Cynk–Hulek local clusters and equivariant derived induction

A. Three complete local cluster models. For a prime-order group a nonfree orbit is a single fixed point. A regular-representation cluster has invariant part generated by its unit. Every invariant function f therefore acts as a scalar on the ENTIRE cluster algebra: $f\cdot 1=a\cdot 1$ and $f(h\cdot 1)=h(f\cdot 1)=a(h\cdot 1)$. In families the invariant unit line is canonically $O_T$, so invariant coordinates define an actual map T to the spectator space. Consequently a local product with a fixed smooth spectator B has cluster scheme B times the following normal-direction cluster scheme, with universal kernel the diagonal of B times the normal universal kernel. This is an equality of functors on base schemes, not just a count of closed points. (A1) For μ2 on A² with (x,y)->(-x,-y), the two cluster charts are y=u x, x²=t; x=v y, y²=s. They are affine planes, with $v=u^{-1},s=u^{2}t$ on the overlap. They give $Tot(O_{P1}(-2))$. In every chart the cluster algebra has the free regular basis(1,x) or(1,y), so the displayed equations define the actual universal flat family. Conversely every length2 regular cluster has such a basis, which proves coverage. General t≠0 is a free orbit, so the whole chart is in the distinguished component. The invariant form extends as dx∧dy = (1/2)dt∧du on the first chart and similarly on the second. Thus the resolution is smooth and crepant, with no hidden singular intermediate model. (A2) For μ3 on A² with (x,y)->(ζx,ζ²y), there are three charts Ux: y=a x², x³=b; U0: x²=e y, y²=f x, xy=ef; Uy: x=c y², y³=d. Their cluster bases are(1,x,x²),(1,x,y),(1,y,y²). On Ux∩U0, e=a^-1, f=a²b, and on Uy∩U0, f=c^-1, e=c²d. Every regular cluster is in one of these basis charts: each nontrivial character has rank one; if both linear character coordinates survive use(1,x,y), and if one is absent its character must be supplied by the square of the other. In the middle chart associativity forces $xy=ef$, so there is no omitted equation or extra embedded component. The quotient invariants x³,y³,xy become(e²f,ef²,ef); hence the target is $uv=w^3$ and the exceptional fibre is the two P1s meeting in the middle origin. Each chart is a smooth affine plane and contains free-orbit points. The volume is dx∧dy = (1/3)db∧da = (1/3)de∧df on the first two charts, with the analogous nonzero expression on Uy. This is the actual A2 crepant resolution. (3s) For scalar μ3 on A³, with all three coordinates of weight ζ², every regular cluster has basis(1,x,x²) for some nonzero direction x: the only linear character present is ζ² and its square must supply ζ. On the chart where the chosen direction has nonzero first coordinate, write y=u x, z=v x, x³=t. The three such charts glue to $Tot(O_{P2}(-3))$. Their universal algebras are free of rank3 with the indicated regular basis. They contain the free orbits t≠0 and cover all clusters, including every supported length3 algebra. The exceptional fibre is P2 and dx∧dy∧dz = (1/3)dt∧du∧dv. Thus the scalar3 model is smooth and crepant.

B. The surface universal kernels are actually fully faithful. Here is the local argument for(A1)/(A2), so there is no appeal to a higher-dimensional BKR theorem. Let $U=A^2$, $K=\mu 2$ orμ3 as above, Y its explicit smooth cluster resolution, C⊂Y×U the universal cluster, and Φ(A)=Rq_*(O_C tensor p^*A). Both kernel-support projections are proper: p is finite flat, and C is closed in Y×(U/K)U, which is proper over U. The cluster algebra is finite free over an affine Y-chart; its coordinate actions are commuting matrices. The Koszul complex of $x-M_x,y-M_y$ on $O_Y[x,y]$ tensor $p_{\ast }{O}_C$ is an explicit length2 relative free resolution. Its dual trace gives a left adjoint Ψ. Residue duality can equally be computed after compactifying A² to P², since the cluster supports avoid infinity; thus Ext²_K(O_Cy,O_Cz)=Hom_K(O_Cz,O_Cy)^* with the invariant dx∧dy, and no projectivity of A² is required. The convolution Q representing ΨΦ is perfect on smooth Y×Y. Its point-Hom is $Ex{t}_K^i(O_{Cy},O_{Cz})$, by the finite Koszul trace and derived fibre restriction. A nonzero equivariant cluster homomorphism sends the invariant cyclic generator1 to a scalar, so is an isomorphism; distinct clusters have Hom0 and, by residue duality, Ext²0. Off the diagonal the only possible degree is1. A minimal free complex for Q there has only one term, a shifted vector bundle. But its support is in Y×(U/K)Y, a proper subset (exceptional product dimension2 in ambient dimension4). Such a vector bundle is zero. Hence Q is supported on the diagonal. For y∈Y, $E=\mathrm{\Psi }\mathrm{\Phi }(k_y)$ is perfect, supported at y and has free amplitude[-2,0]. The counit $E->k_y$ is the identity on degree-zero scalar Hom. Its degree-one tangent map is injective: a module deformation of a regular cluster has a unique invariant generator up to a scalar unit, so a trivial module deformation makes the actual quotient ideal deformation trivial. The long Hom sequence of the counit triangle and minimality give $H^0(E)=k_y$. Finally a length2 free complex over the regular two-dimensional local ring with finite-length homology and $H0=k$ is acyclic in positive homological degrees: its top kernel is finite length in a free module and zero; the next boundary is free of depth2, so a nonzero next homology socle would lift through $Ex{t}^1(k,boundary)=0$ to a socle in a submodule of a free module, impossible. Thus $E\simeq k_y$. The counit cone kernel has zero derived fibre everywhere, so derived Nakayama makes it zero. This proves the ACTUAL local unit equivalence for the universal kernel. Only this full-faithfulness/unit conclusion is needed for the global argument.

