Proof. Riemann–Roch gives \[\dim A=\sum_{i=2}^n h^0(C,K_C^i) =\sum_{i=2}^n(2i-1)(g-1)=R.\] The trace-free Higgs deformation complex has vanishing zeroth and second hypercohomology, by stability and Serre duality. Its Euler characteristic is \(-2R\), so the tangent space has dimension \(2R\) at every point.

We recall why the fiber bound holds also over the singular Hitchin values. Higgs-field scaling makes \(X\) semiprojective: properness of \(h\) extends each scaling orbit at zero, since its Hitchin value extends there, and the fixed locus is closed in the proper fiber \(h^{-1}(0)\). For the Białynicki–Birula decomposition (Białynicki-Birula 1973) in this semiprojective setting, see (Hausel and Rodriguez-Villegas 2015, sec. 1.2). At a fixed point, the symplectic form pairs a tangent weight \(w\) with weight \(1-w\). The weights are integers, so the nonpositive tangent weights account for exactly half the tangent dimension. Consequently each repelling stratum, including its fixed component, has dimension \(R\).

The union of these strata is \(h^{-1}(0)\). Indeed, a point with a limit at infinity must have zero Hitchin value, because every Hitchin coordinate has positive scaling weight. Conversely, an orbit in \(h^{-1}(0)\) has a limit at infinity because that fiber is proper. Thus \(\dim h^{-1}(0)=R\). Pulling \(h\) back along the weighted scaling curve from any \(a\in A\) to zero, upper semicontinuity of fiber dimension for a proper morphism gives \(\dim h^{-1}(a)\le R\). The dimension inequality for a morphism from a smooth \(2R\)-dimensional variety to an \(R\)-dimensional variety gives the reverse inequality on each nonempty fiber. Surjectivity therefore proves the asserted equidimensionality. It follows that \(\dim(X\times_A X)=R+2R=3R\). ◻

Proof. The finite group \(\Gamma=\operatorname{Pic}^0(C)[n]\) acts on \(X\) by tensor product. Averaging over \(\Gamma\) decomposes rational cohomology into its invariant summand and its complementary variant summand. Let \(p\) be the smallest prime divisor of \(n\), and put \[c=n(n-n/p)(g-1).\] Proposition 1.4 of (M. A. A. de Cataldo et al. 2022), which applies to arbitrary rank, gives \[H^i_{\mathrm{var}}(X)=P_{i-c}H^i_{\mathrm{var}}(X).\] Here \(c\ge n^2/2\ge8\), so (5) makes the variant summand vanish in degrees \(0\), \(1\), and \(2\).

The invariant summand is the rational cohomology of the corresponding \(\mathrm{PGL}_n\) moduli space. Markman’s generation theorem (Markman 2002, Theorem 7), in the projective-bundle formulation of (Maulik and Shen 2024, Theorem 2.1(i)), states that this algebra is generated by the Künneth components of the projective Chern characters with indices \(j\ge2\). These components have degrees \(2j\), \(2j-1\), and \(2j-2\) on the moduli space. There are therefore no degree-one generators and only one possible degree-two generator, namely a nonzero normalization of the expression defining \(e\). This proves \(H^0(X)=\mathbb Q\), \(H^1(X)=0\), and \(H^2(X)=\mathbb Q e\), with nonvanishing still to be checked.

Choose an ample line bundle on \(X\). By Lemma 2, a Hitchin fiber is projective of positive dimension, and hence contains an integral projective curve. The ample bundle has positive degree on that curve, so its first Chern class is nonzero in \(H^2(X)\). The one-dimensional upper bound just proved now gives \(e\ne0\) and the final assertion. ◻

Proof. The filtration shift follows from (6) applied to the cup-product morphism. Choose an ample class on \(X\), which is also relatively ample for the proper morphism \(h\), making \(h\) projective. The decomposition theorem and relative hard Lefschetz (Beilinson et al. 1982, Theorems 6.2.5 and 6.2.10) give (8) for that class: the perverse degrees of \(Rh_*\mathbb Q_X[2R]\) have center zero, and the shift to \(K\) moves the center to \(R\). The decomposition theorem also makes the perverse cohomology spectral sequence degenerate, so these sheaf-theoretic isomorphisms give the displayed isomorphisms on associated-graded cohomology. Lemma 3 replaces the ample class by a nonzero scalar multiple of \(e\), without changing invertibility. ◻

Proof. Fix either associated graded, and denote its raiser by \(E\). The Lefschetz decomposition expresses it as a sum of primitive strings \[v,Ev,\ldots,E^\ell v,\qquad \deg v=r-\ell.\] On each string set \[F_0(E^jv)=j(\ell-j+1)E^{j-1}v.\] If \(H_0\) acts by \(k-r\) on grade \(k\), direct calculation gives \[[E,F_0]=H_0,\qquad [H_0,E]=2E,\qquad [H_0,F_0]=-2F_0.\] Thus \((E,H_0,F_0)\) is an \(\mathfrak{sl}_2\)-triple, and \([[F_0,E],E]=-2E\).

We claim that a degree \(-2\) endomorphism satisfying this last identity must equal \(F_0\). On the endomorphism space, the adjoint \(\mathfrak{sl}_2\)-action identifies grading degree \(-2\) with \(H_0\)-weight \(-2\). Decompose this finite-dimensional representation into irreducibles. In any irreducible summand containing weight \(-2\), two raising steps map that weight space nontrivially to weight \(2\). Consequently \[(\operatorname{ad}E)^2: \operatorname{End}(\operatorname{Gr}^A H)_{-2} \longrightarrow \operatorname{End}(\operatorname{Gr}^A H)_{2}\] is injective. Applying this to \(\operatorname{Gr}^A(f)-F_0\) and (9) proves the claim.

It follows that the single operator \(h_0=[e,f]\) on \(H\) preserves both filtrations and acts on their grade \(k\) by \(k-r\). Choose integers \(a\le b\) containing all their nonzero grades. For either filtration, \[(h_0-(k-r))A_k\subseteq A_{k-1}.\] Applying these factors successively from \(k=b\) down to \(k=a\) gives \[\prod_{k=a}^b(h_0-(k-r))=0.\] The polynomial has distinct roots, so \(h_0\) is semisimple. Every \(h_0\)-stable subspace is therefore the direct sum of its intersections with the eigenspaces. If a nonzero eigenvector first belongs to \(A_j\), its image in \(\operatorname{Gr}^A_jH\) shows that its eigenvalue is \(j-r\). Since each \(A_k\) is \(h_0\)-stable, this proves \[ A_k=\bigoplus_{j\le k}\ker\bigl(h_0-(j-r)\bigr). \tag{10}\] The right side is independent of the choice of filtration, proving \(F_k=G_k\). In the additionally graded setting, \(h_0\) has degree zero, so its eigenspaces and (10) decompose degree by degree. ◻

Proof. Choose homogeneous Jordan-string decompositions for the two operators, which are nilpotent because the gradings have finite support. The tensor product of any pair of strings is a graded invariant direct summand, with invariant complement, for the sum operator. Each Lefschetz isomorphism is block diagonal with respect to these summands, so it is an isomorphism on each of them.

