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Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and $Y=g^{-1}(0)$, then is true that $\pi_1(\mathbb{C}P^{n-1}\setminus X^*)$ is isomorphic the $\pi_1(\mathbb{C}P^{n-1}\setminus Y^*)$? (here $\pi_1$ denotes the first fundamental group and $A^*$ is the image of $A$ by canonical projection $p:\mathbb{C}^n\setminus\{0\}\to \mathbb{C}P^{n-1}$).

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  • $\begingroup$ Do you worry that your homeomorphism may not take a generator of $\pi_1(\Bbb C^*)$ to a similar class? $\endgroup$ Mar 10, 2015 at 19:57
  • $\begingroup$ @AlexDegtyarev, sorry, I did not understand, but $h$ can not be homogeneous complex. $\endgroup$
    – userX10
    Mar 10, 2015 at 20:59
  • $\begingroup$ I suggest you have a look at Dimca's book "Singularities and the Topology of Hypersurfaces", which has a chapter on fundamental groups of hypersurface complements. $\endgroup$
    – user69091
    May 8, 2015 at 7:50
  • $\begingroup$ Am I missing something? Doesn't $h$ induce a homeomorphism $\mathbb{CP}^{n-1} \to \mathbb{CP}^{n-1}$ that sends $X^*$ to $Y^*$, so induce a homeomorphism on the complements? $\endgroup$ May 10, 2015 at 13:57
  • $\begingroup$ @LennartMeier, how $h$ induce a homeomorphism $\mathbb{CP}^{n-1} \to \mathbb{CP}^{n-1}$ that sends $X^*$ to $Y^*$? because $h$ and $h^{-1}$ are only continous. $\endgroup$
    – userX10
    May 10, 2015 at 23:48

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