Classifying Calabi–Yau Threefolds Using Infinite Distance Limits

Abstract

We present a novel way to classify Calabi–Yau threefolds by systematically studying their infinite volume limits. Each such limit is at infinite distance in Kähler moduli space and can be classified by an associated limiting mixed Hodge structure. We then argue that such structures are labeled by a finite number of degeneration types that combine into a characteristic degeneration pattern associated to the underlying Calabi–Yau threefold. These patterns provide a new invariant way to present crucial information encoded in the intersection numbers of Calabi–Yau threefolds. For each pattern, we also introduce a Hasse diagram with vertices representing each, possibly multi-parameter, decompactification limit and explain how to read off properties of the Calabi–Yau manifold from this graphical representation. In particular, we show how it can be used to count elliptic, K3, and nested fibrations and determine relations of elliptic fibrations under birational equivalence. We exemplify this for hypersurfaces in toric ambient spaces as well as for complete intersections in products of projective spaces.

Appendices

A Limiting mixed Hodge structures

In this appendix we briefly introduce the mathematical notion of a limiting mixed Hodge structure. It should be stressed, however, that our exposition is short and incomplete. We refer the reader to the original papers [25, 26] and the review [56]. Also [8] contains a concise summary of some of the relevant aspects.

Let us first define a pure Hodge structure and its associated Hodge filtration. Let V be a rational vector space. A pure Hodge structure of weight w describes a decomposition of the complexification $V_{{\mathbb {C}}} = V \otimes {\mathbb {C}}$ as

$$\begin{aligned} V_{\mathbb {C}} = {\mathcal {H}}^{w,0} \oplus {\mathcal {H}}^{w-1,1} \oplus \ldots \oplus {\mathcal {H}}^{1,w-1} \oplus {\mathcal {H}}^{0,w}\; , \end{aligned}$$

(55)

with the subspaces satisfying ${\mathcal {H}}^{p,q} = \overline{{\mathcal {H}}^{q,p}}$ with $w=p+q$. The complex conjugation on $V_{\mathbb {C}}$ is defined with respect to the rational vector space V. The ${\mathcal {H}}^{p,q}$ can also be used to define a Hodge filtration by setting $F^p = \oplus _{i\ge p} {\mathcal {H}}^{i,w-i}$. These spaces are filtered and satisfy

$$\begin{aligned} V_{\mathbb {C}} = F^0 \ \supset \ F^1 \ \supset \ \ldots \ \supset \ F^{w-1}\ \supset \ F^w = {\mathcal {H}}^{w,0}\; , \end{aligned}$$

(56)

such that ${\mathcal {H}}^{p,q} = F^p \cap {\bar{F}}^q$. A crucial additional property arises from demanding that the ${\mathcal {H}}^{p,q}$ define a polarized pure Hodge structure. This necessitates the existence of a bilinear form $S(\cdot , \cdot )$ on $V_{\mathbb {C}}$, such that the conditions

$$\begin{aligned} S({\mathcal {H}}^{p,q}, {\mathcal {H}}^{r,s})= & {} 0\; , \qquad p \ne s,\ q \ne r\; , \end{aligned}$$

(57)

$$\begin{aligned} i^{p-q} S(v,{\bar{v}})> & {} 0\; , \qquad \text {for all}\ \ 0\ne v \in {\mathcal {H}}^{p,q}\; , \end{aligned}$$

(58)

are satisfied. Note that these definitions define a fixed (p, q)-splitting. One can then ask the question how such a structure can vary consistently over a complex base space ${\mathcal {M}}$ and define families of polarized pure Hodge structures. This is captured by the theory of variation Hodge structures. In particular, one demands that with respect to the flat connection $\nabla$ on the family of Hodge structures varying over ${\mathcal {M}}$, the $F^i$ are holomorphic sections and satisfy $\nabla F^i \subset F^{i-1} \otimes \Omega _{\mathcal {M}}$. Let us note that these definitions are essentially algebraic. The application to geometric settings arises, for example, when using them to describe the (p, q)-cohomology $H^{p,q}(Y_3,{\mathbf {C}})$ of a Calabi–Yau threefold $Y_3$ and how it varies over the complex structure moduli space.

Let us next turn to the definition of a mixed Hodge structure. The crucial new ingredient is the so-called monodromy weight filtration $W_i$. This filtration is induced by the action of a nilpotent matrix N on the vector space V. Concretely, one defines the rational vector subspaces $W_{j} (N) \subset V$ by requiring that they form a filtration

$$\begin{aligned} W_{-1}\equiv 0\ \subset \ W_0\ \subset \ W_1\ \subset \ ...\ \subset \ W_{2w-1}\ \subset \ W_{2w} = V\; , \end{aligned}$$

(59)

with the properties

$$\begin{aligned}&1.) \quad N W_i \subset W_{i-2} \; , \end{aligned}$$

(60)

$$\begin{aligned}&2.) \quad N^j : Gr_{w+j} \rightarrow Gr_{w-j}\ \ \text {is an isomorphism,}\quad Gr_{j} \equiv W_{j}/W_{j-1} \; . \end{aligned}$$

(61)

Here the quotients $Gr_i$ are equivalence classes of elements of $W_i$ that differ by elements of $W_{i-1}$. One can show that the filtration $W_i$ with the above properties is unique for a given N.