For a string whose lowest and highest grades are \(p\) and \(q\), call \((p+q)/2\) its center. The average of the lowest and highest grades of a tensor product of two strings is the sum of their centers. Its Lefschetz symmetry forces that sum to equal \(r\), for every pair of strings. Fixing one string in either factor shows that all strings in the other factor have the same center. A Jordan string is Lefschetz about its own center, which proves the conclusion. ◻

Proof. The commutator-product map is a submersion at every irreducible tuple. We give the differential check. For a pair \(a,b'\) set \(c=ab'a^{-1}b'^{-1}\). Its right-translated differential has terms \[(1-\operatorname{Ad}_{ab'a^{-1}})u,\qquad (\operatorname{Ad}_a-\operatorname{Ad}_c)v.\] Using the trace pairing, an annihilating vector \(z\) is fixed by \(ab'a^{-1}\) and satisfies \(\operatorname{Ad}_{a^{-1}}z=\operatorname{Ad}_{b'}z\). It is therefore fixed by \(ab'\), hence by \(a\) and \(b'\). For a product of commutators one applies this argument to the testing vector conjugated by each prefix. Since the vector is fixed by the current commutator, it stays unchanged from one pair to the next. It consequently centralizes every matrix. Irreducibility forces a scalar, and the trace-free condition forces zero.

The space of a matrix in a flag Borel, with its flag, is smooth over its ordered diagonal torus, with fibers of dimension \(2b\). Pulling this space back along the submersion and then taking the free quotient gives smoothness. The relative dimension is \[2g(n^2-1)+2b-2(n^2-1)=2R+2b.\] At \(q\), the section \(\zeta^{-1}\nu-I\) takes values in the nilradical bundle over the flag space. Its zero locus is \(Z^{\mathrm B}\), and submersivity proves that this section is regular. The trace pairing identifies the nilradical with the dual of the flag tangent space, so its associated bundle is \(T_\pi^*\). Self-intersection and the projection formula now give \[\mathsf A\mathsf B(\alpha) =\pi_*\bigl(c_b(T_\pi^*)\pi^*\alpha\bigr) =(-1)^b\chi(\mathrm{Fl}_n)\alpha =(-1)^b n!\alpha.\] All constructions descend through \(\mathrm{PGL}_n\) and apply to full cohomology. ◻

Proof. Consider first a final stratum \(F=(\mathbb C^*)^s\times\mathbb A^a\), or a vector bundle over such a stratum. Its homotopy type is that of a torus. Each connected component \(\widetilde F\) of its finite étale preimage corresponds to a finite-index subgroup of \(\mathbb Z^s\). The pullback \[H^*(F,\mathbb Q)\longrightarrow H^*(\widetilde F,\mathbb Q)\] is therefore an isomorphism: both rings are the exterior algebras of the rational duals of these lattices. Since the map is algebraic, it is an isomorphism of mixed Hodge structures. Both spaces are smooth of the same dimension, and finite-degree trace together with Poincaré duality gives the same assertion for compact supports. These isomorphisms commute with cup product by the pulled-back class.

We apply this argument to each connected component separately. The cohomology of a disconnected preimage is their direct sum, and all components have the same Lefschetz center. Preimages of the closed filtration of the original stratification still give a finite closed filtration. In the compact-support localization sequences, strictness of mixed Hodge morphisms (Deligne 1971, Theorem 2.3.5) permits passage to weight-gradeds. Comparing these exact sequences by a fixed power of the common class and using the five lemma passes the Lefschetz isomorphisms from the strata to the entire auxiliary space. The same sequences show that its cohomology is Hodge–Tate.

The pulled-back auxiliary bundle has the same rank \(b\). Its Thom isomorphism shifts cohomological degree and actual weight by \(2b\). Thus the compact-support middle weight drops from \(\delta+2b\) to \(\delta\). Smooth Poincaré duality sends weight \(w\) to \(2\delta-w\), so ordinary cohomology has the same middle weight \(\delta\), or center \(\delta/2\) in half weights. The relative product stratification proves the Hodge–Tate assertion on every fiber, including repeated eigenvalues, by the same componentwise argument and the affine-bundle Thom isomorphism.

For integral local constancy, use the relative stratification and choose a small contractible analytic open set \(U\subset V\). A finite cover of \(F\times U\) is pulled back from a finite cover of \(F\), since \(\pi_1(F\times U)=\pi_1(F)\). The same statement holds for the total space of a vector bundle over \(F\times U\), which is homotopy equivalent to its base. Topologically the bundle may be identified, in the \(U\) direction, with the pullback of its restriction over one point. Equivalently, one may use the integral Thom isomorphism for its complex orientation. It follows that the compact-support cohomology sheaves of each covered stratum are locally constant finitely generated abelian groups on \(U\). Localization triangles pass this conclusion through the finite stratification: kernels, cokernels, and extensions of locally constant sheaves are locally constant. The auxiliary affine-bundle trace removes its shift.

Write \(f\) for the covered smooth family of relative dimension \(\delta\). The complex \(K=Rf_!\mathbb Z\) is locally perfect: its cohomology is bounded and finitely generated, and \(\mathbb Z\) has finite global dimension. Smooth Verdier duality gives \[Rf_*\mathbb Z\simeq R\mathcal Hom(K,\mathbb Z_V)[-2\delta].\] The right side is locally constant and perfect, so taking its stalk commutes with this duality. Compact-support base change and the fiberwise cup–trace pairing identify its stalk with \(R\Gamma(f^{-1}(t),\mathbb Z)\). This identification is precisely the natural ordinary base-change morphism: both are adjoint to the cup product followed by the smooth orientation trace, whose formation commutes with compact-support base change. This proves the ordinary assertion, without using a nonproper smooth base-change principle. Finally, write \(p:Y^{\mathrm B}\times D_1\to Y^{\mathrm B}\) for the projection and \(s\) for its section at the identity of \(D_1\). For ordinary direct images, \(s^*p^*=\mathrm{id}\) gives the required summand. For compact supports use the Gysin map of the closed section, followed by integration over \(D_1\): these maps have degrees \(4g\) and \(-4g\) and compose to the identity. Equivalently, the integral compact-support Künneth complex of \(D_1\) has its top class as a split summand. Thus both local-constancy assertions pass to \(Y^{\mathrm B}\), and natural ordinary base change passes through the first splitting. The ordinary splitting is algebraic on each fiber, so it also proves the asserted Hodge–Tate property for \(Y^{\mathrm B}\). ◻

Proof. First we identify the underlying ordinary complex. Open-cover descent computes the ordinary direct image of \(\mathscr H\to V\) from the spaces \(U_I\times H_I\). Projection to \(H_I\) induces a quasi-isomorphism on direct images over \(V\): over any open subset of \(V\), its fiber in this product is the contractible space \(U_I\). These projections commute with the transition maps. Their derived limit therefore identifies the rational realization of \(K_{\mathscr H}\) with the ordinary direct image of \(\mathscr H\to V\). The same argument works with integral coefficients.

