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user17778
user17778

Let $k$ be an algebraically closed field and $G_m$ an algebraic one-dimensional torus. Let $X$ be a smooth irreducible variety over $k$, $Y$ an affine scheme of finite type over $k$ and $f\colon X\rightarrow Y$ a proper morphism.

Assume that there exist actions of $G_m$ on $X$ and $Y$ such that $f$ is $G_m$-equivariant. Moreover, the fixed point locus $Y^{G_m}$ of $Y$ consists of only one point $y.$

Edit (after Allen's answer): Moreover, we assume that for any point $x\in X$ there exists $\lim_{t\rightarrow 0} t\cdot x.$

Question: Are $X$ and $f^{-1}(y)$ homotopy equivalent?

Let $k$ be an algebraically closed field and $G_m$ an algebraic one-dimensional torus. Let $X$ be a smooth irreducible variety over $k$, $Y$ an affine scheme of finite type over $k$ and $f\colon X\rightarrow Y$ a proper morphism.

Assume that there exist actions of $G_m$ on $X$ and $Y$ such that $f$ is $G_m$-equivariant. Moreover, the fixed point locus $Y^{G_m}$ of $Y$ consists of only one point $y.$

Question: Are $X$ and $f^{-1}(y)$ homotopy equivalent?

Let $k$ be an algebraically closed field and $G_m$ an algebraic one-dimensional torus. Let $X$ be a smooth irreducible variety over $k$, $Y$ an affine scheme of finite type over $k$ and $f\colon X\rightarrow Y$ a proper morphism.

Assume that there exist actions of $G_m$ on $X$ and $Y$ such that $f$ is $G_m$-equivariant. Moreover, the fixed point locus $Y^{G_m}$ of $Y$ consists of only one point $y.$

Edit (after Allen's answer): Moreover, we assume that for any point $x\in X$ there exists $\lim_{t\rightarrow 0} t\cdot x.$

Question: Are $X$ and $f^{-1}(y)$ homotopy equivalent?

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user17778
user17778

Homotopically Homotopy equivalent spaces and $G_m$-actions

Let $k$ be an algebraically closed field and $G_m$ an algebraic one-dimensional torus. Let $X$ be a smooth irreducible variety over $k$, $Y$ an affine scheme of finite type over $k$ and $f\colon X\rightarrow Y$ a proper morphism.

Assume that there exist actions of $G_m$ on $X$ and $Y$ such that $f$ is $G_m$-equivariant. Moreover, the fixed point locus $Y^{G_m}$ of $Y$ consists of only one point $y.$

Question: Are $X$ and $f^{-1}(y)$ homotopicallyhomotopy equivalent?

Homotopically equivalent spaces and $G_m$-actions

Let $k$ be an algebraically closed field and $G_m$ an algebraic one-dimensional torus. Let $X$ be a smooth variety over $k$, $Y$ an affine scheme of finite type over $k$ and $f\colon X\rightarrow Y$ a proper morphism.

Assume that there exist actions of $G_m$ on $X$ and $Y$ such that $f$ is $G_m$-equivariant. Moreover, the fixed point locus $Y^{G_m}$ of $Y$ consists of only one point $y.$

Question: Are $X$ and $f^{-1}(y)$ homotopically equivalent?

Homotopy equivalent spaces and $G_m$-actions

Let $k$ be an algebraically closed field and $G_m$ an algebraic one-dimensional torus. Let $X$ be a smooth irreducible variety over $k$, $Y$ an affine scheme of finite type over $k$ and $f\colon X\rightarrow Y$ a proper morphism.

Assume that there exist actions of $G_m$ on $X$ and $Y$ such that $f$ is $G_m$-equivariant. Moreover, the fixed point locus $Y^{G_m}$ of $Y$ consists of only one point $y.$

Question: Are $X$ and $f^{-1}(y)$ homotopy equivalent?

Source Link
user17778
user17778

Homotopically equivalent spaces and $G_m$-actions

Let $k$ be an algebraically closed field and $G_m$ an algebraic one-dimensional torus. Let $X$ be a smooth variety over $k$, $Y$ an affine scheme of finite type over $k$ and $f\colon X\rightarrow Y$ a proper morphism.

Assume that there exist actions of $G_m$ on $X$ and $Y$ such that $f$ is $G_m$-equivariant. Moreover, the fixed point locus $Y^{G_m}$ of $Y$ consists of only one point $y.$

Question: Are $X$ and $f^{-1}(y)$ homotopically equivalent?