A CONDITIONAL ABELIANITY THEOREM FOR SPECIAL FOURFOLD PAIRS
WITH A HALF-WEIGHT DIVISOR

0. INTRODUCTION

Let X be a connected smooth projective complex fourfold, and let D be a
nonempty smooth connected reduced divisor on X.  The order-two root
orbifold of the pair (X,D) is written X~.  Locally along D = {z1 = 0}
it is given by z1 = w1^2, zj = wj for j > 1, with the group Z/2 acting
by w1 -> -w1.  The coarse map is pi: X~ -> X.

A line bundle on X~ means an equivariant holomorphic line bundle in
these charts, with compatible gluing.  For an orbifold line bundle L,
its Iitaka dimension kappa(L) is the largest dimension of the images of
its complete plurisection maps (or -infinity if all positive powers
have no sections).  The orbifold X~ is special if there is no integer
1 <= p <= dim X~ and no nonzero sheaf map L -> Omega^p_{X~} with
kappa(L) = p.

The orbifold fundamental group is
G = pi_1(X \ D, x) / <<gamma^2>>,
where gamma is a positively oriented meridian of D.

THEOREM.  Assume the following input from the companion paper:
every connected smooth compact Kaehler manifold which is special in the
above sense has virtually abelian fundamental group, in every complex
dimension.  Then, if X~ is special, the group G is virtually abelian
(i.e. it contains an abelian subgroup of finite index).

The proof has two parts.  First we construct a connected smooth
projective eightfold Y with a map g: Y -> X satisfying four simple
geometric conditions.  Second we show that for any such Y, specialness
of X~ forces Y to be special, and pi_1(Y) surjects onto G.  Then the
manifold theorem applied to Y gives the result.

1. PRELIMINARIES: DIFFERENTIAL LINES AND SATURATION

Lemma 1.1 (Closed-form argument).  Let V be a connected smooth
projective manifold, and let i: L -> Omega^p_V be a nonzero line-bundle
map, with 1 <= p <= dim V.  If s0,...,sr are sections of the same
positive power L^m with s0 != 0, then
    d(sa/s0) ^ i(L) = 0
at the generic point.  In particular kappa(L) <= p.  If kappa(L) = p,
there are sections s0,...,sp such that the image line of i is spanned by
d(s1/s0) ^ ... ^ d(sp/s0) != 0.

Proof.  Adjoin m-th roots of the sections in a finite cover, resolve,
and use that holomorphic forms on a compact Kaehler manifold are
closed.  On the open set where t0 != 0 we have alpha_a = (t_a/t0)
alpha_0, so d(t_a/t0) ^ alpha_0 = 0.  Taking m-th powers gives the
first claim.  If r independent covectors wedge to zero with a nonzero
p-form, expansion shows r <= p.  The last statement follows by taking
p algebraically independent ratios.

Lemma 1.2 (Saturation).  Let E be a holomorphic vector bundle on a
smooth complex manifold.  The intersection with E of a rank-one
meromorphic subspace of its meromorphic sections is an invertible
sheaf K with a nonzero map K -> E.  Moreover, a meromorphic tensor in
the corresponding line inside E^m is holomorphic if and only if it is a
holomorphic section of K^m.  Both assertions hold equivariantly in
smooth root charts.

Proof.  In a local frame clear the denominators of a meromorphic
generator and divide out all common irreducible factors of its entries.
Smooth local rings are factorial, so the resulting vector v has
holomorphic entries with no common height-one factor.  If av is
holomorphic for a meromorphic scalar a, a pole of a at a prime divisor
would force that prime to divide every entry of v.  Thus a has no
divisorial pole and is holomorphic.  This proves that the saturated
module is freely generated by v.  At every prime divisor some entry of
v is a unit at its generic point.  The corresponding entry of v^m has
the same property.  The identical pole test proves the tensor-power
assertion.  Holomorphic extension across codimension two gives the
statements on the whole chart.  The construction is intrinsic to the
meromorphic direction and hence respects equivariance and changes of
chart.

