The determinantal representation conjecture for Calabi–Yau potentials

Let $(X,L)$ be a polarized manifold, let $\boldsymbol{\omega}_{0}$ be a Kähler metric in $c_{1}(L)$, and let $dV$ be the volume form in the Calabi–Yau equation

$$(\omega_{0}+i\partial\bar{\partial}\varphi)^{n}=dV.$$

For each $k$, let $N_k$ be the dimension of the relevant space of sections and let $S^{(k)}$ be an orthonormal basis, with $C_k$ chosen so that $\int_X\varphi^{(k)}dV=0$.

Determinantal representation conjecture. The unique smooth normalized solution $\varphi$ may be represented as a limit in $L^{1}(X)$:

$$\varphi=\lim_{k\rightarrow\infty}\varphi^{(k)},\qquad \varphi^{(k)}=\frac{1}{k}\frac{\int_{X^{N_k-1}}\log\left\Vert\det S^{(k)}(\cdot,x_2,\ldots,x_{N_k})\right\Vert^{2}\,dV^{\otimes(N_k-1)}}{dV}-C_k.$$

This conjecture proposes a zero-temperature determinantal approximation to the solution of the Calabi–Yau equation. It is suggested by interchanging the limits in the preceding Gibbs-measure construction; the source gives no resolution of the conjecture.