72 problems

Let $X$ be a Fano manifold of Picard number $1$, and let $f:X\to X$ be a surjective endomorphism. Here, $\deg f$ denotes the degree of $f$, and $\mathbb{P}^n$ denotes projective…

Let $V$ be a Fano manifold, and let stable mean stable in the sense of Geometric Invariant Theory. Yau's conjecture. The manifold $V$ admits a Kähler–Einstein metric if and only if…

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 $X$ 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…

Bondal's conjecture. For every $k\in\{0,1,\dots,r-1\}$, $D_{2k}(\pi)$ has an irreducible component of dimension at least $2k+1$.

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 $(X,\Delta)$ 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 a log Fano manifold admitting a Kähler–Einstein metric. Let $Z_{\boldsymbol{m}_{N}}$ be the K-reduced partition function, let $\mathcal{M}$ be the Mabuchi funct…

Let $(X,\Delta)$ be a log Fano manifold with vanishing Futaki character. Let $\gamma(X,\Delta)^{\mathcal{G}}$ denote the reduced Gibbs stability threshold and let…

Let $(X,\Delta)$ be a log Fano manifold. Gibbs polystability conjecture. $(X,\Delta)$ is Gibbs polystable if and only if it is uniformly Gibbs polystable if and only if it is K-pol…

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…

Quantized volume comparison conjecture. For every $m\geq 1$,

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…

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…

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…

Fujita's cohomological characterization conjecture. Then $M\cong\mathbb{P}^n$. This is the strongest of the three related conjectures introduced by Fujita. The source gives no evid…

Chain-length characterization conjecture. 1. If

Positive Chern character conjecture. If ${\rm ch}_k(X)$ is positive for every $k\le n$, then

Let $X$ be a Fano manifold. Let $S(z)$ be the quantum differential-equation fundamental solution, $\mu$ the grading operator, $c_1$ act by quantum multiplication, and…

Let $X$ be a Fano manifold. Let $S(z)$ be the quantum differential-equation fundamental solution, $\mu$ the grading operator, $c_1$ act by quantum multiplication, and…

Let $X$ be a Fano manifold satisfying Property $\mathcal O_A$, where $T_{\mathrm{A,con}}$ is a simple rightmost eigenvalue of $\hat c_1$. Let $A_X$ be the principal asymptotic clas…

For a Fano manifold $X$, let $G_X(t)=\sum_{n=0}^{\infty}a_nt^n/n!$ be its quantum period and define the A-model conifold value by … Let $\hat c_1$ denote quantum multiplication by…

Folklore conjecture. If $X$ admits a non-isomorphic surjective endomorphism, then $X$ is a projective space.

Let $X$ be a Fano manifold and let $\mathcal{A}$ be the vector space of flat sections $s(z)$ over the positive real axis such that $e^{T/z}s(z)$ has at most polynomial growth as…