13 problems

Chern-class vanishing conjecture. All Chern classes of $\mathfrak{E}$ vanish.

Bruzzo and Graña Otero conjecture. The Higgs bundle $\mathfrak{E}$ is semistable with respect to some polarization $H$ and $\Delta(E)=0$.

Let $E$ be a rank $2$ torsion-free sheaf on $\mathbb{P}^k$ that is the cohomology of a special monad. Special-monad semistability conjecture. Then $E$ is semistable. This extends t…

Let $k$ be the equal-characteristic residue field and let an overconvergent $(F,\nabla)$-crystal be defined over the local field $k((t))$. Such a crystal is quasi-unipotent if it b…

Let $X$ be an $n$-dimensional smooth connected complex projective variety, and let $\mathfrak E=(E,\phi)$ be a Higgs bundle on $X$. For a coherent sheaf $F$ of positive rank, write…

Curve-semistability characterization. The following are equivalent:

Let $k$ be a field of positive characteristic, and let a Higgs bundle on a smooth projective variety over $k$ be nilpotent if its Higgs field is nilpotent, with its exponent define…

Let $X$ be a smooth complex projective variety, let $\mathcal O_X(1)$ be ample, and let $E$ be a slope-semistable locally free sheaf. Choose a finitely generated model…

Tensor-product semistability conjecture. The tensor product of two semistable Euclidean lattices is again semistable.

Let $G$ be a 1-ended CAT(0) group, meaning a group acting geometrically on a proper CAT(0) space. Its end invariant $\operatorname*{pro}$-$\pi_{1}(\varepsilon(G))$ is defined throu…

Let $G$ be a 1-ended finitely presented group, and let $\varepsilon(G)$ denote its end. Write $\operatorname*{pro}$-$H_{1}(\varepsilon(G);\mathbb{Z})$ for the inverse system of fir…

Let $M$ be a closed aspherical manifold of dimension greater than $1$, and let $\widetilde{M}$ denote its universal cover. A space is strongly connected at infinity when its end ha…

The Condition C conjecture. Every connected complex projective manifold satisfies Condition C.