14 problems

Let $X$ be a fixed smooth projective variety and let $n$ be a positive integer. For a commutative differential graded algebra $A$, let $\mathbb{R}\underline{Vect}_{n}(X)(A)$ be the…

Categorical non-Abelian Hodge conjecture. There are mappings

Derived-stack universal vector bundle conjecture. These choices satisfy the assumptions introduced for the moduli stack of vector bundles and its universal vector bundle.

Compatibility conjecture. This isomorphism is compatible with the $(n-2)$-shifted symplectic structures on both sides. The mapping stack $\mathrm{Map}(C_{\mathrm{dR}},X)$ is known…

Let $X$ be a derived Artin stack, let $B$ be a derived commutative algebra over the de Rham stack $X_{DR}$, and write $X_B=X\mathbin{\times}_{X_{DR}}\operatorname{Spec}(B)$. Let…

Let $X$ and $Y$ be derived stacks, and let $\boldsymbol{\operatorname{IndCoh}}(Y)$ and $\boldsymbol{\operatorname{QCoh}}(Y)$ denote the ind-coherent and quasi-coherent categories o…

Let $\mathbb{A}$ be an augmentation category, and let $\mathcal{G}_n^\mathrm{sm}(\textbf{dAff})$ and $\mathbb{A}\text{-}\mathcal{G}_n^\mathrm{sm}(\textbf{dAff})$ denote the homotop…

Let $L$ be a shifted Poisson stack. A symplectic realization of $L$ is an $n$-shifted Lagrangian morphism $L\to X$ lifting the given $(n-1)$-shifted Poisson structure along the for…

Lagrangian–Poisson correspondence conjecture. The map $f\mapsto\pi_f$ provides an equivalence between the space of $n$-shifted Lagrangian morphisms $f:X\to Y$ such that…

Filtered shifted-structure conjecture. There exists a relative $n$-shifted symplectic structure $\widetilde{\omega}$ on $\widetilde{Y}_f$ over $[\mathbb{A}^1/\mathbb{G}_m]$ such th…

Polyvector-field conjecture. The resulting $\mathbb{P}_{n+1}^{gr}$-algebra structure on $\Gamma\big(X,S(\mathbb{T}_X[-n])\big)$ is equivalent to the $\mathbb{P}_{n+1}^{gr}$-algebra…

Purity conjecture. The Hodge structure on $H_i^{BM}(\mathcal{C}oh(X)_P^{H-\operatorname{ss}})$ is pure for any $H$, and, in general, $H_i^{BM}(\mathcal{X})$ is pure for any algebra…

Let $\mathrm{Corr}_{/\Omega^{2,\operatorname{cl}}[n]}$ be the $(\infty,2)$-category of correspondences of derived stacks equipped with a closed degree $n$ two-form. The canonical a…

Shifted quadraticity conjecture. Under suitable hypotheses on $X$, the derived moduli stack $\mathsf{Sympl}(X;n)$ is a shifted quadratic derived stack.