).

The cohomology modules of \(C_R^\bullet\) are finite over \(R\), by proper coherence for a projective scheme over a Noetherian affine base [39]. A finite module over the regular Noetherian finite-dimensional ring \(R\) has a finite resolution by finite projective modules [39]. The truncation triangles, applied to the finitely many cohomology modules of \(C_R^\bullet\), consequently make its derived object perfect. After localizing, it has the stated finite free model.

For such a model the dimension of the \(i\)-th fiber cohomology is \[ \operatorname{rank}K^i- \operatorname{rank}(d^i\otimes\kappa(b))- \operatorname{rank}(d^{i-1}\otimes\kappa(b)). \tag{23}\] Ranks of matrices are lower semicontinuous by their minors; (23) proves upper semicontinuity. Finally, at a term below \(c\), the first remaining differential is injective on the fiber. Invert a maximal minor, split off this term, and replace the next term by its locally free cokernel. At a term above \(e\), do the dual operation using a fiberwise surjective last differential. Repeating at a chosen point and shrinking its neighborhood gives a model in \([c,e]\). ◻

Proof of Proposition 17. The polarized elliptic limit. We first realize the elliptic product with its product principal polarization as the central Jacobian of a smoothing. This identifies the central twice-theta line with the degree-two product lines tested in Lemma 19.

Start with the stable compact-type curve \[ C_0=R\cup E_1\cup E_2\cup E_3,\qquad R\simeq\mathbb P^1, \tag{24}\] where the elliptic tails \(E_i\) meet \(R\) at three distinct nodes \(e_i\) and have no other intersections. We choose a projective one-parameter smoothing \(\mathcal C\to S\) over a smooth pointed complex algebraic curve \((S,0)\) with regular total surface and with smooth nonhyperelliptic noncentral fibers.

Here is the existence check, for the unmarked stable curve. The local presentation of \(\Omega_{C_0}\) at a node has length one, its local \(\mathcal Ext^1\) is supported at the nodes, and the local-to-global sequence gives \(\operatorname{Ext}^2(\Omega_{C_0},\mathcal O_{C_0})=0\). The global tangent contribution is the sum of the \(H^1\) of the tangent lines on the normalized components, with their node points removed. It is zero for \(T_R(-e_1-e_2-e_3)=\mathcal O_{\mathbb P^1}(-1)\), and has dimension one for each \(T_{E_i}(-e_i)=\mathcal O_{E_i}(-e_i)\). The three node-smoothing parameters add three more dimensions. Thus the unmarked versal germ is smooth of dimension six. The algebraic input is the smooth Deligne–Mumford stable-curve stack and density of its smooth-curve locus [39]; it gives an algebraic versal chart here. Smooth hyperelliptic genus-three curves have a five-dimensional locus, as seen from eight branch points on \(\mathbb P^1\) modulo \(\operatorname{PGL}_2\). Its closure is proper in that chart. A general algebraic curve through the chosen point is not contained in this closure and has nonzero first derivative in each node parameter. The local total-space equation at each node is then \(uv=t\) times a unit, so the total surface is regular. Shrinking \(S\) gives the asserted noncentral fibers. Choose a smooth point on \(R\) only now. Smoothness near it lifts it, after an étale localization of \(S\), to a section \(\sigma\). This point was not included in the six-dimensional unmarked germ.

Let \(\mathcal J\) be the componentwise degree-zero relative Picard space of \(\mathcal C/S\). The exact Picard input used here is [13]: for a stable family this open group space is separated, smooth, and of finite type, and its fibers are semiabelian. The normalization sequence for a connected nodal curve with dual graph \(\Delta\) gives \[ 1\longrightarrow\mathbb G_m^{b_1(\Delta)} \longrightarrow\operatorname{Pic}^0(C_0) \longrightarrow\prod_v\operatorname{Pic}^0(C_v) \longrightarrow1. \tag{25}\] Indeed, its kernel is the set of gluing scalars at the nodes modulo rescaling on each component; one scalar remains per graph cycle. Our graph is a tree and \(\operatorname{Pic}^0(R)=0\), so \(\mathcal J_0=\prod_i\operatorname{Pic}^0(E_i)\). All noncentral fibers are smooth Jacobians. Thus every geometric fiber is proper and connected. Deligne’s [13] states both that a separated flat finite-presentation space with proper connected geometric fibers is proper, and that a generic polarization on an abelian space over a normal integral base extends uniquely and makes that space projective. Consequently \(\mathcal J\) is an abelian scheme and the generic canonical polarization extends to \(\lambda_{\mathcal J}:\mathcal J\to\widehat{\mathcal J}\). Its degree is locally constant and is one generically, so it is principal. We use only the componentwise degree-zero Picard space; no separatedness of the unrestricted relative Picard functor is asserted.

The decomposition of the central Jacobian as an abelian variety does not yet identify its polarization. To obtain the product polarization needed for the vanishing lemma, we compute a theta line by a determinant of cohomology. After an étale localization, the relative Picard space has a universal line bundle; this is the precise universal-line input from [11]. Restrict it to \(\mathcal C\times_S\mathcal J\) and rigidify it along \(\sigma\); denote it by \(\mathcal U\). With \(p_C\) the projection to \(\mathcal J\), put \[ \mathcal T= \left(\det Rp_{C,*}\bigl(\mathcal U\otimes \mathcal O_{\mathcal C}(2\sigma)\bigr)\right)^{-1}. \tag{26}\] The base \(\mathcal J\) is regular, and this is a flat projective curve family with a line bundle. Lemma 20 and the determinant construction following it define (26) and its base changes. Every fiber has Euler characteristic zero and cohomology in degrees zero and one; locally it has a two-term equal-rank model, with determinant section. On a smooth nonhyperelliptic fiber the section vanishes exactly on the effective degree-two locus \(W_2\). Its generic multiplicity there is one: at an effective degree-two line the groups \(H^0\) and \(H^1\) have dimension one, and the first-order differential is cup product \[H^1(\mathcal O_C)\longrightarrow \operatorname{Hom}(H^0(N),H^1(N)).\] Its dual is multiplication \(H^0(N)\otimes H^0(\omega_C N^{-1})\to H^0(\omega_C)\), which is nonzero on nonzero sections. Splitting the invertible blocks of a local two-term model shows that this nonzero first-order differential is the linear term of its remaining one-by-one determinant. Thus the determinant section has a simple zero there. It follows that the homomorphism \(\phi_{\mathcal T}\) associated to \(\mathcal T\) is the generic canonical polarization. It equals \(\lambda_{\mathcal J}\) over \(S\), since the two morphisms agree on the schematically dense generic fiber and the target is separated.

On \(C_0\), the rigidification makes \(\mathcal U\) trivial on \(R\times\mathcal J_0\); its restrictions to the elliptic components are the Poincaré lines \(\mathcal P_i\), normalized at \(e_i\). The normalization exact sequence for \(\mathcal U\otimes\mathcal O(2\sigma)\) factors (26), up to constant lines from \(R\) and the nodes, as \[ \mathcal T_0\simeq \mathop{\boxtimes}_{i=1}^3 \left(\det R\operatorname{pr}_{\widehat E_i,*}(\mathcal P_i)\right)^{-1}. \tag{27}\] For an elliptic curve the \(i\)-th factor is \(\mathcal O_{\widehat E_i}(0)\). Its fiber cohomology vanishes off the origin. At the origin the one-dimensional \(H^0\) and \(H^1\) have the nonzero first-order cup-product differential \(H^0(\mathcal O_{E_i})\otimes H^1(\mathcal O_{E_i}) \to H^1(\mathcal O_{E_i})\); the same determinant argument gives a simple zero. Equation (27) therefore proves \[ (\mathcal J_0,\lambda_{\mathcal J,0}) =\prod_{i=1}^3(\operatorname{Pic}^0(E_i),\lambda_i) \tag{28}\] as principally polarized abelian varieties. In particular, the polarization on this product is the product polarization.

Let \(\mathcal P_{\mathcal J}\) be the normalized relative Poincaré line and set \[ \mathcal D=(1,\lambda_{\mathcal J})^*\mathcal P_{\mathcal J} \quad\text{on }\mathcal J. \tag{29}\] The biextension identity gives \(\phi_{\mathcal D}=\lambda_{\mathcal J} +\lambda_{\mathcal J}^{\vee}=2\lambda_{\mathcal J}\). Concretely, translating the graph in (29) by \(x\) produces the two variable line factors \(\mathcal P_{\mathcal J}(-,\lambda_{\mathcal J}x)\) and \(\mathcal P_{\mathcal J}(x,\lambda_{\mathcal J}(-))\); the remaining factor is a constant fiber. These represent \(\lambda_{\mathcal J}x\) and \(\lambda_{\mathcal J}^{\vee}x\). On a smooth fiber it follows that \(\mathcal D_s\otimes\mathcal O(-2\Theta_s)\in\operatorname{Pic}^0\): their associated homomorphisms, equivalently their integral Riemann forms, agree. Centrally, (28) and the product Poincaré bundle give \(\mathcal D_0=D_1\boxtimes D_2\boxtimes D_3\), up to a constant line, with \(\deg D_i=2\). Every line in \(\operatorname{Pic}^0(\mathcal J_0)\) is the external product of three degree-zero lines.

The flat Abel limit and the torsion orbits. We next construct a flat family of Abel images whose central support is the elliptic star. Support is enough for the vanishing lemma, since the ideal of any thickening of that star is contained in its reduced ideal.

The pointed Abel map on the smooth fibers, \(q\mapsto\mathcal O(q-\sigma)\), extends to a morphism \(\alpha:\mathcal C\to\mathcal J\). We give the extension argument to record the support of its limit. The map is a rational \(S\)-map, defined on the smooth-fiber open. Elimination of indeterminacy for a regular surface and a proper target [39] gives a sequence of blowups at closed points of the central fiber on which it becomes a morphism. Each reduced fiber of this sequence over an indeterminacy point is a connected tree of rational curves. Every morphism \(\mathbb P^1\to\mathcal J_0\) is constant: pullback of the translation-invariant one-forms is zero, and those forms span every cotangent space, so the differential is zero in characteristic zero. The resolved map is therefore constant on each such tree. Its schematic graph image in \(\mathcal C\times_S\mathcal J\) is integral, proper and birational over \(\mathcal C\), with one underlying point in each fiber. A proper morphism with finite fibers is finite [39]. A finite integral birational morphism to the normal surface \(\mathcal C\) is an isomorphism, by integral closedness in its function field. This descends the resolved map to \(\alpha\). The relative first Abel morphism and its tail twists are also given by [11] for a regular smoothing of a stable compact-type curve; the principal component in the present star is \(R\).

We compute the specialization directly. For a smooth \(q_0\in E_i\), lift it étale-locally to a section \(q\). The divisor \(E_i\) is Cartier on the regular total surface and \[ \mathcal O_{\mathcal C}(E_i)|_{E_i}=\mathcal O_{E_i}(-e_i), \quad \mathcal O_{\mathcal C}(E_i)|_R=\mathcal O_R(e_i), \quad \mathcal O_{\mathcal C}(E_i)|_{E_j}=\mathcal O_{E_j}\quad(j\ne i). \tag{30}\] Thus \(\mathcal O_{\mathcal C}(q-\sigma+E_i)\) has componentwise degree zero and agrees generically with the Abel point. Separatedness of \(\mathcal J\) identifies its special value with \(\alpha(q)_0\). The restriction on \(E_i\) is \(\mathcal O_{E_i}(q_0-e_i)\), all other elliptic restrictions are trivial, and \(\mathcal O_R(e_i-\sigma_0)\) is trivial. There is no gluing torus in (25). A section through \(R\) maps to zero, and continuity includes the nodes. Identifying \(\operatorname{Pic}^0(E_i)\) with \(E_i\) using \(e_i\), we get \[ \alpha_0(R)=0,\qquad \alpha_0(q\in E_i)=(0,\ldots,\mathcal O_{E_i}(q-e_i),\ldots,0), \qquad \alpha_0(|C_0|)=|Z_0|. \tag{31}\] Here \(Z_0\) is exactly the three-axis star of Lemma 19.

