2 Type II $_b$ degenerations

Let $\pi _0\colon S_0\to C_0$ be an elliptic fibration over an irreducible, nodal, arithmetic genus one curve $C_0$ with smooth fiber over the node, and $\chi _{\mathrm {top}}(S_0)=12$ . Such a fibration has a Weierstrass form $\{y^2=4x^3-a_0x-b_0\}$ with $a_0\in H^0(C_0,\mathcal {O}_{C_0}(4P_0))$ and $b_0\in H^0(C_0,\mathcal {O}_{C_0}(6P_0))$ for some point $P_0\in (C_0)_{\mathrm {sm}}$ . See Figure 1.

Figure 1 A Type II $_b$ surface $S_0$ with double locus D and section s.

Let $C_0\hookrightarrow {\mathcal C}$ be a smoothing over $(B,0)$ to a genus $1$ curve, with smooth total space, and let ${\mathcal P}$ be an extension of $P_0$ to a section of $\rho \colon {\mathcal C}\to (B,0)$ . Then, for any $k>0$ , Cohomology and Base Change [Reference HartshorneHar77, III.12.11] implies that $\rho _*{\mathcal O}_{{\mathcal C}}(k{\mathcal P})$ is a rank k vector bundle over B. In particular, $a_0$ , $b_0$ extend locally to sections a, b of $\rho _*{\mathcal O}_{\mathcal C}(4{\mathcal P})$ , $\rho _*{\mathcal O}_{\mathcal C}(6{\mathcal P})$ , and so we can smooth the elliptic fibration $S_0\hookrightarrow {\mathcal S}$ over $(B,0)$ . The resulting total space ${\mathcal S}$ is smooth with $S_0$ reduced normal crossings. The double locus D is the smooth elliptic curve fibering over the node of $C_0$ .

The subscript b indicates that the base degenerates. The terminology is motivated by a similar terminology in the classification of one-parameter degenerations of K3 surfacesdue to Kulikov and Persson-Pinkham [Reference KulikovKul77, Reference Persson and PinkhamPP81]. They classify their $K_{\mathcal S}$ -trivial, reduced normal crossing degenerations into Types I, II, III depending on the depth of the singularity stratification of $S_0$ . Here, we instead have $K_{\mathcal S}={\mathcal O}_{\mathcal S}({\mathcal F})$ for a relative fiber ${\mathcal F}\to (B,0)$ .

As a reduced normal crossing degeneration, the Picard-Lefschetz transformation $T\colon H^2(S_t,{\mathbb {Z}})\to H^2(S_t,{\mathbb {Z}})$ is unipotent and has a logarithm $N:=\log T$ . Furthermore, there is a formula for N which can be deduced from the Picard-Lefschetz transformation for a nodal degeneration of curves, or from [Reference ClemensCle69, Thm. 5.6].

Let $\gamma _t\subset C_t$ denote the vanishing $1$ -cycle of the node of $C_0$ . Since the fiber over the node of $C_0$ is smooth, the restriction of the elliptic fibration $\pi _t\colon S_t\to C_t$ to the curve $\gamma _t$ is a topologically trivial $2$ -torus bundle. Trivialize it, and let $\alpha , \beta $ be oriented generators of the homology of some fiber. Define $u:=[\gamma _t\times \alpha ]\in H^2(S_t,{\mathbb {Z}})$ , $v:=[\gamma _t\times \beta ]\in H^2(S_t,{\mathbb {Z}})$ . Then,

Here $u,v\in \{s,f\}^\perp $ because $s,f$ are classes of line bundles on the total space ${\mathcal S}$ , and hence monodromy-invariant. So the classes $u,v$ determine a rank $2$ isotropic lattice $I:=({\mathbb {Z}} u\oplus {\mathbb {Z}} v)^{\mathrm {sat}}\subset I\!I_{2,10}$ .

Let $U_I$ be the unipotent subgroup of $\mathrm {Stab}_\Gamma (I)$ acting trivially on I and $I^\perp /I$ . From the theory of toroidal compactifications [Reference Ash, Mumford, Rapoport and TaiAMRT75] (see also [Reference LooijengaLoo03, Sec. 1A], [Reference Alexeev and EngelAE23, Prop. 4.16] for the case of Type IV domains), the unipotent quotient

$$ \begin{align*}{\mathbb D}/U_I\hookrightarrow A_I\end{align*} $$

embeds as a punctured disk bundle inside a ${\mathbb {C}}^*$ -bundle $A_I\to I^\perp /I\otimes {\mathcal E}$ . Here ${\mathcal E}$ is the universal elliptic curve over ${\mathbb {C}}\setminus \mathbb R$ whose fiber over $\tau \in {\mathbb {C}}\setminus \mathbb R$ is the elliptic curve ${\mathbb {C}}/{\mathbb {Z}}\oplus {\mathbb {Z}}\tau $ . Since $T\in U_I$ the period map P induces a holomorphic period map $B^*\to {\mathbb D}/U_I$ .

We enlarge $A_I\hookrightarrow {\overline A}_I$ to a line bundle and define $({\mathbb D}/U_I)^{\mathrm {II}}$ as the closure of ${\mathbb D}/U_I$ in ${\overline A}_I$ . This closure is a holomorphic disk bundle over $I^\perp /I\otimes {\mathcal E}$ . The nilpotent orbit theorem [Reference SchmidSch73, Thm. 4.9] (the case at hand follows as in [Reference FriedmanFri84, Thm. 4.2])implies that the period map from $B^*$ extends to a holomorphic map $P\colon (B,0)\to ({\mathbb D}/U_I)^{\mathrm {II}}$ sending $0$ into the boundary divisor $\Delta :={\overline A}_I\setminus A_I$ . As the zero-section of the line bundle, the boundary divisor is naturally isomorphic to

$$ \begin{align*}\Delta\simeq I^\perp/I\otimes {\mathcal E}.\end{align*} $$

Note that $I^\perp /I$ is an even, negative-definite, unimodular lattice of rank $8$ , which uniquely determines it to be $I^\perp /I=E_8$ .

There is also a direct construction of the period point $P(0)\in E_8\otimes {\mathcal E}$ from the singular surface $S_0$ described as follows. Let $X\to \mathbb {P}^1$ be the rational elliptic surface normalizing $S_0\to C_0$ and denote the section and fiber classes again by s and f. Then $\{s,f\}^\perp \subset H^2(X,{\mathbb {Z}})$ is isomorphic to $E_8$ . Let $X_p$ and $X_q$ be the two elliptic fibers glued to form the double locus D of $S_0$ . A class $\gamma \in \{s,f\}^\perp $ defines a line bundle ${\mathcal L}_\gamma \in \mathrm {Pic}(X)$ , and we declare

(2.1) $$ \begin{align} \psi_{S_0}(\gamma):={\mathcal L}_\gamma\big{|}_{X_p} \otimes {\mathcal L}_\gamma\big{|}_{X_q}^{-1}\in E:=\mathrm{Pic}^0(X_p),\end{align} $$

where we have used the gluing isomorphism $X_p\to X_q$ to form the tensor product of these two restrictions.

Then $\psi _{S_0}$ defines a homomorphism $\psi _{S_0}\in \mathrm {Hom}(E_8,E)\simeq E_8\otimes E$ . Fixing an identification of $\{s,f\}^\perp $ with a fixed copy of the $E_8$ lattice, then deforming $S_0$ in moduli of Type II $_b$ surfaces, we get a local holomorphic period map

$$ \begin{align*}P^{\mathrm{II}}\colon \mathrm{Def}_{S_0}\to \mathrm{Hom}(E_8, {\mathcal E}),\end{align*} $$

which is identical to the extension of P coming from the nilpotent orbit theorem. The equivalence of these two definitions of the period map follows fromCarlson’s description [Reference CarlsonCar85] of the mixed Hodge structure on $S_0$ ; see Section 6 and Proposition 6.6. From this description of the boundary periodmapping, we see the following:

1. 1. To prove that P is dominant, it suffices to show that $P^{\mathrm {II}}$ is dominant from the moduli of Type II $_b$ elliptic surfaces to $\mathrm {Hom}(E_8,{\mathcal E})$ .

