Lemma 9 ([Bounded degeneration characters]). In the semistable construction above, let R_i be the Stein block covers, of dimensions less than n , over the final curve field. Let c be the marked point, u its uniformizer, and g a …

blocks. Let the block have dimension s . On its projective semistable model over a disc, the direct image of the relative log top line is the extended Hodge line F^s in degree s . This is the semistable comparison theorem for logarithmic de Rham cohomology and the canonical extension with nilpotent residue . The normalized form is a local frame: it is a regular relative log form but cannot be divided by u while remaining regular, because its order on one component is zero. Its eigenvalue on the central fibre of F^s is λ_i . The integral local system in degree s has rank 2s s . Choose a coordinate in which the finite base action is a rotation. On the universal cover of the punctured disc, the lifted transformation, combined with parallel transport, acts by an integral matrix A commuting with the unipotent monodromy. A power of A is a power of monodromy, because the original transformation has finite order. Passing from flat frames to canonical-extension frames multiplies A by the exponential of a scalar multiple of the logarithm of monodromy. This is a commuting unipotent factor, so it does not change the eigenvalues. Hence a primitive root of the order of λ_i occurs among the eigenvalues of an integral matrix of rank 2s s . Its cyclotomic degree is at most that rank. There are only finitely many possible orders, uniformly for s<n . blocks: a good special-fibre model. Let s be the block dimension, write η for its normalized form, and let G be the finite cyclic group generated by the transformation being considered. We will bound the character by applying Lemma 4 ([Finite-order log volume characters]) to a normal component of the special fibre. This requires a degree-one log generator on that component and a bounded power of the transformation preserving it. We first construct an equivariant good model on which the normalized logarithmic form generates along the whole special fibre; we then use structure-sheaf cohomology to find the required power. We claim that, near the marked point, there is a G -equivariant projective model Y such that Y is klt, T=Y_c is reduced Cartier, (Y,T) is dlt, K_Y is semiample over the curve. The geometric generic fibre need only remain birational to the block. Start from an equivariant smooth projective compactification Q of the semistable model, preserving its SNC special fibre. A relative K_Q -MMP is also a (K_Q+Q_c) -MMP, because Q_c is a fibre. For existence and termination, take the finite quotient Q=Q/G and its branch boundary Δ , so that K_Q is the finite pullback of K_ Q +Δ . The quotient is -factorial and (Q,Δ) is klt. Its adjoint is pseudo-effective over the quotient curve: an effective pluricanonical form on the generic fibre upstairs gives an invariant one after multiplying its finitely many translates. Choose a divisor H of sufficiently large degree on the quotient curve, represented by many general points with small rational coefficients. The pair obtained by adding its pullback stays klt, and its adjoint is pseudo-effective absolutely, since a generic section extends after a sufficiently positive base twist. Choose also H>2 Q . The good-model theorem of , followed by a terminating MMP with scaling to that good model, now applies. Every step is over the curve: otherwise the extremal-ray length bound for (Q,Δ) would give a horizontal generating curve C_0 with -(K_ Q +Δ)·C_0≤2 Q , while the pullback of H has degree greater than this on C_0 . Inductively the base pullback and the morphism to the curve persist. Lift each step by normalization in the upstairs function field, using Stein factorization for the contractions. Finite pullback preserves the relative negativity and ampleness needed for flips. No divisors are extracted, and the branch-canonical equality remains valid in codimension one. This gives the finite-group equivariant MMP; invariant -factoriality suffices upstairs . The steps factor over the upstairs curve as well, by normalization in its function field. They preserve dlt for the pair with the special fibre added. Special-fibre multiplicities remain one because there is no extraction. The klt total space is Cohen--Macaulay, so its Cartier fibre has no embedded associated components; generic reducedness therefore makes the fibre reduced scheme-theoretically. Relative semiampleness pulls back from the downstairs good model. This proves. degree-one log generator. The logarithmic divisor of η∧du/u on Y is effective near T : this held on the semistable model, and the program extracts no divisors. The generic fibre has Kodaira dimension zero. Consequently the semiample contraction of K_Y has zero-dimensional image on the generic fibre; its Stein image over the curve is the curve itself. Thus K_Y , and also K_Y+T , is rationally linearly trivial over a neighbourhood of c . The effective log divisor has no horizontal part, since its restriction to the generic fibre is effective and rationally trivial. A principalization of a multiple uses a rational function with vertical divisor, hence a base function. The log divisor is therefore aT for some a∈_ ≥0 . If a>0 , every weight quotient is at least a : it is a plus the log discrepancy of (Y,T) divided by the fibre multiplicity. This contradicts w_u(η)=0 . Hence _ (η∧du/u)=0. In particular K_Y+T , and therefore K_Y , is Cartier near T . Adjunction makes T Gorenstein with trivial dualizing line, generated by the residue of. The transformation acts on that generator by λ : the residue of its action on du/u is one. If a prime component S⊂T is preserved by a power of the transformation, the residue on S generates K_S+Θ_S , where Θ_S is its different. Dlt adjunction makes (S,Θ_S) normal lc and effective. Moreover Θ_S is integral, since the divisor of this degree-one rational form is integral and equals -Θ_S . Its coefficients are at most one, so it is reduced. The character on this component generator is the corresponding power of λ . Thus a bounded power preserving one component will reduce the character bound to Lemma 4 ([Finite-order log volume characters]). We obtain such a power from the structure-sheaf cohomology of T . a preserved component. The family is flat over the smooth curve. Its central fibre is slc by, and its nearby fibres are klt after shrinking; these fibres are Du Bois. Cohomology and base change for proper flat Du Bois families makes h^j(_ Y_t ) locally constant . The nearby fibres are birational to the block and have rational singularities. Their structure-sheaf cohomology is therefore that of a smooth resolution of the block. Extension of differential forms on klt spaces, together with Hodge symmetry, identifies these dimensions with the form algebra of the block . For an irreducible Calabi--Yau block, the only nonzero groups H^j(T,_T) occur in degrees 0 and s , and each has dimension one. Serre duality and the log generator give alternating trace 1+(-1)^sλ^ -1 . If λ has order greater than two, this is nonzero. In the symplectic case there is one dimension in each even degree and none in odd degree. For some power of the transformation of exponent at most s+1 the trace is nonzero. Indeed, if the power sums of the finitely many nonzero eigenvalues all vanished through their number, Newton's identities would force their product to vanish. The coherent Lefschetz theorem implies that a finite-order transformation with nonzero alternating trace has a fixed point . Its vanishing statement when there are no fixed points applies to the possibly singular projective scheme T , for example by equivariant embedding and coherent Lefschetz localization. At a fixed point at most s+1 components of T meet: intersections of components of a dlt boundary are lc strata, with the generic SNC codimension description. A further power of exponent dividing (s+1)! therefore preserves a prime component through that point. Its reduced log Calabi--Yau pair has dimension s<n and the degree-one generator just constructed. Lemma 4 ([Finite-order log volume characters]), which follows from L_s , bounds the order of its character. This bounds the order of λ as well. Taking a common multiple over all s<n and both types of block proves the lemma.