Let \(\mathcal A\subset\mathcal J\) be the scheme-theoretic closure of the generic Abel image. Its ideal is the contraction of the generic ideal. At a stalk over any base DVR its quotient embeds in its localization obtained by inverting a uniformizer, and is therefore torsion-free over that DVR, hence flat. The ideal is coherent by Noetherianity. The morphism \(\alpha\) is proper, and \(\mathcal C_\eta\) is dense in \(\mathcal C\). Its closed topological image is therefore the closure of its generic image, so (31) gives \[ |\mathcal A_0|=|Z_0|. \tag{32}\] No reducedness of \(\mathcal A_0\) is claimed. On the punctured base the relative Abel map is a closed immersion. Indeed, it is proper with finite fibers, since every smooth-fiber Abel map is an embedding, so it is finite by the cited finite-fiber criterion. Finite pushforward commutes with base change by its affine description, and the fiber maps \(\mathcal O_{\mathcal J_s}\to\alpha_{s,*}\mathcal O_{\mathcal C_s}\) are surjective. Nakayama’s lemma makes the total map surjective. Its image is flat there and thus has a saturated ideal, so it agrees with the restriction of the generic closure. Hence the noncentral \(\mathcal A_s\) are the actual Abel curves schemewise.

Put \(H_n=(\mathbb Z/n)^2\). Choose isomorphisms \(H_n\simeq E_i[n]\) and embed \(H_n\) diagonally in \(\mathcal J_0[n]=\prod_iE_i[n]\). Denote its two cyclic direct factors by \(G_1,G_2\). Multiplication by \(n\) on the abelian scheme is proper, has invertible differential in characteristic zero and finite geometric fibers, so its kernel is finite étale. After an étale localization, the chosen torsion points extend to sections with their group relations. They remain distinct, since equality of two sections of a finite étale scheme is open and closed. Thus the two cyclic factors retain order \(n\) and trivial intersection. The entire central \(H_n\) projects injectively into each \(E_i\).

For one cyclic factor, the translates \(t_g\mathcal A\) are flat over \(S\), and their central supports are the disjoint stars of Lemma 19. A pairwise intersection is closed in the proper scheme \(\mathcal J/S\), so its image in \(S\) is closed and misses \(0\). Removing finitely many such images makes all these translates disjoint. Their disjoint union \(\mathcal Z\) is then a closed flat subscheme of \(\mathcal J\), with actual Abel orbits on noncentral fibers and \[ |\mathcal Z_0|=\left|\bigcup_g Z(g)\right|. \tag{33}\] Apply the same argument to the negative closure \(-\mathcal A\) for the other factor; its central support is still \(Z_0\). This reasoning uses the closure and its support, and does not assume that scheme-theoretic image or union formation commutes with the closed-fiber base change.

Vanishing for every Picard twist. We now have flat orbit closures and a line \(\mathcal D\) representing twice the principal polarization on every fiber. It remains to make the central vanishing uniform on a neighborhood of the smoothing parameter. The ideal sequence for \(\mathcal Z\) is exact after every base change to a fiber: both \(\mathcal O_{\mathcal J}\) and \(\mathcal O_{\mathcal Z}\) are \(S\)-flat. It also makes \(I_{\mathcal Z}\) \(S\)-flat. On \(\mathcal J\times_S\widehat{\mathcal J}\) the coherent sheaf \[ \operatorname{pr}_{\mathcal J}^* (I_{\mathcal Z}\otimes\mathcal D)\otimes\mathcal P_{\mathcal J} \tag{34}\] is flat over \(\widehat{\mathcal J}\): locally it is the base change of an \(S\)-flat module, followed by tensoring with a line. The projection to the regular scheme \(\widehat{\mathcal J}\) is projective. Lemma 20 makes the locus \[ \mathcal V=\{N\in\widehat{\mathcal J}_s\text{ for some }s: H^0(\mathcal J_s,I_{\mathcal Z_s}\otimes\mathcal D_s\otimes N) \ne0\} \tag{35}\] closed. On the central fiber the exact ideal sequence embeds \(I_{\mathcal Z_0}\) in \(\mathcal O_{\mathcal J_0}\), and (33) gives \(I_{\mathcal Z_0}\subset I_{\bigcup_gZ(g)}\). Lemma 19, applied to every external degree-two triple, therefore makes \(\mathcal V\cap\widehat{\mathcal J}_0\) empty. Properness of \(\widehat{\mathcal J}/S\) makes the image of \(\mathcal V\) closed. Removing it leaves a neighborhood of \(0\) on which the vanishing holds for every Picard twist on every fiber. Repeat for the negative orbit.

Choose a noncentral complex point of this neighborhood and put \(C=\mathcal C_s\), \(p=\sigma_s\). Under the group identification \(\operatorname{Pic}^0(C)\simeq X\), the positive Abel curve is \(C_p\). The negative Abel point \(\mathcal O_C(3p-q)\) differs from \(\omega_C(-p-q)\) by the fixed degree-zero line \(\mathcal O_C(4p)\otimes\omega_C^{-1}\); hence the negative curve is a translate of \(\Sigma_p\). The difference between \(\mathcal D_s\) and \(H^{\otimes2}\) is absorbed by the arbitrary Picard twist. The two factors thus give exactly (16)–(17). ◻

Proof. At an intersection point choose a regular sequence \(f_1,f_2\) for \(D_1\). Its Koszul resolution restricted to the regular one-dimensional local ring of \(D_2\) has injective left arrow: at least one \(\bar f_i\) is nonzero because the intersection is zero-dimensional, and that local ring is a domain. Thus \(\operatorname{Tor}_k(\mathcal O_{D_1},\mathcal O_{D_2})=0\) for \(k\geq2\). The two ideal sequences give \[\operatorname{Tor}_k(I_{D_1},I_{D_2}) =\operatorname{Tor}_{k+2}(\mathcal O_{D_1},\mathcal O_{D_2})=0 \quad(k>0)\] and \(\operatorname{Tor}_1(\mathcal O_{D_1},I_{D_2})=0\). Tensoring the first ideal sequence with \(I_{D_2}\) proves the injection and product image in (40). Away from the intersection the product ideal equals the reduced-union ideal, proving the assertion about its kernel. For disjoint unions, at any point at most one component from each part is present, so the same local calculation applies. This is the local fact behind [25], in the form needed here. ◻

Proof. The subscheme \(A\times M\) is \(M\)-flat, and the subscheme defined by \(\mathscr J\) is isomorphic to \(\Sigma\times M\) under \((y,x,N)\mapsto(y+x,x,N)\). Their ideal sequences make \(\mathscr I\) and \(\mathscr J\) \(M\)-flat. Locally along \(A\times M\), its regular codimension-two embedding gives the ideal resolution \[ 0\longrightarrow\mathcal O \longrightarrow\mathcal O^{\oplus2} \longrightarrow\mathscr I\longrightarrow0, \tag{43}\] whose restriction to each fiber is the corresponding curve-ideal resolution. Tensor (43) with \(\mathscr J\). On every fiber the left arrow is injective by Lemma 21.

We use the following precise local flatness criterion. For a local map \(R\to S'\) with \(S'\) Noetherian, a finite \(S'\)-module \(Q\), an \(R\)-flat module \(P\), and an \(R\)-linear map \(u:Q\to P\), injectivity modulo the maximal ideal of \(R\) implies injectivity of \(u\) and \(R\)-flatness of its cokernel [39]. Its mechanism is useful here: flatness compares the maximal-ideal filtrations and gives injectivity modulo every power; Krull intersection gives injectivity. Repeating modulo every ideal of \(R\) and using the Tor criterion gives flatness of the cokernel. Apply it at a total-space stalk to \(\mathscr J\to\mathscr J^{\oplus2}\), with \(R=\mathcal O_{M,m}\). It proves that the tensor of (43) is exact on the left and has \(M\)-flat cokernel. Where \(\mathscr I=\mathcal O\) the conclusion is immediate. Thus the derived tensor is the ordinary \(M\)-flat tensor.

Its multiplication map to \(\mathcal O_{X\times M}\) is injective on every fiber by Lemma 21. The same criterion, now with the \(M\)-flat target \(\mathcal O_{X\times M}\), gives an injection with \(M\)-flat quotient. Tensoring either exact sequence with any \(\mathcal O_T\) stays exact because its quotient is \(M\)-flat. This proves multiplication and arbitrary base change before pushforward. The fiber formula follows from \(t_x^*H=H\otimes a(-x)\) and \(\mathcal P|_{X\times\{N\}}=N\). Finally \(M\) is smooth and regular and \(\pi\) is projective. Lemma 20, applied to the actual flat \(\mathscr F\), gives the local finite model and (42). ◻

Proof. For \(m=(x,N)\), Proposition 22 gives \[ 0\longrightarrow\mathscr F_m\longrightarrow\mathcal M_{x,N} \longrightarrow\mathcal O_{Z_x}\otimes\mathcal M_{x,N} \longrightarrow0. \tag{46}\] The line \(\mathcal M_{x,N}\) is ample: it is \(H^{\otimes2}\) tensored by a topologically trivial line. Kodaira vanishing applies on the smooth projective abelian threefold, whose canonical bundle is trivial, and gives \(H^k(X,\mathcal M_{x,N})=0\) for \(k>0\) [22]. Hirzebruch–Riemann–Roch for a line bundle on a smooth projective variety [20], with the trivial tangent bundle of \(X\), gives \[ h^0(X,\mathcal M_{x,N}) =\chi(X,\mathcal M_{x,N}) =\int_X\frac{(2\Theta)^3}{3!}=8. \tag{47}\] Since \(I_{Z_x}\subset I_A\), (17) gives \(H^0(\mathscr F_m)=0\). Since \(Z_x\) has one-dimensional support, (46) gives \[ H^3(\mathscr F_m)=0,\qquad H^2(\mathscr F_m)= H^1(Z_x,\mathcal M_{x,N}|_{Z_x}). \tag{48}\]

Each difference image \(\Sigma_j-C_i\) is the closed image of the proper surface \(\Sigma_j\times C_i\), and has dimension at most two. Hence \[ M^\circ= \left(X\setminus\bigcup_{i,j}(\Sigma_j-C_i)\right)\times\widehat X \tag{49}\] is nonempty and open. For \(m\in M^\circ\), \(Z_x\) is the disjoint union of \(2n\) smooth genus-three curves. By (13), \(\mathcal M_{x,N}\) has degree six on each. Formula (15) gives zero \(H^1\) on each component since \(6>2\cdot3-2\). Thus (48) vanishes on \(M^\circ\), for every \(N\). Equation (42) identifies all these cohomology groups with the cohomology of the derived fibers of \(G\).

The last assertion of Lemma 20 now gives a local model \[ K^1\xrightarrow{u}K^2 \tag{50}\] for \(G\). On \(M^\circ\) its fiber map is surjective. Thus \(u^*:(K^2)^*\to(K^1)^*\) is generically injective. The variety \(M\) is integral, and the source of \(u^*\) is locally free; its generically zero kernel must therefore be zero. Dualizing (50) and shifting as in (44) leaves its sole cohomology in degree zero, with resolution \[ 0\longrightarrow(K^2)^*\xrightarrow{u^*}(K^1)^* \longrightarrow E\longrightarrow0. \tag{51}\] These local sheaves are the sole cohomology sheaf of the globally defined object \(D_M(G)[-1]\), so they glue. This constructs the actual sheaf, rather than assuming \(E=(\mathcal H^1G)^*\).

Finally, on \(M^\circ\), each curve restriction in (46) has Euler characteristic \(6+1-3=4\). Equations (46)–(47) give \[ \chi(\mathscr F_m)=8-2n\cdot4=-8d. \tag{52}\] The Euler characteristic of the finite model is \(-\operatorname{rank}K^1+\operatorname{rank}K^2\). The rank of (51) is therefore \(8d\). ◻

Proof of Proposition 25. For \(\beta:\mathcal O_\Delta\to\mathcal O_\Delta[2]\), write \(a_A(\beta)=L\Delta^*\beta:H_A\to H_A[2]\). Naturality of the adjunction unit, with its canonical shift isomorphisms, is the equality \[\Delta_*a_A(\beta)\circ\eta_\Delta =\eta_\Delta[2]\circ\beta.\] Evaluate this equality of diagonal morphisms on \(Q\). The projection formula gives \[ (1_Q\otimes a_A(\beta))\circ AH_Q =AH_Q[2]\circ\chi_Q(\beta). \tag{84}\] This is an identity of derived morphisms before any trace is taken.

