6 problems

Let $X$ be a compact Riemann surface, let $P\subset X$ be a finite set, and let $(V,\nabla_V)$ be a semisimple holomorphic flat bundle on $X\setminus P$. Let $(H,D_V,h)$ be the tam…

Let $X$ be a Kähler manifold or a Kähler orbifold with first Betti number $b_1=0$ and second Betti number $b_2\geq 2$. Suppose that $X$ contains disjoint complex curves spanning…

Unitary representation conjecture. Representations of this group approximate, as $N\to+\infty$, complex curves in the abelian variety $(\mathbb{C}/(\mathbb{Z}+i\mathbb{Z}))^2$. Thi…

Curve critical-point conjecture. For every $k$, the functional $\mathcal{F}_{k}$ admits a critical point.

Fang's conjecture. The upper bound is achieved precisely by the Fubini–Study metric on $L$, up to scaling.

Gillet–Soulé's conjecture. The determinant of $\Delta_{\bar{\partial}}$, viewed as a functional on the space of all smooth Hermitian metrics on $L$, is bounded from above.