A finite limit of bounded complexes with locally constant cohomology again has locally constant cohomology. Its integral cohomology sheaves have finite type. Point pullback commutes with finite limits, and the termwise base-change maps are isomorphisms by (13). The corresponding maps in mixed Hodge modules are isomorphisms because rational realization is conservative. It follows that \(i_t^*K_{\mathscr H}\) computes the derived diagram of the algebraic fibers \(H_{I,t}\) and that its underlying cohomology is \(H^*(\mathscr H_t,\mathbb Q)\).

Put \(v=\dim V\). Since the ordinary cohomology sheaves of \(K_{\mathscr H}\) are locally constant, its perverse cohomology \({}^pH^{j+v}(K_{\mathscr H})\) is a smooth mixed Hodge module. Smooth mixed Hodge modules, after the dimension shift, are admissible graded-polarizable variations (Saito 1990, sec. 3.2, Equation (3.2.1)). Applying point pullback to the perverse truncations shows that the fiber of this variation is \(H^j(i_t^*K_{\mathscr H})\): a smooth perverse object has only one nonzero fiber degree, namely \(-v\).

Finally, mixed Hodge structures whose weight gradeds have only Hodge types \((r,r)\) are closed under subobjects, quotients, and extensions. The long exact sequence of a cone thus shows that complexes with Hodge–Tate cohomology are closed under finite limits. Applying this observation to the fiber diagram proves the last assertion. Algebraic maps of the diagrams give the stated functoriality. A common good refinement compares different choices of cover and trivializations: its comparison maps are isomorphisms after ordinary descent, hence also in mixed Hodge modules by conservativity. ◻

Proof. For a product, the diagram is constant. Its derived limit is \[C^*(N\mathcal U,\mathbb Q(0))\otimes R\Gamma(M,\mathbb Q_M^H),\] where \(N\mathcal U\) is the finite nerve of the good cover. The first complex has a copy of \(\mathbb Q(0)\) for each simplex, with its ordinary simplicial differential. All its cohomology therefore has weight zero. The external-product comparison on the cover gives exactly the ordinary Künneth map. Evaluation on a simplicial fundamental cycle defines \[C^*(N\mathcal U,\mathbb Q(0))\longrightarrow \mathbb Q(0)[-d_S].\] This is a morphism of complexes of mixed Hodge structures and realizes ordinary integration over \(S\). It has no Tate twist. Projection pullback is induced by the unit of the same cochain complex. This proves the first assertion; functoriality of algebraic pullbacks proves the second.

For the third assertion, fix \(t\) and choose simultaneous trivializations of the map. On each algebraic fiber map \(u_{I,t}:H_{I,t}\to H'_{I,t}\), smooth dualizing complexes and the proper counit give the Gysin morphism \[ Ru_{I,t*}\mathbb Q^H_{H_{I,t}}\longrightarrow \mathbb Q^H_{H'_{I,t}}[2c](c). \tag{15}\] Its degree and Tate twist give the displayed weight shift. Complex orientations are preserved by the algebraic transition isomorphisms, and the proper counit is natural. Consequently (15) gives a coherent map of the finite diagrams and hence of their derived limits.

It remains to identify its ordinary realization. On \(U_I\times H_{I,t}\) the map is the identity on \(U_I\) times \(u_{I,t}\). The orientation trace for this product is the fiber trace, tensored with the identity on \(U_I\). It is compatible with restriction to intersections and with the transition isomorphisms. Open-cover descent therefore identifies the realized map with the ordinary topological Gysin map of the total spaces. In particular, the construction gives the actual Gysin map, not merely a map with the same source and target. ◻

Proof. Each nonzero weight graded is a polarizable variation of pure type \((r,r)\) and weight \(2r\). After the conventional sign, its flat polarization is positive definite on the underlying real vector space. Let \(\mathcal L\) be the torsion-free quotient of the integral cohomology local system. Intersecting \(\mathcal L\) with the rational weight sub-local systems and taking successive quotients gives an invariant full lattice in each rational graded piece. These intersections are saturated because the subspaces are rational.

The automorphisms of a lattice preserving a positive definite form constitute a finite group: in a fixed lattice basis, every column of such an automorphism has a prescribed bounded norm and hence lies in a finite set. Thus each graded monodromy image is finite. Only finitely many cohomological degrees and weights occur, which proves the combined assertion.

Choose \(M\) killing the action of the specified loop on all weight gradeds. Then \[(T^M-1)W_a\subset W_{a-1}=W_{a-2}\] whenever \(a\) is even; the identical conclusion for odd \(a\) follows because successive odd and even steps coincide. The operator \(T^M-1\) is nilpotent, since the weight filtration is finite. Its finite logarithm has the same lowering bound.

Parallel transport preserves cup products. For every integer \(k\geq0\), the polynomial operator \(\exp(kMN)=T^{Mk}\) is therefore a ring automorphism. For cohomology classes \(\alpha,\beta\), the identity \[\exp(kMN)(\alpha\beta) =\exp(kMN)(\alpha)\exp(kMN)(\beta)\] is an equality of polynomials in \(k\). Comparing linear coefficients proves \(N(\alpha\beta)=N(\alpha)\beta+\alpha N(\beta)\). Monodromy preserves cohomological degree, so \(N\) does as well. ◻

Proof. By (17), tensoring fixed-pole data by \(J\) changes its determinant to \(\mathcal O_C(d x(s))\), up to a base line, and changes its determinant operator to the prescribed one. At \(x_0\), the residue \(+\lambda d/n\) of \(J\) cancels the residue \(-\lambda d/n\) of the original rank-\(n\) connection.

This cancellation also holds where the moving and fixed sections meet. In a coordinate \(z\) with moving section \(z=a(s)\), the scalar polar part contributed by \(J\) is \[-\frac{\lambda d}{n}\frac{dz}{z-a(s)} +\frac{\lambda d}{n}\frac{dz}{z}.\] Adding the fixed-pole form leaves only the first term. The identity is an identity of relative meromorphic forms over the whole local base, including \(a(s)=0\). Thus the tensor product has a single logarithmic pole at the moving point. The inverse construction tensors by \(J^{-1}\).

Base-line twists do not change the induced map to coarse moduli, and \(\deg J_s=0\) preserves stability. When \(\lambda=0\), the Higgs field of \(J\) is zero, so the characteristic polynomial is unchanged. ◻

Proof. We first establish zero limits for scaling. Given an unflagged object, keep its bundle fixed and scale its operator. This extends the orbit to the stack of all logarithmic \(\lambda\)-connections, without imposing semistability on the special fiber. Semistable reduction (Fernandez Herrero and Zhang 2025, Proposition 4.48) replaces this DVR family by a semistable one after a finite extension, without changing its generic fiber. Its hypotheses hold for \(\mathrm{GL}_n\) over \(\mathbb C\).

