A conditional abelianity theorem for special fourfold pairs with a half-weight divisor

The construction and the two transfers

Proposition 4.1 constructs a connected smooth projective eightfold $Y$ and a surjection $g:Y\to X$. Its general fibre $F$ is a product of four genus-one curves, so $\Omega_F^1$ is trivial. Moreover, $g^*D=2E$ and $g$ lifts to $f:Y\to\mathcal X$. Suitable submersion points over general points of every divisor are essential for the next step.

Proposition 4.1 · p. 6

Proposition 3.1 shows that a differential line bundle on $Y$ with $\kappa(L)=p$ would descend to $\mathcal X$. Specialness of $\mathcal X$ therefore implies specialness of $Y$. A separate topological argument uses even ramification multiplicities and connected general fibres to obtain $\pi_1(Y)\twoheadrightarrow G$.

Proposition 3.1 · pp. 5–6

Apply the input to the entire group and pass to its quotient

Since $Y$ is smooth projective, it is compact Kähler. Apply Assumption 1.2 to this eight-dimensional $Y$ to obtain an abelian finite-index subgroup $A\subset\pi_1(Y)$. Its image in $G$ is abelian and has index at most $[\pi_1(Y):A]$. This proves the full conclusion of the main theorem. The argument does not assume that the root space admits a finite manifold cover.

[OpenAI26a, Theorem 1.1, p. 2] Proof of Theorem 1.3 · p. 9

Descend ratios of sections to the base

Suppose $L\to\Omega_Y^p$ is nonzero with $\kappa(L)=p$. Choose sections $s_0,\ldots,s_p$ of a power $L^m$ with algebraically independent ratios $u_a=s_a/s_0$. Lemma 2.1 identifies the generic direction of $L$ with the wedge of the differentials of these ratios. Its input is closedness of holomorphic forms on a resolution of the cover obtained by simultaneously adjoining $m$th roots of the sections.

Lemma 2.1 · p. 3 [Dem12, VI, Proposition 8.9, p. 312]

On a general fibre, the cotangent sequence is $0\to\mathcal O_F^{\dim X}\to\Omega_Y^1|_F\to\Omega_F^1\to0$. Its exterior powers have a filtration with trivial graded pieces. A component of the first nonzero graded projection of $L|_F$ gives a section $t$ of $L^{-1}|_F$. On a general fibre where $t^m s_0$ is a nonzero constant, all ratios $s_a/s_0$ are constant. Rational descent along connected general fibres then gives $u_a=g^*v_a$. Thus $\sigma=dv_1\wedge\cdots\wedge dv_p\ne0$, and $p\le\dim X$.

Lemma 2.4 · p. 4

Detect poles at submersion points in root charts

Write $\pi:\mathcal X\to X$ and saturate the direction of $\tau=\pi^*\sigma$ inside $\Omega^p_{\mathcal X}$ to obtain a line bundle $K$. Lemma 2.2 uses primitive generators in codimension one and Hartogs extension; it does not assume that the resulting map is a subbundle everywhere.

Lemma 2.2 · pp. 3–4

First write the images of $s_a$ as $h_a(g^*\sigma)^m$. Over an open set where $\sigma$ is holomorphic and nonvanishing and $g$ is a submersion, $h_a$ is holomorphic and therefore constant on compact connected fibres. Lemma 2.4 gives $h_a=g^*b_a$, so the images are $g^*b_a(g^*\sigma)^m$. The submersion points in conditions 3 and 4 admit local sections. Any divisorial pole downstairs would therefore be detected after pullback to $Y$. Consequently $(\pi^*b_a)\tau^m$ is holomorphic on every root chart and extends across codimension two. Independence of the ratios $v_a$ gives $\kappa(K)\ge p$. On the other hand, $f^*K\to\Omega_Y^p$ and Lemma 2.1 give $\kappa(K)\le\kappa(f^*K)\le p$. This produces the forbidden line bundle with $\kappa(K)=p$ on $\mathcal X$.

