19 problems

Let $X$ be a projective variety of dimension $n$, and let $D$ be a strictly nef divisor on $X$, meaning that $D\cdot C>0$ for every irreducible curve $C$ in $X$. Serrano's conjectu…

CSC metric–polystability conjecture. The projective bundle $P(E)$ admits a CSC Kähler metric if and only if $E$ is polystable.

Projective bundle mirror theorem conjecture. Under these assumptions,

Let $V=\bigoplus_{i=0}^n L_i$ be a vector bundle and let $\mathbb P(V)_{1,(\nu,0)}$ denote the moduli space of degree $(\nu,0)$ stable maps with one marked point. Let $e$ be the ev…

Let $X$ be a smooth projective variety and let $V=\bigoplus_{i=0}^n L_i$ be a direct sum of line bundles, where $L_i$, $i=1,\ldots,n$, are nef and $-K_X-c_1(V)$ is ample. Let…

Uniqueness conjecture. All weakly Bochner-flat Kähler metrics on $P(\mathcal O\oplus\mathcal O(k_1,k_2))\to\mathbb{C}P^{d_1}\times\mathbb{C}P^{d_2}$ are given by the preceding coro…

Morrison's conjecture. For suitable $m$, the polarised projective bundle $(\mathbb P(E),L_m^{\otimes k})$ is Chow stable for $k\gg 0$.

Let $V=\oplus^r_{i=1} \mathcal O(m_i)$ over $\mathbb P^n$, where $m_i>0$ for every $i$, and let $h$ and $\xi$ denote the standard generators used for the quantum cohomology of…

Let $(X,L)$ be as in $(\diamond)$, with $X=\mathbb P(\mathcal E)$ an $n$-dimensional Fano bundle over $Y$, and with $\operatorname{rk}(\mathcal E)\geq\dim Y$ (equivalently,…

Let $(S,g_S,\omega_S)$ be a compact CSCK Hodge $2(m-1)$-manifold with non-negative scalar curvature, and let $E$ be a holomorphic line bundle such that … Set…

Let $(X,L)$ be a polarized manifold of dimension $n egeq 3$ with $operatorname{rk}langle KX,Lrangle=2$. Its Hilbert curve $Gamma$ is associated with the polarized manifold $(X,L)$,…

Relative Yau–Tian–Donaldson conjecture. For any polarization … on $M$, the following two conditions are equivalent:

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth ample divisor. Suppose that $p:Y\to Z$ exhibits $Y$ as a $\mathbb{P}^d$-bundle over a $b$-dimensional smooth…

Let $D$ be a smooth proper Deligne–Mumford stack with projective coarse moduli space, and let $L$ be a line bundle on $D$. Put … Write $\pi:Y\to D$ for the natural projection, and…

Let $(X,J)=P(E)$ be a projective bundle over a compact curve $\Sigma$ of genus at least $2$. A subbundle is stable in the usual slope-stability sense. Apostolov–Calderbank–Gauducho…

Unboundedness conjecture.

Extension conjecture. Then $d\ge b-1$, and $(X,L)\cong (\mathbb{P}(\mathcal{E}),\xi_{\mathbb{P}})$ for an ample vector bundle $\mathcal{E}$ on $Z$, with $p$ equal to the restrictio…

Extremal metric decomposition conjecture. The projective bundle $(M,J)=P(E)$ admits an extremal Kähler metric in some Kähler class if and only if $E$ decomposes as a direct sum of…

Compatibility conjecture. Every such extremal Kähler metric $\omega_t$ is compatible with the bundle structure.