What tools can I use to compute the cohomology of the fibers of a Lefschetz Pencil? I'm learning about Lefschetz pencils and vanishing cycles and have looked at a few sources:


*

*http://www.math.purdue.edu/~dvb/preprints/sheaves.pdf

*http://www3.nd.edu/~lnicolae/Morse2nd.pdf

*Voisin's Complex Algebraic Geometry and Hodge Theory II


I want to be able to start computing explicit examples of vanishing cycles and monodromy, but I need to first learn how to compute the singular homology of complex varieties. What are some tools I can use to compute cohomology of some smooth projective varieties over $\mathbb{C}$? For example, how can I study the cohomology of some Lefschetz pencil for the projective scheme
$$
\begin{matrix}
\textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(x^8 + y^8 + z^8 + w^8 +x^2y^2z^2w^2)} \right)
\end{matrix}
$$
 A: Your example is a smooth hypersurface, which is much easier to understand than a general smooth projective variety.  You ask for a computation of the cohomology of a Lefschetz pencil.  This could mean either:


*

*The cohomology groups of the total space of the pencil.

*A description of the Gauss-Manin local system on $\mathbb P^1-\Delta$ (e.g. in terms of monodromy action of the generators for $\pi_1$). 


The Hodge numbers of a hypersurface $X\subset \mathbb P^n$ can be computed using Lefschetz hyperplane away from the middle row, and then Hirzebruch's generating function for the primitive middle Hodge numbers (see http://www.math.purdue.edu/~dvb/preprints/book-chap17.pdf for details).  In your example, the Hodge diamond has middle row $(35,232,35)$.  The total space of a Lefschetz pencil is the blow up of $X$ at the base locus of the pencil (8 points in your example), so it has $b_2 = 310$.
The local system description is more difficult.  Let $C_p$ be a smooth octic curve over a general point $p\in \mathbb P^1$.  The local system has rank $h^1(C_p)=2g(C_p)= 42$.  There are $392$ nodal curves in the pencil.  Since $\pi_1(\mathbb P^1 - 392,p)$ is free on 391 generators, you need to find the monodromy for each generator.  By the Picard-Lefschetz formula, the monodromy around a given nodal fiber is given by
$$\tau(x) = x- \langle x,e \rangle e$$
where $e\in H^1(C_p,\mathbb Z)$ is the vanishing cycle, represented by an embedded $S^1$ which gets contracted to the node.
