Unsolved Problems
Showing 1-9 of 9 problems
Category
Problem Set
Status
Hodge Conjecture
On a projective non-singular algebraic variety over $\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles....
The Standard Conjectures on Algebraic Cycles
A collection of conjectures about algebraic cycles on smooth projective varieties, including Lefschetz standard conjecture and Künneth standard conjec...
The Abundance Conjecture
For a minimal model $X$ of non-negative Kodaira dimension, the canonical divisor $K_X$ is semi-ample....
Smale's 5th Problem: Height Bounds for Diophantine Curves
Find effective uniform bounds for the heights of rational points on algebraic curves....
Hilbert's 15th Problem: Schubert's Enumerative Calculus
Rigorously justify Schubert's enumerative geometry....
The Hodge Conjecture
On a projective algebraic variety, is every Hodge class a rational linear combination of classes of algebraic cycles?...
The Birch and Swinnerton-Dyer Conjecture
For an elliptic curve $E$ over the rationals, does the rank of its group of rational points equal the order of vanishing of its $L$-function at $s=1$?...
The Tate Conjecture
For varieties over finite fields, are the $\ell$-adic representations arising from étale cohomology related to algebraic cycles in the expected way?...
Abundance Conjecture
If the canonical bundle of a variety is nef, must it be semiample?...