Hodge conjecture
Some topological features of a complex algebraic shape look as though they should come from smaller algebraic shapes inside it. Must they?
Algebraic geometry
Polynomial equations define spaces whose shapes, singularities and algebraic cycles pose different open problems.
Subjects: Algebraic geometry · Topology
Problems in this collection
8 problemsHartshorne’s complete intersection conjecture
Must a smooth complex projective variety of sufficiently small codimension be defined by exactly that many polynomial equations?
Nagata conjecture
How low can a curve’s degree be when it must pass repeatedly through many very general points?
Zariski multiplicity conjecture
Does a complex hypersurface singularity’s local topological shape determine its multiplicity?
Zariski cancellation for affine three-space
If adding one affine coordinate makes a complex variety look like four-dimensional affine space, was it already affine three-space?
Abundance conjecture
Does numerical nonnegativity of a minimal model’s canonical divisor guarantee a globally defined geometric map?
Lefschetz standard conjecture
Can a basic reversible cohomology operation on a complex algebraic variety always be reversed using algebraic cycles?
Green–Griffiths–Lang conjecture
Must all entire complex curves in a general-type projective variety lie in one proper algebraic subset?