Define \[h_Q=\operatorname{Tr}_{Q,H_A}(AH_Q)\in\mathbb H^0(A,H_A), \qquad \sigma_Q^H(v) =\operatorname{Tr}_{Q,H_A[2]}(AH_Q[2]\circ v) \in\mathbb H^2(A,H_A).\] The perfect partial trace commutes with a morphism of coefficients: for \(f:C\to C'\) and \(w:Q\to Q\otimes C\), \[ \operatorname{Tr}_{Q,C'}((1_Q\otimes f)\circ w) =f\circ\operatorname{Tr}_{Q,C}(w). \tag{85}\] This follows directly from its coevaluation–braiding–evaluation definition: \(f\) can be placed after evaluation since it acts only on the coefficient factor. Apply it to (84). We obtain \[\sigma_Q^H(\chi_Q(\beta))=a_A(\beta)\circ h_Q.\] There is no odd exchange sign, since the degree here is two.

By (81) and (85), \[\mathbf I_A(h_Q)=\operatorname{ch}_H(Q),\qquad \mathbf I_A(\sigma_Q^H(v)) =\operatorname{Tr}_Q \left(\left(\sum_{j=0}^d\frac{a_Q^j}{j!}\right)[2] \circ v\right).\] In the second equality the component with coefficient \(\Omega_A^j[j+2]\) is \(\sigma_{Q,j}\); those with \(j>d-2\) vanish because their sheaf cohomology degree exceeds \(d\). Take \(\beta=\operatorname{HKR}_A(\alpha)\) and apply (77). This proves (70). For the component statement, contraction of \(\operatorname{ch}_{H,j+b}(Q)\in H^{j+b}(A,\Omega_A^{j+b})\) removes \(b\) holomorphic forms and adds cochain degree \(a\). Its target is \(H^{j+b+a}(A,\Omega_A^j)=H^{j+2}(A,\Omega_A^j)\). This gives (71). In particular the factors \(1/b!\) in evaluation, \(1/j!\) in \(\sigma_{Q,j}\), and \(1/(j+b)!\) in the Chern component have already been accounted for; there is no further scalar. ◻

Proof of Proposition 26. The first equality in (88) follows from (87), (82), and the definition of \(\Phi^{HT}\).

For the second square, we verify compatibility with the cap action and the categorical character. If \(\beta\in HH^2(A)\), \(h\in HH_0(A)\), and \(\delta\in\mathscr T^{-2}(A)\), the definition of cap and the trace of a shifted perfect object give \[\langle\beta\cap h,\delta\rangle_{A,-2} =\langle h,\delta[2]\circ\beta\rangle_{A,0}.\] The shift is even, so its supertrace sign is \(+1\). Applying this on \(A\) and \(B\), and using (93) and (95), gives \[\begin{split} \langle\Phi_*^{HH}(\beta\cap h),\Psi^{-2}\delta\rangle_{B,-2} &=\langle\beta\cap h,\delta\rangle_{A,-2}\\ &=\langle h,\delta[2]\circ\beta\rangle_{A,0}\\ &=\langle\Phi_*^{HH}h, (\Psi^{-2}\delta)[2]\circ\Phi^{HH}\beta\rangle_{B,0}\\ &=\langle\Phi^{HH}\beta\cap\Phi_*^{HH}h, \Psi^{-2}\delta\rangle_{B,-2}. \end{split}\] The tests \(\Psi^{-2}\delta\) exhaust \(\mathscr T^{-2}(B)\). Nondegeneracy proves the needed transport, and only in this degree: \[ \Phi_*^{HH}(\beta\cap h) =\Phi^{HH}(\beta)\cap\Phi_*^{HH}(h), \qquad \beta\in HH^2(A),\quad h\in HH_0(A). \tag{97}\]

Let \(\mathfrak h_Q\in HH_0(A)\) correspond, under (90), to \(h_Q=\operatorname{Tr}_{Q,H_A}(AH_Q)\). We verify its trace characterization using the same \(AH_Q\) as in Proposition 25. For \(\delta:\mathcal O_\Delta\to\Sigma_A\), let \(\delta':H_A\to\omega_A[d_A]\) be its adjoint, so \(\delta=\Delta_*\delta'\circ\eta_\Delta\). The adjunction \(\Delta_!\dashv L\Delta^*\), defined by Serre duality above, is trace-compatible. Thus (85) and the standard partial-trace expression of the Serre trace give \[ \begin{split} \langle\mathfrak h_Q,\delta\rangle_{A,0} &=\operatorname{Tr}_{A,\mathcal O_A}(\delta'\circ h_Q)\\ &=\operatorname{Tr}_{A,\mathcal O_A} \left(\operatorname{Tr}_Q ((1_Q\otimes\delta')\circ AH_Q)\right)\\ &=\operatorname{Tr}_{A,Q}(\delta_Q). \end{split} \tag{98}\] The last equality uses evaluation of \(\Delta_*\delta'\circ\eta_\Delta\). It includes the supertrace when \(Q\) is shifted. Because the pairing is perfect, this characterizes \(\mathfrak h_Q\) uniquely; it is the categorical Chern character in \(HH_0(A)\). Its image under \(\mathbf I_A\) is \(\operatorname{ch}_H(Q)\) by (81).

For a test \(\delta_B=\Psi^0\delta_A\), equations (95), (98), (94), and (92) yield \[\begin{split} \langle\Phi_*^{HH}\mathfrak h_Q,\delta_B\rangle_{B,0} &=\langle\mathfrak h_Q,\delta_A\rangle_{A,0}\\ &=\operatorname{Tr}_{A,Q}((\delta_A)_Q)\\ &=\operatorname{Tr}_{B,\Phi Q}((\delta_B)_{\Phi Q})\\ &=\langle\mathfrak h_{\Phi Q},\delta_B\rangle_{B,0}. \end{split}\] The tests exhaust \(\mathscr T^0(B)\), so uniqueness gives \[ \Phi_*^{HH}(\mathfrak h_Q)=\mathfrak h_{\Phi Q}. \tag{99}\]

Now take \(\alpha\in HT^2(A)\). By (77), the class \(c_Q(\alpha)\) is the \(\mathbf I_A\)-image of \(\operatorname{HKR}_A(\alpha)\cap\mathfrak h_Q\). Equations (97) and (99) take its homological transport to \[\Phi^{HH}(\operatorname{HKR}_A(\alpha)) \cap\mathfrak h_{\Phi Q}.\] Apply \(\mathbf I_B\) and (77) on \(B\). The result is \(c_{\Phi Q}(\Phi^{HT}\alpha)\), as required. The three comparison maps in (88) are isomorphisms. Taking kernels proves (89). ◻

Proof. The fundamental-class formula is (13), proved there from the Jacobian lattice. Translation acts trivially on cohomology, and inversion acts by \(+1\) on \(H^4\), so it holds for every Abel curve in the stated sense.

For the inclusion \(i:D\hookrightarrow X\), use Grothendieck–Riemann–Roch in its smooth proper form \[\operatorname{ch}(i_*F)\operatorname{td}(X) =i_*\bigl(\operatorname{ch}(F)\operatorname{td}(D)\bigr)\] for a closed immersion of smooth projective varieties; this is the form of the Riemann–Roch theorem recorded by Borel and Serre [3]. The tangent bundle of \(X\) is trivial, and \(\operatorname{td}(D)=1+\tfrac12c_1(T_D)\). Since \(\deg T_D=2-2\cdot3=-4\) and \([\mathrm{pt}]_{\mathrm B}=\theta_{\mathrm B}^3/6\), it follows that \[\operatorname{ch}_{\mathrm B}(\mathcal O_D) =i_*\operatorname{td}(D) =[D]_{\mathrm B}-2[\mathrm{pt}]_{\mathrm B} =\frac{\theta_{\mathrm B}^2}{2} -\frac{\theta_{\mathrm B}^3}{3}.\] For disjoint \(Z\), one has \(\mathcal O_Z=\bigoplus_j\mathcal O_{D_j}\). Additivity in \(0\to I_Z\to\mathcal O_X\to\mathcal O_Z\to0\) gives (100).

It remains to calculate the normal sections. For the original Abel embedding the normal sequence is \[ 0\longrightarrow T_C\longrightarrow \mathcal O_C\otimes H^1(C,\mathcal O_C) \longrightarrow N_{C/X}\longrightarrow0. \tag{101}\] Here the middle bundle is \(T_X|_C\). The dual of the first arrow is the evaluation map \(H^0(C,K_C)\otimes\mathcal O_C\to K_C\): differentiating the Abel integral sends a tangent vector \(v\) to the functional \(\omega\mapsto\omega(v)\). Consequently, under Serre duality on \(C\), the dual of the map \[H^1(C,T_C)\longrightarrow H^1(C,\mathcal O_C)\otimes H^1(C,\mathcal O_C)\] in the long exact sequence of (101) is the multiplication map \[H^0(C,K_C)\otimes H^0(C,K_C)\longrightarrow H^0(C,K_C^2).\] This is the usual cohomological Serre duality [36], applied to the dual evaluation map.

For completeness, multiplication is onto here by an elementary genus-three argument. For an effective divisor \(D_2\) of length two, curve Riemann–Roch, in the form \(h^0(L)-h^0(K_C\otimes L^{-1})=\deg L+1-g\), gives \[h^0(K_C-D_2)=h^0(D_2)=1.\] The last equality uses nonhyperellipticity. Two independent sections of a degree-two effective line, after removing their common base divisor, would give a map to \(\mathbb P^1\) of degree at most two. Degree one would make \(C\) rational, and degree two would make it hyperelliptic. Thus \(K_C\) separates every length-two subscheme. It has three sections and degree four, so it embeds \(C\) as an integral plane quartic. Such a curve is not contained in a conic. Restriction of quadrics therefore makes \(\operatorname{Sym}^2H^0(K_C)\to H^0(K_C^2)\) injective. Both spaces have dimension six, the second by curve Riemann–Roch and \(\deg K_C^2=8>2g-2\). The multiplication map is consequently onto, and its dual is injective.

Since \(H^0(C,T_C)=0\), the long exact sequence of (101) now identifies \(H^0(C,N_{C/X})\) with the middle bundle’s constant sections \(H^1(C,\mathcal O_C)=H^0(X,T_X)\). This is exactly normal restriction of translation-invariant vector fields. Translation and inversion carry the normal sequence and this isomorphism to every Abel curve. ◻

Proof. At a point of \(Z\), write its regular codimension-two ideal in the regular local ring \(R=\mathcal O_{X,x}\) as \(I=(f,g)\), where \(f,g\) are part of a regular system of parameters. It has the resolution \[0\longrightarrow R\xrightarrow{(-g,f)}R^2 \xrightarrow{(f,g)}I\longrightarrow0.\] If an endomorphism sends \(f,g\) to \(u,v\), its relation is \(gu=fv\). Regularity of \(f,g\) gives \(u=fr\), \(v=gr\) for a unique \(r\in R\), so \(\operatorname{Hom}_R(I,I)=R\). Applying \(\operatorname{Hom}_R(-,I)\) to the resolution shows that the degree-one cokernel is \(I/I^2\). Under a change of generators, the dual of the leftmost generator acquires the inverse determinant. Consequently the global degree-one sheaf is \[(I_Z/I_Z^2)\otimes\det(I_Z/I_Z^2)^{-1} \simeq (I_Z/I_Z^2)^\vee=N_{Z/X}.\] The rank-two isomorphism sends \(v\otimes\ell^\vee\) to the functional \(w\mapsto\ell^\vee(v\wedge w)\), so it is compatible with changes of generators. We view \(N_{Z/X}\) here as a sheaf on \(X\) supported on \(Z\). Away from \(Z\) the ideal is \(\mathcal O_X\), and the resolution has length one. Thus, globally, \[ \mathcal{H}om_X(I_Z,I_Z)=\mathcal O_X,\qquad \mathcal{E}xt_X^1(I_Z,I_Z)=N_{Z/X},\qquad \mathcal{E}xt_X^q(I_Z,I_Z)=0\quad(q\geq2). \tag{103}\]

The standard local-to-global spectral sequence \[E_2^{p,q}=H^p\!\left(X,\mathcal{E}xt_X^q(I_Z,I_Z)\right) \Longrightarrow \operatorname{Ext}^{p+q}_X(I_Z,I_Z)\] is the hypercohomology spectral sequence of \(R\mathcal{H}om_X(I_Z,I_Z)\). Its five-term sequence and (103) give \[ 0\longrightarrow H^1(X,\mathcal O_X) \longrightarrow\operatorname{Ext}^1_X(I_Z,I_Z) \xrightarrow{\,e\,}H^0(Z,N_{Z/X}) \longrightarrow H^2(X,\mathcal O_X). \tag{104}\] The first arrow is also the map induced by the perfect unit \(\mathcal O_X\to R\mathcal{H}om(I_Z,I_Z)\): the perfect trace composed with this unit is multiplication by the rank, which is one.

