Combining the results described in Sects. 2.3 and 4.1, we can produce twisted connected sum \(G_{2}\)-manifolds from matching pairs of Fano 3-folds. We will apply the methods developed in [10] to the problem of finding matchings between Fanos of rank 1 and 2. In this section, we summarise the results from [10, §6] on finding matchings with a prescribed “configuration” of the Picard lattices of a pair of Fano 3-folds, reducing the problem to a combination of problems in lattice arithmetic and deformation theory. The main result here—Proposition 5.8—improves on [10, Proposition 6.18] to deal more clearly with skew configurations.
Configurations and matching
Let \(\Sigma _\pm \subset Y_\pm \) be smooth anticanonical divisors of a pair of Fanos, and
a matching. Let \(h_+\) be a marking of \(\Sigma _+\), i.e. an isometry \(h_+ : H^2(\Sigma _+) \rightarrow L\) where L is a copy of the K3 lattice (the unique unimodular lattice of signature (3,19)). Then
is a marking of \(\Sigma _-\). The images of \(H^2(Y_\pm ) \subset H^2(\Sigma _\pm )\) under \(h_\pm \) are a pair of primitive sublattices \(N_\pm \subset L\), isometric to the Picard lattices. This pair is well-defined up to the action of the isometry group O(L), and plays a crucial role.
Definition 5.1
Given a pair of Fanos with Picard lattices \(N_+\) and \(N_-\), call a pair of primitive embeddings \(N_+, N_- \hookrightarrow L\) a configuration. Two such pairs of embeddings are considered equivalent if they are related by the action of O(L).
We call a configuration orthogonal if the reflections of \(L(\mathbb {R})\) in \(N_+\) and \(N_-\) commute. If in addition \(N_+ \cap N_-\) is trivial then we call the configuration perpendicular. If the configuration is not orthogonal then we call it skew.
We saw in Sect. 2.4 that the homeomorphism invariants of the twisted connected sum M resulting from the matching depend on the configuration (e.g. \(H^2(M) = N_+ \cap N_-\)), and in Theorem 3.6 that the generalised Eells–Kuiper invariant \(\mu (M)\) does too. We therefore ask:
Given a pair \(\mathcal {Y}_+\), \(\mathcal {Y}_-\) of deformation types of Fano 3-folds, which configurations of embeddings \(N_\pm \subset L\) of their Picard lattices arise from some matching of elements of \(\mathcal {Y}_+\) and \(\mathcal {Y}_-\)?
We see below that it is not too hard to answer this when one of the types has Picard rank 1, and we will be able to say quite a lot when both types have Picard rank 2. In general the question is quite difficult, but in any case a first step in simplifying it is to rephrase it as a problem of finding suitable triples of classes in \(L(\mathbb {R}) := L \otimes \mathbb {R}\). Recall that the period of a marked K3 surface \((\Sigma , h)\) is an oriented two-plane \(\Pi \subset L(\mathbb {R})\), the image under \(h : H^2(\Sigma ; \mathbb {R}) \rightarrow L(\mathbb {R})\) of the real and imaginary parts of classes in \(H^{2,0}(\Sigma ; \mathbb {C})\).
Lemma 5.2
Let \(Y_\pm \) be a pair of Fano 3-folds, and let \(N_\pm \subset L\) be the images of primitive isometric embeddings of the respective Picard lattices. Then the pair \((N_+, N_-)\) is the configuration of some matching of \(Y_+\) and \(Y_-\) if and only if there exist
-
an orthonormal triple \((k_+, k_-, k_0)\) of positive classes in \(L(\mathbb {R})\),
-
anticanonical divisors \(\Sigma _\pm \subset Y_\pm \),
-
markings \(h_\pm \) of \(\Sigma _\pm \),
such that the oriented plane \(\langle k_\mp , \pm k_0\rangle \) is the period of \((\Sigma _\pm , h_\pm )\), \(h^{-1}_\pm (k_\pm )\) is the restriction of a Kähler class on \(Y_\pm \), and \(N_\pm \) is the image of the composition \(H^2(Y_\pm ) \rightarrow H^2(\Sigma _\pm ) \rightarrow L\).
