16 problems

Let $n>1$, fix $z\in\mathbb{C}^n$, and let … where $z=(\xi+w\overline{\xi})/(1-|w|^2)$, so that $X_\varepsilon$ is the image of a twistor-space neighborhood of the ball…

Let $M$ be a Riemann surface with twistor space projection $\pi$, and let $\mathcal{O}$ denote the structure sheaf of the twistor space. Then $R^1\pi_*\mathcal{O}$ is the first hig…

Let $X$ be an irreducible holomorphic symplectic $4$-fold, let $E$ be a nef line bundle on $X$, and let $L^*$ denote the map in the sheaf Lefschetz action. The sheaf Lefschetz inje…

Let $X$ be a smooth projective manifold, let ${\cal F}$ be a locally free sheaf on $X$, and write $HP_k(X,{\cal F})$ for polar homology with coefficients in ${\cal F}$. Let $K_X$ d…

Corrected Hitchin conjecture. For a generic complex structure on $X$,

Hitchin's conjecture. For a generic complex structure on $X$,

Let $m,n,p$ be nonnegative integers, and consider the punctured space … Let $\mathcal O$ be the sheaf of smooth functions on this space that are constant along $\mathbb R^m$ and ho…

Let $k$ be an algebraically closed field and let $i\colon X=V(f)\hookrightarrow\mathbb{P}^n_k$ be a closed embedding of a smooth irreducible hypersurface of degree $d$. For a bound…

Banach cohomology and comparison conjecture. There is an isomorphism of topological $K$-vector spaces

Let $A$ be an abelian torsion group and $X$ be a smooth quasi-projective variety defined over a field $k$. Consider the identity continuous morphism of sites … and let…

Let $\mathbf{D}$ denote the square disk with product representation $\mathbf{D}\simeq[0,1]\times[0,1]$, and let $H^p([0,1],\widetilde{\mathcal{F}})$ denote the sheaf cohomology gro…

Let $\mathcal{A}$ be a hyperplane arrangement in a vector space of dimension $\ell$, let $L_0$ be its intersection lattice with the minimal element removed, and let $\mathscr{D}$ d…

Non-vanishing conjecture. The stated non-vanishing should hold for every such generalised Grassmannian.

Let $X \subseteq \operatorname{iGr}(n,2n)$ be the big affine cell, or a ball or a polydisc inside it, and let $Z$ be the corresponding twistor space in the stated double fibration.…

The singular BGG complex is constructed on the twistor space $Z'$ associated with a suitably chosen, but arbitrarily small, open subset $X'$. Local exactness conjecture. The conclu…

Let $L\subset\mathbb{R}^n$ be a tropical linear space of dimension $d$. For $p,q\geq 0$, let $H^q(L;W^p)$ and $H_c^q(L;W^p)$ denote the cohomology and compactly supported cohomolog…