The determinant of the replacement and the prescribed determinant agree generically, hence agree over the DVR by separatedness of the rank-one moduli space. A generic flag extends because the flag variety of the extended bundle at the pole is proper. Residue preservation and the ordered diagonal residue equations are closed, and therefore persist at the special fiber. Coprimality makes the limiting semistable pair stable. The same reasoning applies to the scalar-residue condition.

A finite extension of the DVR is enough for a limit in the original coarse space. Indeed, embed this quasiprojective space in projective space as a locally closed subvariety. The original orbit has a unique projective limit. The limit after finite base change is that point and belongs to the coarse moduli space, so the original orbit extends there. This proves the required zero-limit property.

We next check smoothness over \(\lambda=u=0\). At such a point put \[\mathcal F=\operatorname{ParEnd}_0(E),\qquad \mathcal G=\mathcal F^\vee\otimes K_C,\] where \(\mathcal F\) is the sheaf of trace-free endomorphisms preserving the flag at \(x(s)\). Trace pairing identifies \(\mathcal G\) with the trace-free logarithmic Higgs fields having strictly flag-lowering residue. The relative deformation complex is \[ \mathcal F\xrightarrow{[\theta,-]}\mathcal G. \tag{18}\]

Here is the lifting argument, including the parameter directions. For a small Artin extension with prescribed lifts of \(s,\lambda,u\), the bundle, determinant isomorphism and flag lift because the obstruction lies in \(H^2(C,\mathcal F)=0\). In local frames respecting the lifted flag, the operator lifts with its prescribed determinant and diagonal residues: these are affine-linear conditions, and trace splits in characteristic zero. Its gluing obstruction lies in \(H^1(C,\mathcal G)\) tensored with the square-zero ideal. Changing the bundle and flag gluing changes this obstruction by the image of \(H^1(C,\mathcal F)\) under (18).

This map on \(H^1\) is surjective. Its Serre dual is, up to sign, the map on \(H^0\) in (18); its kernel consists of trace-free automorphisms of the stable Higgs pair preserving the flag, hence is zero. Thus every such lifting problem has a solution. Since \[\chi(\mathcal F)=-(n^2-1)(g-1)-b=-R-b,\] the relative tangent dimension is \(-2\chi(\mathcal F)=2R+2b\). For the scalar unflagged family, replace \(\mathcal F\) by \(\operatorname{End}_0(E)\) and obtain dimension \(2R\) in the same way. Finite scalar stabilizers act trivially on these deformations, so the calculation also proves smoothness of the coarse spaces.

The nonsmooth locus of the morphism is closed and scaling invariant. If it contained a point, it would contain that point’s zero limit, contradicting the calculation over \(\lambda=u=0\). This proves smoothness everywhere.

A scaling-fixed point has \(\lambda=u=0\). Its logarithmic Hitchin coefficients vanish, because their scaling weights are positive. The logarithmic Hitchin map is proper; its base has coefficients in \(H^0(C,K_C(x(s))^{\otimes i})\) as in (Cataldo and Fernandez Herrero 2024, sec. 2.4, Equation (4)). The fixed locus is closed in its nilpotent fiber after adding flags and imposing determinant and residue conditions. Since flags and \(S\) are proper, that fixed locus is proper. Together with the zero limits, this proves semiprojectivity. The inverse image of a linear subspace through zero is smooth by base change and inherits both properties.

Finally \(\pi\) is the full flag bundle of the universal projective bundle. On the \(u=0\) family, subtracting the prescribed scalar from the residue gives a section of the bundle of strictly flag-lowering matrices. Its zero scheme is \(\mathcal Z\). The smoothness and dimensions just proved show that it is a regular zero scheme of codimension \(b\). Trace pairing identifies this residue bundle along \(\mathcal Z\) with the cotangent bundle of the flag fibers, giving \(N_\iota\simeq T_\pi^*\). ◻

Proof. Recall that the core of a smooth semiprojective variety is the union of the sets whose scaling orbits have limits as the parameter tends to infinity. Its inclusion induces a cohomology isomorphism (Hausel and Rodriguez-Villegas 2015, Theorem 1.3.1). Because every coordinate on \(\Lambda\) has positive weight, a point of \(\mathcal Y\) with such a limit must lie over zero. Hence \(\mathcal Y\) and its inverse image \(\mathcal Y_\ell\) of any line \(\ell\) through zero have the same core. Their restrictions to the core commute, so \(H^*(\mathcal Y)\to H^*(\mathcal Y_\ell)\) is an isomorphism.

The map \(\mathcal Y_\ell\to\ell\) is smooth and surjective. For surjectivity, its image is a scaling-invariant open subset containing zero, and therefore contains the whole affine line. Apply (Hausel and Rodriguez-Villegas 2015, Corollary 1.3.3): for a smooth surjection from a smooth semiprojective variety to the affine line, equivariant for a positive base weight, restriction to each fiber is an isomorphism. This proves (19) and its line version. The same proof applies to \(\mathcal X\), \(\mathcal O\) and \(\mathcal Z\).

Pullback compatibility is functorial. For Gysin maps, the families are smooth over the line, the scalar embedding is transverse to fiber restriction, and the flag and product projections are smooth proper. The transverse base-change identities for Gysin maps give the claimed compatibilities. ◻

Proof. The local exponents of a connection are \(r_i=-d/n+u_i\). For the regular-singular correspondence and logarithmic lattices, including Manin’s extension theorem and Katz’s nonresonance corollary, see Deligne (Deligne 1970, II, Proposition 5.4, Remark 5.5(ii), Corollary 5.6, and Theorem 5.9). We spell out the nonresonant local construction to track the ordered flags and dependence on the moving parameters. In a disc coordinate \(z\), write a logarithmic connection as \(d+A(z)\,dz/z\), with \(A(0)=A_0\). The regular-singular reduction to \(d+A_0\,dz/z\) by a gauge equal to the identity at zero is obtained recursively using the operators \(k+\operatorname{ad}(A_0)\), \(k\geq1\). Their eigenvalues are \(k+r_i-r_j\), so they are invertible under (20), even when \(A_0\) has repeated eigenvalues or a nonzero nilpotent part. On a small parameter neighborhood their inverse norms are \(O(1/k)\) for large \(k\). The usual power-series majorant for the analytic coefficients therefore gives convergence on a common smaller disc and continuous dependence on parameters.

In this constant-residue model a residue-invariant flag gives a monodromy-invariant flag. Conversely, the prescribed exponents select one logarithm for each monodromy eigenvalue: equal exponential values have equal selected exponents, by (20). Functional calculus on each generalized eigenspace then gives a logarithm of monodromy preserving its invariant flag. This constructs the inverse logarithmic extension with the specified ordered residues. Coordinate and logarithm changes in the constant-residue model act by functions of the residue, and hence preserve the flag; the construction is intrinsic. In rank one the same construction gives exactly \((\mathcal O_C(d x(s)),d_{d x(s)})\), so the determinant is the required one. The subproduct condition defining \(V\) makes the representation irreducible, and thus ensures stability.