Proof
Necessity is trivial, setting
for the Kähler classes
that appear in Definitions 2.5, and \(k_0\) corresponding to a generator of
, all normalised to unit length. Sufficiency relies on the Torelli theorem, cf. [10, Proposition 6.2]. \(\square \)
To study how the matching problem depends on the choice of configuration, let us first set up some notation for various lattices.
-
\(W := N_+ + N_-\) (this need not be primitive in L),
-
\(T_\pm \subset L\) the perpendicular of \(N_\pm \),
-
\(T := T_+ \cap T_-\), or equivalently the perpendicular of W,
-
\(W_\pm := T_\mp \cap N_\pm \), and
-
\(\Lambda _\pm \subset L\) the perpendicular to \(T \oplus W_\mp \), or equivalently the perpendicular to \(W_\mp \) in the primitive overlattice of W.
Remark 5.3
\(N_\pm \subseteq \Lambda _\pm \), with equality if and only if \(N_+\) and \(N_-\) “intersect orthogonally”, i.e. when \(W(\mathbb {R}) = W_+(\mathbb {R}) \oplus W_-(\mathbb {R}) \oplus (N_+(\mathbb {R}) \cap N_-(\mathbb {R}))\); equivalently the configuration is orthogonal in the sense of Definition 5.1.
Necessary conditions
Note that in Lemma 5.2 we must obviously have \(k_\pm \in N_\pm (\mathbb {R})\). On the other hand, \(N_\pm \) is contained in the Picard group of the marked K3 \((\Sigma _\pm , h_\pm )\), which is the subgroup of L orthogonal to the period; the marked K3 is automatically \(N_\pm \)-polarised. Thus \(k_\mp \) and \(k_0\) must both lie in \(T_\pm (\mathbb {R})\). Hence
$$\begin{aligned} k_\pm \in W_\pm (\mathbb {R}), \quad k_0 \in T(\mathbb {R}) . \end{aligned}$$
(5.1)
Now we come to the heart of how the difficulty of the matching problem depends on the configuration one tries to achieve: (5.1) implies that the period \(\langle k_\mp , \pm k_0\rangle \) is orthogonal to all of \(\Lambda _\pm \), so the marked K3 divisors used in a matching with the given configuration are forced to be \(\Lambda _\pm \)-polarised.
The significance is that the Picard group of a generic K3 divisor in a generic member of a deformation type \(\mathcal {Y}_\pm \) of Fano 3-folds will be precisely the Picard lattice \(N_\pm \) of that type. To find matchings for a configuration where \(\Lambda _\pm \) is strictly bigger than \(N_\pm \), we therefore require non-generic K3 divisors in members of \(\mathcal {Y}_\pm \) (the moduli space of \(\Lambda _\pm \)-polarised marked K3 surfaces forms a subspace of the \(N_\pm \)-polarised K3s, whose codimension is \({{\,\mathrm{rk}\,}}\Lambda _\pm - {{\,\mathrm{rk}\,}}N_\pm \)).
For configurations where \(\Lambda _\pm = N_\pm \), we deduce in Sect. 6 the existence of matchings between some elements of \(\mathcal {Y}_+\) and \(\mathcal {Y}_-\) from a general fact (due to Beauville [2]) that a generic \(N_\pm \)-polarised K3 appears as an anticanonical divisor in some member of \(\mathcal {Y}_\pm \). In view of Remark 5.3, this comparatively easy case corresponds to orthogonal configurations. To apply a similar argument for skew configurations (where \(\Lambda _\pm \supset N_\pm \)), we first need to show for those specific \(\Lambda _\pm \) that generic \(\Lambda _\pm \)-polarised K3s appear as anticanonical divisors in members of \(\mathcal {Y}_\pm \). Even when it is true, it is something that we can so far only verify case by case. We refer to this process as ‘handcrafting’.