Translate the \(n\) curves independently. The locus in \(X^n\) where these translated proper curves remain disjoint is an open neighborhood of the original tuple. Over it their union is a disjoint union of flat translated families. Its ideal is flat as well, by the exact sequence from the structure sheaf of the union. Restriction of this family to first-order tangent vectors gives a linear map \[s:\bigoplus_{j=1}^nH^0(X,T_X) \longrightarrow\operatorname{Ext}^1_X(I_Z,I_Z).\] Indeed a flat deformation over \(\mathbb C[\epsilon]/(\epsilon^2)\) gives the extension \(0\to I_Z\to I_{Z,\epsilon}\to I_Z\to0\). For a vector field \(\xi\), the moving generators are \(f-\epsilon\xi(f)\) and \(g-\epsilon\xi(g)\). Lifting the displayed relation \((-g,f)\) gives \[(-g)\bigl(f-\epsilon\xi(f)\bigr) +f\bigl(g-\epsilon\xi(g)\bigr) =\epsilon\bigl(g\xi(f)-f\xi(g)\bigr).\] Thus the local Ext cocycle is represented by \(\overline{g\xi(f)-f\xi(g)}\in I/I^2\), up to the single sign in the chosen Ext boundary convention. Choose \(\ell=\bar f\wedge\bar g\) and \(\ell^\vee(\ell)=1\). Under the written determinant identification \(v\otimes\ell^\vee\mapsto (w\mapsto\ell^\vee(v\wedge w))\), this displayed representative sends \(\bar f,\bar g\) to \(-\overline{\xi(f)},-\overline{\xi(g)}\), respectively. Reversing the boundary convention reverses both signs. Consequently the edge map \(e\) sends the geometric deformation to its normal restriction up to one fixed sign, independent of the vector field and the component. By Lemma 27, \(e\circ s\) is therefore the direct sum of the normal-restriction isomorphisms multiplied by that fixed sign. It follows that (104) reduces to the split short exact sequence \[0\longrightarrow H^1(X,\mathcal O_X) \longrightarrow\operatorname{Ext}^1_X(I_Z,I_Z) \xrightarrow{e}H^0(Z,N_{Z/X})\longrightarrow0,\] with splitting given by \(s\) after this signed normal-restriction identification.

This splitting is equivariant. Translation acts trivially on \(H^1(X,\mathcal O_X)\), as is seen from translation-invariant Dolbeault representatives, and it acts as the identity on \(H^0(X,T_X)\). It sends the deformation of \(D_j\) to the deformation of its translated component. Since a group of order \(n\) acts transitively on \(n\) components, that permutation action is regular. This proves (102); its invariant summands have dimensions three and three.

We also use Serre duality for perfect complexes on a smooth projective threefold: \[\operatorname{Ext}^i(F,G)^\vee \simeq\operatorname{Ext}^{3-i}(G,F\otimes\omega_X).\] This is the extension of the usual locally free Serre duality [36] by finite locally free resolutions. Choose a nonzero translation-invariant holomorphic three-form on \(X\). For \(F=G=I_Z\), the resulting perfect pairing is \[\operatorname{Ext}^1(I_Z,I_Z)\otimes \operatorname{Ext}^2(I_Z,I_Z) \longrightarrow H^3(X,\mathcal O_X)\simeq\mathbb C.\] It is \(\Gamma\)-invariant: translations preserve the chosen form, composition, and the perfect trace. Thus these two Ext groups are dual \(\Gamma\)-representations. Averaging over the finite group in characteristic zero identifies invariants of the dual with the dual of the invariants, so the invariant \(\operatorname{Ext}^2\) also has dimension six. Finally (103) gives \(\operatorname{Hom}(I_Z,I_Z)=H^0(X,\mathcal O_X)=\mathbb C\), with trivial action. ◻

Proof. The ideal sheaves are perfect by their local length-one resolutions. The perfect Künneth isomorphism for their derived global Hom complexes is \[\mathbf R\operatorname{Hom}_{X^2}(\mathcal I,\mathcal I) \simeq \mathbf R\operatorname{Hom}_X(I_\Sigma,I_\Sigma) \otimes_{\mathbb C}^{\mathbf L} \mathbf R\operatorname{Hom}_X(I_A,I_A).\] Taking degree two gives \[\begin{align*} \operatorname{Ext}^2_{X^2}(\mathcal I,\mathcal I) \simeq{}& \operatorname{Ext}^2_X(I_\Sigma,I_\Sigma) \otimes\operatorname{Hom}_X(I_A,I_A) \\ &\oplus\operatorname{Ext}^1_X(I_\Sigma,I_\Sigma) \otimes\operatorname{Ext}^1_X(I_A,I_A) \\ &\oplus\operatorname{Hom}_X(I_\Sigma,I_\Sigma) \otimes\operatorname{Ext}^2_X(I_A,I_A). \tag{105}\end{align*}\] Here the isomorphism follows by taking cohomology of the tensor product over \(\mathbb C\) of the two derived global Hom complexes; over the field \(\mathbb C\) there are no additional Tor terms. It is equivariant for the independent actions of \(G_1,G_2\). Averaging in each factor gives \((V_1\otimes V_2)^{G_1\times G_2} =V_1^{G_1}\otimes V_2^{G_2}\). Applying Lemma 28 to the three summands of (105) gives \(6\), \(6\cdot6\), and \(6\). Because (105) is a decomposition of derived Ext itself, the count retains every local-to-global contribution. ◻

Proof. An element of \(HT^1(X)=\mathsf U\oplus\mathsf T\) is a pair \((\eta,\xi)\). Using (107), its projections under \(c_{I_Z}^1\) to \(H^1(\mathcal O_X)\) and \(H^2(\Omega_X^1)\) are respectively \[ \eta,\qquad -n\,j(\xi)\wedge\theta_{\mathrm H}. \tag{108}\] The first uses the rank term of the character; the second uses \(-n\theta_{\mathrm H}^2/2\). There is no contribution from \(\xi\) to the first projection or from \(\eta\) to the second, because the codimension-one character is zero. The signs in (108) are those of the specified left-interior convention.

The map \(v\mapsto v\wedge\theta_{\mathrm H}\) on \(\mathsf U\) is injective. If \(v\ne0\), choose bases with \(v=u_1\) and \(\theta_{\mathrm H}=\sum_{i=1}^3u_i\otimes w_i\), absorbing the fixed contraction signs in the dual basis. The terms of \(v\wedge\theta_{\mathrm H}\) with \(i=2,3\) are linearly independent nonzero tensors. Therefore the two projections in (108) determine \(\eta\) and \(\xi\), proving the first assertion.

The only summands of \(H\Omega_{-2}(X)\) are \[ H^2(X,\mathcal O_X)=\wedge^2\mathsf U,\qquad H^3(X,\Omega_X^1)=\wedge^3\mathsf U\otimes\mathsf W. \tag{109}\] Each has dimension three. The \(H^2(\mathcal O_X)=\wedge^2\mathsf U\) block of \(HT^2(X)\) maps identically to the first summand, using the rank term \(f_{n,0}=1\). The \(H^1(T_X)=\mathsf U\otimes\mathsf T\) block has no contribution there, since \(f_{n,1}=0\), and its projection to the second summand is \[ \eta\otimes\xi\longmapsto -n\,\eta\wedge j(\xi)\wedge\theta_{\mathrm H}. \tag{110}\] For each \(k\in\{1,2,3\}\), take \(\eta\) and \(j(\xi)\) to be the two basis elements \(u_i,u_j\) complementary to \(u_k\). Their wedge with \(\theta_{\mathrm H}\) is a nonzero multiple of \((u_1\wedge u_2\wedge u_3)\otimes w_k\). Hence (110) spans the second summand. These two source blocks already give rank six; the target in (109) has dimension six. This proves surjectivity. The bivector block \(H^0(\wedge^2T_X)\) is not needed for this lower bound. ◻

Proof. Künneth gives the domain decomposition \[HT^2(X^2)= \bigl(HT^2(X)\otimes HT^0(X)\bigr) \oplus\bigl(HT^1(X)\otimes HT^1(X)\bigr) \oplus\bigl(HT^0(X)\otimes HT^2(X)\bigr).\] The product character is \(f_n\boxtimes f_n\), and \(HT^0(X)=\mathbb C\) acts by \(c_{I_Z}^0(1)=f_n\ne0\). The written left-action and product formulas (64) and (65), applied to the Künneth decomposition, give the external-product contraction rule with its total-complex signs. They therefore send the three blocks onto \[\operatorname{im}(c_{I_\Sigma}^2)\otimes\mathbb C f_n,\qquad \operatorname{im}(c_{I_\Sigma}^1)\otimes \operatorname{im}(c_{I_A}^1),\qquad \mathbb C f_n\otimes\operatorname{im}(c_{I_A}^2).\] Their dimensions are \(6,36,6\), by Lemma 30. The target has the direct Künneth decomposition \[H\Omega_{-2}(X^2) =\bigoplus_{r+s=2}H\Omega_{-r}(X)\otimes H\Omega_{-s}(X).\] The first image lies in the factorwise Hodge-imbalance summand \((2,0)\), the second in \((1,1)\), and the third in \((0,2)\). Here the imbalance of \(H^q(X,\Omega_X^p)\) is \(q-p\); the two-factor Hodge Künneth decomposition is a direct sum for these ordered pairs. Hence the three images cannot cancel, regardless of their Künneth signs. Their ranks add to \(6+36+6=48\). ◻

Proof. Proposition 25 proves for every perfect \(Q\) the exact identity \[\sigma_Q\circ o_Q=c_Q^2\] with the definitions just given. In particular it is an identity for the perfect trace on derived Ext, with the same contraction convention as the rank computation. The components with \(q>\dim B-2\) vanish because \(H^{q+2}(B,\Omega_B^q)=0\).

Translations act trivially on \(HT^2(X^2)\): they fix the invariant tangent fields and the invariant Dolbeault representatives of \(H^a(\mathcal O_{X^2})\). The translation-invariant diagonal-kernel HKR construction identifies this with their trivial action on \(HH^2(X^2)\). Apply the diagonal-kernel evaluation square (87) to each translation equivalence, and identify its translate of \(\mathcal I\) with \(\mathcal I\) by the canonical linearization. Naturality of evaluation then gives \(g\cdot o_{\mathcal I}(\alpha)=o_{\mathcal I}(g\cdot\alpha) =o_{\mathcal I}(\alpha)\) for \(g\in G_{\mathrm{src}}\). Hence \[\operatorname{im}(o_{\mathcal I}) \subset W_{\mathrm{inv}} :=\operatorname{Ext}^2_{X^2}(\mathcal I,\mathcal I)^{G_{\mathrm{src}}}.\] Proposition 29 gives \(\dim W_{\mathrm{inv}}=48\), while Lemma 31 and the exact identity give \[48=\operatorname{rank}(c_{\mathcal I}^2) \leq\operatorname{rank}(o_{\mathcal I}) \leq\dim W_{\mathrm{inv}}=48.\] Thus the inclusion is equality and both maps have rank \(48\). The identity also gives \(\ker(o_{\mathcal I})\subset\ker(c_{\mathcal I}^2)\); equality of ranks on their common domain makes the kernels equal. If \(w=o_{\mathcal I}(\alpha)\in W_{\mathrm{inv}}\) and \(\sigma_{\mathcal I}(w)=0\), then \(c_{\mathcal I}^2(\alpha)=0\), so the kernel equality gives \(w=0\). This proves (111) and source injectivity.

Apply Proposition 26 to the fixed enhanced equivalence \(Q\mapsto\widetilde\Phi(Q)[-3]\). It includes tensoring by \(H\boxtimes H\), the fixed Fourier–Mukai kernel, and the displayed shift. That proposition gives isomorphisms \[u:HT^2(X^2)\xrightarrow{\sim}HT^2(M),\quad v:\operatorname{Ext}^2(\mathcal I,\mathcal I) \xrightarrow{\sim}\operatorname{Ext}^2(\mathcal G,\mathcal G), \quad \gamma:H\Omega_{-2}(X^2)\xrightarrow{\sim}H\Omega_{-2}(M)\] such that \[ v\,o_{\mathcal I}=o_{\mathcal G}\,u,\qquad \gamma\,c_{\mathcal I}^2=c_{\mathcal G}^2\,u. \tag{113}\] The map \(\gamma\) is the HKR expression of the enhanced Hochschild-homology isomorphism: its cap and character naturality, including the shift sign, is exactly what gives the second equality. No identification of \(\gamma\) with a separately normalized Betti integral transform is used.