Holonomy along local moving generators, together with this local normal form, makes the forward maps continuous in the bundle and in \(s,u\). Regard the Betti family as pulled back to \(U\) via \(t\); there is no assertion that \(t\) is globally injective. The forward map is a continuous bijection between manifolds of equal real dimension, by Proposition 14 and the Betti smoothness. Invariance of domain gives its inverse continuity. For a fixed point of \(S\) it is holomorphic on the moduli fibers, so it preserves their complex orientations.

At \(u=0\), scalar residues correspond to scalar monodromy, and forgetting or retaining the flag commutes with this construction. The proper maps in the correspondence have their usual complex orientations in the algebraic fibers and the same orientation on \(S\). Consequently their topological Gysin maps agree under these homeomorphisms. ◻

Proof. Away from the fixed and moving poles, the nth power of the connection on \(J\) is the canonically trivial connection. Its holonomies therefore lie in \(\mu_n\). Locally in \(S\), take a disc containing \(x_0\) and the moving point, and choose the handle generators outside that disc. Their line holonomies depend continuously on \(s\) and take values in the finite group \(\mu_n\), so are locally constant. Tensoring multiplies the handle matrices by these constant scalars, an algebraic map on the representation variety. The argument also applies in collision neighborhoods. Thus the algebraic product identification of the scalar connection family is compatible with the locally constant algebraic Betti transitions. ◻

Proof. Divide out the kernel of the action. Smoothness and connectedness make \(B\) irreducible. Removing the fixed loci of the nonidentity elements gives a dense connected Zariski open \(B^\circ\) on which the action is free. For \(a\in g^{-1}(B^\circ)\) there is a unique \(\gamma(a)\) with \(f(a)=\gamma(a)g(a)\). The free finite quotient is a covering there, so \(\gamma(a)\) is locally constant. It is constant because this open set is connected. Continuity extends the equality over its closure, which is all of \(A\). ◻

Proof. The full fixed-determinant nonabelian-Hodge diffeomorphism, including its \(\Gamma\)-equivariance, is part of the twisted construction in (Hausel and Thaddeus 2003, sec. 2 and 5). We compare its cohomology map with the particular logarithmic family above.

For \((E,\theta)\in X\), choose a harmonic metric \(H\) normalized to a fixed metric on its determinant. Let \(\partial_H\) denote the \((1,0)\) part of its Chern connection. For real \(0\leq\tau\leq1\) consider the holomorphic structure and logarithmic \(\tau\)-operator \[ \bar\partial_E+\tau\theta^{\dagger_H},\qquad \tau\partial_H+\theta+ \frac{\tau}{n} \bigl(d^{1,0}_{d x_0}-\partial_{\det H}\bigr)I. \tag{22}\] Its projectivization is the usual harmonic preferred section (Simpson 1997, sec. 4). We verify the scalar adjustment. The trace of \(\theta\) and of \(\theta^{\dagger_H}\) is zero, so the determinant holomorphic structure stays fixed and the determinant operator is exactly \(\tau d_{d x_0}\). The trace-free holomorphicity equation follows from \(\bar\partial_E\theta=0\), \(\partial_H\theta^{\dagger_H}=0\) and the projective Hitchin equation. The scalar equation is that of the prescribed logarithmic determinant connection. The only nonregular term is the scalar logarithmic term at \(x_0\).

For \(\tau\ne0\), divide the operator by \(\tau\). Its puncture monodromy is \(\zeta I\). A proper invariant subspace of rank \(r\) would have \(\zeta^r=1\) by taking determinants of the commutator relation, contrary to \(\gcd(n,d)=1\). Thus these connections are irreducible and stable. At \(\tau=0\) we recover the original Higgs pair. Hence (22) lies in \(\mathcal X\) throughout.

This construction is a continuous homotopy of maps \(X\to\mathcal X\). One may work in local families of stable Higgs pairs. The normalized harmonic metrics vary smoothly: their trace-free metric linearization is self-adjoint elliptic, and its kernel consists of infinitesimal symmetries, which stability kills. This is also the parameter dependence used for preferred sections in (Simpson 1997, sec. 4). To pass to the analytic topology of coarse moduli, choose a sufficiently positive twist near any one parameter so the underlying bundles are generated and have vanishing \(H^1\). The kernels of the corresponding Dolbeault operators then give continuously varying bases of sections. Their evaluations give continuous quotient presentations; the logarithmic operators, viewed as morphisms from the first jet bundle with fixed symbol, also have continuous coefficients in such presentations. Passing to the stable quotient gives the asserted continuity, independently of the local choices.

At \(\tau=0\) the homotopy is the inclusion \(X\hookrightarrow\mathcal X\). At \(\tau=1\) it factors through \(X_1\), and then through Riemann–Hilbert to a map \(F:X\to X^{\rm B}\). The pullback \(F^*\) is therefore precisely (21), because both restrictions from \(\mathcal X\) are isomorphisms. Projectivizing \(F\) gives the usual projective nonabelian-Hodge map.

Let \(\Phi:X\to X^{\rm B}\) be the fixed-determinant nonabelian-Hodge diffeomorphism. The maps \(F\) and \(\Phi\) agree after projectivization. Two determinant-one lifts of the same projective representation differ on each handle generator by an nth root of unity; thus their classes differ by the finite group of determinant-preserving characters, identified with \(\Gamma\). The low-degree calculation of Section 2 gives \(H^0(X^{\rm B})=\mathbb Q\), so \(X^{\rm B}\) is connected; it is smooth in the coprime setting. Lemma 18 therefore gives \(F=\gamma\Phi\) for one \(\gamma\in\Gamma\). Its action on \(X^{\rm B}\) is algebraic and preserves Deligne weights. Consequently \(F^*\) and \(\Phi^*\) transport the weight filtration identically on every cohomology group. No averaging or projection to invariants occurs.

Finally choose \(M\in\operatorname{Pic}^0(C)\) with \(M^{\otimes n}\simeq L\otimes\mathcal O_C(-d x_0)\). Tensoring by \(M\) identifies the two Higgs moduli spaces algebraically over the same Hitchin base. Their projective Higgs bundles, and therefore their projective harmonic correspondences, are unchanged. Applying the same finite-character argument to the two fixed-determinant nonabelian-Hodge maps shows that this identification also preserves the transported weight filtration. It preserves the perverse filtration because it is an isomorphism over the Hitchin base. ◻

Proof. For each single modification, choose a local frame adapted to its current residue eigenspaces. The off-diagonal coefficients involving the chosen eigenline vanish at the pole, so the singly modified lattice remains preserved by the logarithmic operator. The preceding calculation gives its new residue eigenvalues. Their imaginary parts remain distinct throughout (23), including after the first step. We can therefore reselect the residue eigenspaces before performing the second modification. The two-step composite preserves the determinant and its operator because the lattice shifts have sum zero. Its inverse performs the opposite single modifications in reverse order, again reselecting eigenspaces between steps. These elementary transformations and their inverses vary algebraically where the successive residue eigenvalues are distinct, giving the asserted continuous family.

