22 problems

Let $L$ be a graded Lagrangian submanifold of a Calabi–Yau $n$-fold $X$, with the phase of the holomorphic volume form chosen so that the cohomological phase satisfies $\phi(L)=0$.…

Let $\check{\mathcal{X}}$ be the mirror geometry, let $L$ be a Lagrangian in $\check{\mathcal{X}}$, and let $\mathcal{E}$ be a holomorphic vector bundle on $\mathcal{X}$. A Bridgel…

Joyce's conjecture. There exists a Bridgeland stability condition $\sigma=(Z,\mathcal{P})$ on $\operatorname{DFuk}(M)$ such that the central charge is induced by

Let $(M^n,\omega,\Omega)$ be a possibly non-compact Calabi–Yau manifold, with Kähler form $\omega$ and holomorphic volume form $\Omega$, suitably convex at infinity. Let…

Let $X$ be a Calabi–Yau $3$-fold with a Lefschetz K3-fibration $\pi:X\rightarrow Y$, and let a gradient cycle be a graph in $Y$ whose edges are decorated by $2$-cycles in the K3 fi…

Let $X$ be a Calabi–Yau $3$-fold with a K3-fibration $\pi:X\rightarrow Y$, let $\omega_X$ and $\omega_Y$ be Kähler forms on $X$ and $Y$, respectively, and let…

Let $X_t$ be a $1$-parameter family of compact Calabi–Yau manifolds with metrics $g_t$ admitting special Lagrangian fibration structures and with bounded diameters, and let $C_t$ b…

Thomas–Yau conjecture. There exists a unique special Lagrangian in the Hamiltonian isotopy class of $L$ if and only if $L$ is stable. This conjecture proposes a stability criterion…

Let $\mathcal{L}$ be the admissible class of Lagrangian objects and let $\mathcal{S}$ be the Solomon functional. A special Lagrangian is a Lagrangian on which the relevant holomorp…

Let $X$ be a Calabi-Yau manifold, and let $L$ and $L'$ be special Lagrangian integral currents with the same phase angle $\hat{\theta}$. Equip them with suitable unobstructed brane…

Let $\mathcal{L}$ be the admissible class of Lagrangian currents and let $\mathcal{S}$ be the Solomon functional. Let $\hat{\theta}$ be the phase determined by the relevant Fukaya-…

Let $\mathcal{L}$ be the class of admissible Lagrangian objects and let $\mathcal{S}$ be the Solomon functional. A minimizer is an element $L\in\mathcal{L}$ attaining the infimum o…

Let $\mathcal{L}$ be the class of quantitatively almost calibrated exact unobstructed Lagrangian objects under consideration, and let $L_0\in\mathcal{L}$ be exact, quantitatively a…

Let $(X,\omega,\Omega)$ be an exact Calabi-Yau manifold, and let $L_0$ be a nontrivial unobstructed exact Lagrangian brane with quantitatively almost calibrated phase function. Let…

Let $X$ be a compact almost Calabi-Yau manifold. Let $L_0$ be a reference Lagrangian, and consider graded immersed unobstructed Lagrangians in the same $D^bFuk(X)$ class as $L_0$.…

Let $X$ be a Calabi-Yau manifold, let $D^bFuk(X)$ denote its derived Fukaya category, and let Joyce's conjectural Bridgeland stability be a stability condition on this category. An…

Let a complex structure determine a Fukaya category, and let special Lagrangians be the relevant geometric objects in that category. Stability conjecture. Complex structures are st…

Let $X$ be a Calabi–Yau manifold, let $L\rightarrow X$ be a holomorphic line bundle, and let $L\hookrightarrow X$ be a Lagrangian submanifold. The objects $L$ are viewed respective…

Let $(X,\omega,\Omega)$ be a Calabi--Yau manifold and let $L$ be a compact zero Maslov Lagrangian in it. A Lagrangian is stable when it satisfies the stability condition under disc…

Let $X$ be a Calabi–Yau manifold with mirror $\check{X}$, let $L$ be a Lagrangian in $\check{X}$, and let $E$ be a holomorphic vector bundle on $X$. Write…

Let $M$ be a Calabi–Yau manifold and let $\Sigma$ be a compact embedded Lagrangian submanifold with zero Maslov class. Simplified Thomas–Yau conjecture. The mean curvature flow of…

Let $(M^{2n},\omega,J,\Omega)$ be a Calabi–Yau manifold, and let $L\subseteq M$ be a zero-Maslov-class, almost-calibrated Lagrangian. Suppose that $L$ is flow-stable, meaning that,…