17 problems

Let $g>0$, and consider surfaces of genus $g$ in $\mathbb{CP}^2$. The Clifford torus conjecture asserts that the Willmore functional is minimized by the Clifford torus … This is th…

Montiel–Urbano conjecture. The Clifford torus achieves the minimum of the functional $\mathcal{W}^-$, and hence of $\mathcal{W}$, either among all tori in $\mathbb{C}P^2$ or among…

Let a minimal immersion of the two-sphere into $\mathbb{CP}^n$ be non-holomorphic, non-anti-holomorphic, and non-totally real, and suppose that it has constant Kähler angle. Bolton…

Classification conjecture. Either $\mathcal{F}$ is convex reduced, or $\mathcal{F}$ is homogeneous, or $\mathcal{F}$ is linearly conjugate to $\mathcal{F}_{1}^{d}$. This conjecture…

Let $\mathbb{C}P^2$ be the complex projective plane, let $\mathcal{W}$ be the Willmore functional, and let a complex curve mean a holomorphic curve in $\mathbb{C}P^2$. For each suc…

Let $\mathbb{C}P^2$ be the complex projective plane, let $\mathcal{W}$ be the Willmore functional, and consider the Veronese sphere … Here an immersed sphere has the same self-cros…

Let $X$ be a complex projective manifold whose universal cover is Stein. Liu–Maxim–Wang conjecture for Stein universal covers. The cotangent bundle $T^*X$ should be nef. The paper…

Let $X$ be an aspherical complex projective manifold. A constructible function on $X$ has an effective characteristic cycle when its characteristic cycle is effective. Uniform non-…

Let $X$ be an aspherical complex projective manifold, and let $T^*X$ denote its holomorphic cotangent bundle. A vector bundle is nef when it is numerically effective. Liu–Maxim–Wan…

The Hodge conjecture. Every element of $H^{p,p}(X;\Q)$ is representable by some algebraic $(n-p)$-cycles with $\Q$-coefficients.

Classification conjecture. $M$ is Hopf and all its principal curvatures are constant if and only if $M$ is locally congruent to one of the examples $A_{+}$, $A_{-}$, $B_0$, $B_{+}$…

Let $(M,g)$ be a complete Kähler–Einstein manifold. A local Kähler immersion is a Kähler immersion defined on neighborhoods, and $\mathbb{C}\mathrm{P}^N$ denotes complex projective…

SIC-POVM existence conjecture. $\mathbb{CP}^{n-1}$ possesses such a configuration for every $n$.

Let $\mathbf{u}_0,\dots,\mathbf{u}_n$ be unit vectors representing the vertices of a simplex in complex projective space, in general position, and let … Let $d_{\min}$ denote the m…

Degree-two foliation alternative. Under these hypotheses, one has an alternative.

Isolation conjecture. Except in $\mathbb C\mathbb P^2$, SIC-POVMs are isolated.

Let $D\subset\mathbb{CP}^2$ be a smooth cubic, let $H$ be a smooth sextic close to $2D$, and let the fibration on $\mathbb{CP}^2\setminus D$ be the one described above. K3 fibratio…