Remark 5.4
Before moving on to existence results for matchings with a prescribed configuration, let us point out some necessary conditions.
-
(i)
Since \(W(\mathbb {R})\) contains a two-dimensional positive-definite subspace (spanned by the orthogonal classes \(k_+\) and \(k_-\)), while its orthogonal complement in the signature (3, 19) lattice L contains a class \(k_0\) with \(k_0^2 > 0\), the quadratic form on W must be non-degenerate of signature \({(2, {{\,\mathrm{rk}\,}}(W)-2)}\).
-
(ii)
\(W_\pm \subset N_\pm \) must contain some ample classes of \(Y_\pm \).
-
(iii)
Since \(\Lambda _+ \cap \Lambda _- \subset {{\,\mathrm{Pic}\,}}\Sigma _\pm \) and is orthogonal to an ample class of \(\Sigma _\pm \), it cannot contain any \((-2)\)-classes.
Remark 5.5
In particular, (ii) implies that any matching involving a Fano with Picard rank \({{\,\mathrm{rk}\,}}N= 1\) must be perpendicular. Moreover, for a configuration of lattices \(N_+\) and \(N_-\) where at least one has rank 2, if the intersection \(N_+ \cap N_-\) is non-trivial then (ii) forces the configuration to be orthogonal in the sense of Definition 5.1. For configurations of Picard lattices of Fanos of rank \(\le 2\) that satisfy the necessary conditions to be realised by a matching, we therefore have the following trichotomy:
-
Perpendicular configurations, i.e. every element of \(N_+\) is orthogonal to every element of \(N_-\).
-
Orthogonal configurations with non-trivial intersection. Then \(N_+ \cap N_-\) must have rank 1.
-
Skew configurations. Then \(N_+ \cap N_-\) must be trivial, but \(N_+\) is not perpendicular to \(N_-\) (the maps \(N_\pm \rightarrow N_\mp ^*\) must have rank 1).
In Sects. 6 and 7, we will consider these cases in turn.
Sufficient conditions
In order to describe the ‘genericity properties’ we require for anticanonical K3 divisors in families of Fano 3-folds, we recall some further terminology. The period domain is the space of oriented positive-definite 2-planes in \(L(\mathbb {R})\). It can be identified with \({\{ \Pi \in \mathbb {P}(L(\mathbb {C})) : \Pi ^2 = 0, \; \Pi \, {\overline{\Pi }} > 0\}}\) in order to exhibit a natural complex structure. Given \(\Lambda \subset L\), the period domain of \(\Lambda \)-polarised K3 surfaces is \(D_\Lambda := \{ \Pi \in \mathbb {P}(\Lambda ^\perp (\mathbb {C})) : \Pi ^2 = 0,\; \Pi \, {\overline{\Pi }} > 0\}\).
Definition 5.6
Given a non-degenerate lattice N, an N-marking of a closed 3-fold Y is a surjective homomorphism \(i_Y : H^2(Y) \rightarrow N\) that is isometric for the anticanonical form of Definition 4.1.
We avoid calling \(i_Y\) an “N-polarisation” since we do not impose any conditions on ample classes. If Y is Fano then the Picard lattice is non-degenerate so \(i_Y\) is simply an isometry.
Definition 5.7
Let \(N \subseteq \Lambda \subset L\) be primitive non-degenerate sublattices of L, and let \({{\,\mathrm{Amp}\,}}_\mathcal {Y}\) be a non-empty open subcone of the positive cone in \(N(\mathbb {R})\). We say that a set \(\mathcal {Y}\) of N-marked 3-folds is \((\Lambda , {{\,\mathrm{Amp}\,}}_\mathcal {Y})\)-generic if there is \(U_\mathcal {Y}\subseteq D_\Lambda \) with complement a countable union of complex analytic submanifolds of positive codimension with the property that: for any \(\Pi \in U_\mathcal {Y}\) and \(k \in {{\,\mathrm{Amp}\,}}_\mathcal {Y}\) there is \(Y \in \mathcal {Y}\), a smooth anticanonical divisor \(\Sigma \subset Y\) and a marking \(h: H^2(\Sigma ) \rightarrow L\) such that \(\Pi \) is the period of \((\Sigma , h)\), the composition \(H^2(Y) \rightarrow H^2(\Sigma ) \rightarrow L\) equals the marking \(i_Y\), and \(h^{-1}(k)\) is the image of the restriction to \(\Sigma \) of a Kähler class on Y.
