2. Complex geometry
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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? -
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. -
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. -
Problem 2.6.
[Eleonora Di Nezza] Let u be a plurisubharmonic function satisfying (i \partial \overline \partial u)^n = \psiwith \psi>0 and smooth. Assume u is C^{1,\alpha} for \alpha > 1 - \frac 2 n, then u\in C^\infty. -
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 = cwhere 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)}>0for all proper subvarieties V \subset M, where p = \mathrm{dim}(V) -
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'?
Cite this as: AimPL: Nonlinear PDEs in real and complex geometry, available at http://aimpl.org/nonlinpdegeom.