Proposition 24 constructs the target action by coherent conjugation of the canonical source action by this same equivalence. Hence \(v\) is an isomorphism of the corresponding representations. The lines in the target line-translation action cancel on derived endomorphisms, and the underlying group is the specified \(\bar G\simeq G_{\mathrm{src}}\). Therefore \[v(W_{\mathrm{inv}}) =\operatorname{Ext}^2_M(\mathcal G,\mathcal G)^{\bar G}.\] The first equality of (113) transports the source image equality to the image equality in (112). Both \(u\) and \(\gamma\) are isomorphisms, so its second equality transports the source kernel equality to the target kernel equality. Finally apply \(\sigma_{\mathcal G}o_{\mathcal G}=c_{\mathcal G}^2\), the target instance of Proposition 25. An invariant \(z\) has the form \(o_{\mathcal G}(\alpha)\); if \(\sigma_{\mathcal G}(z)=0\), the target kernel equality gives \(z=0\). This proves injectivity on the full target invariant Ext group. ◻

Proof. Perfect duality is an anti-equivalence and turns the given line-twisted action into its inverse-line action. The quasi-isomorphism from \(D(\mathcal G)[-1]\) to the degree-zero sheaf \(E\) is canonical in the derived category and is equivariant. Transpose therefore gives the asserted invariant isomorphism in even degree.

The sign can be computed in a finite complex of free modules. The dual Atiyah class is the negative transpose of the original class. Transposing a product reverses both its Ext-degree-one factors and its one-form factors. The two reversal Koszul signs cancel: equivalently, an \(\Omega^1[1]\)-valued Atiyah operation has even combined parity. The Ext-degree-two operation \(v\) is also even, so its movement past these factors gives no further sign. Supertrace is invariant under transpose, while the shift \([-1]\) changes supertrace by \(-1\). The \(j\) negative dual Atiyah factors and this last sign give (135). Evaluation, transpose, and the Atiyah construction respect changes of finite free model, so the local calculation glues. The diagonal operator with entries \((-1)^{j+1}\) is invertible, which proves the final assertion. ◻

Proof. We construct the twisted sheaf before comparing its derived endomorphisms. Choose an evenly covered good analytic cover of \(Y_0\), refined so that the finitely many transition lines are trivial on the required sheet intersections. Take every sheet over every cover member, so the sheets \(U_i\) cover \(M\), and retain its image \(V_i\) as a labelled member of the cover. Transport \(E|_{U_i}\) to a sheaf \(B_i\) on \(V_i\). On \(V_i\cap V_j\) for \(i<j\), the deck transformation comparing the two sheets and a local trivialization of its line \(L_h\) give an isomorphism \(\rho_{ij}:B_j\to B_i\). Use the unit and inverse composition data to impose \(\rho_{ii}=\operatorname{id}\) and \(\rho_{ji}=\rho_{ij}^{-1}\). On a triple intersection, the coherent composition law cancels the translations; the remaining line trivializations differ by an invertible scalar function. Thus \(\rho_{ij}\rho_{jk}\rho_{ki}=c_{ijk}\operatorname{id}\).

We use the determinant functor for perfect complexes in its elementary local form: a finite free complex has the alternating tensor product of the determinants of its terms. It commutes with restriction and isomorphism, and the determinant of \(c\operatorname{id}\) on a rank-\(r\) perfect complex is \(c^r\). Each \(B_i\) is perfect, so \(\det B_i\) is a line. Trivialize these lines and let \(\eta_{ij}\) be the determinant of \(\rho_{ij}\); then \(\eta_{ii}=1\) and \(\eta_{ji}=\eta_{ij}^{-1}\). After a further good-cover refinement, choose an \(r\)-th root \(\widetilde\eta_{ij}\) of this nowhere-zero holomorphic function for \(i<j\), and impose \(\widetilde\eta_{ii}=1\) and \(\widetilde\eta_{ji}=\widetilde\eta_{ij}^{-1}\). Set \(\rho'_{ij}=\widetilde\eta_{ij}^{-1}\rho_{ij}\). These maps have determinant one and satisfy \(\rho'_{ii}=\operatorname{id}\) and \(\rho'_{ji}=(\rho'_{ij})^{-1}\). The coherent composition law gives an invertible holomorphic scalar \(\epsilon_{ijk}\) defined by \[\rho'_{ij}\rho'_{jk}=\epsilon_{ijk}\rho'_{ik}.\] Taking determinants gives \(\epsilon_{ijk}^r=1\), so these scalars are locally constant with values in \(\mu_r\). Associativity on a fourfold intersection gives the exact Cech cocycle identity \[\epsilon_{ijk}\epsilon_{ikl}=\epsilon_{jkl}\epsilon_{ijl}.\] The repeated-index multipliers are one; scalar multiplication is faithful because \(B_i\) has positive rank on a smooth manifold. The resulting \(\mu_r\)-cocycle defines \(\alpha\), and the sheaves \(B_i\) with transitions \(\rho'_{ij}\) define \(\mathcal B\). Their determinant trivializations glue, so \(\det\mathcal B\simeq\mathcal O_{Y_0}\). Rank, coherence, and perfection are local, and hence pass from \(E\) to \(\mathcal B\). This is the determinant-normalized form of the local construction in [25]; it uses the perfect determinant, not an exterior power of a locally free sheaf.

Let \(\theta_i:(q^*B_i)|_{U_i}\xrightarrow{\sim}E|_{U_i}\) be the local identification from transport. On an actual overlap \(U_i\cap U_j\), the comparing deck transformation is the identity, so the coherent line action gives an invertible scalar \(l_{ij}\) with \[\theta_i(q^*\rho'_{ij})\theta_j^{-1}=l_{ij}\operatorname{id}_E.\] The normalized transitions give \(l_{ii}=1\), \(l_{ji}=l_{ij}^{-1}\), and \(l_{ij}l_{jk}=(q^*\epsilon_{ijk})l_{ik}\). Use \(l_{ij}\) as the transition from the \(j\)-th trivial line to the \(i\)-th. These lines define the weight-one \(q^*\alpha\)-twisted line \(\mathcal L\) with the same multiplier as \(q^*\mathcal B\), and the local identifications give (136). On translated intersections, conjugation by these identifications is the original conjugation action on the derived endomorphisms of \(E\).

For clarity, a twisted sheaf here is a coherent weight-one sheaf on \(\mathfrak Y_0\), equivalently the just described Cech data. Weight-zero sheaves and complexes descend uniquely to \(Y_0\). A coherent twisted sheaf on the smooth manifold is absolutely perfect: in a local trivialization of the gerbe it is a finite module over a regular analytic local ring and has a finite free resolution. Write \[\mathcal C_{\mathcal B} =R\mathcal{H}om_{\mathfrak Y_0}(\mathcal B,\mathcal B)\] for its weight-zero descent. It is a perfect complex on \(Y_0\), locally \(\mathcal B^\vee\otimes^L\mathcal B\).

We use the following derived base-change identity. For an absolutely perfect \(P\), any complex \(Q'\), and any morphism \(f:T\to V\), \[ Lf^*R\mathcal{H}om_V(P,Q') \simeq R\mathcal{H}om_T(Lf^*P,Lf^*Q'). \tag{139}\] Indeed, locally the left side is \(Lf^*(P^\vee\otimes^L Q')\); derived pullback is symmetric monoidal and carries the dual of a perfect complex to the dual of its pullback. This also proves the identity on gerbe trivializations. The unit and trace, being coevaluation and evaluation, commute with the same derived pullbacks. This is an assertion about derived pullback; ordinary pullback under a nonflat map is not being substituted.

Apply (139) to the etale map \(q\), where ordinary and derived pullback agree. Line cancellation in (136) gives \[ q^*\mathcal C_{\mathcal B} \simeq R\mathcal{H}om(q^*\mathcal B,q^*\mathcal B) \simeq R\mathcal{H}om_M(E,E). \tag{140}\] Both the line factors and their scalar gluing cancel. The descent datum on the right is consequently the honest conjugation action of \(\bar G\), although the action on \(E\) itself is line-twisted.

For a finite covering, \(q_*\) on sheaves is exact: over an evenly covered open set it is the direct sum of the finitely many sheet restrictions. Etale descent gives \(\mathcal C_{\mathcal B}\simeq(q_*q^*\mathcal C_{\mathcal B})^{\bar G}\). Over \(\mathbb C\), invariants are the split summand defined by \(|\bar G|^{-1}\sum_{h\in\bar G}h\). Applying derived global sections to that summand and using exactness of \(q_*\) gives \[R\Gamma(Y_0,\mathcal C_{\mathcal B}) \simeq R\Gamma(M,q^*\mathcal C_{\mathcal B})^{\bar G}.\] Taking cohomology and using (140) proves the full Ext comparison in (137).

The perfect unit and trace are equivariant maps \[ \mathcal O_{Y_0}\xrightarrow{u}\mathcal C_{\mathcal B} \xrightarrow{\operatorname{tr}}\mathcal O_{Y_0}, \qquad \operatorname{tr}u=r\operatorname{id}. \tag{141}\] The last equality is the alternating-rank calculation in a finite free model. Thus \(u/r\) splits the trace. If \(\mathcal C_{\mathcal B,0}=\operatorname{fib}(\operatorname{tr})\), then \[\mathcal C_{\mathcal B}\simeq \mathcal O_{Y_0}\oplus\mathcal C_{\mathcal B,0},\qquad H^iR\Gamma(Y_0,\mathcal C_{\mathcal B,0}) =\ker\bigl(\operatorname{Ext}^i(\mathcal B,\mathcal B) \to H^i(\mathcal O_{Y_0})\bigr).\] The splitting commutes with (140), since derived pullback and tensoring by an invertible line preserve evaluation and coevaluation. Descent of this summand proves the trace-free comparison. In particular, the calculation retains the higher sheaf-Ext terms: it does not replace \(H^2R\Gamma(\mathcal C_{\mathcal B,0})\) by \(H^2(\mathcal End_0(\mathcal B))\).

It remains to compare the semiregularity maps. The cotangent target on a \(\mu_r\)-gerbe is that of its underlying manifold. Equivalently, the scalar triple-gluing functions are locally constant, so their differentials vanish in first jets. The pullback and tensor rules for Atiyah classes, with the differential \(q^*\Omega_{Y_0}^1\simeq\Omega_M^1\), give \[ q^*a_{\mathcal B}=a_E+a_{\mathcal L}\operatorname{id}_E, \qquad a_{\mathcal L}=-\frac{a_{\det E}}r=-\lambda_H. \tag{142}\] The second equality follows by tracing the first and using \(\det\mathcal B=\mathcal O_{Y_0}\); the trace of \(a_E\) is the Atiyah class of its perfect determinant. The pullback compatibility is also [4]; the tensor rule follows from the Leibniz rule for first jets on a local perfect model. The scalar \(-\lambda_H\) commutes with the Atiyah operations in the combined Ext/form grading. Expanding the exponential and applying the natural perfect trace yields \[q^*\sigma_{\mathcal B}(v) =e^{-\lambda_H}\sigma_E(\phi v),\] which is (138). Multiplication by \(\lambda_H\in H^1(\Omega_M^1)\) raises both Hodge indices and is nilpotent on \(\mathcal H_{-2}(M)\); \(e^{-\lambda_H}\) therefore has inverse \(e^{\lambda_H}\). Full invariant injectivity on the right implies full injectivity on the left. The trace-free comparison also shows that injectivity restricted to the invariant trace-free group would suffice: the \(j=0\) component of \(\sigma_{\mathcal B}\) is the perfect trace itself, so its vanishing first puts a class into the trace-free summand. ◻

Proof. We first realize the exact finite-band class \(\alpha\). Set \(\Lambda=H_1(Y_0,\mathbb Z)\simeq\mathbb Z^{12}\). The universal coefficient theorem for the torus gives \[H^2(Y_0,\mu_r)=\operatorname{Hom}(\mathop{\bigwedge}\nolimits^2\Lambda,\mu_r);\] there is no Ext term because \(H_1(Y_0,\mathbb Z)\) is free. Let \(b\) be the alternating pairing corresponding to \(\alpha\), and put \(G_r=\Lambda/r\Lambda\). For an ordered basis \(e_1,\ldots,e_{12}\) of \(\Lambda\), define \[c(x,y)=\prod_{i<j}b(e_i,e_j)^{x_i y_j}\quad(x,y\in G_r).\] This is a well-defined bimultiplicative cocycle with \(c(x,y)c(y,x)^{-1}=b(x,y)\). On the \(m=|G_r|=r^{12}\) dimensional vector space with basis \(\{v_y:y\in G_r\}\), set \[T_xv_y=c(x,y)v_{x+y}.\] Then \(T_xT_y=c(x,y)T_{x+y}\), so \(x\mapsto[T_x]\) is a projective representation. For \(r=1\) this is the trivial one-dimensional representation.