To be able to prove that a set \(\mathcal {Y}\) of Fano 3-folds satisfies the definition we typically take \(\mathcal {Y}\) to be a deformation type, but to make sense of the definition we do not need to remember any additional structure on \(\mathcal {Y}\) (cf. Remark 4.7).
Meanwhile, when applying the next proposition we typically want all elements of the sets \(\mathcal {Y}_\pm \) to be Fano 3-folds (or building blocks) that are topologically the same, so that we have some control over the topology of the \(G_{2}\)-manifolds resulting from the matchings produced; essentially this means that all elements of \(\mathcal {Y}_\pm \) should belong to the same deformation type.
Proposition 5.8
Consider a configuration of primitive non-degenerate sublattices \(N_+, N_- \subset L\), and let \(\mathcal {Y}_\pm \) be a pair of sets of \(N_\pm \)-marked 3-folds. Define W, \(W_\pm \) and \(\Lambda _\pm \) as above. Suppose that there exist non-empty open cones \({{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm } \subseteq N_\pm (\mathbb {R})^+\) such that
-
(i)
the sets \(\mathcal {Y}_\pm \) are \((\Lambda _\pm , {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm })\)-generic,
-
(ii)
\(W_\pm \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm } \ne \emptyset \).
Then there is an open dense subcone \(\mathcal {W} \subseteq (W_+(\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_+}) \times (W_-(\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_-})\) such that for every \((k_+, k_-) \in \mathcal {W}\) with \(k_+^2 = k_-^2\) there exist \(Y_\pm \in \mathcal {Y}_\pm \), anticanonical K3 divisors \(\Sigma _\pm \subset Y_\pm \) and Kähler classes
such that
, with a matching
of
and
whose configuration is the given pair of embeddings \(N_\pm \subset L\).
Proof
The argument is essentially the same as [10, Proposition 6.18], even though the conclusion stated here is slightly stronger.
Let \(T = W^\perp \) as before. Denote the ranks of W and \(W_\pm \) by r and \(r_\pm \). Then \(W_\pm (\mathbb {R})\) and \(T(\mathbb {R})\) are real vector spaces of signature \((1, \, r_\pm - 1)\) and \((1, \, 21 -r)\) respectively
In view of Lemma 5.2 and (5.1), matchings correspond to certain triples of classes \((k_+, k_-, k_0)\) such that \(k_\pm \) and \(k_0\) belong to the positive cones \(W_\pm (\mathbb {R})^+\) and \(T(\mathbb {R})^+\) respectively. Consider therefore the real manifold
$$\begin{aligned} D= \mathbb {P}\bigl (W_+(\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (W_-(\mathbb {R})^+\bigr ) \times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\, . \end{aligned}$$
Below, we need the open subset \(\mathcal {A}= \mathcal {A}_+ \times \mathcal {A}_- \times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\), where \(\mathcal {A}_\pm := \mathbb {P}(W_\pm (\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_\pm })\) is non-empty by hypothesis (ii). We have two Griffiths period domains
$$\begin{aligned} D_{\Lambda _\pm } = \{\text {positive-definite planes }\Pi \subset \Lambda _\pm ^\perp (\mathbb {R}) \} , \end{aligned}$$
and projections
$$\begin{aligned} \text {pr}_\pm :D \rightarrow D_{\Lambda _\pm }, \; (\ell _+, \ell _-,\ell ) \mapsto \langle \ell _\mp , \pm \ell \rangle . \end{aligned}$$
As stated before Definition 5.7, \(D_{\Lambda _\pm }\) can be regarded as an open subset of \(\mathbb {P}(C_\pm )\), where \(C_\pm \) is the null cone in \(\Lambda _\pm ^\perp \otimes \mathbb {C}\); if \(\alpha , \beta \) is an oriented orthonormal basis of \(\Pi \in D_{\Lambda _\pm }\) then \(\Pi \mapsto \langle \alpha +i\beta \rangle \in \mathbb {P}(C_\pm )\). Given a choice \(\alpha \) and \(\beta \), we can identify \(T_\Pi D_{\Lambda _\pm }\) with pairs (u, v) of vectors in the orthogonal complement of \(\Pi \) in \(\Lambda _\pm ^\perp (\mathbb {R})\). Then the complex structure on \(T_\Pi D_{N_\pm }\) is given by \(J : (u, v) \mapsto (-v, u)\).
