8 problems

Projective bundle mirror theorem conjecture. Under these assumptions,

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…

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…

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…

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…