Finally, let us define a mixed Hodge structure (V, W, F). Let $W_i$ be a monodromy weight filtration defined by an N as above and $F^q$ a filtration satisfying (56) on the vector space V.⁷ We require that N is compatible with the $F^q$-filtration and acts on it horizontally, i.e. $N F^p \subset F^{p-1}$. The defining feature of a mixed Hodge structure is that each $Gr_{j}$ defined in (61) admits an induced Hodge filtration

$$\begin{aligned} F^p Gr_j^{\mathbb {C}} \equiv ( F^p \cap W_j^{\mathbb {C}} )/ ( F^p \cap W_{j-1}^{\mathbb {C}})\; , \end{aligned}$$

(62)

where $Gr_j^{\mathbb {C}} = Gr_j \otimes {\mathbb {C}}$ and $W_i^{\mathbb {C}} = W_i \otimes {\mathbb {C}}$ are the complexification. Referring back to (55), this implies that we can split each $Gr_j$ into a pure Hodge structure ${\mathcal {H}}^{p,q}$ as

$$\begin{aligned} Gr_j = \bigoplus _{p+q=j} {\mathcal {H}}^{p,q} \; ,\qquad {\mathcal {H}}^{p,q} = F^p Gr_j \cap \overline{F^q Gr_j}\; , \end{aligned}$$

(63)

where we recall that $w=p+q$ is the weight of the corresponding pure Hodge structure. The operator N is a morphism among these pure Hodge structures. Using the action of N on $W_i$ and $F^p$, we find $N Gr_j \subset Gr_{j-2}$ and $N {\mathcal {H}}^{p, q} \subset {\mathcal {H}}^{p-1,q-1}$. Note that this induces a jump in the weight of the pure Hodge structure by $-2$, while the mixed Hodge structure is preserved by N. The natural next step is to introduce a polarized mixed Hodge structure. This again uses the bilinear form $S(\cdot ,\cdot )$. We first define the primitive subspaces ${\mathcal {P}}_{l} \subset Gr_{l+w}$, by setting ${\mathcal {P}}_{l} = ker(N^{l+1} : Gr_{w+l} \rightarrow Gr_{w-l-2})$. The mixed Hodge structure is polarized if for all l the restriction of the pure Hodge structure (63) to the primitive subspaces ${\mathcal {P}}_l$ is polarized with respect to $S_l(\cdot ,\cdot ) = S(\cdot ,N^l\cdot )$.

With this definition at hand, we can now introduce a limiting mixed Hodge structure. The introduction of this structure is needed due to the fact that a pure Hodge structure at certain limits of ${\mathcal {M}}$ can degenerate and no longer describe the splitting of $V_{{\mathbb {C}}}$. Let us describe a one-parameter degeneration limit $t\rightarrow i \infty$. At such a limit one can introduce a nilpotent matrix N from the monodromy transformation as discussed in the main part of the paper. One can then split off the singular part of the pure Hodge filtration defining

$$\begin{aligned} F_{\infty }^p = \lim _{t\rightarrow i \infty } e^{- t N} F^p \; . \end{aligned}$$

(64)

While the $F_{\infty }^p$ in general do not describe a pure Hodge structure, they can be used to define a mixed Hodge structure. This mixed Hodge structure is defined with respect to the limit $t \rightarrow i \infty$ and hence known as a limiting mixed Hodge structure.

B Enhancement diagrams obtained in KS and CICY scan

We give all distinct diagrams obtained in our scans of the Kreuzer–Skarke and CICY data sets up to $h^{1,1}=3$ for simplicial and non-simplicial Kähler cones in Figs. 16 and 17, respectively. Each diagram is accompanied by the number of times it occurred in both these data sets.

Fig. 16

Enhancement diagrams obtained via scans of the Kreuzer–Skarke and CICY databases up to $h^{1,1}=3$, including only simplicial Kähler cones. The numbers below each diagram indicate how often it was encountered in the Kreuzer–Skarke and CICY scans respectively

Fig. 17

Enhancement diagrams obtained via our scan of the Kreuzer–Skarke database for $h^{1,1}=3$, including only non-simplicial Kähler cones, where the number below each diagram indicates its multiplicity in this scan