Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

2. Complex geometry

    1. Problem 2.1.

      [Heather Macbeth] Find a toric proof of the toric case of the orbifold compactness theorem proved by Chen-Weber.
        •     A complex manifold is called almost Kähler if it admits a complex structure J, not necessarily integrable, and a Hermitian metric g such that g(\cdot,J\cdot) defines a closed 2-form. One says that g and J are compatible.

          Problem 2.2.

          [Casey Kelleher] (Goldberg Conjecture) Is a compact, almost Kähler, Einstein manifold necessarily Kähler?
              Solved by Sekigawa for non-negative Einstein constant.
            • Problem 2.3.

              [Gábor Székelyhidi] (Question of Donaldson) Find smooth solutions to the equation \ddot{\varphi}- \frac 12 |\nabla \dot \varphi|^2_{\omega_\varphi}=-\lambda R_{\omega_\varphi}
              for \lambda>0 on M \times [0,1] with \varphi|_{M\times\{0\}}=0, \varphi|_{M\times\{0\}}=\varphi_0.
                  It was pointed out by Donaldson that such a solution would prove the uniqueness of cscK metrics
                • Problem 2.4.

                  [Eleonora Di Nezza] M Kähler, \theta big cohomology class. Can we define a family of distances generalizing the Finsler geometry of Kähler potentials and compatible with weak Mabuchi geodesics?
                    • Problem 2.5.

                      [Valentino Tosatti] Find a counterexample to the maximal rank conjecture.

                      Conjecture (Maximal Rank Conjecture)

                      Let A\subseteq\mathbb C be the annulus A = \{1<|w| < e | w \in \mathbb C \} . Given (M,\omega) n-dimensional compact Kähler manifold, \varphi_0 and \varphi_1 potentials, the geodesic on M \times A is obtained as the weak limit of solutions to the family of degenerating complex Monge-Ampère equations \begin{cases} (\pi^*\omega + i \partial \overline \partial \Phi_\delta)^{n+1} = \delta (\pi^*\omega + i dw\wedge d\overline w)^{n+1} \\ \Phi_\delta|_{M\times { 0 } }=\varphi_0 \\ \Phi_\delta|_{M\times { 1 } }=\varphi_1. \end{cases}
                      Then (\pi^*\omega+i\partial \overline \partial \Phi_\delta)|_{M\times\{\mathrm{pt.}\}} \geq \varepsilon \omega for \varepsilon >0 independent of \delta.
                          May try using the counterexample of Ross-Witt Nyström from Hele-Shaw flow to produce such a counterexample.
                        • Problem 2.6.

                          [Eleonora Di Nezza] Let u be a plurisubharmonic function satisfying (i \partial \overline \partial u)^n = \psi
                          with \psi>0 and smooth. Assume u is C^{1,\alpha} for \alpha > 1 - \frac 2 n, then u\in C^\infty.
                              Solved for \alpha>1 - \frac 1 n by Li-Li-Zheng.
                            •     Let (M,\chi) compact Kähler, and \omega another Kähler metric on M. There are necessary and sufficient conditions for solving \mathrm{tr}_{\omega_\varphi} \chi = c
                              where c = n \frac {[\omega]^{n-1} \cap [\chi]} {[\omega]^n} is a constant (Song-Weinkove) in terms of a positivity condition. However, this may be hard to check in general.

                              Problem 2.7.

                              [Ben Weinkove] (Conjecture of Lejmi-Székelyhidi) There exists a solution if and only if \int_V{(c \omega^p - p\omega^{p-1}\wedge \chi)}>0
                              for all proper subvarieties V \subset M, where p = \mathrm{dim}(V)
                                  Condition is necessary, by Donaldson. Solved by Collins-Székelyhidi in the toric case.
                                • Problem 2.8.

                                  [Eleonora Di Nezza] Can one establish regularity properties of geodesics in the space of Kähler potentials for singular Kähler varieties?
                                    • Problem 2.9.

                                      [Jiewon Park] Let (M,\omega) and (M',\omega') be compact Kähler-Einstein manifolds which are \varepsilon-close in Gromov-Hausdorff distance. Is there a holomorphic family M_t with M_0 = M and M_1=M'?
                                          By Colding, M and M' are necessarily diffeomorphic for \varepsilon>0 sufficiently small.

                                          Cite this as: AimPL: Nonlinear PDEs in real and complex geometry, available at http://aimpl.org/nonlinpdegeom.