5. Boundedness and moduli theory
From the point of view of birational geometry, there are three main classes of algebraic varieties: Fano varieties, Calabi–Yau varieties, and canonical models. These are distinguished based on the behavior of their canonical divisor $K_X$: It is anti-ample, numerically trivial, and ample, respectively.In dimensions 1 and 2, each of these classes is well understood. In particular, Fano and Calabi–Yau surfaces admit a very explicit classification, while canonical models admit a qualitative description based on some key invariants.
In higher dimensions, the picture becomes more complicated. Due to the cumulative effort of many mathematicians, including works of Hacon–McKernan–Xu and Kollár, canonical models admit a qualitative description and well behaved moduli theory in arbitrary dimensions. Similarly, due to work of Kollár–Miyaoka–Mori and Birkar, we have strong finiteness results for Fano varieties in arbitrary dimensions. Furthermore, recent developments in $K$-stability provide a good theory to contruct moduli of certain classes of Fano varieties.
In this picture, little is known for Calabi–Yau varieties in dimensions 3 or more. For this reason, many interesting questions regarding the finiteness of deformation classes and the moduli theory are open already in low dimensions.
Definition: Let $\mathfrak P = \{X_i\}_{i \in I}$ be a collection of projective varieties. We say that $\mathfrak P$ is bounded if there exists schemes of finite type $\mathcal{X}$ and $T$ and a projective morphism $\pi \colon \mathcal X \to T$ such that, for every $X_i$, there exists a closed point $t \in T$ such that $X_i \simeq \mathcal{X}_t$ holds.
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Calabi–Yau varieties are not endowed with a natural polarization. For this reason, given a bounded class of smooth (or klt) Calabi–Yau varieties, there could be many different possible moduli spaces to consider. Said otherwise, a moduli space of smooth Calabi–Yau varieties may admit many meaningful compactifications. These possible moduli spaces carry a natural $\mathbb Q$-line bundle, called the Hodge line bundle, which is known to be big. In the case of K3 surfaces or abelian varieties, the Baily–Borel compactification of the moduli problem is minimal among the possible ones and it coincides with the ample model of the Hodge bundle. In general, we may pose the following question:
Conjecture 5.1.
Let $\mathcal{M}$ be a moduli space of smooth Calabi–Yau varieties. Does there exist a compactification $\mathcal{M}\hookrightarrow \mathcal{M}^*=\mathcal{M}\sqcup\coprod_i\mathcal{M}_i$ such that for every $\mathcal{M}\hookrightarrow\overline{\mathcal{M}}$ SNC, we have $\text{Proj}\oplus_{n\ge0}H^0(\overline{\mathcal{M}},n\overline{\mathcal{L}}_{\text{Hodge}})=\mathcal{M}^*$? -
An important problem in the study of Calabi–Yau varieties is to understand how their (birational) automorphism group acts on the nef (rsp. movable) cone. In particular, it is expected that this action admits a polyhedral fundamental domain. Under some suitable assumptions, expectations from mirror symmetry relate the toric variety associated with this cone to a local description of the boundary of the moduli space of Calabi–Yau varieties. In particular, we may ask:
Problem 5.2.
Near a cusp of $\mathcal{M}^*$ prove that there is a mirror $X^{\text{mir}}$ of the general fibres $\mathcal{X}\longrightarrow\mathcal{M}$ such that \[ \text{Nef}(X^{\text{mir}})/\Gamma= \sigma \] where $\sigma$ is the fan modeling the neighborhood of the cusp. -
Given a log Calabi–Yau pair $(X,D)$ with $D$ ample and reduced, one may degenerate it to the cone over $D$ via the degeneration to the normal cone. Under these assumptions, one may investigate the relationship between the moduli problem to which $(X,D)$ belongs and the deformation theory of the cone over $D$:
Problem 5.3.
If we degenerate a log CY pair $(X,D)$ coming from a Fano to the normal cone $\text{Cone}_N(D)$, does the deformation theory of the normal cone recover the moduli space? More precisely, do we get an isomorphism: \[ \mathbb{C}^*\curvearrowright\text{Def}(\text{Cone}(D))^-\simeq \text{Cone}_{\text{CM}}(\mathcal{M}_{(X,D)}) \] -
In dimension 2, the moduli theory of Calabi–Yau surfaces has strong relations with Hodge theory. It is interesting to investigate these aspects in higher dimensions and the in the case of pairs, where MHS naturally appear:
Problem 5.4.
Let $(X,D)$ be a log Calabi–Yau 3-fold. Is the period map \begin{align*} (X, D) & \mapsto \text{MHS}(H^3(X \backslash D, \mathbb{Z})) \\ \mathcal{M}_{(X, D)} & \mapsto \mathbb{D} / \text{Monodromy} \end{align*} injective? -
Upon the choice of an ample $\mathbb Q$-Cartier divisor, we may apply the moduli theory of stable pairs to consturct moduli spaces for Calabi–Yau varieties. For instance, this perspective has been investigated by works of Alexeev–Engel in the case of K3 surfaces. More generally, one can pose the following question:
Problem 5.5.
Consider $(X,L)\in F_{2d}$, the universal family $\mathcal{X}\overset{\pi}{\longrightarrow}F_{2d}$, and $\mathbb{P}(\pi_*nL)\simeq\mathbb{P}^g$-bundle over $F_{2d}$ with a section $R$. Can we find a semi-toric compactification $F_{2d} \hookrightarrow \overline{F}^{\mathcal{F}}_{2d}$ and $\overline{\mathcal{X}}$ its universal family with $(\overline{\mathcal{X}},\epsilon R)$ slc stable over $\overline{F}^{\mathcal{F}}_{2d}$? -
In general, it is easier to treat Calabi–Yau varieties under additional geometric assumptions. For instance, it is interesting to consider Calabi–Yau varieties $X$ admitting a fibration $X \to Y$ given by a line bundle $L$. Then, it is natural to ask whether nearby deformations of $X$ preserve this property. More precisely, we may ask:
Problem 5.6.
Let $X$ be a Calabi–Yau variety, and let $L$ be a semi-ample Cartier divisor on $X$. Consider the universal family $(\mathcal X ,\mathcal L) \to \mathrm{Def}(X,L)$ of deformations of $(X,L)$. Is $\mathcal{L}$ relatively semi-ample over $\mathrm{Def}(X,L)$? -
Abelian fibered Calabi–Yau varieties constitute an important case of fibered Calabi–Yau varieties. For instance, they naturally appear in the study of primitive symplectic varieties, as well as strict Calabi–Yau varieties. In general, given an abelian fibration $X \to Y$, one can define a new fibration, called relative Albanese fibration, whose generic fiber over $Y$ is the Albanese variety of the generic fiber $X_{K(Y)}$. This new fibration admits a rational section and may thus be understood via the moduli theory of abelian varieties. To this end, it is interesting to investigate the relationship between $X$ and its relative Albanese variety:
Problem 5.7.
Is the relatie Albanese variety of a Calabi–Yau variety itself Calabi–Yau? How many abelian fibered Calabi–Yau varieties admit the same relative Albanese fibration?
Cite this as: AimPL: Higher-dimensional log Calabi-Yau pairs, available at http://aimpl.org/higherdimlogcy.