Observe that the real analytic embedded submanifold \(\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr ) \hookrightarrow D_{\Lambda _\pm }\) is totally real: for \(w \in W_\mp \) and \(t \in T(\mathbb {R})\), the tangent space \(\mathcal {T}\) to \(\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\) at \(\Pi = \langle w,t\rangle \) corresponds to (u, v) such that \(u \in w^\perp \subseteq W_\mp (\mathbb {R})\) and \(v \in t^\perp \subseteq T(\mathbb {R})\), so \(J(\mathcal {T})\) is transverse to \(\mathcal {T}\).
Crucially, this totally real submanifold has maximal dimension:
$$\begin{aligned}&\dim _\mathbb {R}\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\\&\quad = (r_\mp -1) + \bigl (22 - r - 1\bigr ) = 20 - r + r_\mp = 20-{{\,\mathrm{rk}\,}}\Lambda _\pm = \dim _\mathbb {C}D_{\Lambda _\pm } \end{aligned}$$
Consequently, its intersection with any positive-codimensional complex analytic submanifold of \(D_{\Lambda _\pm }\) is a positive-codimensional real analytic submanifold of \(\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\). Hence the pre-image in \(\mathbb {P}\bigl (W_\mp (\mathbb {R})^+\bigr )\times \mathbb {P}\bigl (T(\mathbb {R})^+\bigr )\) of the subset \(U_{\mathcal {Z}_\pm } \subset D_{\Lambda _\pm }\) from Definition 5.7 is open and dense. Because \(\text {pr}_\pm \) is a projection of a product manifold onto a factor the same is true for \(\text {pr}_\pm ^{-1}(U_{\mathcal {Z}_\pm }) \subset D\). In turn,
$$\begin{aligned} \bigl (\mathcal {A}_+ \times \mathcal {A}_- \times \mathbb {P}(T(\mathbb {R})^+)\bigr ) \cap \text {pr}_+^{-1}(U_{\mathcal {Z}_+}) \cap \text {pr}_-^{-1}(U_{\mathcal {Z}_-}) \end{aligned}$$
is open and dense in \(\mathcal {A}_+ \times \mathcal {A}_- \times \mathbb {P}(T(\mathbb {R})^+)\), and hence the image \(\mathcal {W}'\) of this subset under projection to \(\mathcal {A}_+ \times \mathcal {A}_-\) is open and dense in \(\mathcal {A}_+ \times \mathcal {A}_-\).
If we let \(\mathcal {W} = \{(k_+, k_-) \in (W_+(\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_+}) \times (W_-(\mathbb {R}) \cap {{\,\mathrm{Amp}\,}}_{\mathcal {Y}_-}) : ([k_+], [k_-]) \in \mathcal {W}'\}\), then for every \((k_+, k_-) \in \mathcal {W}\) such that \(k_+^2 = k_-^2\) there is a \(k_0 \in T(\mathbb {R})^+\) such that Lemma 5.2 applies to \((k_+, k_-, k_0)\). \(\square \)