# Historical proof of Leschetz Hyperplane Theorem

I browse in Phillip Griffiths' Slides on historical development of Hodge-theory and these include a sketch of the original approach with Lefschetz used to study complex surfaces in his famous hypersurface theorem.

The part where I need some clarifications in presented on pages 22-24.

The idea was to study embedded complex surfaces $$S = \{f(x,y,z)=0 \} \subset \mathbb{P}^3_{\mathbb{C}}$$

($$y \in \mathbb{C}$$ or $$\mathbb{P}^1$$ if we compactify $$S$$) by inductive methods on their dimension and so consideringing the fibers

$$C_y = \{f(x,y,z)=0, y = \text{ constant } \}$$

of the projection map to first coordinate which in a modern framework corresponds to map $$F:\overline{S} \to \mathbb{P}^1$$ which is given by Lefschetz fibration which arise as blow up along the base locus of the pencil. The next argument on slide 24 (of see below for the picture) I not understand.

Historically Lefschetz' idea was to introduce slits like below in the base space $$\mathbb{P}^1$$ where $$y_1,..., y_N$$ correspond to points having singular fiber $$C_{y_i}$$ and $$C_{y_0}$$ is a fixed smooth reference curve.

The next step is confusing. It says that since the complement of the above configuration $$R:= \bigcup_{i=0}^N \overline{y_0 y_i}$$ is contractible in sphere $$\mathbb{P}^1 \cong S^2$$, so the whole surface retracts onto the part lying over the slits in the complex plane. Here the important slide:

Why? The last sentence on the right in the image doesn't make any sense. The existence of such retraction fails already in the base space, since there is no retraction of whole $$S^2$$ onto $$R$$, so so why should exist a 'lifted' retraction of $$S$$ into $$F^{-1}(R)$$.

Maybe I misunderstand to idea. Does anybody see what is here meant? I think that this slides are going to elaborate the gaps of the unclear sketch of Lefschetz original proof from wiki , but this retraction statement confuses me.

• There is a nice exposition of Lefschetz's proof from a more modern point of view by Klaus Lamotke (Topology 20 (1981) 15-41.) I'm not sure if it directly addresses your questions but it is rather detailed and so perhaps it would be helpful. Apr 28 at 19:38
• Thank you for the reference. The strategy in §5 (34) of this paper is quite similar. Essentially one important step there exploits that if we remove the point $\infty$ at infinity from $\mathbb{P}^1 \cong S^2$ then of couse $\mathbb{P}^1 - \infty$ retracts to $R:= \bigcup_{i=0}^N \overline{y_0 y_i}$ and the proof of this paper shows that the restricted lifts admit also such retraction. Maybe implicitely Griffiths assumed in the image above the infinity point also to be already removed, otherwise the second sentence below the graph in the image doesn't make any sense. Apr 28 at 20:26