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
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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?
Hartshorne’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?
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?