Lemma 1.3 (Rational descent).  Let g: Y -> X be a dominant morphism of
connected smooth projective varieties with connected general fibre.  A
rational function on Y which is constant on general fibres, wherever
defined, is the pullback of a rational function on X.

Proof.  Let Z subset X x P^1 be the closure of the joint image of g
and the function, on its regular locus.  The actual image is
constructible and dense in Z, so it contains a dense open of Z.  Its
fibres over general points of X have at most one point.  The
fibre-dimension theorem therefore makes Z -> X generically finite.
After shrinking X, its generic degree is realized by distinct points,
since the characteristic is zero, and all those points belong to the
actual-image open.  The degree is consequently one.  Hence C(Z) =
C(X), giving the descent.  Connectedness of the general fibre ensures
that the constant is a single value on that fibre.

2. THE TRANSFER CRITERION

Let g: Y -> X be a surjective morphism with Y connected smooth
projective.  Assume:

(i) g^*D = 2E for an effective integral divisor E on Y.
(ii) Over a nonempty Zariski open B in X\D, the map g is smooth with
     connected fibres F, and Omega^1_F is trivial.
(iii) Above a general point of each prime divisor P != D of X, there
     is a point where g is a submersion.
(iv) Above a general point of D, there is a point where a local lift
     to the root chart is a submersion.

PROPOSITION 2.1 (Transfer).  If X~ is special, then Y is special.  Also
there is a surjection pi_1(Y) -> G.

Proof of specialness.  Suppose L -> Omega^p_Y contradicts specialness
of Y.  Choose sections s0,...,sp of L^m as in Lemma 1.1, and put
ua = sa/s0.

For a general smooth fibre F over B, the exact cotangent sequence
    0 -> O_F^n -> Omega^1_Y|F -> Omega^1_F -> 0
gives a filtration of Omega^p_Y|F with trivial graded pieces.  Choose
F where i|F and s0|F are nonzero.  The first nonzero projection of i|F
to a graded piece, followed by a coordinate projection, gives a nonzero
section t of L^{-1}|F.  Then t^m s0|F is a nonzero constant (F is
compact and connected).  Hence t and s0|F have no zeros, and every ua
is constant on F.  By Lemma 1.3, ua = g^* va for rational functions va
on X.

Their independent differentials imply p <= n and
    sigma = dv1 ^ ... ^ dvp != 0.
By Lemma 1.1, the generic image of L is the line of g^* sigma.

Next descend the plurisections through their images as differential
tensors.  Write i^m(sa) = ha (g^* sigma)^m with ha rational on Y.
Shrink B so that sigma is holomorphic and nowhere zero.  Since g is a
submersion there, g^* sigma is nowhere zero.  Hence ha is holomorphic
on g^{-1}B.  Compact connected fibres force ha constant on each fibre,
so ha = g^* ba for rational functions ba on X.

Let tau = pi^* sigma be the meromorphic form on X~.  By Lemma 1.2,
tau determines an orbifold line bundle K -> Omega^p_{X~} with generic
direction tau.  We claim that every tensor
    (pi^* ba) tau^m
is holomorphic.  Its pullback to Y is holomorphic by construction.
At the general point of any ordinary base divisor, condition (iii)
supplies a local section of a submersive lift.  At D, condition (iv)
does the same in the root chart.  Pulling back along such a section
shows there is no pole at that divisorial point.  Any polar chart
divisor belongs to the invariant global polar locus; its finite image
is a base divisor, so the preceding tests exclude every polar
component.  Extension across codimension two proves the claim.

By Lemma 1.2, these tensors are sections of K^m.  Their ratios are the
independent va, so kappa(K) >= p.  Conversely, f^*K -> Omega^p_Y is a
nonzero line map (on the dense ordinary open it is the differential
pullback by a dominant map), so kappa(K) <= kappa(f^*K) <= p.  Thus K
contradicts specialness of X~.  This proves that Y is special.

