18 problems

Collins–Jacob–Yau conjecture. The dHYM equation admits a solution if and only if, for every proper subvariety $V\subset X$,

Stability conjecture. When these stability-type cohomological obstructions vanish, the deformed Hermitian–Yang–Mills equation admits a solution.

Let $X$ be a compact Kähler manifold with Kähler form $\omega$, let $[\alpha]$ be a real $(1,1)$-cohomology class, and for every irreducible analytic subvariety $V\subseteq X$ defi…

Collins–Yau's conjecture. The following are equivalent:

Collins–Jacob–Yau hypercritical-phase conjecture. The set $\mathcal{H}_{\Omega}$ is non-empty and $\Omega$ has hypercritical phase if and only if, for every irreducible subvariety…

Let $X$ be a complex manifold of complex dimension $n$, and suppose that existence of a solution to the deformed Hermitian Yang–Mills equation imposes a necessary Chern number ineq…

Let $(X,\alpha^{\mathbb{C}})$ be a complexified Kähler manifold in the hypercritical range, and let $\mathcal{H}$ denote the space of admissible potentials. C^{1,1} geodesic conjec…

dHYM–Bridgeland stability conjecture. The existence of a dHYM instanton $\nabla$ on ${\bf E}$ should be equivalent to Bridgeland stability of ${\bf E}$. This conjecture proposes an…

Let $X$ be the compact Kähler manifold under consideration, and let $L$ be a holomorphic line bundle on $X$. A metric solving the deformed Hermitian–Yang–Mills equation is a metric…

Let $X$ be the variety under consideration, let $L\in\operatorname{Pic}(X)$ be a line bundle, and set $[\psi]=c_1(L)$. Conjectural dHYM–Bridgeland correspondence. The existence of…

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, let $[\alpha]\in H^{1,1}(X,\mathbb R)$ be a real cohomology class, and let $\hat\theta$ be a supercritical p…

Let $(X,\omega)$ be a Kähler threefold and let $L\rightarrow X$ be a holomorphic line bundle. Suppose that the Chern number inequality holds, the lifted angle is well-defined with…

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 $M^n$ be a Kähler manifold with Kähler metrics $\chi$ and $\omega_0$, and fix $\hat\theta\in(\frac{n\pi}{2}-\frac{\pi}{4},\frac{n\pi}{2})$. For each $p$-dimensional subvariety…

Collins–Jacob–Yau's conjecture. Solvability of the deformed Hermitian–Yang–Mills equation should be equivalent to a notion of stability. The equation arises as a large-parameter li…

Let $(X,\omega)$ be a Kähler manifold, let $L$ be a line bundle or let $[\alpha]$ be a real $(1,1)$ class, and let $Z_V(L)$ denote the central charge associated to each analytic su…

Let $(X,\omega)$ be a Kähler $3$-fold and let $L\to X$ be a holomorphic line bundle. Assume that $L$ admits a solution of the deformed Hermitian–Yang–Mills equation with lifted ang…

Let $(X,\omega)$ be a Kähler $4$-fold, let $L$ be a line bundle on $X$, and write $a.b=\int_X a\wedge b$. Suppose that $L$ admits a solution of the deformed Hermitian–Yang–Mills eq…