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5. Boundedness and moduli theory


    1. Add remark

      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, $\text{Proj}\oplus_{n\ge0}H^0(\overline{\mathcal{M}},n\overline{\mathcal{L}}_{\text{Hodge}})=\mathcal{M}^*$? What is the connection between $\mathcal{M}^*$ and the Cone Conjecture?

      Points of $\mathcal{M}^*$ parametrize CY cone of the degeneration.


        • Add remark

          Problem 5.2.

          Connection between conjectural $\mathcal{M}^*$ and the Cone Conjecture / Mirror Symmetry?

          Near cusp of $\mathcal{M}^*$ have a mirror $X^{\text{mir}}$ (of the general fibre $\mathcal{X}\longrightarrow\mathcal{M}$).

          $\text{Nef}(X^{\text{mir}})/\Gamma=$ fan modeling neighborhood of cusp.

          Walls of $\mathcal{L}\in\text{Nef}(X^{\text{mir}})/\Gamma\longleftrightarrow\text{Strata} $, where fibres of $|\mathcal{L}|=(\text{CY cone})^{\text{mir}}$.


            • Add remark

              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?

              $\mathbb{C}^*\curvearrowright\text{Def}(\text{Cone}(D))^-=\text{Cone}_{\text{CM}}(\mathcal{M}_{(X,D)})$?


                • Add remark

                  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?


                    • Add remark

                      Problem 5.5.

                      $(X,L)\in F_{2d}$, $\mathcal{X}\overset{\pi}{\longrightarrow}F_{2d}$, $\mathbb{P}(\pi_*nL)\simeq\mathbb{P}^g$-bundle over $F_{2d}$ with a section $R$.

                      $$\begin{tikzcd} F_{2d}\arrow[r,hook] & \overline{F}^{\mathcal{F}}_{2d}\\ \mathcal{X}\arrow[u]\arrow[r]&\overline{\mathcal{X}}\arrow[u] \end{tikzcd}$$ where $\overline{F}^{\mathcal{F}}_{2d}$ semi-toric and $\overline{\mathcal{X}}$ slc K-trivial varieties.

                      $(\overline{\mathcal{X}},\epsilon R)$ slc stable over $\overline{F}^{\mathcal{F}}_{2d}$.

                      Take: $F_{2d}\hookrightarrow F^{\text{slc}}_{2d}=$ slc K-trivial limits of K3s.

                      What stability conditions cut out nice moduli spaces (K-moduli, KSBA moduli, ...)?

                          Cite this as: AimPL: Higher-dimensional log Calabi-Yau pairs, available at http://aimpl.org/higherdimlogcy.