show/hide this revision's text 3 fix more

From a post to The Jouanolou trick:

Are all topologically trivial (contractible) complex algebraic varieties other then affine lines necessarily affine? Are all there examples of them rationalthose not birationally equivalent to an affine space?

The examples that come to my mind are like similar to a singular $\mathbb P^1$ without a point given by equation $x^2 = y^3$. This particular curve is also clearly birationally equivalent to affine line.

Perhaps the affine "affine" part would follow follows from a comparison between Zariski cohomology and complex cohomology?
show/hide this revision's text 2 reworded

From a post to The Jouanolou trick:

Are there examples of a all topologically trivial (contractible) complex algebraic varieties other then affine lines necessarily affine? If yes, are Are all of them necessarily affinerational?

This looks

The examples that come to my mind are like some kind of a singular $\mathbb P^1$ without a point given by equation $x^2 = y^3$. This particular curve is also birationally equivalent to affine line.

Perhaps the affine part would follow from a comparison between Zariski cohomology and complex cohomology.?
show/hide this revision's text 1

Topologically contractible algebraic varieties

From a post to The Jouanolou trick:

Are there examples of a topologically trivial (contractible) complex algebraic varieties other then affine lines? If yes, are all of them necessarily affine?

This looks like some kind of comparison between Zariski cohomology and complex cohomology.