2. 2. On Type II $_b$ surfaces, the period map $P^{\mathrm {II}}$ is constructed by comparing the restriction of a line bundle in $\{s,f\}^\perp \subset \mathrm {Pic}(X)$ to the two glued fibers.

Observe that (1) follows from the observation at the beginning of this section that every Type II $_b$ elliptic surface is smoothable to the interior of F, so the Zariski closure of $\mathrm {im}(P)\subset ({\mathbb D}/\Gamma )^{\mathrm {II}}$ must contain $\mathrm {im}(P^{\mathrm {II}})$ .

3 Dominance of the period map

Fix a smooth cubic $D\subset \mathbb {P}^2$ and let $\gamma \in PGL_3({\mathbb {C}})$ be generic. Then D and $\gamma (D)$ generate a pencil of cubics with $9$ distinct base points. Blowing up at the nine base points $D\cap \gamma (D)=\{p_1,\dots ,p_9\}$ of this pencil, we get a rational elliptic surface $X\to \mathbb {P}^1,$ together with an isomorphism $\gamma \colon D \to \gamma (D)$ between two of its fibers. The nine blow-ups give rise to nine exceptional sections $F_1,\dots ,F_9$ of the resulting elliptic fibration. Let $t\colon D\to D$ be an arbitrary translation and consider the surface $S_0$ which results from gluing our two fibers of $X\to \mathbb {P}^1$ by the isomorphism

$$ \begin{align*}\gamma \circ t \colon D\to \gamma(D).\end{align*} $$

This construction defines a family of singular surfaces ${\mathcal S}\to U$ over a Zariski open subset $U\subset PGL_3({\mathbb {C}})\times E$ where $E:=\mathrm {Pic}^0(D)$ .

A very general surface over $(\gamma ,t)$ does not have a section, as there are only countably many sections of $X\to \mathbb {P}^1$ ; for a sufficiently general translation t, none of these will glue to a section of the singular surface. Still, for all such surfaces, there is a periodhomomorphism $\psi _{S_0}\colon H^2(X,{\mathbb {Z}})\to E$ defined by (2.1). It descends to the rank $9$ quotient $L:=H^2(X,{\mathbb {Z}})/{\mathbb {Z}} f$ because $f|_D= \mathcal {O}_D$ and $f|_{\gamma (D)}= \mathcal {O}_{\gamma (D)}$ . There is a translation action of $t\in E$ on U given by $ (\gamma _0,\,t_0)\mapsto (\gamma _0,\,t_0\circ t)=:(\gamma _0',t_0')$ . It acts on the period homomorphism as follows:

(3.1) $$ \begin{align}\psi_{S_0'}(v)=\psi_{S_0}(v)+(v\cdot f) t.\end{align} $$

From this formula, we deduce that the dominance of the period map for Type II $_b$ elliptic surfaces is equivalent to dominance of the more general period map

(3.2) $$ \begin{align} PGL_3({\mathbb{C}})\times E\dashrightarrow \mathrm{Hom}(L,E). \end{align} $$

Consider the codimension one subtorus of $\mathrm {Hom}(L,E)$ for which $\psi _{S_0}(h)=0\in E$ , where h is the pullback of the hyperplane class on $\mathbb {P}^2$ . The inverse image of this subtorus contains, as a component, the locus of $(\gamma ,t)$ for which $t=0$ , because under a projective linear identification $\gamma $ , we have $\gamma ^*{\mathcal O}_{\gamma (D)}(1)={\mathcal O}_D(1)$ . Thus, the dominance of (3.2) is implied by the dominance of

(3.3) $$ \begin{align} PGL_3({\mathbb{C}})\dashrightarrow \mathrm{Hom}(H^2(X,{\mathbb{Z}})/{\mathbb{Z}} f+{\mathbb{Z}} h,E).\end{align} $$

This follows because the action of $t\in E$ on $\mathrm {Hom}(L,E)$ described by (3.1) is translation by an elliptic subcurve transverse to the codimension $1$ subtorus of $\mathrm {Hom}(L,E)$ appearing on the right-hand side of (3.3).

Finally, ${\mathbb {Z}}^9 \simeq \textrm {span}\{F_i\,\big {|}\,i=1,\dots ,9\}=h^\perp $ surjects onto $H^2(X,{\mathbb {Z}})/{\mathbb {Z}} f+{\mathbb {Z}} h$ . Pulling back the period map to this lattice, we get a map

(3.4) $$ \begin{align} \begin{aligned} PGL_3({\mathbb{C}})&\dashrightarrow \mathrm{Hom}({\mathbb{Z}}^9,E)/\mathfrak S_9 \\ \gamma&\mapsto\{\psi_{S_0}(F_1),\dots,\psi_{S_0}(F_9)\}.\end{aligned} \end{align} $$

Here, the base points $D\cap \gamma (D)$ , and hence the exceptional curves $F_i$ , are not canonically ordered; they are permuted by the monodromy of the universal family. This is why we must quotient the target by the symmetric group $\mathfrak S_9$ . Since $\sum _{i=1}^9 [F_i] = 3h-f$ in $H^2(X,{\mathbb {Z}})$ , the image of the period map (3.4) lands in

$$ \begin{align*}\{(e_1,\dots,e_9)\in E^9\,\big{|}\,e_1+\cdots +e_9=0\}/\mathfrak S_9=A_8\otimes E/W(A_8)\simeq \mathbb{P}^8.\end{align*} $$

The last isomorphism follows from a well-known theorem of Looijenga [Reference LooijengaLoo76]. Applying the definition of $\psi _{S_0}$ gives a very explicit construction of (3.4):

Definition 3.1. Fix a smooth cubic $D\subset \mathbb {P}^2$ . Define $E:=\mathrm {Pic}^0(D)$ and let $A\colon \mathrm {Sym}^9E\to E$ denote the addition map. For a generic $\gamma \in PGL_3({\mathbb {C}})$ , set $D\cap \gamma (D)=\{p_i\}_{i=1}^9$ and $q_i:=\gamma ^{-1}(p_i)\in D$ . We define

(3.5) $$ \begin{align} \begin{aligned} \Psi\colon PGL_3({\mathbb{C}}) & \dashrightarrow A^{-1}(0)\simeq \mathbb{P}^8 \\ \gamma &\mapsto \{{\mathcal O}_D(p_i-q_i)\}_{i=1}^9.\end{aligned}\end{align} $$

Proof. Let $G\subset PGL_3({\mathbb {C}})$ be the finite subgroup for which $g(D)=D$ . We claim that $\Psi $ extends, as a morphism, from U to $PGL_3({\mathbb {C}})\setminus G$ . This is easy: the map $\Psi $ extends continuously because $D\cap \gamma (D)$ is still a finite set for all $\gamma \in PGL_3({\mathbb {C}})\setminus G$ . Normality of $PGL_3({\mathbb {C}})\setminus G$ implies that a continuous extension is algebraic.

We choose D and $\gamma $ carefully so that the set $D\cap \gamma (D)$ has only three elements. Concretely, consider the extremal cubic pencil $X_{9111}\to \mathbb {P}^1_{[\lambda :\mu ]}$ in the notation of [Reference Miranda and PerssonMP86], given by the equation

$$ \begin{align*}\lambda(x^2y+y^2z+z^2x)+\mu(xyz)=0.\end{align*} $$

See Figure 2. Let $D:=D_{[\lambda :\mu ]}$ be a generic fiber, and let $\gamma = \mathrm {diag}(1,\zeta _3,\zeta _3^2)$ where $\zeta _3$ is a primitive third root of unity. Then $\gamma (D)=D_{[\zeta _3\lambda :\mu ]}$ , and so D and $\gamma (D)$ generate the pencil. The intersection multiset $D\cap \gamma (D)$ is $\{3p_1,3p_2,3 p_3\}$ where

$$ \begin{align*}p_1=[1:0:0],\,p_2=[0:1:0],\, p_3=[0:0:1].\end{align*} $$

Since this $\gamma \in PGL_3({\mathbb {C}})$ fixes $p_1$ , $p_2$ , $p_3$ , the period $\Psi (\gamma )=\{0,\dots ,0\}\in \mathrm {Sym}^9E$ vanishes. To prove that $\Psi $ is dominant, it suffices to show that there is no small deformation $\gamma '\in PGL_3({\mathbb {C}})$ of $\gamma $ for which $\Psi (\gamma ')=\{0,\dots ,0\}$ .

