Stability on listed higher-dimensional CY examples

On every required product of elliptic curves choose degree-one numerical classes and the F2 condition. Its invariance includes the diagonal groups in the Cynk–Hulek constructions, and includes the source factorwise surface actions and every permutation of their repeated two-dimensional blocks. For an abelian surface quotient $S=(C_1\times C_2)\mathrm{/}G$ in (i), the free quotient being abelian forces $G$ to act by translations: its invariant two-dimensional one-form space equals the whole product one-form space, so every derivative is the identity; an elliptic-product automorphism with identity derivative is a translation. Thus it too is factorwise and is covered. Enriques and bielliptic surface actions are the explicitly stated factorwise affine automorphisms. The orbit-lattice construction in F2 gives a genuinely equivariant numerical support lattice for each finite wreath group, not a possibly order-dependent Liu lattice declared invariant.

Construct the surface and repeated-product categorical comparisons. A free quotient $A\to S$ has the exact descent equivalence $D^b(S)\simeq D_G^b(A)$ from F4, with its natural line-bundle linearizations. For the Enriques case, first use the explicit A1 Kummer resolution kernel from F5 for $A=C_1\times C_2$ and $⟨(-1,-1)⟩$. The residual free involution $\tau$ acts naturally on that universal-cluster kernel, so finite free descent gives $D^b(S)\simeq D_{⟨-1,\tau ⟩}^b(A)$. Its freeness is the actual source Enriques condition: both two-torsion parameters are nonzero, as proved in F6. Taking the exterior tensor product of the actual Fourier–Mukai kernels preserves the residual actions and has the canonical permutation/Koszul coherence. F3 then supplies $D^b(S^{[n]})\simeq D_{S_n}^b(S^n)\simeq D_{G^n⋊S_n}^b(A^n),$ with the appropriate surface group $G$ in each case. This uses the actual all-n Hilbert kernel, not the surface or threefold McKay theorem outside its scope.

In (iv), the F2 condition on $C_1\times \cdots \times C_n$ is invariant under the actual diagonal group $G_n$ for $m=2$ or $3$. F4 induces a condition on its equivariant category. The F5 universal-kernel induction is an actual equivalence of that category with $D^b(X_n)$, for the source recursive Cynk–Hulek resolution, so transports the condition with its HN filtrations, local finiteness and numerical support. F5 independently proves smooth projectivity, crepancy and strict Hodge vanishing for this precise named construction. No stability on an arbitrary product or arbitrary-dimensional McKay theorem has been inserted.

Apply finite equivariant induction in F4 to the F2 wreath-invariant product condition, and transport along the equivalence of step 1.2. This gives actual locally finite support-property conditions on the named surfaces themselves (take $n=1$) and on every required $S^{[n]}$. The support remains a finite-rank numerical coimage: forgetting an equivariant structure preserves numerical equivalence by its induction adjunction, and the actual Fourier–Mukai equivalence preserves Euler pairings. These are full stability conditions with finite HN filtrations, not only invariant central charges.

For the abelian $S$ in (i), translation gives the actual isogeny-trivialization $S^{[n]}{\times }_{S,[n]}S\simeq K_n(S)\times S,(Z,a)\mapsto (Z-a,a)\mathrm{.}$ The fibre is smooth by the differential of translation, and connected because its symmetric-product image is the quotient of the connected kernel of $\sum :S^n\to S$ and Hilbert–Chow has connected fibres. For every finite-order flat line $L$ on $S$, ${\alpha }_n^{\ast }L$ is its Hilbert norm line: their characters are both the product of the point-loop characters. The exact F3/F4 norm kernel identity turns this tensor into the external product of the pullback line on $C_1\times C_2$, which splits into finite-order flat lines on its elliptic factors. The actual F2 Pic0 slicing invariance therefore makes the constructed Hilbert condition invariant under these ${\alpha }_n^{\ast }L$. F6's actual fibre-restriction theorem now gives a locally finite support-property condition on $K_n(S)$. The cases where this fibre is a point have the ordinary vector-space condition directly. No strict-CY label is imposed on this generalized Kummer family.

In (ii), F6 constructs the canonical universal double cover $\pi :\widetilde{S^{[n]}}\to S^{[n]}$ and its strict Calabi–Yau geometry for $n\ge 2$. Its character line is the canonical line ${\omega }_{S^{[n]}}$. Under the actual equivalence of step 1.2, the Serre-functor trace compatibility identifies tensor by that line with tensor by the canonical character of the equivariant elliptic product. Its underlying canonical bundle is trivial; permutations of two-dimensional surface blocks have determinant $+1$, and the residual Enriques involution has canonical character $-1$. The induced condition is invariant under ANY character twist because its underlying forgotten object and charge do not change. Thus the canonical character line preserves the actual slicing downstairs. F4's complete finite étale lifting construction supplies the required stability upstairs, with HN, local finiteness and support. This proves the Enriques and even-dimensional strict-CY assertions without using equality of Chern characters as a slicing uniqueness argument.

In (iii), F6 constructs a finite abelian étale cover $\pi :X\times E\to S^{[n]}$, with $E$ a finite elliptic cover of $\mathrm{Alb}(S)$ and $X$ strict Calabi–Yau. Every character line of this cover is a finite-order Hilbert norm $N_n(M)$ from the surface. For the source free quotient $C_1\times C_2\to S$, the pullback of $M$ splits into finite-order flat factor lines, by the direct-sum elliptic fundamental lattice and the actual finite character cover construction. F3/F4 intertwine tensor by $N_n(M)$ with tensor by their external product, and F2 preserves it as a SLICING. Thus F4 lifts the Hilbert condition to $X\times E$. For a finite-order line $L$ on $E$, extend its character across the finite-index isogeny lattice to a finite-order line $M_B$ on $B=\mathrm{Alb}(S)$. Then ${\mathrm{pr}}_E^{\ast }L={\pi }^{\ast }{f}^{\ast }{M}_B$. Projection formula and the verified norm-line invariance downstairs make the lifted condition invariant under these factor lines. Apply F6's actual product-fibre restriction to obtain a locally finite support-property condition on $X$. All source bielliptic inputs are retained; no isogeny between their two original elliptic factors was introduced.

The preceding constructions give exactly the four named classes in the Statement. Each output condition is obtained by a verified finite equivariant induction, actual Fourier–Mukai equivalence, finite étale lifting or scheme-level fibre restriction, each preserving the stated support/local-finiteness/HN data. F6's strict factors and F1's separate convention retain the original distinction between strict Calabi–Yau manifolds and generalized Kummer varieties. Nothing is concluded for an unlisted higher-dimensional variety.