I have been reading BBD and Geordie Williamson's “An illustrated guide to perverse sheaves”. The latter has been tremendously helpful to get some intuition for the former.

Let $K$ be some field, and for future use we fix a prime number $\ell$ that is invertible in $K$. (I am particularly interested in finitely generated fields. But I think that this does not matter for what follows.)

Let $f \colon X \to D$ be a Lefschetz pencil of relative dimension $n$. With this I mean: $D = \mathbb{P}^1_K$; $f$ is smooth of relative dimension $n$ over a non-empty open $j \colon U \hookrightarrow D$, and over $i \colon S = (D\setminus U) \to D$ the fibres of $f$ have exactly one ordinary double point and no other singularities, and have dimension $n$. $\def\QQl{\mathbb{Q}_\ell}\def\pR{{}^p\!R}$

Such Lefschetz pencils have been studied in detail, and Picard–Lefschetz theory gives a detailed description of the sheaves $R^qf_*\QQl$. However, I now want to understand the perverse sheaves $\pR^qf_*(\QQl[n+1])$.

Q1. Is there a description of these perverse sheaves in the literature?

The perverse sheaves $F^q = \pR^qf_*(\QQl[n+1])$ are complexes $F^q_{-1} \to F^q_0$, since they are perverse sheaves over a curve. Also, $F^q_0$ is supported on $S$.

For $q \ne n, n+1$ the classical sheaf $R^qf_*\QQl$ is constant. I have the strong feeling that this means that for $q \ne -1,0,1$ the perverse sheaf $F^q$ is isomorphic to $(R^{n+q}f_*\QQl)[1]$. In other words $F^q_{-1}$ is the constant sheaf $(R^{n+q}f_*\QQl)$ and $F^q_0 = 0$.

For $q = -1,0,1$ my guesses are:

  • $F^q$ is also constant for $q = \pm1$. We have $F^{\pm1}_0 = 0$. The sheaf $F^{-1}_{-1}$ is $R^{n-1}f_*\QQl$, whereas $F^1_{-1}$ is $j_*j^*R^{n+1}f_*\QQl$.
  • For $q = 0$ the description depends on whether the vanishing cycles are trivial or not.
    • If the vanishing cycles are non-trivial, then $F^0_0 = 0$ and $F^0_{-1}$ is $R^nf_*\QQl$. In this case $L = j^*R^nf_*\QQl$ is a local system with non-trivial monodromy, and $F^0 = R^nf_*\QQl = j_{!*}L[1]$.
    • If the vanishing cycles are trivial, then $n$ is odd and $F^0_0 = \bigoplus_{s \in S} \QQl(\tfrac{n+1}{2})_s$. The sheaf $F^0_{-1}$ is isomorphic to $R^{n}f_*\QQl$ which in this case is constant.

So far this picture is only based on my (probably misleading) intuition and on what I know about the classical picture.

Q2. How much of my guess is correct? If it is wrong, how should I fix it? And what is the proper way to prove the correct picture?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.