16 problems

Let $X=\Gamma\backslash G$ be a nilmanifold with invariant complex structure $J$, and let $\mathfrak{g}_{\mathds{C}}$ be the complexification of its Lie algebra. The Dolbeault coho…

Dolbeault cohomology conjecture. The Dolbeault cohomology of $M$ coincides with its left-invariant Dolbeault cohomology:

Invariant-form conjecture. The Dolbeault cohomology of $G/\Gamma$ can be computed in terms of invariant forms on $G$, i.e. in terms of the pair $({\mathfrak g},J)$.

Polar de Rham theorem. The mapping $\rho$ is an isomorphism. Equivalently,

Let $X$ be a smooth projective variety of dimension $n$, let $E$ be an ample vector bundle of rank $e$ on $X$, and let $L$ satisfy the preceding positivity hypotheses. For an integ…

Invariant Dolbeault cohomology conjecture. The inclusion $\iota$ is a quasi-isomorphism; equivalently, it induces an isomorphism

Let $N=\Gamma\backslash G$ be a nilmanifold, equipped with an invariant complex structure. The Dolbeault cohomology of $N$ should be computable using invariant forms. This is an an…

Invariant Dolbeault cohomology conjecture. The map $\iota$ is an isomorphism for every complex nilmanifold.

Left-invariant Dolbeault cohomology conjecture. The Dolbeault cohomology of $X$ is computed by left-invariant forms.

Let $X$ be a compact Kähler manifold and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor in $X$. Let $L$ be a holomorphic bundle over $X$ such that … where…

Conjecture on balanced metrics and the $\partial\overline{\partial}$-Lemma. The metric balanced condition and the cohomological $\partial\overline{\partial}$-Lemma property are str…

Localness and extension conjecture. Any continuous $G'$-homomorphism

Single-centered index y-independence conjecture. For a generic superpotential, $\Omega_{\rm S}(\gamma,y,t)$ is independent of $y$.

Console–Fino conjecture. These maps of complexes are quasi-isomorphisms, equivalently,

Let $Z$ be an analytic space with regular part $Z_{reg}$, let $q$ be a nonnegative integer, and suppose that the codimension conditions … hold, equivalently the conditions…

Let $G$ be a reductive Lie group, let $Y=G/L$ be a flag manifold with its equivariant complex structure, and let $\chi$ be a representation of the reductive subgroup $L$. The twist…