Proof of group surjection.  Set Y0 = Y \ g^{-1}(D) and X0 = X\D.
The inclusion Y0 -> Y surjects on pi_1, with kernel normally generated
by meridians of the components of g^{-1}(D).  Consider the composition
pi_1(Y0) -> pi_1(X0) -> G.  At the general point of a component of
g^{-1}(D), its meridian maps to gamma^{2e}, where 2e is the
multiplicity of g^*D.  All these die in G because gamma^2 = 1 in G.
Hence the map factors through pi_1(Y).

Over B the map g is proper and smooth, hence a differentiable fibre
bundle by Ehresmann.  Connectedness of the fibres gives a surjection
pi_1(g^{-1}B) -> pi_1(B).  Since B is a dense open in X0, general
position gives pi_1(B) -> pi_1(X0) surjective.  Therefore the map
pi_1(Y) -> G is surjective.

3. CONSTRUCTION OF THE EIGHTFOLD Y

Choose an ample line bundle M on X such that M and M(-D) are globally
generated.  Let T be the root line on X~: T^2 = pi^* O_X(D).  Consider
the orbifold vector bundle
    V = O_{X~}^2 + T + T
(meaning two copies of O and two copies of T).  Let P = P_{X~}(V) be
its projectivization; it is a P^3-bundle over X~.  The Z/2 action on a
fibre is
    [z1:z2:z3:z4] -> [z1:z2:-z3:-z4].

Take four independent copies of P over X~, and call the product P^4.
On each factor j = 1,...,4 choose two diagonal quadratic equations
    q_{j,nu} = sum_{i=1}^4 a_{j,nu,i} z_i^2 = 0,   nu = 1,2,
where the coefficients are sections:
    a_{j,nu,i} in H^0(X,M)     for i = 1,2,
    a_{j,nu,i} in H^0(X,M(-D)) for i = 3,4.
(The factor T^2 = O(D) explains why the last two coefficients are
twisted by -D.)  Let A be the affine space of all coefficients, and
for a in A let S_a be the common zero locus of the eight quadrics in
P^4.

Lemma 3.1 (Parameter choice).  For general a in A, the following hold:
(1) S_a is smooth in the root charts.
(2) S_a has no nontrivial stabilizers under the Z/2 action.
(3) There is a closed subset Z of X of codimension at least two such
    that for every x not in Z, every matrix obtained from any C_j(x)
    by deleting one column has rank two.  Here C_j(x) is the 2x4 matrix
    of coefficients of the two quadrics in factor j at x.
(4) On a nonempty open B in X\D, every 2x2 minor of every C_j(x) is
    nonzero.

Proof.  Take a finite trivializing cover of X for the bundles involved,
with smooth root charts.  The eight equations have independent
evaluations on the parameter space, so the universal incidence over the
chart product is smooth.  Generic smoothness in characteristic zero
then gives a dense open of parameters where S_a is smooth.  This
proves (1).

Only points over D can have nontrivial stabilizer.  There the
involution acts as above; its fixed locus is the union of the two
eigenspace lines.  If a fixed point satisfies both equations, then
either the first two columns or the last two columns of C_j(x) are
dependent.  Each such determinant condition is a proper hypersurface in
the space of 2x4 matrices.  A fixed point in the fourfold product
requires one such condition in each of the four independent factors,
so its parameter codimension is four.  The bad incidence over D
therefore has dimension at most dim A + dim D - 4 = dim A - 1.
General parameters avoid it.  This proves (2).

For (3), fix one 3-column submatrix.  Rank at most one has codimension
two in the space of 2x3 matrices.  Surjective evaluation implies the
incidence over X has dimension at most dim A + 2.  For a general
parameter the bad base locus has dimension at most two, by the
fibre-dimension theorem.  There are finitely many choices of factor and
deleted column; their union is closed and gives (3).

