7 problems

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

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

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…

Clifford torus energy minimization conjecture. The minimum of the energy functional is attained on the Clifford torus.

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…

Let $D\subset\mathbb{CP}^2$ be a smooth elliptic curve, and equip $\mathbb{CP}^2\setminus D$ with the holomorphic volume form having poles along $D$. Special Lagrangian torus fibra…