Figure 2 The pencil generated by two cubics, shown in red and black, with set-theoretic base locus three blue points.

Suppose, to the contrary, that there were. Since $\Psi (\gamma ')=\{0,\dots ,0\}$ , every base point in $D\cap \gamma '(D)$ is fixed by $\gamma '$ . If $|D\cap \gamma '(D)|\geq 4$ , then $\gamma '$ must fix a line in $\mathbb {P}^2$ . This is impossible for a small deformation of $\gamma $ , which has isolated fixed points. Conversely, $|D\cap \gamma '(D)|\geq 3$ because each of $p_1$ , $p_2$ , $p_3$ deforms to some fixed point of $\gamma '$ . Hence, $\gamma '$ fixes exactly three points $p_1'$ , $p_2'$ , $p_3'$ . Furthermore, $D\cap \gamma '(D)=\{3p_1',3p_2',3p_3'\}$ as a multiset, again because $\gamma '$ is near $\gamma $ , and the map

$$ \begin{align*}PGL_3({\mathbb{C}})\setminus G \to \mathrm{Sym}^9(D)\end{align*} $$

sending $\gamma ' \mapsto D\cap \gamma '(D)$ with multiplicities is continuous.

Since ${\mathrm {mult}}_{p_i'}(D\cap \gamma '(D))\geq 2$ , we deduce that $\gamma '$ preserves the tangent direction $T_{p_i'}D$ and the corresponding tangent line $L_i'$ . Thus, $\gamma '\in PGL_3({\mathbb {C}})$ fixes the point $L_i'\cap L_j'\in \mathbb {P}^2$ . But, as we noted before, $\gamma '$ only fixes three points (this holds not just on D but in the ambient plane $\mathbb {P}^2$ ). Using that $\gamma '$ is a small deformation of $\gamma $ , we deduce that

$$ \begin{align*}L_1'\cap L_2'=p_2',\,L_2'\cap L_3'=p_3',\,L_3'\cap L_1'=p_1'.\end{align*} $$

Write $p_i'=p_i+t_i$ for a translation $t_i$ . By the addition law on a cubic, we have

$$ \begin{align*}2p_1'=-p_2',\,2p_2'=-p_3',\,2p_3'=-p_1'\end{align*} $$

from which we can conclude that $t_1=(-2)^3t_1$ i.e. $t_1$ is $9$ -torsion. But since $t_i$ are small, we conclude that $t_1=t_2=t_3=0$ and so $p_i'=p_i$ .

Thus, $\gamma '$ fixes $(p_1,p_2,p_3)$ , implying that $\gamma '\in ({\mathbb {C}}^*)^2\subset PGL_3( {\mathbb {C}})$ lies in the maximal torus associated to the coordinates $[x:y:z]$ . Furthermore, $\gamma '$ preserves the base locus scheme $D\cap \gamma '(D)$ , as this is the unique subscheme of D which has length $3$ at each of $p_1,p_2,p_3$ . So $\gamma '$ induces an automorphism of the pencil generated by D and $\gamma '(D)$ . Since the automorphism group of a rational elliptic surface is discrete, and $\gamma '$ is a small deformation of $\gamma $ , the automorphism $\gamma '$ must have order $3$ . But no nontrivial small deformation of $\gamma =\mathrm {diag}(1,\zeta _3,\zeta _3^2)$ within the torus $({\mathbb {C}}^*)^2$ has order $3$ . This is a contradiction.

4 Type II $_f$ degenerations

We consider in this section degenerations of $S\to C$ that keep the base C constant. These are never of Type II $_b$ because in all such degenerations, $j(C)\to \infty $ .

Take a one-parameter deformation of $a,b\in H^0(C,{\mathcal O}_C(4p)), H^0(C,{\mathcal O}_C(6p))$ over $(B,0)$ until the discriminant $4a_0^3+27b_0^2=0\in H^0(C,{\mathcal O}_C(12p))$ vanishes identically. For instance, we can take the fiber over $0\in B$ to be

$$ \begin{align*}y^2=x^3-3r^2x+2r^3\end{align*} $$

with $r\in H^0(C,{\mathcal O}_C(2p))$ . The degeneration

$$ \begin{align*}\overline{{\mathcal S}}\to C\times B\to (B,0)\end{align*} $$

of elliptic surfaces has a central fiber ${\overline S}_0\to C$ whose generic fiber is irreducible nodal, with two cuspidal fibers over the zeroes of r. In particular, the normalization ${\overline S}^\nu _0:=X\to C$ is the smooth $\mathbb {P}^1$ -bundle $X=\mathbb {P}_C({\mathcal O}\oplus L)$ , and ${\overline S}_0$ is reconstructed from gluing a bisection D of $X\to C$ , branched over the two zeroes of r. This bisection D is glued along the involution switching the two sheets of $\nu \colon D\to C$ .

For future reference, note that $\mathrm {NS}(X)\simeq H^2(X,{\mathbb {Z}})$ is spanned by the $\mathbb P^1$ -fiber class f and the class of the section $s_{\infty } = \mathbb P_C({\mathcal O}\oplus 0)$ , with intersection form

$$ \begin{align*}f\cdot f = 0,\,\,\,\, s_\infty\cdot f = 1, \,\,\,\, s_\infty\cdot s_{\infty} = -1,\end{align*} $$

and $K_X = -f-2s_\infty $ . The other natural section $s_0 = \mathbb P_C(0\oplus L)$ has class $f+s_\infty $ .

The bisection $D\subset X$ has genus 2, being a double cover of C branched over two points. Thus, its cohomology class is $[D]=2f+2s_\infty = -K_X+f = 2s_0$ . Note that $[D]^2=4$ and $[D]\cdot K_X = -2$ . The section s that is present on the smooth surfaces in the family $\mathcal S$ limits to $s_\infty $ , which is the unique section of X disjoint from D.

Proof. Consider a deformation of the Weierstrass equation

$$ \begin{align*}y^2=x^3-(3r^2+\epsilon g_4)x+(2r^3+\epsilon g_4r+\epsilon^2g_6),\end{align*} $$

where $g_d\in H^0(C,{\mathcal O}_C(dp))$ has degree d. The discriminant $\Delta = 4a^3+27b^2$ is

$$ \begin{align*}\Delta = 9r^2(12rg_6 - g_4^2)\epsilon^2+{\mathcal O}(\epsilon^3).\end{align*} $$

Thus, the Zariski closure of the discriminant divisor is

$$ \begin{align*}\lim_{\epsilon\to 0} \mathrm{div}(\Delta) = 2\cdot \mathrm{div}(r)+\mathrm{div}(12rg_6-g_4^2).\end{align*} $$

For fixed r, the sections $rg_6$ form a linear subspace $\mathbb {P}^5\subset \mathbb {P}^7=\mathbb {P}H^0(C,{\mathcal O}(8p))$ of codimension $2$ . The sections $g_4^2\in \mathbb {P}H^0(C,{\mathcal O}(8p))$ are the image of the degree $2$ Veronese embedding, followed by a linear projection

$$ \begin{align*}v_2\colon \mathbb{P}^3\hookrightarrow \mathbb{P}^9= \mathbb{P}\mathrm{Sym}^2H^0(C,{\mathcal O}(4p))\dashrightarrow \mathbb{P}^7.\end{align*}$$

The inverse image of $\{\mathrm {div}(rg_6)\}=\mathbb {P}^5\subset \mathbb {P}^7$ is a copy of $\mathbb {P}^7\subset \mathbb {P}^9$ under the linear projection. Thus, the vanishing loci of linear combinations are represented geometrically as the join of the projective subvarieties $v_2(\mathbb {P}^3)$ , $\mathbb {P}^7\subset \mathbb {P}^9$ . This join is all of $\mathbb {P}^9$ . Thus, we can realize any divisor in $|8p|$ as $\lim _{\epsilon \to 0}\mathrm {div}(\Delta )-2\cdot \mathrm {div}(r)$ .

