18 problems

Let $\Omega\subset \mathbb C^n$ be a bounded homogeneous domain, let $m\in\mathbb N^*$, and let $\Omega_{m,s}$ be the associated Hartogs domain, with $s\neq 0$. Bergman–Kähler–Ricc…

Let $X$ be a compact Kähler manifold undergoing the normalized Kähler–Ricci flow, and suppose that, in the finite-time singularity case, there is a proper holomorphic map with conn…

Let $X$ be a smooth Fano manifold admitting a Kähler–Ricci soliton, and let $Y:=K_X^{\times}$ be the complement of the zero section in its canonical bundle. A Calabi–Yau cone struc…

Existence conjecture. The conclusions of Theorem 1 hold for $M$, depending on the sign of $c_1(\det E\otimes K_B^{\vee})$. If $d_\ell=0$ and $\det E\otimes K_B^{\vee}\to B$ is triv…

Let $X$ be a smooth Fano variety. A Kähler-Ricci soliton on $X$ is a Kähler metric satisfying the Kähler-Ricci soliton equation, and the anticanonical link is the link associated w…

Let $(M,J,g,X)$ be a simply connected complete steady gradient Kähler–Ricci soliton, meaning that $g$ is a complete Kähler metric and $X$ is a vector field generating the soliton.…

Properness conjecture. The moment map $\bm\mu$ is proper.

Generalized Yau–Tian–Donaldson conjecture. The manifold $M$ admits a Kähler–Ricci soliton if and only if $M$ is modified $K$-stable.

Let $X$ be a Fano manifold. The regression procedure described in the source produces a maximal model, denoted $(X_{KE},\omega_{KE})$ when $R(X)=1$ and $(X_{KR},\omega_{KR})$ when…

Let $X$ be a Fano manifold. Its greatest Bakry–Émery–Ricci lower bound is … where the supremum is taken over the possible values of $\beta$. Greatest Bakry–Émery–Ricci lower bound…

Modified F-functional geodesic stability conjecture. The following are equivalent:

Let $X$ be a smooth toric variety, and let $\Delta_t$ be a family of toric $\mathbb{Q}$-divisors with coefficients less than $1$, affine in $t\in(0,1]$, converging to a reduced div…

Sasaki–Einstein circle-bundle conjecture. If there exists a Kähler–Ricci soliton on $M$, then the $S^1$-bundle associated to $K_M$ admits a Sasaki–Einstein metric with a suitable c…

Generalized Hamilton-Tian conjecture. As $s\to\infty$, the flow starting from $\omega$ converges in the Gromov-Hausdorff sense to a Kähler-Ricci soliton $(X_\infty,\omega_{KR})$. T…

Let $M$ be a compact Kähler manifold with $c_1(M)>0$, and let $\omega_t$ solve … with $\omega+\sqrt{-1}\partial\bar\partial\varphi>0$. Suppose that $M$ admits no Kähler–Einstein me…

Let $X$ be a Fano manifold and let $g(t)$ be a normalized Kähler-Ricci flow. Set $\tau=e^t-1$, and let $\bar g$ be the corresponding $S^1$-invariant metric on $X\times\mathbb C$ sa…

Folklore convergence conjecture. As $t$ tends to infinity, $g(t)$ converges to a Kähler-Ricci soliton, possibly with mild singularities along a subvariety of complex codimension at…

Kähler–Ricci soliton stability conjecture. There exists an analogous version of Theorem 3.1 for general Kähler–Ricci solitons.