45 problems

Strominger–Yau–Zaslow conjecture. The mirror $Y^{\vee}$ is the moduli space of certain A-branes in $Y$, and the mirror functor in homological mirror symmetry is given by a fiberwis…

Let $X$ be a smooth projective variety, and let $\operatorname{FM}\operatorname{Perf}X$ denote the set of its Fourier–Mukai partners. Kawamata's finiteness conjecture. … The source…

Let $(X,\alpha)$ and $(Y,\beta)$ be twisted K3 surfaces, with $\alpha$ and $\beta$ killed by $p$. Let $B$ and $B'$ be B-field lifts of $\alpha$ and $\beta$, respectively, and let…

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

Let $(A,\mathrm{\Theta})$ be a principally polarized abelian variety, let $C\subset A$ be a curve, and let $\mathcal I_C(2\Theta)$ denote the ideal sheaf of $C$ twisted by twice th…

Let $X$ be a smooth projective variety of dimension $d$ with generically finite Albanese map $a:X\rightarrow A$, let $P$ be a Poincaré bundle on $X\times\widehat A$, and let…

Let $X$ and $Y$ be smooth projective varieties, and write ${\mathbf D}_{\mathrm{coh}}^b(X)$ for the bounded derived category of coherent sheaves on $X$. Let…

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 $X$ be the derived-category target space, let $\mathbf{D}(X)$ be its derived category, and let $\mathsf{A},\mathsf{B}\in\mathbf{D}(X)$ represent D-branes. Write…

Kontsevich's conifold monodromy conjecture. For the monodromy around the conifold point, the kernel $\Psi$ is the complex

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 $2n$, let $\mathcal E$ be the twisted vector bundle used in the source, and let $\widetilde{\mathfrak F}$ and…

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…

Conjecture B. The object $E$ extends to a bounded complex of vector bundles on $A^{NC}$ if and only if the Fourier–Mukai transform $F(E)$ is supported in dimension at most $1$.

Let $C$ be a nonsingular projective curve of genus $g\ge 2$, and let $M_{r,d}$ be the moduli space of semistable Higgs bundles on $C$ with Hitchin fibration $h:M_{r,d}\to A$. Write…

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 $\mathcal T$ 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 $\mathcal T$ be a triangulated category with $X \in \operatorname{FM} \mathcal T$. Two smooth projective varieties are $K$-equivalent when they are related by a crepant biratio…

Let $X$ be a smooth projective variety, and let $X'$ be a Fourier–Mukai partner of $X$, meaning that $D^b(X')$ and $D^b(X)$ are equivalent. Let $\operatorname{Alb}(X)$ and…

Let $X$ be a smooth projective curve of genus $g\geq 3$, let $L$ be a fixed line bundle of odd degree on $X$, and let $M$ be the moduli space of rank-$2$ stable vector bundles on…

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…