For general $g_4$ and $g_6$ , the punctured family over $B\setminus 0$ has smooth total space. The threefold $\overline {{\mathcal S}}$ is a double cover branched over the vanishing locus of the cubic $x^3-(3r^2+\epsilon g_4)x+(2r^3+\epsilon g_4r+\epsilon ^2g_6)$ , so it can only be singular where two of the roots of the cubic coincide. This shows that the singular locus $\overline {{\mathcal S}}_{\mathrm {sing}}\subset V(y,x-r,\epsilon )$ is contained in the singularities of the fibers of ${\overline S}_0\to C$ .

Since $\epsilon ^2\mid \mid \Delta $ , the local equation of the double cover is generically $y^2=u^2+\epsilon ^2$ along the nodes of ${\overline S}_0\to C$ . So the nodes form a family of $A_1$ -singularities in $\overline {{\mathcal S}}$ . At the nodes on the fibers lying over $\mathrm {div}(12rg_6-g_4^2)$ , the local equation is rather $y^2=u^2+v\epsilon ^2$ . Thus, to find a semistable model ${\mathcal S}\to (B,0)$ , we simply blow up the double locus of ${\overline S}_0$ in the total space $\overline {{\mathcal S}}$ .

The resulting central fiber is $S_0=X\cup _DV$ for a ruled surface $V\to C$ , which contains D as a bisection and has $8$ reducible fibers over the points in $\mathrm {div}(12rg_6-g_4^2)$ ; see Figure 3. Thus, $V\sim Bl_{p_1,\dots ,p_8}X$ is deformation-equivalent to the blow-up of X at $8$ points on D, with the double locus on V identified with D via the strict transform. It is only deformation-equivalent because $V\to C$ could be the projectivization of a non-split extension of L by ${\mathcal O}$ . Regardless, we can identify

$$ \begin{align*}H^2(V,{\mathbb{Z}})=H^2(X,{\mathbb{Z}})\oplus_{i=1}^8{\mathbb{Z}} E_i\end{align*} $$

and $[D] = 2s_0 - [E_1]-\cdots -[E_8]=-K_V+f$ .

Figure 3 A Type II $_f$ surface $S_0 = X\cup _D V$ with the genus $2$ double locus D shown in red, the section s in green, limits of $8$ nodal fibers in blue, and limits of pairs of nodal fibers dashed.

From Section 6 and Proposition 6.6, the mixed Hodge structure of a Type II $_f$ surface has a period map to $E_8\otimes {\mathcal E}$ which can be described as follows. Consider the sublattice $\{K_V,f\}^\perp \subset H^2(V,{\mathbb {Z}})$ . This is isometric to the root lattice

$$ \begin{align*}D_8 = \{(a_1,\dots,a_8)\in {\mathbb{Z}}^8 \,\big{|}\, a_1+\cdots+a_8\in 2{\mathbb{Z}} \}\end{align*} $$

via the map $(a_1,\dots ,a_8) \mapsto \sum _{i=1}^8 a_i[E_i] - \left (\frac {1}{2} \sum _{i=1}^8 a_i\right ) f$ . When this isometry is understood, we will refer to $\{K_V,f\}^\perp $ simply as $D_8$ .

Let $E:=\mathrm {Pic}^0(D)/\mathrm {Pic}^0(C)$ be the Prym variety of the double cover $\nu \colon D\to C$ . We define a period homomorphism

(4.1) $$ \begin{align} \begin{aligned} \psi_{S_0}\colon D_8& \to E \\ \gamma & \mapsto \mathcal{L}_\gamma\big{|}_D\textrm{ mod }\mathrm{Pic}^0(C) \end{aligned}\end{align} $$

by lifting an element $\gamma \in D_8$ to an element ${\mathcal L}_\gamma \in \mathrm {Pic}(V)$ . These lifts form a $\mathrm {Pic}^0(C)$ -torsor, and thus, the image of ${\mathcal L}_\gamma \big {|}_D\in \mathrm {Pic}^0(D)$ under the map to E is well-defined.

Proof. The period point $\psi _{S_0}$ and limit mixed Hodge structure of ${\mathcal S}$ are encoded, up to a finite map, in the data $(\nu \colon D\to C,\{r_i\}_{i=1}^8)$ consisting of

1. 1. a degree $2$ map $\nu \colon D\to C$ from a genus $2$ to a genus $1$ curve, and

2. 2. a multiset of $8$ points $\{r_1,\dots ,r_8\}\subset C$ .

Let $\iota \colon D\to D$ be the involution switching the sheets of $\nu $ and let $\{p_i,q_i\}=\nu ^{-1}(r_i)$ . Then ${\mathcal O}_D(p_i-q_i)\in \mathrm {Pic}^0(D)$ gives, upon quotienting by $\mathrm {Pic}^0(C)$ , the period

$$ \begin{align*}\psi_{S_0}(F_i-F_i')=[{\mathcal O}_D(p_i-q_i)]\in E,\end{align*} $$

where $F_i+F_i'$ is a reducible fiber of the ruling $V\to C$ . Ranging over the eight reducible fibers, the tuple

$$\begin{align*}({\mathcal O}_D(p_i-q_i)\textrm{ mod }\mathrm{Pic}^0(C))_{i=1}^8\in E^8\end{align*}$$

encodes $\psi _{S_0}$ up to torsion because $\bigoplus _{i=1}^8{\mathbb {Z}}(F_i-F_i')\subset D_8$ has finite index.

Let $\{r_9, r_{10}\}\in C$ be the branch points of $\nu $ . Then $\nu $ is determined by the monodromy representation $\rho \colon \pi _1(C\setminus \{r_9,r_{10}\},*)\to {\mathbb {Z}}_2$ . Let $\mathrm {Prym}^2{\mathcal C}$ be the moduli space of Prym data $(C,\{r_9,r_{10}\},\rho )$ over the universal genus $1$ curve $\mathcal {C}\to \mathcal {M}_1$ . It is a Deligne-Mumford stack of dimension $2$ , one dimension for $j(C)$ and another for the element $r_9-r_{10}\in \mathrm {Pic}^0(C)$ , well-defined up to sign. The data of $\rho $ is finite.

A point $r_i\in C$ determines $p_i$ up to switching $p_i\leftrightarrow q_i$ which acts by negation on the image of ${\mathcal O}_D(p_i-q_i)$ in E. Thus, we globally get a well-defined map

(4.2) $$ \begin{align}\begin{aligned} \Psi\colon \mathrm{Sym}^8{\mathcal C} \times_{\mathcal M_1} \mathrm{Prym}^2{\mathcal C} &\to {\mathbb{Z}}^8 \otimes {\mathcal{E}}/\mathfrak S_8^{\pm} \\ (C, \{r_1,\dots,r_8\} ,\{r_9,r_{10}\},\rho)& \mapsto \{{\mathcal O}_D(p_i-q_i)\textrm{ mod }\mathrm{Pic}^0(C)\}_{i=1}^8,\end{aligned}\end{align} $$

where ${\mathcal E}$ is the universal elliptic curve. Since the image of each ${\mathcal O}_D(p_i-q_i)$ in E is only well-defined up to sign, and the reducible fibers of $V\to C$ are unordered, we must quotient the target by the signed permutation group $\mathfrak S_8^{\pm }$ .

