Strict covers in the named Enriques and bielliptic Hilbert families

1. Fundamental groups and the actual isotrivial covers. For every smooth connected projective surface S and $n\ge 2$, pi1(S^[n]) = H1(S,Z). The configuration/simple-double-collision van Kampen argument is supplied in [F5], Proof 1.4, “Why every finite cover character on a Hilbert scheme is such a norm”: deleting diagonals from $S^n$ does not change the pure pi1 (real codimension4), unordered configurations add Sn, and the transverse A1 Hilbert boundary kills the transpositions. The deeper collision locus has complex codimension $\ge 2$ in the smooth Hilbert scheme and changes no pi1. Thus all point-factor loops are identified and commute. This is the actual abelianization, with its natural point-loop map. Let $A=E1\times E2$ and let $S=A\mathrm{/}G$ be the source FREE bielliptic quotient, with translations on E1 and automorphisms on E2. Its Albanese B is an elliptic curve isogenous to the TRANSLATION factor E1. Indeed G-invariant one-forms are the E1 one-form; the nontrivial linear holonomy on E2 eliminates the other. The E1 action by translation commutes with G and descends to S, with a nonzero isogeny $a:E1\to B$. After fixing origins the induced map f:S^[n] -> Sym^n(S) -> Sym^n(B) -> B satisfies $f(t\cdot Z)=f(Z)+na(t)$. Consequently it is smooth (differentiate the translation action), and pullback by the finite étale isogeny $na:E1\to B$ is EXACTLY S^[n] ×_B E1 = F×E1, (Z,t) <-> (-t·Z,t), F=f^-1(0). The fibre F is connected. To identify the base and its fibres explicitly, let H be the finite translation image of G on E1. The quotient map $S\to E1\mathrm{/}H$ has fibre E2/G0, where G0 is the kernel of that translation image. Freeness forces G0 to act on E2 by translations (an elliptic automorphism with nontrivial linear part has a fixed point), so this fibre is a connected elliptic curve. The base E1/H is the Albanese: its one-form pulls back to all invariant one-forms, and any residual nontrivial isogeny between it and Alb(S) would disconnect the displayed fibres. Thus the Albanese map $S\to B$ has connected elliptic fibres; on symmetric products the fibres over a cycle are products of symmetric powers of these connected curves, while ${\mathrm{Sym}}^n(B)\to B$ has fibre the complete linear system of a degree-n line bundle, namely $P^{n-1}$ (curve RR gives $h0=n,H1=0$). Hilbert–Chow has connected fibres because it is proper birational onto a normal symmetric product. The composite therefore has connected fibres. F is thus smooth connected projective of dimension 2n-1. The group H1(S,Z) has rank2: S has a finite torus cover, so its rational first cohomology is the finite-group invariant part, dimension2 by the one-form calculation and Hodge conjugacy. Its torsion is finite by the finite-CW theorem. The exact sequence of the locally trivial fibration f, with $\pi 2(B)=0$, therefore identifies pi1(F) with a FINITE ABELIAN subgroup of $\pi 1(S^{[}n])$. Surjectivity to pi1(B) follows from connectedness of the fibre. Local triviality here is provided by the displayed isogeny product, not an unproved effectivity theorem.

