26 problems

Let $f$ and $g$ be meromorphic functions on ${\mathbb{C}}$, and let $T(r,f)$ and $T(r,g)$ denote their Nevanlinna characteristic functions. Suppose that $f$ and $g$ share three dis…

Griffiths–Lang conjecture. There exists a proper Zariski closed subset $Z_D\subsetneq X$ such that, for all holomorphic curves $\mathbf{x}:\mathbb{C}\to X$ with…

For a meromorphic function $f$ and meromorphic functions of slow growth serving as moving targets, the relevant notions of value distribution are understood in the setting of Nevan…

Let $X$ be a compact complex manifold, and let a Nevanlinna current be a positive closed $(1,1)$-current arising from an entire curve. By Siu's decomposition, write such a current…

Let $f$ be an entire function of finite order, and let $\{\delta_n\}$ be the sequence of its deficiencies, where $\delta(a,f)$ denotes the Nevanlinna deficiency at…

In tropical Nevanlinna theory, let the truncated Nevanlinna counting functions be obtained by truncating the multiplicities in the usual second main theorem for tropical meromorphi…

Irreducibility conjecture. For every $cepsilon>0$, there exists a nonzero negatively twisted invariant logarithmic $2$-jet differential $comega$ in this space with

Let $X$ be a smooth quasi-projective variety. Define the special loci … as follows: $\mathrm{Sp}_{\mathrm{sab}}(X)$ is the Zariski closure of the union of images of non-constant ra…

Griffiths' conjecture. (1) For every holomorphic curve $f:\mathbb{C}\to X$ with Zariski-dense image,

Griffiths' conjecture. For any $\epsilon>0$, the inequality

Cao and Zheng's tropical Griffiths conjecture. The defect sum satisfies

Chen et al.'s conjecture. The conclusion that every such solution satisfies $\rho(f)=\deg(Q)=1$ also holds for $n=2$.

Rong and Xu's conjecture. Case (iii) can be removed when $N(r,1/f)=S(r,f)$ for $n=2,3$.

Let $X$ be a smooth projective variety, let $D$ be a simple normal crossing divisor on $X$, and let $A$ be an ample line bundle on $X$. Let $f:\mathbb{C}\to X$ be a Zariski dense e…

Let $X=\mathbb{P}^n$ and let $f:\mathbb{C}\rightarrow\mathbb{P}^n$ be an entire holomorphic curve whose image is not contained in any hyperplane. Let $T_f(r)$ be the characteristic…

Let $D$ be a domain and let $\mathcal{F}$ be a family of analytic functions on $D$ satisfying, for every $f\in\mathcal{F}$ and every $z\in D$, $f(z)\neq 0$ and $f'(z)\neq 1$. Monte…

Truncated tropical Cartan conjecture. The displayed truncated second main theorem is proposed as the tropical counterpart of the truncated Cartan second main theorem. The source gi…

Truncated tropical second main theorem conjecture. These truncated inequalities and equality are conjectured to hold as stated. They are tropical analogues of the truncated classic…

Tropical Griffiths-type defect conjecture. One should have

Let $f$ be a transcendental meromorphic function, let $c\in\mathbb{C}$, and let $e_1,e_2$ be three distinct values together with $\infty$. Suppose that $f$ and … share the three di…

Let $f$ be the holomorphic curve and let $T(r,f)$ and $m_k(r,f)$ be the characteristic and higher proximity functions defined in the paper. For $k\in[1,n]$, set … In the case $n=2$…

Let $X_1={\mathbb P}(T_X)$, let $f$ be the parabolic curve under consideration, and let $T[f']$ be the associated lifted Nevanlinna current. Let $\omega$ be a fixed reference metri…

Let $X$ be a projective manifold, let $f$ be the parabolic curve under consideration, and suppose the stability hypothesis stated immediately before the claim holds. Let $\omega_k$…

Let $X$ be a projective manifold, let $k$ and $m$ be positive integers, and let $E_{k,m}T_X^\star$ denote the relevant bundle of jet differentials. Let $L\to X$ be a line bundle sa…

Representation conjecture. An arbitrary tropical meromorphic solution $y$ may be represented as a linear combination of shifts of $y_1$ and $y_2$.