Observe that $\mathrm {Sym}^8{\mathcal C} \times _{\mathcal M_1} \mathrm {Prym}^2{\mathcal C}$ is ten-dimensional. There is a single condition ensuring that a point in the domain of $\Psi $ arises from a degeneration of surfaces in F: If $L\to C$ is the Hodge bundle, then $r_9+r_{10}\in |2L|$ and so by Proposition 4.1, $\{r_1,\dots ,r_8\},\{r_9,r_{10}\}$ can arise so long as $r_1+\cdots +r_8\in |8L|$ (i.e., the relation

(4.3) $$ \begin{align} r_1+\cdots+r_8- 4(r_9+r_{10})=0\in \mathrm{Pic}^0(C) \end{align} $$

is satisfied). So the Type II $_f$ limits of degenerations from F are described by

$$ \begin{align*}Z= \{\textrm{elements of }\mathrm{Sym}^8{\mathcal C} \times_{\mathcal M_1} \mathrm{Prym}^2{\mathcal{C}}\,\big{|}\,r_1+\cdots+r_8-4(r_9+r_{10})=0\}.\end{align*} $$

Our goal is to prove the dominance of the map $\Psi \big {|}_Z\colon Z\to {\mathbb {Z}}^8 \otimes {\mathcal E}/\mathfrak S_8^{\pm }.$

Fix an elliptic curve fiber E of $\mathcal {E}$ , consider the point $\{0,\dots ,0\}\in \mathrm {Sym}^8E$ , and let $\ker _E(\Psi ):= \Psi ^{-1}(\{0,\dots ,0\})$ . It suffices to prove that $Z\cap \ker _E(\Psi )$ contains, as a component, some zero-dimensional scheme. Let $L_E\subset \mathrm {Prym}^2\mathcal {C}$ be the sublocus of Prym data whose Prym variety is E. It is a curve inside the surface $\mathrm {Prym}^2{\mathcal C}$ . Then, $\mathrm {ker}_E(\Psi )$ contains, as a component, an unramified double cover $M_E\to L_E$ on which $r=r_1=\cdots = r_8$ and $r\in \{r_9,r_{10}\}$ because the morphism $D\to E$ sending $p\mapsto {\mathcal O}_D(p-\iota (p))\textrm { mod }\mathrm {Pic}^0(C)$ is surjective.

The defining equation (4.3) of Z restricts to $M_E$ to give the equation

$$ \begin{align*}4(r_9-r_{10})=0\in \mathrm{Pic}^0(C)\end{align*} $$

(i.e., $r_9-r_{10}\in \mathrm {Pic}^0(C)[4]).$ The locus in $L_E$ on which $r_9-r_{10}$ is $4$ -torsion is finite and nonempty. So the theorem follows.

The proofs of Theorems 3.2 and 4.4 suggest a rather wild conjecture:

The existence of such a birational involution would give a geometric explanation for why degenerations of Type II $_b$ and II $_f$ can have the same periods, even though $j(C)\to \infty $ in the former, while $j(F)\to \infty $ in the latter.

5 A family losing dimension

Let $F^{\mathrm {cusp}}\hookrightarrow F$ be the closure of the sublocus of elliptic fibrations $S\to C$ which have six cuspidal (Kodaira type II) fibers. These fibrations are isotrivial and have a Weierstrass form $y^2=x^3+b$ for some $b\in H^0(C,{\mathcal O}_C(6p))$ . There is a fiber preserving automorphism $\sigma \colon S\to S$ , given by

$$ \begin{align*}\sigma\colon (x,y)\mapsto (\zeta_3x,-y),\end{align*} $$

and $\sigma ^*\Omega _S=\zeta _6\Omega _S$ acts nontrivially on the holomorphic $2$ -form by a primitive sixth root of unity. Furthermore, since $\sigma $ preserves s and f, it defines an element $\sigma ^*\in \Gamma =O(I\!I_{2,10})$ which is easily checked to fix only the origin of $I\!I_{2,10}$ . So $\sigma ^*$ endows $I\!I_{2,10}$ with the structure of a Hermitian lattice of hyperbolic signature $(1,5)$ over the Eisenstein integers ${\mathbb {Z}}[\zeta _6]$ , and

$$ \begin{align*}{\mathbb B}:=\mathbb{P}\{x\in I\!I_{2,10}\otimes {\mathbb{C}}\,\big{|}\,x\cdot \overline{x}>0,\, \sigma^*x=\zeta_6x\}\subset {\mathbb{D}}\end{align*} $$

is a Type I Hermitian symmetric subdomain (a complex ball), of dimension $5$ . Letting $\Gamma _0:=\{\gamma \in \Gamma \,\big {|}\,\gamma \circ \sigma ^* = \sigma ^*\circ \gamma \}$ be the group of Hermitian isometries, we get a period map to a $5$ -dimensional ball quotient

$$ \begin{align*}F^{\mathrm{cusp}}\to {\mathbb B}/\Gamma_0.\end{align*} $$

But $\dim F^{\mathrm {cusp}}=1+5=6$ with parameters corresponding to $j(C)$ and the relative locations of the six cuspidal fibers. Thus, $P\big {|}_{F^{\mathrm {cusp}}}$ has positive fiber dimension.

It seems likely that $P\big {|}_{F^{\mathrm {cusp}}}$ is surjective, with generic fiber dimension $1$ . Regardless, this gives a second example, after Ikeda’s [Reference IkedaIke19], proving that P is not a finitemap, even though it is generically finite by Theorem 1.1:

6 Mixed Hodge Structures

MHS of a normal crossings surface

Let $S_0$ be a reduced normal crossings surface with smooth double locus and no triple points. Our goal in this section is to explicitly describe the mixed Hodge structure on $H^2(S_0)$ . Let $S_0 = \bigcup _{i=1}^m S_i$ with the double curve $D_{ij} = S_i\cap S_j$ a smooth, possibly disconnected or empty curve for all $i<j$ . Let $D:=\bigcup _{i<j} D_{ij}$ . The Mayer-Vietoris sequence associated to a covering of $S_0$ by neighborhoods of the irreducible components $S_i$ reads

(6.1) $$ \begin{align} \bigoplus_{i=1}^m H^1(S_i) \overset{\iota^*}\to \bigoplus_{i< j} H^1(D_{ij}) \to H^2(S_0) \to \bigoplus_{i=1}^m H^2(S_i)\overset{\mathrm{res}}\longrightarrow \bigoplus_{i< j} H^2(D_{ij}).\end{align} $$

Here, $\iota ^*$ and $\mathrm {res}$ are signed restriction maps. Let $K\subset \bigoplus H^2(S_i)$ be the kernel of the morphism $\mathrm {res}$ – that is, $K=\{(\alpha _i\in H^2(S_i))\,\big {|}\,\alpha _i\cdot D_{ij} = \alpha _j\cdot D_{ij}\}$ . Define

$$ \begin{align*}J:= \mathrm{coker}(\iota^*).\end{align*} $$

By exactness of the sequence (6.1), we obtain a short exact sequence

$$ \begin{align*}0 \to J \to H^2(S_0) \to K \to 0.\end{align*} $$

In fact, it is a short exact sequence of mixed Hodge structures with left-hand term J pure of weight $1$ , and the right-hand term K pure of weight $2$ .

Proposition 6.1. If $p_g(S_i)=0$ for all components $S_i\subset S_0$ (equivalently, K is Hodge-Tate of weight $2$ ), then the Carlson classifying map [Reference CarlsonCar85]

$$ \begin{align*}\phi:K \to \mathrm{Jac}(J)\end{align*} $$

of the extension coincides with the Abel-Jacobi map. More precisely, an element of K is a tuple $(\alpha _i\in H^2(S_i,{\mathbb {Z}}))$ represented by line bundles $\mathcal {L}_i$ such that for each $i<j$ , we have $c_1(\mathcal {L}_i|_{D_{ij}}) -c_1(\mathcal {L}_j|_{D_{ij}}) =0\in H^2(D_{ij})$ . Then $\phi = \pi \circ \mathrm {AJ} \circ \psi $ , where

$$ \begin{align*} (\alpha_i\in H^2(S_i,{\mathbb{Z}}))&\overset{\psi}\mapsto \textstyle \bigoplus_{i<j} \mathcal{L}_i|_{D_{ij}} \otimes\mathcal{L}_j|_{D_{ij}}^{-1}\in \mathrm{Pic}^0(D), \end{align*} $$

$\mathrm {AJ}\colon \mathrm {Pic}^0(D) \to \mathrm {Jac}(D)= \mathrm {Jac}(H^1(D))$ is the classical Abel-Jacobi isomorphism, and $\pi \colon \mathrm {Jac}(D) \to \mathrm {Jac}(J)$ is the projection map.

