35 problems

Generalized Bloch conductor formula. If $p:X\to S$ is an arithmetic $S$-scheme with $S$-proper singular locus $Z$, then

Let $X\subseteq \mathbb P^N$ be a smooth projective variety of dimension $n$, let $Y=X\cap H$ be a smooth hyperplane section, let $U\subseteq (\mathbb P^N)^*$ be the open set of sm…

Quantum Naturality Conjecture. Suppose that $T:X\rightarrow X_z$ is a small transition in any dimension and $\phi:H^*(X_z,\mathbb{R})\rightarrow H^*(X,\mathbb{R})$ is defined previ…

Geometric-cycle conjecture. Because of the symmetry in the example, $\delta^{l_p}$ is the cycle around

Let $f(z_0,\ldots,z_n)=z_0^{a_0}+z_0z_1^{a_1}+\cdots+z_{n-1}z_n^{a_n}$ with $n\geq1$. Define $r_k=a_0a_1\cdots a_k$, $r_{-1}=1$, and let $c_0,\ldots,c_\mu$ be the coefficients in…

Morsification conjecture for topological exceptional collections. The ordered collection of vanishing cycles obtained from a topological exceptional collection of geometric vanishi…

Let … be a $(-1)$-shifted exact oriented Lagrangian correspondence. Here $\phi_M$ and $\phi_N$ are the perverse sheaves of vanishing cycles associated with the oriented $(-1)$-shif…

Let $X$ be a smooth hypersurface and suppose that $n=\dim(X)$ is odd. Let $T_v$ be the connected component of the analytic covering space of the smooth hyperplane-section parameter…

Vanishing conjecture. The $W$-invariant parabolic induction functor maps central objects to vanishing objects; equivalently, there is a braided monoidal functor

Let $X$ be a smooth variety over an algebraically closed field $k$ of characteristic $p\neq 2$, let $\mathcal{F}\in D(X)$, and let $(x,\xi)$ be a smooth point of $SS\mathcal{F}$. F…

Let $X$ be a smooth variety over an algebraically closed field $k$ of characteristic $p\neq 2$, and let $\mathcal{F}\in D(X)$. Write $SS\mathcal{F}$ for its singular support and le…

Let $S$ be a strictly henselian trait with closed point $s$, generic point $\eta$, and geometric generic point $\bar\eta$. Let $p:U\to S$ be flat, with $U$ regular and purely of re…

Let $f_d:M_d(\widetilde{Q}')\to\mathbb C$ be the lifted trace-of-potential function on the representation atlas described in the paper, and let $[\phi_{f_d}]$ denote its motivic va…

Jacobi-type theorem. There is some positive integer $k$ such that the topological Abel--Jacobi mapping

Strong Tube Theorem. The image of the topological Abel--Jacobi homomorphism

One-elementary-vanishing-cycle generation conjecture. Only one elementary vanishing cycle is enough to generate $H_{2n-1}^{\operatorname{van}}(X,\mathbb{Q})$.

Let $\boldsymbol{Y}$ be a $-2$-shifted symplectic derived scheme over $\operatorname{Spec}{\mathbb C}$, equipped with an orientation, and set $Y=t_0(\boldsymbol{Y})$. Let…

Let $\boldsymbol{\tau} \colon \boldsymbol{L} \to \boldsymbol{X}$ be an oriented Lagrangian of an oriented $-1$-shifted symplectic derived Artin stack. Write…

In the situation of the Picard–Lefschetz formula, let $c_1=w_1c_0$ and $c_2=w_2c_0$ with $w_1,w_2\in W$, and let $W_\chi^0$ be the specified subgroup of $W$. Assume that the cosets…

Equality conjecture.

Vanishing-cycle W-algebra action conjecture. There is a $W_\kappa({\mathfrak gl}_r)$-action on this direct sum identifying it with the $W$-vacuum module.

Let $\mathfrak{M}$ be a quasi-smooth QCA derived stack, let $\mathcal{N}=t_0(\Omega_{\mathfrak{M}}[-1])$ be its associated d-critical locus, and let $\mathcal{N}_{\mathrm{st}}$ be…

Let $Q_L^n$ be the Quot schemes associated with the local line $L\subset\mathbb A^3$, and let…

Let $Y$) be a smooth threefold containing a fixed curve $C$, and let $Q_C^n$ denote the Quot scheme parametrising the corresponding curves-and-points configurations. The wall-cross…

Let $\widetilde f$ be the Berglund--Hübsch transpose of the chain-type invertible polynomial, and let $\chi_n$ denote the matrix appearing in the Orlik--Randell description of the…