12 problems

Conjecture 2. There exist real or complex numbers $y_1,\ldots,y_n$ such that

Let $V$ be a nonsingular complex projective variety of complex dimension $n$, viewed as a complex manifold, with fundamental class $[V]\in H_{2n}(V,\mathbb Z)$ and total Todd class…

Let $P:\mathbb C^2\rightarrow\mathbb C^2$ satisfy $J(P)\equiv 1$. Suppose there exists an algebraic function defined on some open set ${\cal U}$ such that $P\circ Q=P$, and suppose…

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

Let $n$ be a positive integer and let $a,b\in\mathbb{C}$ satisfy $|a|\neq 1$. Set $h(z,w)=z^n+aw^n+b$ and $h^{\ast}$ for the polynomial obtained by conjugating the coefficients and…

Let $f:{\mathbb{C}}^n\rightarrow{\mathbb{C}}^n$ be a polynomial map, meaning that each component $f_i:{\mathbb{C}}^n\rightarrow{\mathbb{C}}$ is polynomial. Jacobi conjecture. If th…

Let $F:\mathbb C^n\to\mathbb C^n$ be a polynomial map, and let $DF$ denote the Jacobian of $F$. Smale's strong Jacobian conjecture. If $DF$ does not vanish, then $F$ is injective.…

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a complex line arrangement, and let $s$ be the number of its lines. A non-pencil complex supersolvable line ar…

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a complex line arrangement. An arrangement is nontrivial if it is neither a pencil nor a near pencil, and it i…

Let $f_1,\ldots,f_{n-1}\colon\mathbb C^n\to\mathbb C$ be polynomial functions and write … Let $B(F_{\widehat n})$ denote the set of points at which this mapping is not a locally tr…

Let $P(z)=(P^1(z),\ldots,P^q(z))$ be a polynomial mapping in $z=(z^1,\ldots,z^n)\in\mathbb C^n$. Suppose that $A(z,\bar z)=\lVert F(z)\rVert^2-\lVert G(z)\rVert^2$ for polynomial m…

Let $P(z)=(P^1(z),\ldots,P^q(z))$ be a polynomial mapping in $z=(z^1,\ldots,z^n)\in\mathbb C^n$, and let $A(z,\bar z)$ be a Hermitian polynomial satisfying … If $r$ denotes the lin…