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I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality.

At this point it has been proven that a non singular affine algebraic variety of complex dimension $k$ has the homotopy type of a CW-complex of dimension less than or equal to $k$. We wish to prove the following:

Let $V$ be an algebraic variety of complex dimension $k$ lying in complex projective space $\mathbb{C} P^n$. Let $P$ be a hyperplane of $\mathbb{C} P^n$ containing all singularities of of $V$. Then the inclusion map $V\cap P \to V$ induces isomorphisms in singular homology in dimensions strictly less than $k-1$. In dimension $k-1$ the induced homomorphism is onto.

The proof goes like this:

Consider the long exact sequence of the pair $(V, V\cap P)$: $$\dots \to H_i (V\cap P) \to H_i (V) \to H_i (V, V \cap P) \to H_{i-1} (V) \to \dots$$ It is sufficient to show that $H_i (V, V \cap P) = 0$ for $i \leq k-1$. By Lefschetz Duality $$H_i(V, V\cap P; \mathbb{Z}) \cong H^{2k-i}(V - (V\cap P);\mathbb{Z})$$ But $V - (V\cap P)$ is a Zariski closed set in the affine space $\mathbb{C} P ^n - P$ and is nonsingular since $P$ contains the singularities of $V$. Hence $V - (V\cap P)$ is a nonsingular affine algebraic variety and so applying our theorem we find $H^{2k-i}(V - (V\cap P);\mathbb{Z})=0$ for $i \leq k-1$.

My confusion is over the application of Lefschetz duality. I was familiar with Poincare duality but not Lefschetz duality so I had to look it up. The statement seems simple enough but I don't understand how what Milnor is saying follows from Lefschetz duality.

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First note that complex manifolds are always orientable. Therefore the Poincaré-Alexander-Lefschetz Duality theorem as stated in "Topology and Geometry" by Glen E. Bredon (Chapter VI, Theorem 8.3) says in this case:

If $M$ is a complex manifold of real dimension $2k$ and $L \subset K$ are compact subsets of $M$, then there is an isomorphism $$H^{2k-r}(K, L; \mathbb{Z}) \cong H_r(M \setminus L, M \setminus K; \mathbb{Z}).$$

On page 42 in Milnor's book (in the proof of Theorem 7.4) the author claims that there exists an open neighbourhood $U \subset V$ of $V \cap P$ which has the same homotopy type as $V \cap P$. Since $V \subset \mathbb{C}P^n$ is compact, the set $V \setminus U$ is a compact subset of $V$. We can therefore apply the duality theorem as stated above with $$M := V, \hspace{2mm} K:= V \setminus U, \hspace{2mm} L:= \emptyset$$ to obtain $$H_r(V, V \cap P) \cong H_r(V, U) \cong H^{2k-r}(V \setminus U) \cong H^{2k-r}(V \setminus (V \cap P))$$ as desired.

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  • $\begingroup$ Thank you for your response. I gave you +1. However, I am concerned about letting $M=V$ because $V$ may have singularities. $V-P$ is certainly a complex manifold but $V$ may not be. Is there a way to fix this? $\endgroup$
    – Seth
    Aug 5 '14 at 13:38
  • $\begingroup$ Also I am not sure why we introduced $U$ as it seems unnecessary. $\endgroup$
    – Seth
    Aug 5 '14 at 13:40

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