2. Algebraic finite covers killing this finite group. We spell out algebraicity so that an analytic universal cover is not silently used as an algebraic supplier. Every finite character of pi1(S) is realized by an algebraic torsion line bundle. The cover $A\to S$ has deck group G and subgroup $\pi 1(A)=\mathrm{\Lambda }=Z^4$. For an integer r, the composite $A--[r]-\to A\to S$ is a finite étale cover whose deck group Gamma consists of all lifts of G through [r]. Such lifts are algebraic affine automorphisms: choose r-th roots of the translation terms, keep their linear parts, and include A[r]. All lifts form a finite extension of G by A[r]. They act freely because an element with nontrivial G-image would give a fixed point in the original free quotient, and a nontrivial kernel element is a torsion translation. Thus Gamma is the actual algebraic deck group, corresponding to pi1(S)/rLambda. A finite character kills rLambda for some r. Its rank-one eigensheaf in the finite free cover algebra is the desired algebraic finite-order flat line M. Norm M on $S^{[}n]$ then realizes the induced abelianized character, by the determinant-norm construction in the companion cover note. Every character of the finite subgroup pi1(F) extends to a finite character of H1(S,Z): a finitely generated abelian group splits as its finite torsion and a free group, and finite characters extend after choosing roots in C*. It is therefore realized by the restriction of such a norm line. Taking the finite product of the associated cyclic torsors and a connected component produces an ALGEBRAIC finite étale cover $X\to F$ with trivial pi1, because the intersection of all character kernels in a finite abelian group is zero. The canonical line of S has finite order: the character of the finite affine action on the constant two-form gives this directly. The completed Hilbert canonical-line supplier makes ${\omega }_{S^{[n]}}$ a norm line of finite order; since F is a smooth fibre over an elliptic curve its normal line is trivial and ${\omega }_F$ is the restriction of that torsion canonical line. On simply connected X every finite-order flat character is trivial, so ${\omega }_X$ is trivial. Thus X×E1 -> F×E1 -> S^[n] is a finite étale algebraic cover by smooth projective varieties. Since $\pi 1(S^{[}n])$ is abelian, it is a torsor under a finite abelian deck group. The elliptic factor E1 maps by a finite étale isogeny to Alb(S).

3. A simply connected symplectic comparison space, constructed directly. Let $M=K_n(A)$ be the zero fibre of the sum morphism $A^{[}n]\to A$. Translation by A makes this smooth and isotrivial after $[n]:A\to A$; it is connected since $\ker (\sum :A^n\to A)=A^{n-1}$ is connected and Hilbert–Chow is connected. The pi1 calculation of the Fundamental groups and the actual isotrivial covers construction identifies $\pi 1(A^{[}n])\to \pi 1(A)$ with the identity of H1(A,Z). As $\pi 2(A)=0$, the fibration exact sequence therefore gives $\pi 1(M)=0$. Here is the symplectic form without a hyper-Kähler classification theorem. At a length-n cluster Z, its module deformation map from the Hilbert tangent space to $\mathrm{Ext}1(O_Z,O_Z)$ is injective: a trivial cyclic module deformation changes its generator only by a unit and cannot change its annihilator ideal. Both spaces have dimension2n. The first dimension is the actual completed Hilbert smoothness theorem. For the second, $\mathrm{End}(O_Z)=O_Z$ has dimension n, local surface residue duality gives Ext2 dimension n, and a length2 regular free resolution has Euler rank zero, so its Hom-into-$O_Z$ Euler characteristic is zero and Ext1 dimension2n. Thus the tangent map is an isomorphism. Serre trace of the Yoneda product gives a nondegenerate alternating pairing on this Ext1 space. The universal finite-flat cluster and its coordinate-matrix Koszul resolutions make this pairing a regular two-form on $A^{[}n]$; dimensions are constant, so the relevant perfect cohomology bundles and base-change maps are obtained by local finite-matrix cancellation. On the distinct-points locus it is sum ${\omega }_A$ on the factors, hence closed there; its differential is regular and vanishes densely, so it is closed globally. This constructs a symplectic form on $A^{[}n]$. On that same open, pairing a translation vector v with a tangent vector w is ${\omega }_A(v,\sum w_i)$. Thus the translation plane is nondegenerate, with pairing $n{\omega }_A$, and its orthogonal complement is precisely the sum-fibre tangent space. The identity extends globally, so restriction gives a symplectic form sigma on M and a nowhere-zero volume ${\sigma }^{n-1}$. The holomorphic-form algebra of M is EXACTLY $C[\sigma ]\mathrm{/}({\sigma }^n)$, with sigma in degree2. To see the bound, pull a form back over the finite free configuration cover from $B0=\ker (\sum :A^n\to A)$. The omitted diagonals have complex codimension2, so coefficient Hartogs extends it to the abelian variety B0. It is Sn-invariant. Its one-forms are $x_i,y_i$ with sum $x_i=\sum y_i=0$, and Sn permutes indices. Averaging is exact, so compute invariant exterior forms before this quotient. A monomial orbit with two unpaired x-indices averages to zero by swapping those indices and the wedge sign; the same holds for two unpaired y-indices. A repeated index has at most the pair $x_i$ wedge $y_i$. Thus every invariant is generated by u=sum x_i, v=sum y_i, w=sum x_i wedge y_i. Indeed $u^a{v}^b{w}^c$ gives each orbit type with a,b in{0,1}; when $a=b=1$ subtract the coinciding-index paired term to isolate the remaining orbit type. On $B0,u=v=0$. The remaining invariants are the powers of w; they are nonzero through exponent n-1 because w is symplectic on the standard-representation tensor C². The previously constructed sigma restricts to w, so its powers realize all those invariants. Odd form degrees vanish.