C. The scalar-threefold universal kernel is an actual equivalence. Put $Y=Tot(O_{P2}(-3))$, $S=C[x,y,z]$, $A=S⋊\mu 3$. The tautological eigensheaves $R_i=p_{\ast }{O}_C[{\chi }^i]$ are $O_Y(i)$, $i=0,1,2$: on a direction chart their frames are1,x,x² and the overlap frames have precisely those transition factors. Let $T_i=R_i^{\mathrm{v}\mathrm{e}\mathrm{e}}=O_Y(-i)$ and $T=\text{direct sum}$ $T_i$. Since the bundle projection $Y->P2$ is affine, Ext^q(T_i,T_j)=direct_sum_(r>=0) H^q(P2,O(i−j+3r))=0, q>0. This follows directly from the standard projective Čech monomial contraction: positive higher cohomology of O(k) on P2 can survive only for $k\le -3$, whereas $i-j+3r\ge -2$. Also Hom(T_i,T_j)=direct_sum_(r>=0) S_(i−j+3r) = e_i A e_j for the Fourier character idempotents, with the opposite-ring convention appropriate to left projective modules $A{e}_i$. All composition maps are polynomial multiplication. Hence $End(T)=A^{\mathrm{o}\mathrm{p}}$. The three line bundles compactly generate. The rank2 section defining the diagonal in P2×P2 has its exact Koszul resolution. Swapping factors and twisting first by O(-2), second by O(2) leaves $O_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}$ unchanged and expresses its identity transform using the three coefficient complexes RΓ(F),RΓ(F(1)),RΓ(F(2)). Therefore their simultaneous vanishing forces $F=0$. Apply this to the conservative affine pushforward of a complex from Y. Compactness follows from the finite affine Čech cover. The usual bar-cell unit/counit argument now identifies the derived category with A-modules: they agree on T and A, extend to free bar resolutions, and compactness makes every compact object a retract of a finite cell stage. For clarity this is the UNIVERSAL-CLUSTER functor, not merely an unrelated tilting equivalence. Dualizing its explicit coordinate-matrix Koszul resolution identifies its left adjoint on $A{e}_i$ with $R_i^{\mathrm{v}\mathrm{e}\mathrm{e}}=T_i$. Its map on Hom is exactly the multiplication matrix just computed. Thus that adjoint agrees on the projective generators with the constructed Morita functor. The algebra A has global dimension3: an equivariant finite S-free resolution has length3, and an S-projective final syzygy is A-projective by averaging an S-linear splitting. Therefore every bounded finite module complex is generated by these projectives. The universal adjoint, and hence Φ, is the actual equivalence. These constructions commute with every commuting automorphism, since coordinate actions and character projections are natural.

D. Global gluing and spectators. For a finite action on a smooth complex variety, average a set of local parameters to eigenparameters at a fixed point. The induced map on cotangent spaces is an isomorphism, and the completed local ring is the equivariant power-series ring in these parameters; this follows by successive power-series substitution in the smooth regular local ring. Thus the completed local cluster functor is the product inA with its fixed spectator formal variables. This includes infinitesimal families, because the invariant-unit calculation inA holds over every base algebra. It proves smoothness of the global distinguished Hilbert component and identifies its local crepant volume byA. The global component is projective: impose group invariance on the projective length-m Hilbert scheme, split the finite flat universal section bundle by averaging, impose its regular multiplicities (an open-and-closed rank condition), then take the reduced closure of the free orbits. All three local chart families are in that closure. The kernel is its universal $O_C$ with its NATURAL residual linearization. Perfection and the global adjoints follow by the coordinate-matrix Koszul resolutions and projective trace/duality. Its unit is a coherent perfect kernel. Complete at each quotient orbit and use the formal product description. It becomes the identity spectator kernel times the unit inB orC, hence an isomorphism. Flat completion and finite affine Čech base change commute with the proper kernel convolution; faithful flatness detects its coherent cone. Thus the global Φ is fully faithful. This uses no arbitrary-dimensional intersection theorem: the only nontrivial normal kernels wereB/C. In our induction the source variety is connected smooth projective with a group-invariant nowhere-zero canonical volume. The target is smooth projective crepant byA, so both Serre functors are[n]. The fully faithful image has both adjoints, and Serre[n] makes its left and right orthogonals equal. The equivariant target category is indecomposable: $O_V\otimes \rho$ is indecomposable for every irreducible G-representationρ because $H0(O_V)=C$; each has a nonzero map to a free-orbit sheaf, and every simple equivariant point-orbit sheaf receives a map from some suchρ. If a coherent sheaf lay in a second orthogonal summand, a nonzero simple point-orbit quotient would contradict orthogonality. A nonzero bounded complex is likewise detected by a point quotient of its highest nonzero cohomology. Therefore the image is the whole category. This proves a GLOBAL equivalence, with actual kernel units/counits, not just pointwise Ext agreement.

