144 problems

Let $X$ be an adjoint variety of type $A_n$, and let $Y\subset X$ be a general hyperplane section. Let ${\rm QH}(X)_{\rm can}$ and ${\rm QH}(Y)_{\rm can}$ denote the restrictions o…

The big quantum cohomology of G/P is generically semisimple.

For a smooth Fano variety $X$ of Hodge-Tate type, the modern form of Dubrovin's conjecture predicts … The refined Gamma conjecture II also identifies categorical and quantum data.…

Let $G\subset \mathrm{SL}_2(\mathbb{C})$ be a nontrivial finite subgroup, let $p:X\to \mathbb{C}^2/G$ be the minimal resolution, and equip the spaces with the relevant…

Injectivity conjecture. With the same notation as in the quantum Lefschetz proposition, there is a natural injective morphism of algebras

Let $X$ be a Fano manifold. Denote by $c_1(X)\star_0$ quantum multiplication by the first Chern class at the origin, and suppose that it has a simple rightmost eigenvalue. Let…

Let $X$ be a smooth Fano variety. The Dubrovin conjecture. The derived category $\mathrm{D}^b(X)$ admits a full exceptional collection if and only if the big quantum cohomology rin…

Let $X$ be a Fano manifold, let $\mathcal{H}_X=H^\bullet(X)\otimes\mathcal{O}_{\mathbb{C}_z}$, and let … be its quantum connection at quantum parameter $\tau=0$. Assume that the qu…

For a Fano manifold $X$, let $QH^\bullet(X):=H^{\rm ev}(X)\otimes_{\mathbb{Q}}\mathbb{Q}[\mathbf{q}]$ be its even quantum cohomology, and let $\hat c_1$ be quantum multiplication b…

Fukaya-category elliptic-curve summand conjecture. The Fukaya category of $X$ should contain a direct summand equivalent to the derived category of sheaves on a genus-one curve who…

Let $\mathcal{X}$ be an orbifold with coarse moduli space $X$, and let $\phi\colon Z\to X$ be a crepant resolution. Iritani's conjecture. There exists an isomorphism … induced by a…

Galkin-Golyshev-Iritani's Conjecture O. Every Fano manifold $X$ satisfies: $\rho$ is an eigenvalue of $\hat c_1$ with multiplicity one; and for every eigenvalue $\lambda$ of…

Let $(M,\omega)$ be a closed symplectic manifold and let $K\subset M$ be a compact subset. The subset $K$ is SH-heavy in the sense of the quantum cohomology ideal-valued quasi-meas…

Noncommutative minimal model conjecture. For any smooth Fano variety $X$, there exist $\tau\in {\mathrm H}^\bullet(X)$, a sector $\mathscr{S}\subset\mathbf{C}^*$, $\epsilon>0$, and…

Let $X$ be a smooth proper Deligne–Mumford stack with projective coarse moduli space, and let $Y$ be a smooth complete intersection in $X$. Let $I^{tw}$ be a point on the twisted L…

Let $X$ be a Fano variety of index $m$ over an algebraically closed field $\boldsymbol{\Bbbk}$ of characteristic zero, and assume that the big quantum cohomology…

The critical-eigenspace conjecture. There is an isomorphism

The filtration conjecture. One has

Eguchi–Hori–Xiong conjecture. The partition function satisfies

Let $Y$ be the blow-up of a smooth projective threefold $X$ along a curve $C$ such that $g(C)\geq 1$, or $g(C)=0$ and $-K_X.C\geq 0$. The associativity equations of the quantum pro…

Projective bundle mirror theorem conjecture. Under these assumptions,

Let $\mathcal{X}$ be an orbifold satisfying the hard Lefschetz condition and admitting a crepant resolution $Y$. Let $F^Y$ and $F^\mathcal{X}$ denote their potential functions, wit…

Let $X$ be a smooth projective variety and let $V=\bigoplus_{i=0}^n L_i$ be a direct sum of line bundles, where $L_i$, $i=1,\ldots,n$, are nef and $-K_X-c_1(V)$ is ample. Let…

Two smooth varieties $X$ and $X'$ are K-equivalent if there exist birational morphisms … from a smooth variety $Y$ such that … Let their quantum cohomology rings be considered over…

The isomorphism conjecture. The algebra homomorphism $\eta$ is an isomorphism. The statement appears as a conjecture in the source, but the supplied material gives no resolution ev…