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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.


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          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}}.


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              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)})?


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                  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?


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                      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.