E. Residual action induction, with all chart coordinates checked. For m2 suppose the old residual odd involution has a fixed divisor and tangent normal weight−1. The new even action on $X_k\times E_{k+1}$ has normal type(-1,-1); its odd residual h acts only on the NEW elliptic coordinate. On the A1 chart $(y=ux,x^2=t)$, h(u,t)=(-u,t). The other chart is identical. Thus the new odd fixed locus is again divisorial, with exactly one normal weight−1. Away the even fixed locus, a residual-fixed free orbit lies either over a fixed new elliptic point or over an old odd-fixed point; the same tangent description follows from the local quotient isomorphism. No other odd-fixed locus occurs. For m3 use induction with odd fixed types B1: codimension1, normal weightζ; B2: codimension2, normal weights(ζ²,ζ²). The new even generator is(old $odd,g_{\mathrm{n}\mathrm{e}\mathrm{w}}^{-1}$). Its fixed normal types are(ζ,ζ²) over B1 and(ζ²,ζ²,ζ²) over B2. The new residual h acts only byζ on the last coordinate. On A2, with x old and y new, h acts on Ux as(a,b)->(ζa,b), on Uy as(c,d)->(ζc,d), and on U0 as(e,f)->(ζ²e,ζ²f). Thus there are exactly the divisorial weightζ pieces and the codimension2 weight(ζ²,ζ²) middle piece. These formulae include the coordinate along the resolved base, not merely a direction in the exceptional fibre. On scalar3, with x,y old and z new, the x-direction chart has (u=y/x,v=z/x,t=x³) -> (u,ζv,t), and similarly for the y chart. The z-direction chart has (u=x/z,v=y/z,t=z³) -> (ζ²u,ζ²v,t). Again the new odd types are precisely B1 andB2. Outside the even fixed locus, a residual-fixed free orbit satisfies $h(v,c)=g_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}^r(v,c)$. If c is not fixed then $r=-1$ and the induced local residual action is the old odd action; if c is fixed and v is not old-fixed it is the new elliptic action. The remaining case is exactly the fixed-point chart calculation. Thus the induction covers ALL fixed points and retains precisely the two stated normal types. The initial case is E1, with its usual involution or order3 automorphism, so induction applies for every n. The forms inA show that every even action preserves the current volume and every resolution is crepant; the odd residual acts on the canonical form by the original root eigenvalue. This also verifies the induction's canonical-line hypotheses.

F. Full Gn comparison and strict CY geometry. Write $H=G_{n-1}$ on the first n-1 factors. It is normal in $G_n$, and the quotient is μm, split by the generator $e_{n-1}-e_n$. Equivariant objects can be computed successively: a $G_n$-linearization is exactly an H-linearization plus a quotient linearization with the actual conjugation coherence. Here the groups are abelian and the splitting is explicit, so this follows directly by writing the cocycle identities. The universal kernel at the previous stage is canonically equivariant for the residual odd action, as it is defined by the Hilbert functor itself. Exterior product with $O_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}(E_n)$ therefore gives Db_Gn(product E_i) ≃ Db_μm(X_(n-1)×E_n) ≃ Db(X_n) byD. Kernel convolution gives the claimed algebraic equivalence with residual action, recursively and with its actual cocycles. The composite quotient resolution is projective birational. The invariant volume on the original product descends and pulls back to the nowhere-zero volume inA at every stage. The quotient is Gorenstein as well: on a quotient affine chart, invariant Noether-normalization parameters give a regular local base over which the smooth cover is finite free; its Reynolds summand is Cohen–Macaulay. Finite free trace duality identifies its canonical module with the invariants of the cover canonical module, which is its invariant ring because the volume is invariant. This elementary calculation works in the actual ambient dimension and is not a derived-McKay theorem. Finally a holomorphic p-form on $X_n$ pulls back on the common finite-cover smooth open to a $G_n$-invariant form on the product. Its complement upstairs has codimension at least2, so its coefficient functions extend by Hartogs on the smooth product. This gives an INJECTION into $H^0(\mathrm{product}{E}_i,{\mathrm{\Omega }}^p{)}^{G_n}$; no general extension theorem for klt differential forms is assumed. The product forms are wedges of the n elliptic differentials. A wedge indexed by a subset I is invariant exactly when its indicator vector is constant modulo m, since $G_n$ is the kernel of sum. For $0<p<n$ this is impossible for m2 or3. Hence the required intermediate Hodge groups vanish. The top volume and connectedness were already proved. Thus these are exactly the source strict-CY resolutions, not only arbitrary smooth models with an assumed canonical line.