124 problems

Kontsevich's homological mirror symmetry conjecture. Under the mirror map $\Greekmath 0124 \longleftrightarrow\Omega^{\vee}$ and $\Omega\longleftrightarrow\Greekmath 0124 ^{\vee}$,…

Let $(X,\omega)$ be a symplectic manifold equipped with a Lagrangian torus fibration admitting a Lagrangian section. Kontsevich–Soibelman associate to this data a rigid analytic sp…

Berglund–Hübsch homological mirror symmetry conjecture. There exists a derived equivalence between $\mathcal{F}(W,G)$ and $\mathcal{MF}(W^T,G^T)$.

Let $X$ be a Fano manifold. A Landau–Ginzburg model for $X$ is a one-parameter family $Y\to\mathbb{A}^1$, with associated Fukaya category $Fuk(X)$, Fukaya–Seidel category $FS(Y)$,…

Open string mirror symmetry conjecture. Under mirror symmetry, there is a quasi-equivalence of triangulated categories

Bridgeland–Joyce's conjecture. This correspondence can be understood as an isomorphism between some quotients of the spaces of stability conditions. Roughly, the space of stability…

Let $A=(a_{ij})$ be an $(n+1)\times(n+1)$ integer matrix with nonzero determinant, and define … Let $\check{{\bf w}}$ be the polynomial associated to $A^T$, let $\Gamma_{\bf w}$ be…

Let $(M,\omega)$ be a symplectic manifold with $c_1(M)=0$, let $(X,J)$ be a complex manifold forming a hypothetical mirror pair with $(M,\omega)$, and let…

Let $f\in S^n$ be an invertible polynomial, and let $\widetilde{f}$ denote its Berglund–Hübsch transpose. Let $L_f$ be the maximal grading of $f$, and let…

Equivariant torus-fibration mirror-symmetry conjecture. There should be equivalences

Let $\Delta$ be a convex lattice polygon containing the origin in its interior. A Laurent polynomial $W$ has Newton polygon $\Delta$, and a pair $(G,D)$ consists of a consistent di…

Let $M \to B$ and $\check{M} \to B$ be dual torus fibrations arising in the SYZ picture, and let a Lagrangian section of $M \to B$ be a Lagrangian submanifold mapping isomorphicall…

Let $B_{1}=F_{1}=B_{2}=F_{2}=S^{1}$, let $E_{i}=B_{i}\times F_{i}$ be elliptic curves, and let $S_{i}=B_{i}\times F_{i}^{\vee}$. Set $X=E_{1}\times E_{2}$ and $Y=S_{1}\times S_{2}$…

Fix a field $k$ and positive integers $p_0,\ldots,p_n>1$. For each $i$, let $W_{p_i}(X_i)\in\mathbb{C}[X_i]$ be a general polynomial of degree $p_i$, and set … Let…

Let $(M,\omega)$ be an integral symplectic manifold, meaning $[\omega]\in H^2(M;\mathbb Z)$. Let $\mathfrak h$ be the upper half-plane, set $M_\tau=(M,-\sqrt{-1}\tau\omega)$ for…

Let $\mathcal F(X)$ be the Fukaya $A_\infty$-category of a Calabi–Yau manifold $X$, and let $\mathbf D\mathcal F_0(X)$ be its derived Fukaya category. Let $QH^*(X)$ denote the quan…

Let $M$ be a Calabi–Yau threefold, and let $D(\operatorname{Coh} M)$ be the derived category of coherent sheaves on $M$. A Pi-stable object is an object of this derived category sa…

Pi-stability monodromy conjecture. Pi-stability leads to such an autoequivalence ${\cal F}_{\cal P}$ for every closed loop ${\cal P}$. This conjecture asserts that transporting Pi-…

Let $(M,\beta)$ be a compact symplectic manifold with a mirror partner $X$ such that Kontsevich's homological mirror symmetry conjecture holds. Let $S$ be a Lagrangian sphere in…

Homotopical homological mirror symmetry conjecture. The $A_\infty$ categories

Let $\mathbf{FIPS}$ be the Fayet–Iliopoulos parameter space, and choose basepoints at the relevant large complex structure limits. Let $D^b(\check X)$ be the bounded derived catego…

Let $\check X$ be a toric variety with $d$ rays in its fan, and let $T$ be an algebraic torus of rank $\dim(\check X)$. Let $V(E_A)$ be the hypersurface cut out by the principal…

Homological Mirror Symmetry conjecture. The bounded derived category of coherent sheaves on $X$ has a full exceptional collection consisting of $k$ objects:

Let $X_{\mathrm{res}}$ be the resolution of the $A_n$ surface singularity, let $D_r$ be the indicated divisor, and let $\sigma$ be a stop. Let $s$ be a formal parameter, with coeff…

Let $X_{\mathrm{res}}$ be the resolution of the $A_n$ surface singularity, let $D_r$ be the indicated divisor, and let $s$ be a formal parameter with coefficient field…