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4. Symplectic and log symplectic pairs


    1. Add remark

      Problem 4.1.

      Let \mathcal{X}\longrightarrow\Delta be a dlt degeneration of Hyperkähler varieties with K_{\mathcal{X}/\Delta}\simeq\mathcal{O_X}. Is

      \Gamma(\mathcal{X}_0) = \begin{cases} \text{point} & \text{if in type I} \\ \sigma_d & \text{if in type II}\\ \mathbb{CP}^d& \text{if in type III ?} \end{cases}


      Given \Delta^*\overset{\phi}{\longrightarrow}\Gamma\backslash\mathbb{D}, what equivalence relation on \phi ensures that \Gamma(\mathcal{X}_0) is well-defined on equivalence class? Eg. Picard–Lefschetz.


        • Add remark

          Problem 4.2.

          Can \text{K3}^{[5]} and \text{OG}_{10} be distinguished via combinatorics of \Gamma(\mathcal{X}_0) for degeneration \mathcal{X}\longrightarrow\Delta?

          "Combinatorics": integral affine structure? Lagrangian Chern class?


            • Add remark

              Problem 4.3.

              Do 1-parameter Hyperkähler degenerations admit K-trivial, SNC models, after a finite base change?

              Note: minimal dlt exist, due to Fujino, Lai. This is true in Type I (KLSV).


                • Add remark

                  Problem 4.4.

                  If (X,D) is log sympletic dlt (or plt) pair and X\backslash X^{\text{SNC}} has codimension \ge3 in X, are the singularities rigid (in the sense of definitions of log sympletic dlt pairs)?


                    • Add remark

                      Problem 4.5.

                      Are Type III log sympletic pairs rational, log rational, or cluster type?


                        • Add remark

                          Problem 4.6.

                          Local deformation theory of dlt log sympletic singularities?


                            • Add remark

                              Problem 4.7.

                              Examples of dual complexes which are not \text{Hilb}^n or \text{Kum}^n of a K3 degeneration?

                              degeneration of cubic 4-fold X\rightsquigarrow V(x_0x_1x_2).

                              V(x_0x_1x_2+\epsilon f_3)\longrightarrow\Delta_{\epsilon}\rightsquigarrow degeneration of FX?

                              V(x_0,x_1,x_2,f_3) smooth? nodal?

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