We check stability, which is not supplied by lattice modification alone. On the Higgs line with \(z\ne0\), a proper invariant saturated subbundle would have a trace Higgs differential with a single possible simple pole. Its residue would be \(z\sum_{i\in I}v_i\) for a proper nonempty subset \(I\). This is nonzero, contrary to the residue theorem. The modified Higgs bundles satisfy the same argument. Distinct residue eigenvalues split the local Higgs operator into formal eigenline summands; shifting these summands and shifting them back proves invertibility.

For \(a>0\), divide the operator by \(a\). A proper invariant monodromy subspace would have determinant monodromy one around the puncture. The imaginary part of the sum of its exponents is \(a^{-1}\sum_{i\in I}\operatorname{Im}v_i\), which is nonzero. Thus no such subspace exists. Differences of exponents have nonzero imaginary part, so no nonzero integral resonance occurs. The inverse modifications are valid and remain stable by the same tests.

The characteristic polynomial on the Higgs line is unchanged away from the pole and hence unchanged as a meromorphic polynomial on \(C\). Finally, modifications of a connection change only its logarithmic lattice at the puncture. At \(a=1\) the exponent changes are integers, so the meromorphic flat bundle and the ordered monodromy eigenlines are unchanged. This is precisely the endpoint identification asserted in the lemma. ◻

Proof. Fix a degree and a parameter \(t_0\). Choose finitely many compact singular cycles forming a rational homology basis of \(M_{t_0}\). A lift of the interval vector field near the compact union of their carriers has a flow for a common short time. It transports these cycles to nearby fibers; the resulting cycles sweep homologies in \(M\). Their images in \(A\) are therefore homologous, so their pairings with the restrictions of a fixed basis of \(H^*(A)\) are constant. The restriction isomorphisms show that the transported cycles remain a basis of the fiber homology.

Applying the continuous map \(bF\) to these swept cycles gives homologies in \(B\). Hence their pairings with every fixed class of \(H^*(B)\) are also constant. These pairings are the matrix coefficients of \(\alpha_t^{-1}F_t^*\beta_t\), relative to the fixed bases. The operator is thus locally constant, including one-sided neighborhoods of interval endpoints, and therefore constant on \(I\). ◻

Proof. The full-flag Betti family has integral lisse direct images, ordinary base change, and Hodge–Tate fibers by Section 3. The moving-puncture transition maps are algebraic and locally constant by Proposition 16. The finite-monodromy statement of Section 4 therefore applies to this very family. Choose \(M\) that kills the monodromy action on every weight-graded piece of its total cohomology. There are finitely many such pieces. Then \(\mathsf T^M-1\) is nilpotent, and (26) is a finite logarithmic sum. The weights are even, so \(N\) lowers them by at least two at \((1,v)\).

The path \((1,av)\), \(0\le a\le1\), stays in the nonresonant inverse image of \(V\). At \(a=0\), the scalar point belongs to \(V\) because \(\zeta\) has order \(n\). For \(a>0\), the imaginary-part argument applies again. Parallel transport preserves the diagram weights and coincides with the restriction identifications. This gives the stated estimate at \((1,0)\).

Lemma 12 makes the logarithm a derivation on the moving-puncture Betti cohomology. The restriction identifications are ring isomorphisms, so this property transports to \(H^*(\mathcal Y,\mathbb Q)\). ◻

Proof. The scalar correspondence and its Gysin maps commute with the restriction identifications of Corollary 15. Thus (27) transports to the same expression at \(\lambda=1\). By Proposition 19, these identifications transport the ordinary Betti weight filtration on the full cohomology of \(X^B\). The possible difference from a chosen nonabelian-Hodge identification is one algebraic \(\Gamma\)-action, which preserves every weight subspace, including those in variant cohomology.

At \(\lambda=1\), the scalar Betti space is the flat algebraic product \(S\times X^B\) of Lemma 17. Its diagram mixed Hodge structure assigns weight zero to the cochains of \(S\) and the ordinary Deligne weights to \(H^*(X^B)\). Hence both \(\mathop{\mathrm{pr}}_1^*\) and \(\mathop{\mathrm{pr}}_{1,*}\) preserve weights. The flag maps are maps of the same flat transition diagrams. Their ordinary topological Gysin maps are the diagram Gysin maps, with weight shifts \(2b\) for \(\mathsf B_1\) and \(-2b\) for \(\mathsf A_1\). These shifts cancel, while \(N_{1,0}^2\) lowers weight by four by Lemma 22. ◻

Proof. Smoothness and the restriction assertions follow from Proposition 14 and Corollary 15. For properness, start with the relative logarithmic Higgs Hitchin morphism over \(S\). Imposing determinant \(\mathcal O_C(dx(s))\) and trace zero gives a closed subspace. Adding a residue-invariant full flag is projective over that subspace. Finally the requirement that the ordered diagonal residues equal \(zv_i\) is closed after adjoining \(z\). The properness of the logarithmic Hitchin morphism from Section 5 therefore implies properness of (28). Coprimality ensures that the semistable limits in this argument remain stable.

For \(z\ne0\) the residual eigenvalues \(zv_i\) are distinct, so the elementary modifications defining \(T_v\) and their inverses are algebraic in \(z\). They remain in the stable moduli by the trace-residue argument of Section 6. A modified Higgs field agrees with the original meromorphic Higgs field away from the pole. Their characteristic coefficients are consequently equal as sections of \(K_C(x(s))^{\otimes i}\), which proves that \(T_v\) is over \(\mathcal E\). Multiplying the Higgs field by \(z\) identifies the fiber at \(1\) with that at \(z\); division by \(z\) is its inverse. This gives the claimed product over \(D^*\). The product need not fix the Hitchin coefficient in \(\mathcal E\); only the translation must do so. ◻

Proof. Smoothness gives the canonical isomorphism \(\mathbb Q_{Y_0}\simeq\psi\mathbb Q_Y\). Compatibility of nearby cycles with proper direct image gives \[Rg_{0,*}\mathbb Q_{Y_0} \simeq Rg_{0,*}\psi\mathbb Q_Y \simeq\psi(Rg_*\mathbb Q_Y).\] Apply nearby cycles to the pullback endomorphism induced by \(T\) on the punctured direct image, and use these isomorphisms to define \(\tau_0\).

We verify its action on global cohomology without interchanging nearby cycles with a nonproper direct image to a point. Put \(K=Rg_*\mathbb Q_Y\) on \(B\times D\). Let \(\widetilde D^*\to D^*\) be the universal cover and write \[\widetilde j_B:B\times\widetilde D^*\longrightarrow B\times D, \qquad i_B:B\times\{0\}\hookrightarrow B\times D, \qquad \widetilde K=\widetilde j_B^*K.\] Set \(\widetilde Y^*=Y|_{D^*}\times_{D^*}\widetilde D^*\). Proper base change identifies \(\widetilde K\) with the direct image of the constant sheaf from \(\widetilde Y^*\). The nearby-cycle formula downstairs is \(\psi K=i_B^*R\widetilde j_{B,*}\widetilde K\). Restriction of this complex gives a natural morphism \[ \begin{split} R\Gamma(\widetilde Y^*,\mathbb Q) &\simeq R\Gamma(B\times D,R\widetilde j_{B,*}\widetilde K)\\ &\longrightarrow R\Gamma(B,\psi K) \simeq R\Gamma(Y_0,\mathbb Q). \end{split} \tag{29}\] The last isomorphism uses proper nearby-cycle compatibility and smoothness of \(Y\to D\). The composite of (29) with restriction from \(R\Gamma(Y,\mathbb Q)\) is ordinary restriction to \(Y_0\). Indeed the specialization unit for \(K\) corresponds, under proper direct image, to the unit \(\mathbb Q_{Y_0}\to\psi\mathbb Q_Y\), which is the canonical smooth isomorphism used above.

