133 problems

Donald's conjecture. One has

Bondal's conjecture. The locus $\mathsf{D}_{2k}(\sigma)$ has an irreducible component of dimension at least $2k+1$ for every such integer $k$. The conjecture predicts that degenera…

Yau's conjecture. The geometric-invariant-theoretic condition should be equivalent to the existence of a Kähler–Einstein metric on $M$.

Let $(M,\omega)$ be a positive monotone symplectic manifold of dimension six admitting a Hamiltonian circle action. A Fano manifold is a compact complex manifold whose anticanonica…

Tian's conjecture. K-stability implies the existence of a Kähler–Einstein metric on $M$.

Toric Fano extremal metric conjecture. Every toric Fano manifold admits an extremal Kähler metric in its first Chern class.

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…

Weak Campana–Peternell conjecture. If the tangent bundle of a Fano manifold is nef, then it is also big.

Let $X$ be a Fano manifold, let $QH^*(X)$ be its quantum cohomology ring, and let $c_1$ denote the first Chern class of $X$. The quantum product by $c_1$ defines a linear operator…

Let $D$ be a divisor in a Fano manifold, and consider conical Kähler–Einstein metrics along $D$ with cone angles tending to zero. The complement of $D$ is equipped with a complete…

Fujita–Odaka conjecture. Uniform K-stability is equivalent to

Let $(M,\omega)$ be a compact Kähler manifold of complex dimension $m$ such that $[\omega]=2\pi c_1(M)$, $\operatorname{Ric}(\omega)\ge R_0>0$, and…

Let $X$ be a quasi-projective manifold and let $\varphi\colon X\rightarrow Y$ be an elementary contraction associated with an extremal ray $R$. Let $E$ be an irreducible component…

Aubin's conjecture. There exist positive constants $A$ and $B$ such that

Let $X$ be a compact complex manifold with positive first Chern class, and let $F_X\colon \operatorname{Lie}(\operatorname{Aut}(X))\to \mathbb{C}$ be its Futaki character. An Einst…

Let $V$ be a smooth Fano manifold of complex dimension $n-1$, and let $I(V)\in\mathbb{N}$ be its Fano index. GMSY's Fano inequality conjecture. … with equality if and only if…

Let $M$ be a Fano manifold of Picard number one and let $\mathcal{E}$ be an ample vector bundle of rank $r$ on $M$. Let $L$, the extremal ray $R_1$, and a minimal rational curve…

Let $V$ be a Fano manifold of Picard number one, let ${\mathcal O}_V(1)$ be an ample generator of ${\rm Pic}(V)$, and suppose that $\dim V=2r+1$ is odd. Let…

Let $X$ be a Fano manifold. Denote its Picard number by $\rho_X$, its pseudo-index by $\iota_X$, and its dimension by $\dim(X)$. Generalized Mukai conjecture. One has … This genera…

Let $X$ be a K-semistable Fano manifold, and let $\omega$ denote the initial Kähler form for the Kähler–Ricci flow. The iterated balancing filtration is the filtration referred to…

Let $X$ be a Fano manifold, and let $QH_{\rm sm}(X)$ be its small quantum cohomology with small Dubrovin connection, flat-section space $\mathcal{S}^{\rm sm}$, and…

Let $(\mathcal{X},\mathcal{D})$ be an arithmetic log pair over $\mathbb{Z}$ such that $-\mathcal{K}_{(\mathcal{X},\mathcal{D})}$ is a relatively ample $\mathbb{Q}$-line bundle, and…

Let $X$ be a Fano manifold with a Kähler–Einstein metric $\omega_0$. Let $\mathcal{H}(X,\omega_0)/\mathbb{R}$ be the space of Kähler potentials modulo constants, and let…

Let $(X,\Delta)$ be a log Fano manifold with vanishing Futaki character. Define the divisorial polystability threshold $\delta(X,\Delta)_{\mathrm{ps}}$ as the infimum of the ratio…

Let $(X,\Delta)$ be the complexification of an arithmetic log Fano variety $(\mathcal{X},\mathcal{D})$, with the arithmetic structure providing a canonical normalization and invari…