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I'm learning about Lefschetz pencils and vanishing cycles and have looked at a few sources:

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 $$ \begin{matrix} \textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(x^8 + y^8 + z^8 + w^8 +x^2y^2z^2w^2)} \right) \end{matrix} $$

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  • $\begingroup$ I found a paper giving an algorithm computing cohomology of a projective variety using D-modules and alexander duality on the arxiv: arxiv.org/pdf/math/0103013v1.pdf I wish alexander duality was included in my algebraic topology course! $\endgroup$
    – 54321user
    Commented Aug 19, 2016 at 2:37
  • $\begingroup$ Did you mean to write $z^6$, not $z^8$? $\endgroup$
    – François
    Commented Aug 22, 2016 at 21:40
  • $\begingroup$ Yes, let me change that... $\endgroup$
    – 54321user
    Commented Aug 22, 2016 at 22:55

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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:

  1. The cohomology groups of the total space of the pencil.
  2. 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.

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  • $\begingroup$ How can I compute the module of vanishing cycles and is there a "nice way" to understand the generators for this module. I'm not too sure what I mean by this, since there are many ways to understand this statement. Also, how can I understand the intersection form on cohomology? $\endgroup$
    – 54321user
    Commented Aug 23, 2016 at 1:10
  • $\begingroup$ Also, how did you get the number of nodal curves in the pencil? $\endgroup$
    – 54321user
    Commented Aug 23, 2016 at 1:55
  • $\begingroup$ The quickest way to get the number of nodes is to compare the Euler characteristic of the total space with the product of the Euler characteristics of the base and fiber. The difference gives you the number of nodal fibers. I had computed it wrong before, but now it's corrected. $\endgroup$
    – François
    Commented Aug 23, 2016 at 5:27
  • $\begingroup$ The intersection form on $H^2(X,\mathbb Z)$ is the unique even unimodular lattice of signature $(71,231)$. $\endgroup$
    – François
    Commented Aug 23, 2016 at 5:32
  • $\begingroup$ To understand the vanishing cycles, you could choose an affine chart $\mathbb C \subset \mathbb P^1$ containing $p$ and $\Delta$, and then draw real line segments from $p$ to each singular point. Topologically, there will be real cones over each segment tracing out the vanishing. $\endgroup$
    – François
    Commented Aug 23, 2016 at 5:39

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