Choose a lift of \(1\in D^*\). The assumed product and the contractibility of \(\widetilde D^*\) identify \(H^*(\widetilde Y^*)\) with \(H^*(Y_1)\) by actual restriction to that lift. Pullback from \(H^*(Y)\) therefore becomes \(r_1\) and is an isomorphism. Since its composite with (29) is \(r_0\), the latter map is also an isomorphism on cohomology.

The comparison was constructed on \(B\times D\) so that it is equivariant for the required operator: because \(T\) is over \(B\times D^*\), it induces an endomorphism of \(K|_{B\times D^*}\) and hence of \(\widetilde K\), \(R\widetilde j_{B,*}\widetilde K\), and \(\psi K\). Every arrow in (29) respects these actions. Restricting at the chosen lift identifies the action on its domain with \(T_1^*\), and its action on the codomain is \(H^*(\tau_0)\). Thus the displayed formula for \(H^*(\tau_0)\) follows. No extension of \(T\) to an automorphism of \(Y\) is asserted or needed. ◻

Proof. Let \(O_0=S\times X\) using the scalar product identification of Lemma 13, and write \(q=\operatorname{id}_S\times h\). Both maps in the scalar flag correspondence of Figure 1 at \(\lambda=0\) are over \(\mathcal E\). The flag projection has relative complex dimension \(b\), and the scalar inclusion has complex codimension \(b\). Thus their pullbacks and proper Gysin maps give, with \[\begin{gathered} F_O=j_*Rq_*\mathbb Q_{O_0},\qquad F_Y=Rg_{0,*}\mathbb Q_{Y_{0,0}},\\ F_Z=R(g_0\circ\iota)_*\mathbb Q_{Z_0}, \end{gathered}\] morphisms \[\mathsf B_0:F_O\longrightarrow F_Y[2b], \qquad \mathsf A_0:F_Y\longrightarrow F_O[-2b].\] Composing these with the appropriate shifts of \(p(\tau_0)^2\) gives an endomorphism of \(F_O\) with zero net shift. Its cohomological action is \(\mathsf A_0N_{0,0}^2\mathsf B_0\). Since direct image by the closed embedding \(j\) is fully faithful, this is an endomorphism of \(Rq_*\mathbb Q_{O_0}\) over \(S\times A\).

Let \(p_S:S\times A\to A\) be the projection, and write \(K=Rh_*\mathbb Q_X\). The product identification gives \[Rp_{S,*}Rq_*\mathbb Q_{O_0} \simeq K\otimes R\Gamma(S,\mathbb Q).\] Push the preceding endomorphism to \(A\), precompose with the projection unit \(K\to K\otimes R\Gamma(S,\mathbb Q)\), and postcompose with the orientation trace \[K\otimes R\Gamma(S,\mathbb Q)\longrightarrow K[-2].\] This constructs (30). On ordinary cohomology the unit is \(\operatorname{pr}_0^*\), the trace is \(\operatorname{pr}_{0,*}\), and the middle map is the flag composite already identified. Hence the resulting operator is exactly \(f'\).

For completeness, set \(K_{\mathrm{norm}}=K[R]\), the normalization used in the statement of the theorem. Functoriality of perverse truncation applied to (30) gives \[{}^p\tau_{\le k}K_{\mathrm{norm}} \longrightarrow{}^p\tau_{\le k}(K_{\mathrm{norm}}[-2]) =({}^p\tau_{\le k-2}K_{\mathrm{norm}})[-2].\] Taking hypercohomology in degree \(m-R\) and then its image in ordinary cohomology proves the stated inclusion. This uses neither a splitting nor multiplicativity of the perverse filtration. ◻

Proof. Work first on \(Y_{0,v}\). Denote by \(i:\Delta\hookrightarrow C\times Y_{0,v}\) the graph of the moving point. Its normal line is the pullback of \(x^*T_C\), so \[\mathcal O(\Delta)|_\Delta\simeq x^*T_C, \qquad c_1(N_\Delta)=t_C.\] The residue eigenvalues \(v_i\) are distinct. Consequently the Higgs operator has uniquely labeled eigenline summands on the formal neighborhood of \(\Delta\). This follows by lifting its simple residual eigenvalues and their idempotent projectors to every finite order. Multiplying a local frame of the logarithmic canonical line by a unit does not change those projectors. Thus the formal summands are intrinsic, and any finite number of their jets suffices for an elementary modification.

The \(k\)-fold translation replaces the \(i\)th formal line lattice by its twist by \(k\ell_i\Delta\). On the \((i,j)\) block of \(\operatorname{End}E\), the lattice shift is therefore \[m_{ij}=k(\ell_i-\ell_j).\] If a line block has restriction \(F\) to \(\Delta\), its K-theory change under an integer shift \(m\) is \[ [\text{new block}]-[\text{old block}] =\sum_{a=1}^{m}[i_*(F\otimes N_\Delta^{\otimes a})]. \tag{33}\] For \(m<0\), the right side means the negative of the sum over \(m<a\le0\); for \(m=0\) it is zero. For positive \(m\), the formula follows by successive quotients of the lattices at pole orders \(1,\ldots,m\). For negative \(m\), apply the same quotient sequence between the smaller lattice and the original one. In particular \(m=-1\) contributes \(-[i_*F]\), as it must for the kernel elementary transformation.

The resulting identity for \(\operatorname{End}E\) is global in K-theory, before applying a characteristic class. Indeed the intrinsic formal idempotents lie in the descended endomorphism algebra, so its \((i,j)\) blocks are global formal line bundles. Identify the original and modified endomorphism bundles away from \(\Delta\), and choose a common sufficiently small lattice inside both. Their quotients by this lattice are supported on a finite infinitesimal neighborhood of \(\Delta\). The formal block projectors and the filtration by pole order give global finite filtrations of these quotients. Their associated graded terms are graph pushforwards of the descended line bundles \(Q_i\otimes Q_j^{-1}\otimes N_\Delta^{\otimes a}\). Subtracting the two quotient classes cancels the common pole orders and leaves exactly the signed sums in (33). Additivity for these filtrations therefore gives the identity in the Grothendieck group of coherent sheaves on \(C\times Y_{0,v}\); the common lattice also cancels.

