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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.