238 problems

A Calabi–Yau manifold is a Kähler manifold equipped with a nowhere-vanishing holomorphic volume form, and a Lagrangian submanifold is stable in an appropriate sense as specified by…

Let $Y$ be a Calabi–Yau manifold, meaning that $Y$ has trivial canonical bundle and $h^1(\mathcal{O}_Y)=0$. Let $\mathrm{Nef}^+(Y)$ denote the convex hull of the rational points of…

Let $f(z)\in\mathbb{C}[z_0,\dots,z_{d+1}]$ be a generic homogeneous polynomial of degree $d+2$, where $d\geq1$, and let … be the resulting maximally degenerate one-parameter family…

Let $X$ and $Y$ be birational Calabi-Yau manifolds, meaning smooth projective varieties with trivial first Chern class. Their quantum cohomology includes the big quantum cohomology…

Let $(M,J,\omega,\Omega)$ be an almost Calabi--Yau $m$-fold, let $X$ be a compact special Lagrangian $m$-fold in $M$ with conical singularities, and let…

Let $Z_{S^2}$ denote the partition function on the two-sphere of a two-dimensional ${\cal N}=(2,2)$ gauged linear sigma model (GLSM) whose infrared description is a non-linear sigm…

An extremal transition is a process connecting topologically distinct Calabi–Yau manifolds through a singular Calabi–Yau variety and its smoothing or resolution. Candelas–de la Oss…

Closed string mirror symmetry conjecture. There are equivalences, as Frobenius manifolds,

Let $\pi: (X,L)\to \mathbb{D}^*$ be a polarised maximal degeneration of compact Calabi-Yau manifolds, and let $(X_t,\omega_{CY,t},\Omega_t)$ denote its fibre with the Calabi-Yau st…

Let $M$ be a compact Kähler manifold of complex dimension $n$, let $\mathcal K$ be its Kähler cone, and define … For $l>0$, let $\mathcal K_l=f^{-1}(l)\subset\mathcal K$ with the i…

Consider the Gross–Siebert construction of a family of Calabi–Yau manifolds from an integral affine manifold with singularities and its associated tropical geometry; let $B$ denote…

Vafa's conjecture. The moduli spaces of A-cycles and B-cycles, together with their correlation functions, should be identified under mirror symmetry.

Let $X$ be a compact Kähler manifold. A class $\alpha \in H^{1,1}(X,\mathbb{R})$ is semi-ample if there exists a holomorphic contraction $\rho:X\to X'$ to a normal analytic space…

A Calabi–Yau manifold is a compactification space for the extra six dimensions of the heterotic string, while an $\mathcal{N}{=}2$ superconformal field theory (SCFT) describes the…

Gross–Pavanelli's mirror conjecture. The mirror of $X$ is $\check{X}$, and the mirror of $\check{X}$ is $Y$.

Let $X_t$ be the smooth Calabi–Yau fibres of the family … A special Lagrangian fibration is a fibration of $X_t$ by special Lagrangian submanifolds. SYZ conjecture. For sufficientl…

Let $X$ and $Y$ be birational Calabi-Yau threefolds, meaning smooth projective varieties with trivial first Chern class. Morrison's conjecture. $X$ and $Y$ have identical small qua…

A compact Calabi–Yau manifold is a compact Kähler manifold with vanishing first real Chern class; its topological type means its homeomorphism type. Yau's finiteness conjecture. In…

Let $(\mathcal{X}_{t_i},g_i)$ converge to $(X_{\infty},g_{\infty})$. Let $(B_i,d_i)$ be non-empty for large $i$ and converge to $(B_{\infty},d_{\infty})$, where $B_i$ are compactif…

Let $M$ be a Calabi–Yau manifold. The Dolbeault cohomology $A(M)=H^{*,*}(M)$ carries a formal Frobenius manifold structure arising from the DGBV algebra…

Precise symplectic SYZ mirror conjecture. For any Calabi–Yau 3-fold $X$, with Calabi–Yau metric $\omega_g$ and holomorphic volume form $\Omega$, there exists a Lagrangian torus fib…

A special Lagrangian fibration is a fibration of a Calabi–Yau manifold by special Lagrangian submanifolds, and a large complex structure limit point is a boundary point in the comp…

A torically constructed family of Calabi–Yau manifolds has a projective normal form Picard–Fuchs equation. A no-apparent-singularities conjecture. No apparent singularities arise i…

Barannikov–Kontsevich mirror symmetry conjecture. The resulting formal Frobenius manifold of the $B$-model can be identified with the quantum cohomology of the mirror dual Calabi–Y…

Let $X_1$ be the cotangent bundle of $S^3$ and let $X_2$ be the $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ bundle of $S^2$. Mirror-duality conjecture. The Calabi–Yau manifolds $X_1$ an…