Grothendieck–Riemann–Roch for the graph (Fulton 1998, sec. 15.2) gives \[\operatorname{ch}(i_*F) =i_*\bigl(\operatorname{ch}(F)\operatorname{td}(N_\Delta)^{-1}\bigr).\] Hence the contribution of a summand in (33) to \(\int_C\operatorname{ch}_2\) is \[c_1(F)+(a-\tfrac12)t_C.\] This also exhibits the sign of the normal-bundle correction: the normal line is the tangent line, and its inverse Todd class starts with \(1-t_C/2\). For the signed sums just specified, the elementary identities \[\sum_{a=1}^{m}1=m, \qquad \sum_{a=1}^{m}(a-\tfrac12)=\frac{m^2}{2}\] hold for every integer \(m\).

Write \(w_i=c_1(Q_i)\) on a universal-bundle chart. Only \(w_i-w_j\) and expressions invariant under common translation of the \(w_i\) will occur. Summing the contributions for \(F=Q_i\otimes Q_j^{-1}\) gives \[T^{k*}e_{\mathcal Y}-e_{\mathcal Y} =\frac1{2n}\sum_{i,j} \left(k(\ell_i-\ell_j)(w_i-w_j) +\frac{k^2}{2}(\ell_i-\ell_j)^2t_C\right).\] Since \(\sum_i\ell_i=0\), \[\sum_{i,j}(\ell_i-\ell_j)(w_i-w_j)=2n\sum_i\ell_iw_i, \qquad \sum_{i,j}(\ell_i-\ell_j)^2=2n\sum_i\ell_i^2.\] For the chosen cocharacter these are \(2ny\) and \(4n\), respectively, proving (31). Since the lattice identity was global on the coarse moduli space before applying Grothendieck–Riemann–Roch, this is a global cohomological identity there.

Let \(M\) be the integer used to define \(N\). Replace \(k\) by \(Mq\) in (31). Since \(\mathsf T^{Mq}=\exp(MqN)\), both sides are polynomial functions of the nonnegative integer \(q\) with values in a finite-dimensional cohomology group. They agree as polynomials. Their linear coefficients are respectively \(MN(e_{\mathcal Y})\) and \(My\), giving (32) on \(Y_{0,v}\). Corollary 15 transports this identity to \(H^*(\mathcal Y)\) and then to all its fibers: the three classes in (31) are restrictions of the global classes just defined. ◻

Proof. The projective-bundle pushforward formula is a sum over the roots, with denominator the product of the other-root differences. Iterating it along the full-flag tower gives the sum over ordered roots, with denominator the tangent Euler class at that ordering. Multiplication by \(c_b(T_\pi)\) cancels this denominator, giving the first identity. One may perform this calculation after passing to a splitting space and inverting root differences; the resulting equality is a polynomial identity in the universal Chern classes, so it holds without the inversion. The projective-bundle pushforward is characterized by its Chern-polynomial relation and the normalization that the top power of the hyperplane class integrates to one; using the complex tangent Euler class fixes the orientation signs in the displayed cancellation.

For (34), a fixed root occurs \((n-1)!\) times at a fixed position, and a fixed ordered pair of distinct roots occurs \((n-2)!\) times at two prescribed distinct positions. Since \(\sum_{i\ne j}\ell_i\ell_j=-\|\ell\|^2\), expansion of the square gives \[\begin{split} &(n-1)!\|\ell\|^2\sum_iw_i^2 -(n-2)!\|\ell\|^2\sum_{i\ne j}w_iw_j\\ &\hspace{15mm} =n(n-2)!\|\ell\|^2 \left(\sum_iw_i^2-\frac{(\sum_iw_i)^2}{n}\right). \end{split}\] Finally, rational pullback from \(B\mathrm{PGL}_n\) to \(B\mathrm{GL}_n\) is injective: on a maximal torus it is the inclusion of symmetric polynomials in the root differences into symmetric polynomials in all roots. Universal characteristic-class identities invariant under common root translation can therefore be checked after this pullback, proving the last assertion. ◻

Proof. Work on the scalar correspondence at \(\lambda=0\). The restriction of \(e_{\mathcal Y}\) along \(\iota:Z_0\hookrightarrow Y_{0,0}\) equals \(\pi^*\operatorname{pr}_0^*e\): the universal projective characteristic class is unchanged by the line twist \(J\) in the scalar product identification. Projection formulas therefore show that the maps outside \(N_{0,0}^2\) in \(f'\) intertwine multiplication by the corresponding classes \(e\) and \(e_{\mathcal Y}\).

For a degree-zero derivation \(N\) and an even class \(a\), write \(m_a\) for multiplication by \(a\). Then \([N,m_a]=m_{N(a)}\), and direct expansion gives \[[[N^2,m_a],m_a]=2m_{N(a)^2}.\] Indeed the first commutator is \(m_{N^2(a)}+2m_{N(a)}N\); the first term commutes with \(m_a\), while the second contributes \(2m_{N(a)^2}\). Thus no vanishing assertion about \(N^2(a)\) is needed. By Lemma 27, \[ [[f',e],e] =2\operatorname{pr}_{0,*}\mathsf A_0 (y^2\cup-)\mathsf B_0\operatorname{pr}_0^*. \tag{37}\]

The normal bundle of \(\iota\) is \(T_\pi^*\). Self-intersection therefore replaces \(\iota^*\iota_*\) in (37) by \[c_b(T_\pi^*)\cup-=(-1)^b c_b(T_\pi)\cup-.\] For a class pulled back from \(X\), the projection formula and Lemma 28 reduce the flag integral to the symmetric class \[\begin{split} \sum_{\sigma\in\mathfrak S_n} (w_{\sigma(1)}-w_{\sigma(2)})^2 &=2n(n-2)! \left(\sum_iw_i^2-\frac{(\sum_iw_i)^2}{n}\right)\\ &=4n(n-2)!\,\widetilde{\operatorname{ch}}_2(E_{x(s)}). \end{split}\] Here \(E_{x(s)}\) denotes the universal projective data restricted to the graph of \(x\); the normalized expression is well defined without an honest universal vector bundle.

Under \(O_0\simeq S\times X\), this characteristic class is the pullback of \(\widetilde{\operatorname{ch}}_2(E)\) from \(C\times X\) along \(x\times\operatorname{id}_X\). Tensoring by \(J\) contributes nothing to it. Integration along \(S\) multiplies integration along \(C\) by \(\delta\): \[\operatorname{pr}_{0,*} \widetilde{\operatorname{ch}}_2(E_{x(s)})=\delta e.\] This follows directly from the degree formula \(x_*[S]=\delta[C]\), or from the \(C\)-degree-two term in the Künneth decomposition. The other curve degrees integrate to zero. Combining this equality, the factor \(2\) in (37), and the self-intersection sign proves (35).

The integer \(\delta\) is positive because \(x\) is a nonempty finite étale cover of connected projective curves. Thus (36) is a nonzero rational rescaling and gives the stated double commutator. The degree of \(f'\) is \(-2\), since the two flag shifts cancel and integration over \(S\) has degree \(-2\). The filtration bounds are Propositions 26 and 23; they are unchanged by the rescaling. ◻