The multiplication map \([r]:Y\to Y\) is a finite etale torsor under \(Y[r]\). Its differential on the relative tangent bundle is multiplication by \(r\); it is proper, and every abelian fiber has \(r^{12}\) inverse images of a point. Thus it is a proper local biholomorphism of that degree. Contractibility of \(S\) trivializes the finite local system \(Y[r]\) as \(G_r\), with its central identification. Form \(Z\) as the quotient of \(Y\times\mathbb P^{m-1}\) by the free diagonal action covering translation by \(G_r\), taking the inverse action on the projective coordinate. If local sections of the torsor satisfy \(s_j=s_i+x_{ij}\), this convention gives \([s_j,z]=[s_i,[T_{x_{ij}}]z]\). Local sections of the finite covering show that \(p:Z\to Y\) is a holomorphic projective-space bundle with these projective transitions.

We must check the class in \(H^2(Y_0,\mu_r)\), not only its image in the analytic Brauer group. Rescale each \(T_x\) to have determinant one. The multiplier of these special-linear lifts is \(\mu_m\)-valued: it is scalar with determinant one. It differs from \(c\) by a scalar coboundary and still has commutator \(b\). The monodromy of the central \([r]\)-torsor is \(\Lambda\to\Lambda/r\Lambda\). For any constant coefficient group \(C'\), \[H^2(Y_0,C')=\operatorname{Hom}(\mathop{\bigwedge}\nolimits^2\Lambda,C').\] In the bar complex the cycle \([e_i|e_j]-[e_j|e_i]\) represents \(e_i\wedge e_j\), and an inflated multiplier evaluates on it as its alternating commutator. Thus the special-linear obstruction class has pairing \(b\). The inclusion \(\mu_m\hookrightarrow\mathbb Q/\mathbb Z\) induces an injection on this \(H^2\), so that class is the image of \(\alpha\) under \(\mu_r\hookrightarrow\mu_m\). On a common good cover, its \(\mu_m\)-cocycle therefore differs from a chosen \(\mu_r\)-cocycle for \(\alpha\) by the coboundary of a \(\mu_m\)-valued one-cochain. Multiplying the special-linear lifts by that cochain gives transitions with exactly the chosen \(\mu_r\)-multiplier and still determinant one. They define \(Q_0\), and their projectivizations give \(Z_0\).

Use the convention that \(\mathbb P(Q_0)\) parametrizes lines in \(Q_0\). The tautological \(\ell_0=\mathcal O(1)\) then has the inverse twist. The twisted factors cancel in the Euler sequence \[0\longrightarrow\mathcal O_{Z_0} \longrightarrow p_0^*Q_0\otimes\ell_0 \longrightarrow T_{Z_0/Y_0}\longrightarrow0.\] Since \(\det Q_0=\mathcal O_{Y_0}\), its determinant is \(H|_{Z_0}=\ell_0^m\). On any projective-space fiber the same anticanonical line is \(\mathcal O(m)\).

On each \(s\), the quotient construction is the algebraic quotient of the projective variety \(Y_s\times\mathbb P^{m-1}\) by a free finite algebraic action. It is projective: tensor the translates of an ample line on the product, use the permutation linearization on that tensor product, descend it along the free finite quotient, and use descent of ampleness under a finite surjection. The map \(p_s\) is algebraic and projective, for instance by its closed graph in a projective product. The global line \(H\) is \(p\)-ample. The composition rule for relative ampleness gives positive \(u,v\) for which \(H^u\otimes p^*\mathcal L_Y^v\) is ample for \(a\circ p\). The class \(\zeta_{B,s}\) is the restriction of the global class \(c_1^B(H)/m\). Restrictions of a total class along a proper smooth submersion are horizontal, and a holomorphic line gives type \((1,1)\). Its fiber degree is one by the calculation of \(H\). The raw equality in (145) is the degree-one comparison (134); centrally it gives \(a_{\ell_0}=\zeta_{H,0}\).

We now verify ordinary semiregularity. The twists of \(p_0^*\mathcal B\) and \(\ell_0\) cancel, so \(F_0\) is ordinary coherent. It need not be locally free. Since \(p_0\) is flat, the derived identity (139) and line cancellation give \[R\mathcal{H}om_{Z_0}(F_0,F_0)\simeq p_0^*\mathcal C_{\mathcal B}.\] Moreover \(Rp_{0*}\mathcal O_{Z_0}=\mathcal O_{Y_0}\). To check this equality, pull back along the finite etale cover \([r]:Y_0\to Y_0\); the bundle becomes \(Y_0\times\mathbb P^{m-1}\), where the assertion is the vanishing of the higher cohomology of \(\mathcal O_{\mathbb P^{m-1}}\). Flat base change for this fixed projective fiber and faithful flatness give the equality on \(Y_0\). The perfect projection formula now yields \[\begin{split} \operatorname{Ext}^2_{Z_0}(F_0,F_0) &\simeq H^2R\Gamma(Z_0,p_0^*\mathcal C_{\mathcal B})\\ &\simeq H^2R\Gamma(Y_0,\mathcal C_{\mathcal B}) =\operatorname{Ext}^2_{\mathfrak Y_0}(\mathcal B,\mathcal B). \end{split}\] This is (146); it uses the full derived Hom complex. Pullback and line tensoring preserve evaluation and coevaluation, so this identification commutes with the unit and perfect trace under \(p_0^*:H^2(Y_0,\mathcal O_{Y_0})\xrightarrow{\sim} H^2(Z_0,\mathcal O_{Z_0})\) and identifies their trace-free kernels. The tensor rule and \(a_{\ell_0}=\zeta_{H,0}\) give (147).

The raw Hodge pullback in that equation is injective. Indeed, for \(\eta_H\in H^{j+2}(Y_0,\Omega_{Y_0}^j)\), put \(\eta_B=(2\pi i)^{-j}\iota_{Y_0}(\eta_H)\), a complex Betti class. Comparison commutes with pullback. With \(f=m-1\), normalized Betti integration gives \[p_{0*}^B\bigl(\zeta_{B,0}^{\,f}p_0^*\eta_B\bigr)=\eta_B, \qquad \int_{\mathbb P^f}c_1^B(\mathcal O(1))^f=1.\] Thus \(p_0^*\eta_H=0\) implies \(\eta_B=0\), and then \(\eta_H=0\). This avoids any choice of normalization for Hodge Gysin. The exponential \(e^{\zeta_{H,0}}\) is invertible on the finite direct sum \(\mathcal H_{-2}(Z_0)\). Equation (147) and semiregularity of \(\mathcal B\) prove full semiregularity of \(F_0\).

Finally, pullback is a morphism of cohomology local systems and \(\zeta_B\) is horizontal, so (148) is horizontal. Each degree-\(2k\) part is rational of type \((k,k)\). Its flat lift \((2\pi i)^kA_{B,k}\) lies in \(F^k\), and its edge in \(R^k(a\circ p)_*\Omega^k_{Z/S}\) is the raw Hodge class to be used in deformation theory. Multiplicativity of the comparison and the one Tate factor contributed by each \(\zeta_H\) give, centrally, \[ \begin{split} A_H(0)&=p_0^*\operatorname{ch}_H(\mathcal B)e^{\zeta_{H,0}} =\operatorname{ch}_H(F_0),\\ A_B(0)&=p_0^*\operatorname{ch}_B(\mathcal B)e^{\zeta_{B,0}} =\operatorname{ch}_B(F_0). \end{split} \tag{150}\] Here \(A_{H,k}\) denotes the edge of \((2\pi i)^kA_{B,k}\), and \(A_H=\sum_kA_{H,k}\). The normalized projection formula and the hyperplane degree give \[\begin{split} p_{s*}^B\bigl(\zeta_{B,s}^{\,m-1}e^{-\zeta_{B,s}}A_B(s)\bigr) &=p_{s*}^B\bigl(\zeta_{B,s}^{\,m-1}p_s^*\kappa_B(s)\bigr)\\ &=\kappa_B(s)\,p_{s*}^B(\zeta_{B,s}^{\,m-1}) =\kappa_B(s). \end{split}\] All exponentials are finite polynomials in cohomology. This proves (149) and the proposition. ◻

Proof. At \(x\in X_0\), put \(B=\mathcal O_{X,x}\) and \(M_E=E_x\). Both are \(A\)-flat, while \(B_0=B\otimes_A\mathbb C\) is regular local of dimension \(n\). Choose a minimal finite-rank free \(B\)-resolution of \(M_E\), initially allowed to be infinite. Tensoring it over \(A\) with \(\mathbb C\) remains exact because its terms and \(M_E\) are \(A\)-flat. It is still minimal over \(B_0\), since maximal-ideal entries remain in the maximal ideal after reduction. The regular-local projective-dimension theorem gives projective dimension at most \(n\) over \(B_0\), so this minimal reduced resolution has no terms in higher degree. Its ranks are the ranks before reduction, and the original resolution therefore stops as well. This proves absolute \(\mathcal O_X\)-perfectness. It is separate from relative Tor-amplitude \([0,0]\), which is precisely the assumed \(A\)-flatness; the nonreduced total space has not been declared regular. For the general fiberwise perfectness criterion, see [4].

Let \(i:X_0\hookrightarrow X\). Flatness of \(\mathcal O_X\) and \(E\) over \(A\), together with derived associativity, gives \[ Li^*E\simeq E\otimes_A^L\mathbb C=F_0,\qquad E\otimes_{\mathcal O_X}^L W_X \simeq E\otimes_A^L W =E\otimes_AW=i_*(F_0\otimes_\mathbb CW). \tag{154}\] In particular the last derived tensor is an ordinary sheaf. We do not use the generally false identity \(Li^*W_X=\mathcal O_{X_0}\otimes_\mathbb CW\). Perfect duality and the projection formula for the closed immersion give \[R\mathcal{H}om_X(E,E\otimes_AW) \simeq E^\vee\otimes^L i_*(F_0\otimes W) \simeq i_*(F_0^\vee\otimes^L F_0\otimes W).\] Derived global sections prove the first identity in (153). The coefficient sheaf \(\Omega^j_{X/A}\otimes_AW\) is the closed pushforward of \(\Omega_{X_0}^j\otimes_\mathbb CW\), proving the second. These are finite direct-sum identities, not higher-direct-image base change for an endomorphism complex.

The relative first-jet Atiyah class pulls back to that of \(F_0\), and evaluation \(E^\vee\otimes E\to\mathcal O_X\) pulls back to central evaluation. Apply these identities to the displayed perfect-Hom formula. Each coefficient semiregularity component becomes the corresponding central component tensored with \(W\). This proves the asserted equality and injectivity. ◻

Proof. We first record the precise coefficient exactness needed for (155). The relative ample line algebraizes every \(X_\nu\), and every coherent sheaf on it, over the finite complex algebra \(A_\nu\). We use projective analytic GAGA in the form of an exact equivalence on coherent sheaves and comparison on their cohomology; local analytification is faithfully flat [37]. The algebraic model is projective and smooth over \(A_\nu\): its analytification is locally a smooth product, and smoothness for these finite-type models is reflected by analytification. Deligne’s proper-smooth theorem [12] applies in characteristic zero, including to nonreduced Artinian bases. It states that \(H^q(X_\nu,\Omega^j_{X_\nu/A_\nu})\) is finite locally free, commutes with arbitrary algebra base change, and that the Hodge-to-de-Rham spectral sequence degenerates.

Here is the finite-module consequence. For a finite \(A\)-module \(W\), apply algebra base change to the split square-zero algebra \(A\oplus W\). On the same underlying space the relative differential sheaf is \(\Omega^j_{X/A}\oplus(\Omega^j_{X/A}\otimes_AW)\). The base-change isomorphism respects the inclusion and augmentation of \(A\oplus W\). Taking the \(W\)-summand gives, naturally in \(W\), \[ H^q(X,\Omega^j_{X/A}\otimes_AW) \simeq H^q(X,\Omega^j_{X/A})\otimes_AW. \tag{156}\] The latter Hodge module is free, so this functor of \(W\) is exact. In particular \(d:J\hookrightarrow K\) stays injective after (156). GAGA transfers the same assertion to analytic sheaves.

