365 problems

For every real parameter $t>1$, does there exist a real-analytic CR-embedding $F_t:M_t^3\hookrightarrow\mathbb{C}^3$ such that $F_t(M_t^3)$ is a compact real-analytic strongly pseu…

Let $X$ be a compact complex manifold of complex dimension $n$. If $X$ admits a balanced Hermitian metric with fundamental form $\omega_{B}$ satisfying $d(\omega_{B}^{n-1})=0$ and…

Iitaka–Severi conjecture. The number of birational equivalence classes of such manifolds $Y$ is finite. This conjecture generalizes the de Franchis–Severi finiteness theorem for ma…

Let $X$ be a compact complex manifold, let $\mathcal{N}\subset H^{1,1}(X,\mathbb{R})$ be its cone of nef $(1,1)$-classes, let $[\alpha]\in\mathcal{N}$, and let $V\subset X$ be a cl…

Let $D$ be a Mumford--Tate domain parameterizing pure, effective, weight $\mathsf{w}$, $Q$-polarized Hodge structures on a finite-dimensional rational vector space $V$. Let…

Let $X$ be the compact complex manifold, let $\theta$ be the closed $(1,1)$-form, and let $\theta$-plurisubharmonic function $\varphi\in\operatorname{PSH}(X,\theta)$ be as above, w…

Rigidity conjecture. Either $T=0$ or $S=0$.

Full-mass conjecture. $e_n(\varphi)=1$ if and only if $u\in\mathcal{E}(\omega_0)$.

Let $X$ be a complex manifold, let $L$ be a semi-positive holomorphic line bundle on $X$, and let $Y$ be a compact Kähler submanifold such that $L|_Y$ is topologically trivial. Let…

Let $A$ be an anti-self-dual connection on a bundle $E\rightarrow T\times\mathbb{C}$, with curvature satisfying $|F_A|\sim O(|w|^{-2})$. A holomorphic structure…

Let $M$ be a compact complex manifold, let $obreaklambda$ be a line bundle on $M$, and let $boldsymbol1\lambda$ denote its identity differential operator. Write…

Invariance of plurigenera conjecture. The plurigenera is invariant under smooth projective deformations; equivalently, for every integer $m\geq 1$, $P_m(X_s)$ is independent of…

Solvmanifold deformation conjecture. (i) Every left-invariant complex structure on $G$ is Stein, respectively biholomorphic to ${\bf C}^n$ in the nilpotent case. (ii) Every small d…

Let $X$ be a projective manifold with split tangent bundle … A subbundle $V\subset T_X$ is integrable if it is closed under the Lie bracket of local sections. Integrability conject…

Let $M\hookrightarrow \mathbb{C} P^N$ be a complete, connected, and full complex submanifold, meaning that it is not contained in a proper hyperplane. Suppose its normal holonomy g…

Let $G$ be a complex simple Lie group and let $P\subset G$ be a maximal parabolic subgroup. Assume that $G/P$ is not a compact Hermitian symmetric space, or, if $G/P$ is a compact…

Generalized spiral-point conjecture. After this normalization, the sections $(p_k\mathbf{w}^k,q_k\mathbf{w}^k)$ satisfy the conditions in Theorem. The construction is supported by…

Strong Green–Griffiths conjecture. There exists a closed subvariety $E\subsetneq X$ such that $E$ contains the image $f(\mathbf{C})$ for every non-constant holomorphic map

Let $X$ be a Stein manifold diffeomorphic to ${\mathbb C}^n$ and having the density property, meaning that the closure of the Lie algebra generated by the complete holomorphic vect…

Let $X$ be a smooth projective variety of general type. A minimal variety is one with no exceptional $(-1)$-curves in the surface case, and more generally with the minimal-model pr…

Let $X$ be a smooth projective variety of general type. A nowhere vanishing holomorphic one-form on $X$ is a holomorphic section of the cotangent bundle $4\Omega_X^14$ with no zero…

Abelian-variety entire-curve conjecture. Every non-constant holomorphic map

Pointed Brody conjecture. If the infinitesimal Kobayashi–Royden pseudometric is degenerate on $T_xX$, then there exists a non-constant holomorphic map

Campana's torus-submersion conjecture. Up to a finite étale cover of $X$, the manifold $X$ admits a torus submersion over a projective manifold $Y$ such that $K_Y$ is ample and the…

Let $X$ and $Y$ be fixed smooth compact complex curves of genus at least $2$, and let the genus of $X$ be $g$. A polynomial bound conjecture. There is a polynomial function $B(g)$…