Proof. Following Carlson’s construction, the classifying map $\phi $ for a weight separated extension of mixed Hodge structures is given by the composition of two splittings. First, choose a left-splitting $a: H^2(S_0) \to J$ over ${\mathbb {Z}}$ . Next, choose a right-splitting $b:K \to F^1 H^2(S_0)_{\mathbb {C}}$ over ${\mathbb {C}}$ , which respects the Hodge filtration. The composition $a_{\mathbb {C}}\circ b: K \to J_{\mathbb {C}}$ gives the classifying map after passing to the Jacobian quotient:

$$ \begin{align*}\phi: K \to J_{\mathbb{C}} / (J_{\mathbb{Z}} + F^1 J_{\mathbb{C}}).\end{align*} $$

For a, it suffices to produce a morphism on homology $\ker (\iota _*) \to H_2(S_0)$ , and then use the universal coefficient theorem to give a map in the opposite direction:

$$ \begin{align*}H^2(S_0) \to H_2(S_0)^* \to \ker(i_*)^* \simeq \mathrm{coker}(\iota^*) = J.\end{align*} $$

To define the morphism $\ker (\iota _*) \to H_2(S_0)$ , choose a basis for $\ker (\iota _*)$ at the singular chain level: tuples of 1-cycles $t_k=(\gamma ^k_{ij}\in \mathcal {Z}_1(D_{ij}))$ such that for each i,

$$ \begin{align*}\sum_j \iota_*(\gamma^k_{ij}) = \partial(\Gamma^{k}_i) \textrm{ for some }\Gamma_i^k\in \mathcal{C}_2(S_i).\end{align*} $$

We use the convention that $\gamma _{ij} = - \gamma _{ji}$ . Choosing such $\Gamma _i^k$ for each $t_k$ in the basis of $\ker (i_*)$ , we construct a $2$ -cycle (see Figure 4),

$$ \begin{align*}T_k=\bigcup_i \Gamma_i^{k} \in \mathcal{Z}_2(S_0).\end{align*} $$

We take the $1$ -cycles $\gamma _{ij}^k$ to be ${\mathbb {Z}}$ -linear combinations of some fixed $2g(D_{ij})$ loops on each $D_{ij}$ , whose union we call $\gamma $ , chosen so that their complement in $D_{ij}$ is a contractible $4g$ -gon. The assignment $t_k \mapsto [T_k]\in H_2(S_0)$ then induces a splitting

$$ \begin{align*}a: H^2(S_0) \to J.\end{align*} $$

Figure 4 Heuristic diagram of irreducible components $S_i$ in black, double curves $D_{ij}$ in red, $1$ -cycles $\gamma _{ij}\subset D_{ij}$ in green, and $2$ -cycles $\Gamma _i\subset S_i$ capping the $1$ -cycles in blue.

To construct a splitting b, we use the Čech-de Rham model of $H^2(S_0,{\mathbb {C}})$ , and its Hodge filtration $F^1$ . An element of $H^2(S_0,{\mathbb {C}})$ is represented by two tuples of differential forms:

$$ \begin{align*}(\omega_i\in {\mathcal Z}^2(S_i))_i\textrm{ and } (\theta_{ij}\in \mathcal{A}^1(D_{ij}))_{i<j}\end{align*} $$

such that for all $i<j$ , we have $\omega _i|_{D_{ij}} - \omega _j|_{D_{ij}} = d\theta _{ij}$ . If furthermore, $\theta _{ij}\in \mathcal {A}^{1,0}(D_{ij})$ for all $i<j$ , then the element lies in $F^1H^2(S_0,{\mathbb {C}})$ .

Given $(\alpha _i)\in K = \ker (\mathrm {res})$ , we know that $\alpha _i|_{D_{ij}} - \alpha _j|_{D_{ij}}=0\in H^2(D_{ij})$ . To define $b:K \to F^1 H^2(S_0,{\mathbb {C}})$ , select a basis for K; for each basis element $(\alpha _i)\in K$ , there exists line bundles $\mathcal {L}_i$ such that $c_1(\mathcal {L}_i)=\alpha _i$ . Since each $S_i$ is projective, we may assume that the $\mathcal {L}_i \simeq \mathcal {O}_{S_i}(C_i - C^{\prime }_i)$ , where $C_i$ and $C_i'$ are ample effective curves on $S_i$ meeting each $D_{ij}$ transversely away from $\gamma $ . We take $\omega _i\in {\mathcal Z}^2(S_i)$ representing $c_1(\mathcal {L}_i)$ and supported on a small neighborhood of $C_i\cup C^{\prime }_i$ . Since $\omega _i|_{D_{ij}} - \omega _j|_{D_{ij}}\in {\mathcal Z}^2(D_{ij})$ integrates to 0, it has a $\overline {\partial }$ -primitive $\theta _{ij}\in \mathcal {A}^{1,0}(D_{ij})$ , unique up to the addition of a holomorphic one-form.

To interpret the composition $\phi =a_{\mathbb {C}} \circ b:K \to J_{\mathbb {C}}$ , we will regard $J_{\mathbb {C}}$ as $\mathrm {Hom}(\ker (\iota _*),{\mathbb {C}})$ . Then $(a_{\mathbb {C}} \circ b)(\alpha _i)$ is the unique homomorphism $\ker (\iota _*)\to {\mathbb {C}}$ which sends $t_k$ to

(6.2) $$ \begin{align} \sum_{i=1}^m \int_{\Gamma_i^k} \omega_i + \sum_{i<j} \int_{\gamma_{ij}^k} \theta_{ij}. \end{align} $$

We henceforth drop the index k as we will consider a single basis vector $t=t_k$ .

We will make two simplifications in order to compare $\phi $ with the Abel-Jacobi map. First, the chains $\Gamma _i$ can be replaced with $\Gamma _i+x_i$ for any $x_i\in \mathcal {Z}_2(S_i)$ such that the tuple of homology classes $(x_i)$ is Poincaré dual to an element of K. By Lefschetz duality, there is a perfect pairing associated to the 4-manifold with boundary

$$ \begin{align*}I:H_2(S_i - N_\epsilon(\gamma))\times H_2(S_i - N_\epsilon(\gamma), \partial) \to {\mathbb{Z}},\end{align*} $$

and we have $\int _{\Gamma _i} \omega _i = I(C_i - C_i', \Gamma _i) \in {\mathbb {Z}}$ . Since $(\alpha _i)$ is primitive in K, one can find $x\in K$ such that

$$ \begin{align*}I(C_i - C_i',x) = -I(C_i - C_i', \Gamma_i).\end{align*} $$

So replacing $\Gamma _i$ with $\Gamma _i+x_i$ , we may assume that the first sum in (6.2) vanishes.

Second, the primitives $\theta _{ij}$ are not closed, so the second integral does not make sense on the homology classes $[\gamma ^k_{ij}]$ . To remedy this, we construct smooth 1-forms $\lambda _{ij}\in {\mathcal Z}^1(D_{ij})$ supported away from $\gamma $ such that $d(\theta _{ij}+\lambda _{ij})=0$ . Let $\ell _{ij}$ be a smooth 1-chain on $D_{ij}\setminus \gamma $ with boundary the signed intersection points:

$$ \begin{align*}\partial \ell_{ij} = (C_i - C_i')\cap D_{ij} - (C_j - C_j')\cap D_{ij}.\end{align*} $$

By Lemma 6.2 below, we may produce a form $\lambda _{ij}$ supported in a neighborhood of $\ell _{ij}$ . This allows us to write the Carlson map for our extension as

$$ \begin{align*}\phi((\alpha_i)) = \left[ t \mapsto \sum_{i<j} \int_{\lambda_{ij}} (\theta_{ij}+\lambda_{ij}) \right] \in J_{\mathbb{C}} / (J_{\mathbb{Z}} + F^1J_{\mathbb{C}}).\end{align*} $$

But for any $\tau \in \Omega ^1(D_{ij})$ , since $\theta _{ij}\in \mathcal {A}^{1,0}(D_{ij})$ we have, again by Lemma 6.2,

$$ \begin{align*}\int_{D_{ij}} (\theta_{ij}+\lambda_{ij})\wedge \tau = \int_{D_{ij}} \lambda_{ij}\wedge \tau = \int_{\ell_{ij}} \tau.\end{align*} $$

Observe that the classical Abel-Jacobi map $\mathrm {AJ}\colon \mathrm {Pic}^0(D)\to \mathrm {Jac}(D)$ indeed sends $[\partial \ell _{ij}] \mapsto \int _{\ell _{ij}}$ . The proposition follows.

