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From a post to The Jouanolou trick:

Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine space?

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

Perhaps the "affine" part follows from a comparison between Zariski cohomology and complex cohomology?

From a post to The Jouanolou trick:

Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine space?

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

Perhaps the "affine" part follows from a comparison between Zariski cohomology and complex cohomology?

From a post to The Jouanolou trick:

Are all topologically trivial (contractible) complex algebraic varieties other then affine lines necessarily affine? Are all of them rational?

The examples that come to my mind are like 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?

From a post to The Jouanolou trick:

Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine space?

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

Perhaps the "affine" part follows from a comparison between Zariski cohomology and complex cohomology?

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.

From a post to The Jouanolou trick:

Are all topologically trivial (contractible) complex algebraic varieties other then affine lines necessarily affine? Are all of them rational?

The examples that come to my mind are like 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?