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I am studying the paper of Dimca and Papadima Hypersurface complements, Milnor fibers and higer homotopy groups of arrangements and try do use the their techniques in order to study some topological properties of the complement manifold of toric arrangements.

However, I have some basic questions in complex algebraic geometry. In particular, I would like to better understand the relationship between the complement manifold of a hypersurface in the projective space and the complement manifold of its affine part. To be more specific, here is my question.

Question Let $F\in\mathbb{C}[x_{0},\ldots,x_{n}]$ be a homogeneous polynomial and let $M(F)$ be the complement manifold of the hypersurface $\{F=0\}$ in the complex projective space $\mathbb{CP}^{n}$. Let $M(F)_{a}$ be the complement manifold in $\mathbb{C}^{n}$ of the affine part of $F$. How the topology of $M(F)$ and $M(F)_{a}$ are related?

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  • $\begingroup$ You might want to flag @AlexSuciu. $\endgroup$
    – Mikhail Katz
    Commented Mar 21, 2017 at 12:30
  • $\begingroup$ Trivial remark: if $D_+(x_i)$ is the affine part $\mathbb C^n$ corresponding to $x_i \neq 0$, then $D_+(F) \cap D_+(x_i) = D_+(F \cdot x_i)$. Thus, it is given by the complement of the (reducible) hypersurface $V(F \cdot x_i)$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 29, 2017 at 16:54

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I would suggest looking at how $\pi_{1}(\mathbb{C}\mathbb{P}^{2}\setminus C)$ depends on the geometry of the curve $C$. If the curve is smooth then the fundamental group is a cyclic group (depending on the degree of the curve). If we allow singularities then it is much more complicated and infinite groups appear.

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