Now, we produce the one-form $\lambda _{ij}$ with the desired properties.

Lemma 6.2. Let C be a Riemann surface and let $\mathcal {L}=\mathcal {O}_C(q-p)$ . There is a hermitian metric h on $\mathcal {L}$ , a $(1,0)$ -form $\theta \in \mathcal {A}^{1,0}(C)$ , and a smooth $1$ -form $\lambda $ supported in a neighborhood of a path $\ell $ from p to q for which

1. 1. $\overline {\partial } \theta = \frac {i}{2\pi }\partial \overline {\partial }\log (h)$ ,

2. 2. $d\lambda = -\overline {\partial } \theta $ , and

3. 3. $\int \lambda \wedge \tau = \int _{\ell } \tau $ for any holomorphic one-form $\tau $ .

Proof. Let z be a chart to ${\mathbb {C}}$ from a neighborhood of $\ell $ . There exists a function $f\colon C\setminus \{p,q\}\to {\mathbb {C}}^*$ of the following form:

Such a smooth interpolation exists because $\frac {z-q}{z-p}$ has winding number zero along the boundary of $N_{\epsilon /2}(\ell )$ . Let $s\in \mathrm {Mero}(C, \mathcal {L})$ be a meromorphic section with a zero at q and a pole at p. Then, there is a hermitian metric h on $\mathcal {L}$ for which $h(s,\overline {s})=|f|^2$ . The associated curvature form is $\tfrac {i}{2\pi }\partial \overline {\partial } \log |f|^2$ , and since $c_1(\mathcal {L})=0$ , we can find a $(1,0)$ -form $\theta $ satisfying (1). Furthermore, $\lambda = -\frac {i}{2\pi }(\overline {\partial } \log (f)-\partial \log (\overline {f}))$ is a $(0,1)$ -form, supported in $A:=N_{\epsilon /2}(\ell )^c\cap N_\epsilon (\ell )$ and satisfying (2).

It remains to check (3). We may write $\tau = dg$ for some holomorphic function $g\colon N_\epsilon (\ell )\to {\mathbb {C}}$ . Applying Stokes’s formula and the residue formula, we have

$$ \begin{align*}\int_C \lambda\wedge \tau &=- \tfrac{i}{2\pi} \int_A\overline{\partial}\log(f) \wedge dg = \tfrac{i}{2\pi} \int_A d(ig\cdot d\log(f)) =\tfrac{i}{2\pi}\int_{\partial A} ig\cdot d\log(f) \\ &= -\tfrac{i}{2\pi}\int_{\partial N_{\epsilon/2}(\ell)}g \cdot d\log(\tfrac{z-q}{z-p}) = -\tfrac{i}{2\pi} (2\pi i)(g(q)-g(p)) = \int_\ell\tau. \end{align*} $$

More generally, the lemma holds for any degree zero line bundle $\mathcal {O}_C(\sum (q_i-p_i))$ , for a union of paths connecting each pair of points $p_i$ to $q_i$ by taking the product of the hermitian metrics, and sum of the corresponding $\theta $ ’s and $\lambda $ ’s.

Remark 6.3. To apply Lemma 6.2 to the proof of Proposition 6.1, our forms $\omega _i$ must be such that $\omega _i|_{D_{ij}}-\omega _j|_{D_{ij}}$ is the two-form $\frac {i}{2\pi } \partial \overline {\partial }\log (h)$ supported in a neighborhood of $\ell _{ij}$ . This is achieved by choosing $\omega _i = \frac {i}{2\pi } \partial \overline {\partial } \log (h_i)$ for hermitian metrics on $h_i$ on $\mathcal {L}_i$ (and similarly for j) so that $h=h_i/h_j$ is the desired hermitian metric on $\mathcal {L}_i|_{D_{ij}}\otimes \mathcal {L}_j|_{D_{ij}}^{-1}$ . Note though that we must allow the two-form $\omega _i$ to be supported in a tubular neighborhood of $C_i\cup C_i'\cup \ell _{ij}$ rather than just $C_i\cup C_i'$ . Since $\ell _{ij}$ is disjoint from $\gamma $ , the argument of Lemma 6.1 is unaffected.

Clemens-Schmid sequence

Let $\mathcal {S}\to (B,0)$ be a degeneration of projective surfaces with smooth total space and reduced normal crossings central fiber $S_0=\bigcup _{i=1}^m S_i$ with smooth double locus. Assume, furthermore, that $p_g(S_i)=0$ for all i.

The monodromy is unipotent by Clemens [Reference ClemensCle69]. So let N be the nilpotent logarithm of the monodromy operator on $H^*(S_t)$ . We have the Clemens-Schmid sequence [Reference MorrisonMor84] relating the integral cohomology of $S_0$ and $S_t$ :

(6.3) $$ \begin{align} 0 \to H^0(S_t) \overset{N}\longrightarrow H^0(S_t) \to H_4(S_0) \to H^2(S_0) \to H^2(S_t) \overset{N}\longrightarrow H^2(S_t). \end{align} $$

Since the monodromy operator acts trivially on $H^0(S_t)$ , the first nilpotent operator in (6.3) is identically $0$ . Using these two observations, the Clemens-Schmid sequence can be shortened to

(6.4) $$ \begin{align} 0 \to H^0(S_t) \to H_4(S_0)\simeq {\mathbb{Z}}^m \to H^2(S_0) \to H^2(S_t) \overset{N} \longrightarrow H^2(S_t). \end{align} $$

The limit mixed Hodge structure $H^2(S_t)$ has a monodromy-weight filtration defined in terms of N: $\{0\} = W_0 \subset W_1 \subset W_2 \subset W_3 = H^2(S_t)$ .

$$ \begin{align*} W_1 H^2(S_t) &= \mathrm{im}(N) ;\\ W_2 H^2(S_t) &= \ker(N) ;\\ W_3 H^2(S_t) &= H^2(S_t). \end{align*} $$

We call $\ker (N)$ the 1-truncated mixed Hodge structure. To describe the 1-truncation explicitly, we combine (6.4) and (6.1) above at their common term $H^2(S_0)$ , with Mayer-Vietoris written horizontally and Clemens-Schmid written vertically.

Here, $\xi _k:= \sum _j [D_{jk}]-[D_{kj}]$ , where $[D_{jk}]\in H^2(S_j)$ and $[D_{kj}]\in H^2(S_k)$ are the fundamental classes of the double loci, and $\Lambda $ is the cokernel of $J\to \ker (N)$ . We have that $\xi _k= c_1(\mathcal {O}_{\mathcal {S}}(S_k)|_{S_0})$ . By Proposition 6.1, we have $\xi _k\in \ker (\phi \colon K\to \mathrm {Jac}(J))$ because the line bundles $\mathcal {O}_{\mathcal {S}}(S_k)\big {|}_{S_i}\simeq \mathcal {O}_{\mathcal {S}}(S_k)\big {|}_{S_j}$ agree on the double locus. Hence, the Carlson extension homomorphism $\phi $ descends to a homomorphism

$$ \begin{align*}\psi_{S_0}\colon \Lambda\to \mathrm{Jac}(J)\end{align*} $$

encoding the 1-truncated mixed Hodge structure.

Application

In this section, we apply the general results above to the mixed Hodge structures associated to the degenerations of Type II $_b$ and II $_f$ , and relate their associated periods to the boundary of the toroidal extension $({\mathbb D}/\Gamma )^{\mathrm {II}}$ .

It is convenient to make an order $2$ base change and resolution to the Type II $_b$ degenerations. The effect is to normalize the first component and insert a second component isomorphic to $\mathbb {P}^1\times E$ where E is the fiber over the node of $C_0$ . This second component is glued to the rational elliptic surface $X\to \mathbb {P}^1$ along the two fibers $X_p, X_q$ .