Proposition 3.1 · pp. 5–6

Even meridian powers yield the group surjection

For $Y\setminus g^{-1}D\to X\setminus D$, a meridian of a component of multiplicity $2e$ in $g^*D$ maps to $\gamma^{2e}$ and hence dies in $G$. The inclusion of this complement in $Y$ induces a surjection on fundamental groups, whose kernel is normally generated by divisor meridians. The map therefore factors through $\pi_1(Y)\to G$. The proper smooth map $g^{-1}B\to B$ has connected compact fibres and is locally trivial, so it is surjective on fundamental groups. General position also gives surjectivity of $\pi_1(B)\to\pi_1(X\setminus D)$, proving the required surjection.

[Ehr52, §1, Proposition, print p. 154 / PDF p. 3] Proposition 3.1 · pp. 5–6

Diagonal quadrics on the root space and avoidance of fixed points

Set $T=\mathcal O_{\mathcal X}(\mathcal D)$, so $T^2=\pi^*\mathcal O_X(D)$. Choose $M$ with both $M$ and $M(-D)$ globally generated. On the fourfold fibre product of $\mathbb P_{\mathcal X}(\mathcal O^{\oplus2}\oplus T^{\oplus2})$, impose two diagonal quadrics in each factor $j=1,\ldots,4$:

For $i=1,2$ the coefficients lie in $H^0(X,M)$, and for $i=3,4$ in $H^0(X,M(-D))$. The latter are evaluated after tensoring with $T^2$; they are not automatically zero along $D$.

Independent coefficient evaluations for the eight equations give smoothness for general parameters. Over $D$, a fixed point of $\mu_2$ lies in one of two eigenspace copies of $\mathbb P^1$ in every factor. A common zero of the corresponding two quadrics requires the relevant $2\times2$ determinant to vanish. The four factors impose four independent conditions. Since $\dim D=3$, general parameters avoid every fixed point. This dimension count explains the use of four factors.

Lemma 4.2 · pp. 7–8

Squaring coordinates provides projectivity

The smooth analytic space $S$ obtained from the free root charts maps finitely, by squaring coordinates, to the ordinary projective bundle product

GAGA algebraizes the finite coherent algebra together with its multiplication and unit. Taking relative Spec gives an algebraic finite morphism to $Q$. Finite morphisms are projective and $Q$ is projective, so $S$ is projective.

Lemma 4.3 · p. 8 [Ser56, §12, Théorèmes 2–3, print pp. 19–20 / PDF pp. 20–21] [StacksM, Lemma 45.16 (Tag 0B3I), p. 108]

Arrange connected fibres and submersion points over divisors

For general parameters there is a nonempty open $B$ where all two-column minors of every $2\times4$ coefficient matrix are nonzero. The intersection of its two quadrics is smooth there. The Koszul sequence and vanishing of intermediate cohomology on $\mathbb P^3$ give $H^0(C,\mathcal O_C)=\mathbb C$, while adjunction gives $K_C\simeq\mathcal O_C$. Thus each curve is connected of genus one.

[StacksC, Lemma 8.1 (Tag 01XT), pp. 16–17] Lemma 4.4 · p. 9

The rank-drop condition for a $2\times3$ matrix obtained by deleting any column has codimension two. The corresponding bad set $Z\subset X$ can therefore be chosen to have codimension at least two. At $x\notin Z$, the kernel contains a vector with every coordinate nonzero. Taking its coordinatewise square roots yields a point where the vertical derivative of each pair of quadrics has rank two. It is a submersion point to the root chart, giving the required condition over general points of every prime divisor, including $D$.

Connected general fibres give a unique component $Y$ of $S$ dominating $X$, and the submersion points above belong to it. Finally, $g^*D=2f^*\mathcal D$ and $\dim Y=4+4=8$. All hypotheses of Proposition 3.1 have now been supplied.

Lemma 4.2 · pp. 7–8 Lemma 4.4 · p. 9