Finally choose a point x0 in X\D.  By evaluation surjectivity we can
make all 2x2 minors nonzero at x0 (for instance take columns of the
form (1, lambda_i) with distinct lambda_i).  This is a nonempty
Zariski-open condition on parameters.  Intersecting with the previous
open sets gives (4).  This completes the lemma.

Lemma 3.2 (Projectivity).  Let S = S_a / (Z/2) for a parameter a
satisfying Lemma 3.1.  Then S is a smooth projective variety.

Proof.  Square the four coordinates in each factor.  The
identification T^2 = pi^* O_X(D) gives a globally defined invariant
map to the ordinary projective variety
    Q = P_X(E_0) x_X ... x_X P_X(E_0)   (four factors),
where E_0 = O_X^2 + O_X(D)^2.  On a root chart this map is the finite
root map W -> U together with the finite projective coordinate-square
maps.  Restriction to the closed zero locus remains finite.  Passing
to the finite-group quotient preserves finiteness: over an affine
target this takes an invariant subalgebra of a finite algebra, which is
again finite.  These invariant algebras glue under the bundle
transitions, since the coordinate-square construction is invariant and
global.  Hence S -> Q is a finite analytic map.  A finite analytic
space over a complex projective variety algebraizes by projective
GAGA.  A finite algebraic morphism is projective.  Thus S is
projective.  Smoothness follows from Lemma 3.1(1) and the freeness of
the action.  This completes the proof.

Fix such a parameter a, and let S be as in Lemma 3.2.  There is a
unique connected component Y of S that dominates X.  More explicitly,
the fibres over B are connected and nonempty, so exactly one component
can dominate X.  The variety Y is a connected smooth projective
eightfold.  Its map to X is called g.

4. VERIFICATION OF THE TRANSFER CONDITIONS

We check the four conditions of Section 2.

(i) g^*D = 2E.  The map g factors through the root orbifold X~.  The
root divisor pulls back to twice an integral divisor, so the same holds
for D on Y.

(ii) Over B = X\D, the map g is smooth with connected fibres.  Each
factor is the common zero set of two quadrics in P^3; it is a smooth
complete intersection of two quadrics, hence a genus-one curve.  The
product of four such curves is a product of four elliptic curves.  The
cotangent bundle of a product of elliptic curves is trivial.

(iii) Above a general point of every prime divisor P != D, there is a
point where g is a submersion.  Take x not in Z (the set from Lemma
3.1).  In each matrix C_j(x), the three columns remaining after any
deletion still span C^2.  Therefore the kernel of C_j(x) is contained
in no coordinate hyperplane, so there is a vector b in the kernel with
all four coordinates nonzero.  Choose square roots z_i of its entries.
The vertical derivative 2 C_j(x) diag(z_i) then has rank two.  Doing
this independently in each factor makes the full vertical derivative
of the eight equations surjective.  The implicit function theorem gives
a submersion from the zero set to the base root chart at that tuple.
This works at any x not in Z, including points of D.

(iv) Above a general point of D, the same construction works because
a general point of D is not in Z (Z has codimension at least two).
Hence there is a point where the lifted map to the root chart is a
submersion.

Thus all the hypotheses of Proposition 2.1 are satisfied.

5. CONCLUSION

By Proposition 2.1, specialness of X~ implies that Y is special.  The
same proposition gives a surjection
    pi_1(Y) -> G.
Now Y is a connected smooth projective eightfold, hence a compact
Kaehler manifold.  By the assumed manifold theorem, pi_1(Y) is
virtually abelian.  Any quotient of a virtually abelian group is
virtually abelian (the image of a finite-index abelian subgroup is
abelian of finite index).  Therefore G is virtually abelian.

This completes the proof of the theorem.

REMARK.  The companion appendix gives an alternative construction of a
14-dimensional smooth projective manifold Y with the same properties,
using an auxiliary branched cover and an action on a product of
elliptic curves.  The eightfold constructed here is sufficient and
more concise.