143 problems

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.…

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…

Let $A=\operatorname{Spec}\bigoplus_{n\geq0}H^0(w\mathbb{P},-nK)$ be the anticanonical affine cone over the weighted projective space $w\mathbb{P}$, and let $\widetilde{A}\to A$ be…

Let $F_n=\mathbb P(\mathcal O\oplus\mathcal O(-n))$ be the Hirzebruch surface over $\mathbb P^1$, in the range $n\geq 3$ considered by the source. Parity conjecture. There are two…

Let $\mathbb P(\mathcal O\oplus\mathcal O(k)\oplus\mathcal O(-2-k))$ and $\mathbb P(\mathcal O\oplus\mathcal O(-1)\oplus\mathcal O(-1))$ denote the corresponding projective bundles…

Let $G/P$ be a partial flag variety, let $Z_P^{\mathfrak h^\vee}$ be the pullback of the mirror family to $(\mathfrak h^\vee)^{W_P}$, and let $\omega$ and $\mathcal F_P$ be respect…

Let $X=\mathcal O_{\mathbb P^2}(-3)$, and let $\operatorname{Stab}^{0}_{n}(X)$ be the normalized connected submanifold of stability conditions defined by $Z(\mathcal O_x)=i$. Let…

Let $R_{\sigma\tau}^{(n)}(t,\lambda)$ be the coefficient of $\hbar^n$ in the asymptotic solution matrix $R_{\sigma\tau}(t,\lambda,\hbar)$ of the equivariant quantum differential eq…

Dubrovin's reduced quantum cohomology conjecture. The variety $X$ has generically semisimple reduced quantum cohomology, that is, $M'$ is generically semisimple, if and only if the…

Let $(W,S)$ be a crystallographic Coxeter system, let $qBE(W,S)$ be the associated quantum algebra, and let $\tilde{P}_w\in {\bf S}(V^{\ast})\otimes {\bf R}[q_s]$ be the polynomial…

Let $(W,S)$ be a crystallographic finite Coxeter system, let $qBE(W,S)$ be the associated quantum algebra, and let ${\bf R}[q_s][\tilde{\theta}_s\mid s\in S]$ denote the subalgebra…

Let $\kappa=\lambda/d/\mu$ be a toric shape, let $S^\kappa$ be its toric Specht module, and let $S^\nu$ denote the irreducible representation of the symmetric group indexed by…

Givental's conjecture. The Floer $\mathcal{D}$-module and the quantum cohomology $\mathcal{D}$-module are the same; equivalently, $F=G$, with the components of $F$ giving appropria…