23 problems

Let $Z_0$ and $X_0$ be smooth projective K3 surfaces, and let … be a Fourier–Mukai equivalence. The induced map on integral cohomology is denoted by … Fix the canonical orientation…

Let $(X,D)$ and $(X',D')$ be log smooth pairs, and let $E$ be a strong Fourier–Mukai kernel on $X\times^{\log}X'$. Let $\pi:X\times^{\log}X'\to X\times X'$ be the projection and re…

Let $X$ and $Y$ be hyper-Kähler varieties of dimension $2n$, and let $\mathfrak{F}$ and $\mathfrak{F}^{-1}$ be the Fourier transforms and their inverses considered in the source, w…

Let $X$ be the Salvetti complex associated with the discriminantal arrangement, let $\Theta$ be its set of vertices, and let $\Pi_1(X,\Theta)$ be the fundamental groupoid. For each…

Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$ of degree $l$, and let $g^{\rm{sp}}$ be the genus of the spectral curve. For integers…

Ito's conjecture. The Fourier–Mukai spectrum of $\mathcal T$ should coincide with its Serre spectrum:

Let $\mathcal T$ be a triangulated category with $X\in\operatorname{FM}\mathcal T$. The proper Fourier–Mukai locus $\operatorname{Spec}^{\mathsf{pFM}}\mathcal T$ and the Serre-inva…

Let $X$ and $Y$ be smooth projective varieties. They are $K$-equivalent if there is a crepant birational map between them, and they are Fourier–Mukai partners if their perfect deri…

Let $\mathcal T$ be a triangulated category with $\operatorname{FM} \mathcal T\operatorname{NE}\emptyset$. The proper Fourier–Mukai locus is…

Let $\mathcal T$ be a triangulated category with $X \in \operatorname{FM} \mathcal T$. The Fourier–Mukai locus $\operatorname{Spec}^{\mathsf{FM}} \mathcal T$ and the Serre-invarian…

Let $X$ be a connected smooth projective curve of genus $g\ge2$, let $r\ge2$ and let $d<r$ be coprime positive integers, let $L$ be a line bundle of degree $d$ on $X$, and let…

Let $(\beta,n)\in\mathbb{Z}^2$ satisfy … Assume $\beta\neq 0$ or $n>0$. Fourier–Mukai wall-avoidance conjecture. For every integer solution $(c,d)$ of … one has … This is a constra…

Fourier–Mukai heart-equivalence conjecture. The Fourier–Mukai transform gives an equivalence of the conjecturally constructed stability-condition hearts:

Fourier–Mukai support-dimension invariance conjecture. If smooth projective varieties $X$ and $Y$ are Fourier–Mukai partners, then

Fourier–Mukai compatibility conjecture. One has

Fourier–Mukai compatibility conjecture. For $1\leq k\leq g-1$, there are suitable complexified ample classes $\Omega_k$ and $\Omega'_k$ such that

Let $Z$ and $X$ be separated schemes of finite type over $k$. Let $\mathcal{A}$ and $\mathcal{B}$ be DG-enhancements of $D(Z)$ and $D(X)$, and let $\bar{S}\in D(Z\times X)$ corresp…

Let $X$ and $Y$ be irreducible smooth projective schemes over $\mathbb{C}$, of dimensions $d_X$ and $d_Y$, respectively, related by the categorical data in the source. Let…

Finiteness conjecture for Fourier–Mukai partners. If $X$ is a quasi-projective variety, then there are only finitely many quasi-projective Fourier–Mukai partners $X'$.

Let $X_1$ and $X_2$ be smooth projective varieties. An exact functor $\mathsf{F}\colon\mathrm{D}^{b}(X_1)\to\mathrm{D}^{b}(X_2)$ is called splitting when it is both right and left…

Let a Fourier–Mukai partner of a smooth projective variety $X$ be a smooth projective variety $Y$ such that $\mathrm{D}^{b}(X)\cong\mathrm{D}^{b}(Y)$. Kawamata's conjecture. The nu…

Uniform Fourier–Mukai kernel bound conjecture. The line bundles $\mathcal A_i$ and equivalences $F_i$ can be chosen so that there exists a finitely supported function…

Let $X$ be an abelian surface with $\operatorname{NS}(X)=\mathbb{Z} H$ and let $v$ be a Mukai vector with … Suppose that $v$ has at least one solution of the numerical equation. Am…