4. Dominant comparison to X and strictness without Bogomolov decomposition. On distinct points with distinct G-orbits, the finite étale map $A\to S$ induces a dominant generically finite rational map $A^{[}n]-\to S^{[}n]$. Compose with the finite isotrivial cover $M\times A\to A^{[}n]$. The Albanese coordinate on $S^{[}n]$ is n a(t1), independent of t2. Restrict a component of its zero fibre that dominates F; it is M×E2 up to a fixed translation on E1. Thus there is a dominant generically finite rational map M×E2 --> F. Resolve its graph by regular-center principalization. Such blowups preserve pi1: their exceptional projective-space fibres are simply connected and the codimension-two deleted centers are handled by the local van Kampen calculation. The resolved domain therefore still has $\pi 1=\pi 1(E2)$, since $\pi 1(M)=0$. Pull back the finite cover $X\to F$. After replacing E2 by its multiplication-r cover for r divisible by the finite deck exponent, this pulled-back finite cover has trivial monodromy. It splits ALGEBRAICALLY: a finite étale cover of a smooth projective variety that is analytically disconnected has the same algebraic components, since each algebraically connected smooth projective component is analytically connected by algebraic curve sections. A degree-one component is an isomorphism over the normal domain. Selecting it gives a dominant rational map M×E2 $-\to X$, with possibly an isogenous E2 but the same form algebra. Pullback injects the holomorphic-form algebra of X into that of M×E2; graph blowups do not change holomorphic forms because the smooth target's coefficient functions extend across codimension2 by Hartogs. Put $d=\dim X=2n-1$. The latter algebra is C[sigma]/(sigma^n) tensor exterior(eta), where eta is an elliptic one-form. Equip simply connected projective X with the Ricci-flat Kähler metric supplied by the ACTUAL completed Calabi existence theorem, since ${\omega }_X$ is trivial. Every holomorphic p-form is parallel: in Kähler normal coordinates the Bochner formula on (p,0)-forms has curvature contraction sum Ricci, so integration with $Ricci=0$ gives $\int \mathrm{\mid }\mathrm{\nabla }\alpha {\mathrm{\mid }}^2=0$. If an even form alpha of degree2j with $0<2j<d$ existed, its pullback would be a nonzero multiple of ${\sigma }^j$. On M×E2 this has kernel exactly the elliptic complex line, since contraction by a nonzero M-vector in a symplectic power ${\sigma }^j$ is injective for $1\le j\le n-1$: write it as $j(i_v\sigma )$ wedge ${\sigma }^{j-1}$ and wedge further by ${\sigma }^{n-1-j}$; the resulting map from one-forms to top-minus-one forms is an isomorphism in a Darboux basis. The rational map is generically finite, so alpha has a one-dimensional kernel generically on X. Parallelness makes its kernel a global parallel complex line subbundle. Its orthogonal complement is parallel as well. Torsion-free parallel distributions integrate to a LOCAL metric product: Frobenius gives product coordinates and the Koszul formula makes the two metric blocks independent. The Ricci curvature on the complex one-dimensional factor is therefore zero, so its connection is flat, including mixed directions. On simply connected X it has a global nonzero parallel vector, hence a nonzero parallel real one-form. Such a form has a nonzero de Rham class on a compact manifold (if df were parallel, f would be harmonic and constant), contradicting $\pi 1(X)=0$. There are NO intermediate even forms. For a nonzero odd p-form with $p<d$, its parallelness and a parallel holomorphic volume Omega give a nonzero parallel holomorphic (d-p)-form by the antilinear metric-volume map $\alpha \to$ $\mathrm{contraction}((\bar{\alpha }{)}^{\mathrm{♯}},\mathrm{\Omega })$. Since d is odd this is an intermediate EVEN degree, already excluded. Thus X has no holomorphic forms in ANY degree $0<p<d$. It is the required strict Calabi–Yau factor. No general Beauville–Bogomolov or de Rham GLOBAL decomposition theorem is a premise.

