21 problems

de Cataldo–Hausel–Migliorini P=W conjecture. The diffeomorphism $\psi$ between the spaces respects these filtrations.

Let $G$ be a reductive group. Write $\operatorname{Higgs}_G$ for the Dolbeault moduli stack of $G$-Higgs bundles, $\operatorname{Higgs}_G^{\operatorname{irred}}$ for its irreducibl…

Categorical non-Abelian Hodge conjecture. There are mappings

Let $\sigma(0):=(E,\bar\partial,\theta)$ be a nilpotent Higgs bundle on $X_0$, and suppose that the non-abelian Kodaira–Spencer map $\Phi^{0,1}_{KS}$ is nonzero at $\sigma(0)$ for…

Let $\lambda$ be a parameter, written in characteristic $p$ as a lifting $(\lambda_0,\lambda_1)$, and let $M_p(\lambda_0,\lambda_1)$ denote the matrix associated with the periodici…

Simpson's conjecture. The non-abelian Hodge locus $\mathrm{NHL}({\mathcal Y}/{\mathcal S},\operatorname{GL}_n)$ is an algebraic variety, and its inclusions into…

Let $\Sigma$ be a smooth surface and let $\mathcal{T}^n$ be the moduli space of higher complex structures of order $n$ on $\Sigma$. Consider the cotangent bundle $T^*\mathcal{T}^n$…

Let $\Sigma$ be a smooth surface, let $n$ be a positive integer, and let $\mathcal{T}^n$ be the moduli space of higher complex structures of order $n$ on $\Sigma$. Write…

Let $K$ be algebraically closed, let $X$ be a smooth proper rigid space over $K$, and let $n$ be a positive integer. Write…

Let $N_{Dol}^*$ be the complement of a large Hitchin-map sublevel set in $\mathcal{M}_{Dol}$, let $N_B^*$ be a punctured tubular neighborhood of the boundary divisor in a log compa…

Let $C$ be a punctured Riemann surface, let $G$ be a linear reductive algebraic group over $\mathbb{C}$, and let $\mathcal{M}_B$ and $\mathcal{M}_{Dol}$ be corresponding Betti and…

P=W conjecture for untwisted character varieties. For every $k$,

PI=WI conjecture. For every $k$,

P=W conjecture for twisted character varieties. For every $k$,

For each index $\alpha$, let $G_\alpha^1$ be a de Rham stratum and let $\big(G_\alpha^1\big)^{\rm VHS}\subset G_\alpha^1$ be the subset of polarized complex variations of Hodge str…

Let $M_{\rm dR}(X,r)$ be the de Rham moduli space with its stratification, and let $G_u^1$ denote the oper stratum. The source identifies $G_u^1$ as the highest stratum. Simpson's…

Let $M$ be a (quasi-)projective variety with a stratification by locally closed subsets … The stratification is nested if there is a partial order $(\Lambda,\leq)$ such that … For…

Let $X$ be a smooth projective complex curve. The pro-completed category…

Let $X$ be a smooth projective complex curve. The derived stack $\mathbf{Coh}_{/\mathbb A^1}^\ast(X_\mathsf{Del})$ parametrizes Deligne's $\lambda$-connections, interpolating betwe…

Let $X$ be a smooth complex projective variety and $G$ an algebraic reductive group. Let $M_D$ be the moduli space of Higgs $G$-bundles, with proper Hitchin map, and let $M_B$ be t…

The $P=W$ phenomenon refers to an exchange between the perverse filtration arising from an algebraically completely integrable system and the weight filtration appearing in the cor…