For the remaining derived calculation, algebraize \(X'\), reduce it to obtain \(X\), and algebraize \(E\); use the same symbols for this projective algebraic pair. Faithfully flat local analytification reflects \(A\)-flatness, and the proof of Lemma 43 gives absolute perfectness on the algebraic model too. The derived foundations used here are the usual tensor and adjunction formalism for modules, cotangent transitivity, the first conormal homology \(H^{-1}(L_{X/X'})=J_X/J_X^2\), and the functorial derived principal-parts construction of the Atiyah class. For a perfect complex the trace is evaluation; its cyclicity and its compatibility with pullback and coefficient maps follow in a finite free model. We also use the characteristic-zero shifted exterior convention, for a cotangent complex \(L\), \(L\operatorname{Sym}^k(L[1])\simeq (L\!\bigwedge^k L)[k]\). These are the standard algebraic cotangent and perfect-duality constructions; no obstruction theorem for a general perfect complex is imported.

Only ordinary relative objects have to be compared back to analysis. For the smooth morphism \(X/A\), its cotangent complex is \(\Omega^1_{X/A}[0]\) and its derived self-product is underived by flatness. The first-principal-parts sequence is split for the input \(\mathcal O_X\)-action, so tensoring it with any coherent \(E\) is exact. Thus the relative derived Atiyah triangle is the ordinary first-jet extension. Ordinary differentials, conormal sequences, and first jets commute with analytification, as follows from finite presentations; perfect trace does too by local free evaluation. For \(Q=E\otimes_AJ\), GAGA for the bounded coherent complex \(E^\vee\otimes^LQ\) compares \(\operatorname{Ext}^2_X(E,Q)\) with the analytic group, and ordinary GAGA compares the Hodge targets. Hence the coefficient map of Lemma 43 is the same one on the model. We do not compare an absolute nonreduced analytic cotangent complex or an \(\operatorname{Ext}^1\) group on \(X'\); the extension below will be constructed algebraically and then analytified.

Define the cotangent extension class \[\delta_X:L_{X/\mathbb C}\longrightarrow L_{X/X'} \longrightarrow H^{-1}(L_{X/X'})[1]=J_X[1].\] The whole complex \(L_{X/X'}\) need not be \(J_X[1]\). Smoothness of \(X'/A'\), restricted to \(X\), gives the exact sequence \[ 0\longrightarrow\mathcal O_X\otimes_AK \longrightarrow\Omega^1_{X'/\mathbb C}\otimes\mathcal O_X \longrightarrow\Omega^1_{X/A}\longrightarrow0. \tag{157}\] The conormal map \(J_X\to\Omega^1_{X'/\mathbb C}\otimes\mathcal O_X\) factors through its first arrow by \(1\otimes d\). It is injective by (155) and \(A\)-flatness. Consequently the ordinary conormal sequence ending in \(\Omega^1_{X/\mathbb C}\) is short exact as well.

Map the transitivity triangle \[Lj^*L_{X'/\mathbb C}\longrightarrow L_{X/\mathbb C}\longrightarrow L_{X/X'}\] to this conormal triangle by the degree-zero truncations. The induced cone map \(L_{X/X'}\to J_X[1]\) is the canonical truncation: on \(H^{-1}=J_X/J_X^2\) it is the identity, and, since \(L_{X/X'}\in D^{\leq-1}\), truncation adjunction determines a map to \(J_X[1]\) by that map on \(H^{-1}\). Now map the conormal sequence to (157), using \(1\otimes d\) on the left and the identity in the middle. Naturality of boundaries and exterior comultiplication give the derived square \[ \begin{array}{ccc} L\!\bigwedge^k L_{X/\mathbb C}&\xrightarrow{\ \rho_k\ }&\Omega^k_{X/A}\\ \big\downarrow{\scriptstyle\delta_{X,k}}&& \big\downarrow{\scriptstyle\nabla'_k}\\ (\Omega^{k-1}_{X/A}\otimes_AJ)[1]& \xrightarrow{\ 1\otimes d\ }& (\Omega^{k-1}_{X/A}\otimes_AK)[1]. \end{array} \tag{158}\] Here \(\rho_k\) projects each factor to relative forms, \(\delta_{X,k}\) contracts one factor with \(\delta_X\) and projects the others, and \(\nabla'_k\) is the exterior boundary of (157). The exterior signs are those of the shifted symmetric convention just specified. This argument does not require a regular immersion.

We next prove the required sheaf obstruction statement. By (154), \[Q=E\otimes_AJ \simeq E\otimes_{\mathcal O_X}^LJ_X\] is an ordinary coherent sheaf. Put \(\mathcal K=Lj^*j_*E\). The Tor sequence for \(0\to J_X\to\mathcal O_{X'}\to\mathcal O_X\to0\) gives \(H^0(\mathcal K)=E\) and \(H^{-1}(\mathcal K)=Q\); lower cohomology is allowed. Its truncation triangle is \[ Q[1]\longrightarrow\tau_{\geq-1}\mathcal K \longrightarrow E\xrightarrow{b_E}Q[2]. \tag{159}\] Derived adjunction identifies \(\operatorname{Ext}^1_{X'}(j_*E,j_*Q)\) with \(\operatorname{Hom}_{D(X)}(\mathcal K,Q[1])\). Because \(Q\) is a sheaf, this Hom factors through \(\tau_{\geq-1}\mathcal K\). Applying \(\operatorname{Hom}(-,Q[1])\) to (159) gives the exact segment \[ \operatorname{Ext}^1_{X'}(j_*E,j_*Q) \longrightarrow\operatorname{Hom}_X(Q,Q) \longrightarrow\operatorname{Ext}^2_X(E,Q). \tag{160}\] Use the cochain convention in which a connecting map is the differential of a lift. The last map sends \(\operatorname{id}_Q\) to \(b_E\). If the first class is represented by \(0\to j_*Q\to E'\to j_*E\to0\), the Tor boundary after restriction sends \(j_0\otimes e\) to \(j_0\widetilde e\), where \(\widetilde e\) lifts \(e\) in \(E'\). This follows directly from the two-term presentation \(J_X\to\mathcal O_{X'}\) of \(\mathcal O_X\), with the same connecting-map convention. It is the map \(Q\to Q\) in (160). Thus \(b_E=0\) precisely when there is an extension \[ 0\longrightarrow E\otimes_AJ\longrightarrow E' \longrightarrow E\longrightarrow0 \tag{161}\] whose kernel identification is the multiplication map \(J_X\otimes_{\mathcal O_X}E\to E'\). Yoneda Ext supplies an actual extension; it is coherent because coherent modules on the noetherian model are closed under extensions.

We identify \(b_E\) with the Atiyah contraction, including its orientation. Keep the two actions on \[\mathcal C=\mathcal O_X^{(1)} \otimes^L_{\mathcal O_{X'}}\mathcal O_X^{(2)},\qquad \mathcal I=\operatorname{fib}(\mathcal C\to\mathcal O_X).\] Tensor over action (1), using action (2) as output. Then \(\mathcal C\otimes^L_{(1)}E=\mathcal K\), \(\mathcal I\in D^{\leq-1}\), and \(H^{-1}(\mathcal I)=J_X\). Universal derivation on action (2) gives the diagonal linearization \(\ell:\mathcal I\to L_{X/X'}\), oriented by \(1\otimes f-f\otimes1\mapsto df\). To check its first homology, resolve factor (2) by a local free DG algebra over \(\mathcal O_{X'}\). A degree-\(-1\) generator \(u\) with \(d_{\rm DG}u=j_0\in J_X\) represents \(j_0\) in \(H^{-1}(\mathcal I)\); its universal differential represents the same class in \(H^{-1}(L_{X/X'})=J_X/J_X^2\). Thus \(\ell\) induces the identity under the canonical first-conormal-homology identifications. Universal derivation makes this check compatible on overlaps.

The homotopy pushout of \(\mathcal I\otimes^L_{(1)}E\to\mathcal K\) along \(\ell\otimes E\) is the derived first-principal-parts triangle \[E\otimes^L L_{X/X'}\longrightarrow P^1_{X/X'}(E) \longrightarrow E \xrightarrow{\operatorname{At}_{X/X'}(E)} E\otimes^L L_{X/X'}[1].\] The boundary is the principal-parts definition of Atiyah. Its sign relative to Buchweitz–Flenner can be read from first jets. For ordinary degree-zero jets put \(\jmath(p)=(1\otimes1)\otimes p\), with \(p\) on action (1) and the module action on action (2). With \(df=1\otimes f-f\otimes1\), \[\jmath(pf)-\jmath(p)f=-p\,df.\] A connection satisfies \(\nabla(pf)=\nabla(p)f+p\,df\), so \(\jmath+\nabla\) is the module splitting. In a free DG model its failure to commute with the differential is \([d_{\rm DG},\nabla]\). This is precisely \(\operatorname{At}_{\mathrm{BF}}\) in [4], not its negative.

Let \(t:L_{X/X'}\to J_X[1]\) be truncation. Since \(t\ell\) is the identity on \(H^{-1}\), truncation adjunction identifies it with \(\mathcal I\to J_X[1]\). After tensoring with \(E\), the analogous assertion has target \(Q[1]\): the tensor cohomology spectral sequence has in degree \(-1\) only \(H^{-1}(\mathcal I)\otimes E=Q\), and all higher Tor terms have lower degree. The derived target is \(Q[1]\) by (154). Naturality of the fiber and truncation triangles therefore identifies \(b_E\) with the principal-parts Atiyah class followed by \(t[1]\). Naturality under \(L_{X/\mathbb C}\to L_{X/X'}\) gives \[b_E=(1_E\otimes\delta_X)[1]\operatorname{At}_{\mathrm{BF},X/\mathbb C}(E).\] Define \(o_E=-b_E\). With \(a=-\operatorname{At}_{\mathrm{BF}}\), we have proved \[ o_E=\langle\delta_X,a_{X/\mathbb C}(E)\rangle. \tag{162}\] It vanishes if and only if (161) exists with its multiplication identification. The use of the two highest cohomology degrees of \(\mathcal K\) depends on \(Q\) being a sheaf. For a general perfect complex, the Hom in (160) need not factor through this truncation.

Absolute perfectness supplies the trace in the next identity. Put \(L=L_{X/\mathbb C}\). Under \(L\operatorname{Sym}^k(L[1])\simeq(L\!\bigwedge^kL)[k]\), the Atiyah morphism \(a:E\to E\otimes L[1]\) is a degree-zero coefficient-valued endomorphism. The map \(\delta_X[1]:L[1]\to J_X[2]\) extends as the derivation contracting each symmetric generator. Its output coefficient has even shift two. Expanding contraction of \(a^k\) gives \(k\) terms with the same supertrace by graded cyclicity: the symmetric algebra accounts for the internal Koszul signs, and rotation of the degree-zero Atiyah operations and the even coefficient leaves no residual sign. Trace commutes with coefficient maps. Dividing by \(k!\) proves \[ \sigma_{k,J}(o_E) =\delta_{X,k}\left(\frac{\operatorname{tr}(a_{X/\mathbb C}(E)^k)}{k!}\right). \tag{163}\] This supplies the particular obstruction and trace identities corresponding to [4].

The multiplication identification also gives \(A'\)-flatness. For completeness, the square-zero flatness criterion says that an \(A'\)-module \(M'\) is flat if \(M'/JM'\) is \(A\)-flat and \(J\otimes_A(M'/JM')\to M'\) is injective. The latter condition is \(\operatorname{Tor}^{A'}_1(A,M')=0\). The change-of-rings Tor sequence then gives \(\operatorname{Tor}^{A'}_1(W,M')=0\) for every \(A\)-module \(W\). Filtering an arbitrary \(A'\)-module by \(JW\subset W\) gives the same vanishing for every \(W\), which is flatness. For (161), the kernel identification gives \(E'/JE'=E\) and the required injection. Apply the criterion to the algebraic stalks. Flat local analytification preserves this \(A'\)-flatness, coherence, and the extension.