5. The source Enriques construction and its universal Hilbert cover. For $A=E1\times E2$ let $i=(-1,-1)$ and $\tau (x,y)=(-x+t1,y+t2)$. In the actual Enriques case both ti must be NONZERO 2-torsion: a tau-fixed orbit modulo i would satisfy $\tau (x,y)=+(x,y)$, forcing $t2=0$, or $\tau (x,y)=-(x,y)$, forcing $t1=0$. Conversely if both are nonzero there is no such orbit, including the resolved exceptional points. The A1 cluster charts construct the Kummer K3 S' and lift tau freely. Its volume transforms by -1. This is precisely the free Enriques case in the target's source hypotheses; zero torsion parameters do not produce an Enriques surface and are not silently included. The Kummer resolution S' is simply connected. The smooth quotient open has orbifold group Lambda semidirect{±1}; filling its16 A1 exceptional neighborhoods kills the elements (lambda,-1) for all lambda mod2Lambda. Their differences generate Lambda, and one of them kills -1, so the resulting group is zero. Thus the free quotient $S=S^{\mathrm{\prime }}\mathrm{/}\tau$ has $\pi 1={\mathbb{Z}}_2$. Its canonical line is nontrivial of order2 by the volume character, and $H^p(S,O)=0$ for $p>0$: forms inject into the finite quotient product forms, with the residual tau eliminating the top form, and Kähler Hodge conjugacy gives the O-cohomology statement. For $Y=S^{[}n]$, the completed Hilbert geometry gives its canonical norm line of order2. It is nontrivial by restriction to the open embedding of S obtained by fixing n-1 points; finite-point removal does not change its divisor class. the Fundamental groups and the actual isotrivial covers construction gives $\pi 1(Y)={\mathbb{Z}}_2$. The canonical double cover is therefore the universal cover. For completeness its strictness follows directly from cohomology, without the original OS decomposition argument. Hilbert–Chow $\rho :Y\to {\mathrm{Sym}}^{n}S$ is crepant with line dualizing target. Nef-and-big adjoint vanishing applied to pullbacks of high ample target twists gives $R^i{\rho }_{\ast }{\omega }_Y=0$ for $i>0$ by Leray/Serre, hence $R^i{\rho }_{\ast }{O}_Y=0$ by the canonical projection formula. The finite quotient $S^n\to {\mathrm{Sym}}^{n}S$ has exact averaging, so $H^p(Y,O)=H^p(S^n,O{)}^{S_n}=0$ for $p>0$. Serre duality then gives $H^p(Y,{\omega }_Y)=0$ for $p<2n$. The canonical-cover algebra $O_Y⨁{\omega }_Y$ gives $H^p(cover,O)=0$ for $0<p<2n$ and a trivial canonical line. Thus its universal cover is a strict CY of dimension2n, as required.

6. Necessary correction of the copied elliptic-factor label. The bielliptic elliptic factor is isogenous to E1, the TRANSLATION factor, not generally the automorphism factor E2. For nonisogenous E1,E2, a finite étale cover X×E2 with strict X would have Albanese isogenous to E2, whereas every finite étale Hilbert cover in this construction has Albanese isogenous to E1. The source OS Theorem3.5 says only "an elliptic curve" and its proof starts with $E=\mathrm{Alb}(S)$. Cheng's fixed C2 label is therefore an actual copied label error. The richest valid named family is retained by replacing that label with "an elliptic finite étale cover E of Alb(S)." No restriction to isogenous elliptic input factors is introduced. The corrected Albanese label retains the entire named bielliptic family and introduces no isogeny assumption between its original elliptic factors.