Examples of the Gysin Maps for Hodge Cohomology

Question

I'm trying to learn about Gysin maps on Hodge cohomology, as defined in the Stacks project (https://stacks.math.columbia.edu/tag/0G8A). Namely, if $X \to S$ is a morphism of schemes and $Z \to X$ is a closed immersion of finite presentation whose conormal sheaf is locally free of rank $c$ satisfying additional regularity properties (e.g. locally being cut out by a Koszul regular sequence), then we get maps $$ \gamma^{p, q}: H^q(Z, \Omega^p_{Z/S}) \to H^{q + c}(X, \Omega^{p+c}_{X/S})$$ which are useful in defining cycle classes in Hodge cohomology.

1. The definition of $\gamma^{p,q}$ given in the link above is somewhat opaque to me; is there a more "down-to-earth" way to understand the construction of the maps $\gamma^{p, q}$ (perhaps with simplifying assumptions on $Z$ and $X$)?

2. Are there computable explicit examples of the $\gamma^{p,q}$ that help illustrate what is really going on here?

Answer 1

Here is one case where it's easy to understand: Suppose that $X\to S$ is smooth and proper, and $Z\subset X$ is a smooth divisor, so $c=1$. Then Gysin is just the connecting map associated to the Poincaré residue sequence $$0\to \Omega_{X/S}^{p+1}\to \Omega_{X/S}^{p+1}(\log Z)\to \Omega_{Z/S}^p\to 0$$ Of course, you can iterate this if $Z$ is a global complete intersection. But I'm not sure if it can be made so concrete in a completely general situation.