381 problems

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

Let $Z$ be the open Batyrev–Borisov complete intersection with skeleton … where $S=\partial\widetilde{C}_{0,\beta}\subset M_\R$ is the codimension-$r$ sphere introduced above, and…

Let $M_N^k$ be the degree-$k$ projective Fano hypersurface considered in the paper, and let $\langle {\cal O}_{h^a}{\cal O}_{h^b}\rangle_0(t^0,t^1,\ldots,t^{N-2})$ denote the pertu…

Let $X$ be a Calabi–Yau manifold. A Lagrangian fibration is a map $\pi:X\to B$ onto a topological manifold $B$ whose fibers are graded with respect to a holomorphic volume form…

The E2-degeneration conjecture. The spectral sequence degenerates at the $E_2$ page.

Let $V$ be the Milnor fiber under consideration, let $W^\vee$ be its Berglund–Hübsch mirror polynomial, and let $\Gamma^\vee$ be the maximal symmetry group of $W^\vee$, … Write…

Let $X$ be the symplectic manifold under consideration. For each Lagrangian $L_i\subset X$, let $\mathcal L_i$ be the corresponding object in the category of $\alpha_X$-twisted she…

Let $z(q)$ be the mirror map for a one-parameter deformation of an algebraic K3 surface arising from an orbifold construction whose Picard–Fuchs equation is third order. A Lian–Yau…

Master Family degeneration conjecture. Vertex algebras arising from Mirror Symmetry for hypersurfaces defined by $\Delta$ and $\Delta^*$ are degenerations of the Master Family of v…

Toric Fano mirror conjecture. There should exist a primitive form for $f$ given by

Barannikov–Kontsevich conjecture. The Frobenius structure $(S',\eta_F|_{S'},\Phi_F|_{S'})$ should be a Frobenius submanifold of the Frobenius manifold constructed by Barannikov–Kon…

Mirror-family conjecture. The pair $M_y+W_y$ is the mirror family of $X_A+Y_A$, with $M_y$ and $W_y$ forming different parts of the B-model moduli space and $X_A$ and $Y_A$ forming…

Let a point of the caustic be fixed, and consider the moduli space of gradient lines and the moduli space of pseudoholomorphic discs associated with the Lagrangian fibration near t…

Let $\{\Delta_1',\ldots,\Delta_r'\}$ and $\{\Delta_1”,\ldots,\Delta_r”\}$ be the translated nef-partitions described in the preceding discussion, with corresponding mirror Calabi–Y…

Let $\Delta$ be a Gorenstein polytope of CY-dimension $n$, so that its stringy E-function $E_{\rm st}(\Delta;u,v)$ and CY-dimension are defined as in the preceding discussion. Stri…

The B-model of a Calabi–Yau threefold has a partition function associated with its monodromy action. B-model partition-function conjecture. The analogue of Theorem holds for the B-…

Projective bundle mirror theorem conjecture. Under these assumptions,

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…

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb{P}^1$ carry the diagonal torus action $\lambda_1=\lambda_2$, and let…

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb{P}^1$ carry the antidiagonal torus action $(\lambda,-\lambda)$ on the bundle, and let $t_k$ denote the Kähler class o…

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 $\pi\colon\mathcal W\to\mathbb P^{1}$ be a family of quintic mirror threefolds. Let $N_g(d)$ be the genus-$g$ Gromov–Witten invariant of degree $d$ of a general quintic threefo…

Equivariant mirror conjecture. A full set of solutions to the $T$-equivariant quantum differential equations of $G/P$ is given by these integrals. This is the $T$-equivariant exten…

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…