After the base change and resolution, we have that in both II $_b$ and II $_f$ degenerations, the central fiber $S_0$ has two irreducible components and reduced normal crossings: $S_0 = S_1 \cup _D S_2$ . The double locus D is a disjoint union of two copies of the same elliptic curve E in Type II $_b$ and a connected, smooth genus $2$ curve in Type II $_f$ . Let $D_1\subset S_1$ and $D_2\subset S_2$ denote the double locus restricted to each component.

In both cases, the divisor D admits a natural involution $\iota $ , and the image of the first map $\iota ^*$ in (6.1) is the $(+1)$ -eigenspace of this involution on $H^1(D)$ . The image of the restriction map $\mathrm {res}$ in (6.1) is a rank 1 subgroup of $H^2(D)\simeq H_0(D)$ , so the Mayer-Vietoris sequence takes the form

(6.5) $$ \begin{align} 0 \to H^1(D)^- \to H^2(S_0) \to H^2(S_1)\oplus H^2(S_2) \overset{\mathrm{res}}\longrightarrow {\mathbb{Z}} \to 0. \end{align} $$

Case II $_{\mathbf {b}}$ . The component $S_1$ is a rational elliptic surface X, with $D_1 = X_p \cup X_q$ a pair of isomorphic elliptic curve fibers. The component $S_2$ is simply ${\mathbb P}^1\times E$ with $D_2=\{0,\infty \}\times E$ . The involution on D swaps the two isomorphic components. Note that since $[X_p] = [X_q]\in H^2(S_1)$ , and similarly for $S_2$ , the two restriction maps $H^2(S_i) \to H^2(D)\simeq H^2(E)^{\oplus 2}$ have the same image – namely, the diagonal.

Case II $_{\mathbf {f}}$ . The component $S_1$ is an elliptic ruled surface $X\simeq {\mathbb P}_C({\mathcal O}\oplus L)$ , with $D_1$ a genus 2 bisection of class $2s_0 = 2(s_\infty +f)$ . The component $S_2$ is the blow-up of (a deformation of) $S_1$ at 8 points along $D_1$ with $D_2$ the proper transform of $D_1$ in the blow-up. The class of $D_2$ is $2s_0 - \sum e_i$ . The involution on D is induced by the double cover map $\nu \colon D \to C$ which comes from the ruling of X. Since D is irreducible, $H^2(D)\simeq {\mathbb {Z}}$ .

In both cases, the Jacobian $\mathrm {Jac}(H^1(D)^-)=E$ is an elliptic curve. In Type II $_b$ , it is $\mathrm {Jac}(E)$ , where E is either of the double curves, while in Type II $_f$ , it is the Prym variety of the double cover map $\nu \colon D\to C$ . Thus, the mixed Hodge structure on $H^2(S_0)$ is encoded by a Carlson extension map $\phi \in \mathrm {Hom}(K,E)$ . By the previous subsection, this extension homomorphism descends to $\psi _{S_0}\in \mathrm {Hom}(\Lambda ,E)$ , where

$$ \begin{align*}\Lambda = K/\mathrm{span}\{\xi_1,\xi_2\} = \ker(H^2(S_1)\oplus H^2(S_2)\overset{\mathrm{res}}\longrightarrow{\mathbb{Z}})/{\mathbb{Z}}(D_1,-D_2).\end{align*} $$

There is a symmetric bilinear form on $H^2(S_0)$ . Let

$$ \begin{align*}p\colon H^2(S_0)\to H^2(S_1)\oplus H^2(S_2)\xrightarrow{\mathrm{PD}} H_2(S_1)\oplus H_2(S_2)\to H_2(S_0)\end{align*} $$

be restriction, followed by the Poincaré duality, followed by inclusion. Then define $\alpha \cdot \beta := \langle \alpha , p(\beta )\rangle $ on $H^2(S_0)$ . The map $H^2(S_0)\to H^2(S_t)$ respects the bilinear forms on the source, and target and the bilinear form descends to $K=\ker (\mathrm {res})$ .

By Poincaré duality and the Hodge index theorem, $H^2(S_1)\oplus H^2(S_2)$ is a unimodular lattice of signature $(2,10)$ , and it is odd since at least one summand contains $(-1)$ -curves. Since $D_1^2+D_2^2 =0$ , the lattice vector $(D_1,-D_2)$ is isotropic, and its orthogonal complement is precisely $\ker (\mathrm {res})$ . Hence, the lattice $\Lambda $ is unimodular of signature $(1,9)$ .

Our degenerating families are polarized by ${\mathbb {Z}} s\oplus {\mathbb {Z}}(s+f)\subset H^2(S_t)$ . The monodromy operator fixes these curve classes, and hence, we have a copy of $I_{1,1}\subset \ker (N)$ . So s, f extend over the singular fiber by (6.3). They can be represented inK as follows: $(s,s)$ , $(f,0)$ for Type II $_b$ and $(s_\infty , 0)$ , $(f,f)$ for Type II $_f$ , respectively. In both cases, they span a sublattice of $\Lambda $ isometric to $I_{1,1}$ whose orthogonal complement we call $\Lambda _0\subset \Lambda $ . We also have $\Lambda _0\simeq \Lambda /I_{1,1}$ canonically.

Proposition 6.4. The lattice $\Lambda _0$ is isometric to $E_8$ in both cases.

Proof. Note that $\Lambda _0$ is unimodular of signature $(0,8)$ , so it suffices to check that it is even. The orthogonal complement of $\{s,f\}$ in $\ker (N)$ is even because $f=K_{S_t}$ and $x\cdot x\equiv x\cdot K_{S_t}\textrm { mod }2$ for any $x\in H^2(S_t)$ . Hence, its image $\Lambda _0$ is even because $\ker (N)\to \Lambda $ preserves the intersection form.

Remark 6.5. The lattice $\Lambda _0$ can be described more directly using one irreducible component (only up to finite index in the Type II $_f$ case). For Type II $_b$ , the sublattice $\{s,f\}^\perp \subset H^2(S_1)$ lies in K and is even, unimodular of signature $(0,8)$ . So it maps isometrically to $\Lambda _0\simeq E_8$ . For II $_f$ , the sublattice $\{D_2,f\}^\perp \subset H^2(S_2)$ lies in K and so maps isometrically to an index two sublattice $D_8\subset \Lambda _0\simeq E_8$ .

We summarize the results of this section in the following proposition:

Proposition 6.6. Let $\mathcal {S}\to (B,0)$ be a degeneration of Type II $_b$ or Type II $_f$ . Let $K = \ker (H^2(S_1)\oplus H^2(S_2)\to H^2(D))$ be the kernel of signed restriction, and let $\Lambda :=K/{\mathbb {Z}}(D_1,-D_2)$ and $\Lambda _0=\{s,f\}^\perp \subset \Lambda $ . Let E be $\mathrm {Pic}^0$ of either double curve in Type II $_b$ and the Prym variety $\mathrm {Pic}^0(D)/\mathrm {Pic}^0(C)$ in Type II $_f$ .

The Carlson extension class $\phi \in \mathrm {Hom}(K,E)$ describing the mixed Hodge structure on $S_0$ descends to $\mathrm {Hom}(\Lambda ,E)$ , and so determines the $1$ -truncated limit mixed Hodge structure of the degeneration. This homomorphism further descends to a period point $\psi _{S_0}\in \mathrm {Hom}(\Lambda _0,E)$ where $\Lambda _0\simeq E_8$ . Explicitly.

1. (II _b ) The period point $\psi _{S_0}$ given by the map sending ${\mathcal L}\in \{s,f\}^\perp \subset \mathrm {Pic}(S_1)$ to ${\mathcal L}\big {|}_{X_p}\otimes {\mathcal L}\big {|}_{X_q}^{-1}\in E$ .

2. (II _f ) The period point $\psi _{S_0}$ is determined up to $2$ -torsion by the map sending $c_1({\mathcal L})\in \{D,f\}^\perp \subset H^2(S_2)$ to ${\mathcal L}\big {|}_D \in \mathrm {Pic}^0(D)/\mathrm {Pic}^0(C)=E$ .