We now relate the obstruction to the assumed horizontal class. The exterior boundary \(\nabla'_k\) in (158) is the restricted Hodge-graded Gauss–Manin map. Indeed, quotient the absolute de Rham complex of \(X'/\mathbb C\) by forms with at least two base differentials. The connecting map of the resulting sequence is base differentiation in product coordinates. Filtering by relative form degree gives on the graded term exactly the exterior boundary of (157). Hodge degeneration identifies the degree-\(2k\) graded term with \(H^k(\Omega^k)\), and (156) identifies its restriction to \(A\). Therefore the restriction of the edge of a flat \(F^k\) class on \(X'/A'\) is killed by \(\nabla'_k\).

The image under \(\rho_k\) of the absolute character in (163) is the relative character of \(E\), by Atiyah naturality. Evaluate (158) on that absolute character and use (163). The result is \[ (1\otimes d)_*\sigma_{k,J}(o_E) =\nabla'_k\bigl(\operatorname{ch}^{\rm rel}_{H,k}(E)\bigr). \tag{164}\] This is the restricted Gauss–Manin square used in [4]. The hypothesis of the lemma kills its right side for \(k\in I\). The left coefficient map is injective by (156), and \(\sigma_{I,J}\) is injective by Lemma 43, since \(\mathfrak m_AJ=0\). Thus \(o_E=0\). Equations (162) and (161), followed by the flatness criterion, give the required analytic flat sheaf lift. ◻

Proof. For \(k=0\) the assertion is constancy of rank on each connected component. Assume \(k\geq1\). The projective model of \(X_\nu\) has an ample line. Repeated Serre global-generation surjections from finite sums of negative powers of that line resolve \(E_\nu\). The local projective-dimension bound in Lemma 43 makes the \(n\)-th syzygy locally free. GAGA gives a global finite holomorphic vector-bundle resolution \(V^\bullet\to E_\nu\). No compatibility of these resolutions as \(\nu\) varies is asserted or needed.

Choose a differentiable trivialization of the proper submersion \(\mathcal X\to S\). Holomorphic \(t\)-jets through order \(\nu\) of relative smooth forms define a morphism \[\Omega^\bullet_{X_\nu/A_\nu} \longrightarrow \mathcal A^\bullet_{X_0}\otimes_\mathbb CA_\nu,\] where \(\mathcal A^\bullet_{X_0}\) is the smooth de Rham complex. Explicitly, after the trivialization take \[\sum_{|\mathbf a|\leq\nu} \frac{(\partial_t^{\mathbf a}\omega)|_{t=\bar t=0}} {\mathbf a!}\,t^{\mathbf a}.\] Leibniz’s rule makes this compatible with products, and jets commute with the fiber differential. The jets of the fiberwise type projections give a bigrading on the target; their integrability identities persist under taking jets. In a local holomorphic product chart the resulting Dolbeault complex is the ordinary one with coefficients \(A_\nu\), up to an invertible formal smooth coordinate change. The coefficientwise Dolbeault lemma therefore makes it a fine resolution of each \(\Omega^p_{X_\nu/A_\nu}\), respecting the form filtration.

Both total complexes resolve the constant sheaf \(A_\nu\). For the holomorphic complex, filter by powers of \(\mathfrak m\); its associated graded complex is \(\Omega^\bullet_{X_0}\otimes_\mathbb C\operatorname{gr}A_\nu\), to which the holomorphic Poincare lemma applies. The jet map is the identity on constants. It therefore realizes the ordinary flat de Rham comparison in the statement.

A holomorphic vector bundle \(V\) on \(X_\nu\) becomes a smooth \(A_\nu\)-bundle under the same jet map, and is smoothly isomorphic to \(V_0^\infty\otimes A_\nu\). Indeed, modulo successive powers of \(\mathfrak m\), the discrepancy between its transition matrices and the central ones is a Cech one-cocycle with values in smooth sections of \(\operatorname{End}(V_0^\infty)\otimes \mathfrak m^a/\mathfrak m^{a+1}\). This sheaf is fine, so a smooth gauge change removes that discrepancy. Induction through the finitely many powers of \(\mathfrak m\) gives the isomorphism.

Choose a smooth right-module connection \(D=\bar\partial_V+\nabla^{1,0}\) on this bundle, and put \(F_D=D^2\). Its curvature has no \((0,2)\)-part. In the Dolbeault resolution the BF commutator \([\bar\partial,\nabla]\), with its form coefficient written after the \((0,q)\)-form, maps under exterior multiplication to \(F_D^{1,1}\). The sign also follows from the first-jet calculation above: for a line with frames \(e_j=e_i g_{ij}\), the BF cocycle is \(-d\log g_{ij}\), while the connection matrices differ by \(d\log g_{ij}\); the Cech–Dolbeault boundary carries this cocycle to \(+\bar\partial A_i\). Thus \(a=-\operatorname{At}_{\mathrm{BF}}\) is represented by \(-F_D^{1,1}\). It follows that the closed form \[\frac{(-1)^k}{k!}\operatorname{tr}(F_D^k)\] belongs to \(F^k\) and that its edge is the raw degree-\(k\) Atiyah character.

This de Rham class is the constant central class. Use a constant connection on \(V_0^\infty\otimes A_\nu\) and the smooth isomorphism just constructed. For \(D_u=D_0+u(D_1-D_0)\), the Chern–Simons identity is \[\partial_u\frac{\operatorname{tr}(F_{D_u}^k)}{k!} =d\,\frac{\operatorname{tr}((D_1-D_0)F_{D_u}^{k-1})}{(k-1)!}.\] Integration in \(u\) proves independence of the connection; the polynomial identity is valid over the finite complex algebra \(A_\nu\). For this right-connection convention, the normalized topological character is \(\operatorname{tr}\exp(-F_D/(2\pi i))\). Hence the signed unscaled form above represents \((2\pi i)^k\operatorname{ch}_{B,k}(V_0)\) in the flat comparison.

Apply the calculation termwise to \(V^\bullet\). A grading-preserving connection represents \(\operatorname{At}_{\mathrm{BF}}\) by its commutator with \(d_{V^\bullet}+\bar\partial\), and \(a\) by the negative of that commutator. In a power, every term involving \(d_{V^\bullet}\) has positive internal degree and zero supertrace. The remaining terms are the alternating sum of the signed bundle curvature traces. Their edge is therefore \(\operatorname{ch}^{\rm rel}_{H,k}(E_\nu)\). By (154), the central restriction of the resolution resolves \(F_0\). Its alternating central character is \(\operatorname{ch}_{B,k}(F_0)\), which proves (165). ◻

Proof of Theorem 42. Let \(\beta_{H,k,\nu}\) be the edge of the restriction of \((2\pi i)^k\beta_{B,k}\) to \(X_\nu/A_\nu\). A flat section is uniquely determined near the origin by its central value. Thus Lemma 45 identifies the relative character of every \(E_\nu\) with \(\beta_{H,k,\nu}\). Starting with \(E_0=F_0\), the assumption that \(\beta_{B,k}\) lies in \(F^k\) provides the flat lift to order \(\nu+1\). Lemma 44 then gives \(E_{\nu+1}\) with its identification on \(X_\nu\). This constructs a compatible tower of flat coherent sheaves. Only the extensions \(A_{\nu+1}\to A_\nu\) and their modules \(J_\nu\) were used, not arbitrary Artinian extensions.

We make the classifying maps compatible before applying analytic approximation. Choose \(c\) large enough that \(F_0(c)=F_0\otimes\mathcal L_{\mathcal X,0}^{c}\) is generated by \(b\) global sections and has \(H^1(X_0,F_0(c))=0\). Let \(\mathcal V=(\mathcal L_{\mathcal X}^{-c})^{\oplus b}\) and fix the central quotient \(\mathcal V_0\twoheadrightarrow F_0\). Given a quotient \(\mathcal V_\nu\twoheadrightarrow E_\nu\), tensor the extension for \(E_{\nu+1}\) by \(\mathcal L_{\mathcal X}^{c}\). Its kernel is \(i_{0*}(F_0(c)\otimes_\mathbb CJ_\nu)\), whose first cohomology vanishes. The global-section map to order \(\nu\) is therefore surjective. Lift the \(b\) sections defining the quotient. Their map to \(E_{\nu+1}\) is surjective by Nakayama’s lemma, because its central restriction is the fixed central quotient. We obtain compatible quotients \(\mathcal V_\nu\twoheadrightarrow E_\nu\) for every \(\nu\).

We use the relative analytic Quot theorem of Pourcin [32]: proper flat quotients of a coherent sheaf on a separated finite-dimensional complex space over a separated finite-dimensional complex base, which may be nonreduced, are represented by a complex space with a universal flat quotient. These hypotheses hold here; in particular quotient supports are proper because \(\mathcal X\to S\) is proper. Let \(T\) be the germ of this Quot space at the central quotient. The compatible quotients give unique compatible \(S\)-maps \(\operatorname{Specan}A_\nu\to T\). Passing to the inverse limit gives a local \(R\)-algebra map \[\mathcal O_{T,0}\longrightarrow\widehat R =\mathbb C[[t_1,\ldots,t_e]],\] that is, a formal section of \(T\to S\).

A graph embedding presents this analytic \(S\)-germ as \[\mathcal O_{T,0} =R\{z_1,\ldots,z_h\}/(g_1,\ldots,g_v).\] The formal section gives zero-constant formal series \(z_j(t)\) solving these finitely many convergent equations. Artin’s analytic approximation theorem [1] says that such a formal solution over \(\mathbb C\) has an exact convergent solution with any prescribed finite-order congruence. Take congruence modulo \(\mathfrak m\). The solution still maps the origin to the central quotient, and, with \(t\) fixed as the parameters, defines an analytic section \(S\to T\). Pullback of the universal quotient gives a coherent \(S\)-flat sheaf \(F\) with central fiber \(F_0\). The convergent solution need not realize the entire chosen formal tower; retaining the central fiber is sufficient.

It remains to identify characters on actual fibers. The total space \(\mathcal X\) is smooth, so \(F\) is absolutely perfect. Separately, \(S\)-flatness and derived associativity give, for every \(s\), \[ Li_s^*F \simeq F\otimes^L_{\mathcal O_{\mathcal X}} (\mathcal O_{\mathcal X}\otimes^L_{\mathcal O_S}k(s)) \simeq F\otimes^L_{\mathcal O_S}k(s) =F|_{X_s}[0]. \tag{166}\] Near \(X_0\), \(F\) has a finite resolution by total holomorphic vector bundles: use relative Serre generation repeatedly and the regular-local projective-dimension bound on the smooth total space, then shrink by properness at the final locally free syzygy. The alternating normalized topological Chern character of these bundles restricts by (166) to \(\operatorname{ch}_B(F|_{X_s})\). Restrictions of a total cohomology class along a proper smooth submersion are flat. Their central value is \(\operatorname{ch}_B(F_0)\), so uniqueness of flat transport proves (152).

Every \(X_s\) is projective algebraic, and projective GAGA algebraizes \(F|_{X_s}\). On the smooth projective fiber a finite locally free resolution identifies its normalized topological character with the cycle class of its rational algebraic Chern character. Its raw degree-\(k\) Atiyah character is the weightwise \((2\pi i)^k\) image of that class. This proves algebraicity and completes the theorem. ◻

Proof. Choose \(k=1+tu\in K^\times\) with \(t\in\mathbb Q\) such that \(\sigma(k)^6\ne\bar\sigma(k)^6\). Such \(t\) exists because \[(1+tu)^6-(1-tu)^6\] is a nonzero polynomial in \(t\), with linear coefficient \(12u\); only finitely many rational \(t\) are excluded. Over \(K\) (or \(\overline\mathbb Q\)) the plane is the direct sum of the two conjugate sixth exterior eigenspaces from the definition of \(\mathcal W_K(A)\). A nonzero rational vector \(w\) has nonzero coordinates in both lines: scalar conjugation exchanges them, so one zero coordinate would force the other to be zero. On the two lines, \(k^*=\eta(k)^*\) has respective eigenvalues \(\sigma(k)^6\) and \(\bar\sigma(k)^6\). The determinant formed by \(w\) and \(k^*w\) in those coordinates is therefore nonzero. They are linearly independent over \(\mathbb Q\), and form a basis of \(\mathcal W_K(A)\).

Choose an integer \(n_k>0\) clearing the denominator of \(\eta(k)\in\operatorname{End}^0(A)\). The resulting \(h=n_k\eta(k)\) is an algebraic endomorphism, and on \(H^6\) one has \(k^*=n_k^{-6}h^*\). Pullback by \(h\) preserves rational algebraic cycle classes, so \(k^*w\) is algebraic when \(w\) is. Both basis vectors, hence the whole rational plane, are algebraic. ◻