Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?

Question

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:

On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a non singular analytic space and purely of dimension $n+1$. Let $D=\{z: |z|< 1 \}$ the unit disc and $D^*=D-\{0\}$, and let $f: X\rightarrow D$ a morphism of analytic spaces such that:

• $f$ is proper

• $f$ is smooth outside of a point $x$ of the special fiber $X_0=f^{-1}(0)$.

• In $x$, $f$ has a non-degenerate quadratic point.

• Let $t\neq 0$ in $D$ and $X_t=f^{-1}(t)$ "the" general fiber.

With the above data he associate some results for the cohomology groups of the fibers.

Then, in section 4.2. He says: There is an analog of (4.1) in abstract algebraic geometry. The disk $D$ is replaced by the spectrum of a henselian discrete valuation ring $A$ with an algebraically closed residue field. Let $S$ be the spectrum, $\eta$ its generic point (spectrum of the field of fractions of $A$), $s$ the closed point (spectrum of the residue field). The role of $t$ is played by the geometric generic point $\overline{\eta}$ (spectrum of the closure of the field of fractions of $A$).

I am wondering why he replaces $t$ by the geometric generic point $\overline{\eta}$? In any sense, there is a one to one correspondence between them?

Answer 1

If you want a canonical description of the homotopy type of the general fiber, you can take the homotopy type of the space $X\times_{\mathbb{D}}\mathbb{H}$ where $\mathbb{H}:=\{z\in\mathbb{C}\mid \Im z>0\}$ is the upper half complex plane, and the map $\mathbb{H}\to\mathbb{D}$ is the exponential map $z\mapsto e^{-z}$. This is because, as a complex manifold, $\mathbb{H}$ is the universal cover of the punctured disc $\mathbb{D}^\times=\mathbb{D}\smallsetminus\{0\}$. Since $\mathbb{H}$ is contractible and $X\smallsetminus X_0\to\mathbb{D}^\times$ is a fibration, then the fiber product $X\times_\mathbb{D}\mathbb{H}$ deformation retracts on $X\times_{\mathbb{D}}\{t\}$ for every $t\neq 0$. The monodromy action is then given by the map $\mathbb{H}\to \mathbb{H}$ sending $z$ to $z-2\pi i$, which is a map over $\mathbb{D}$ (this is exactly the deck transformation corresponding to the standard generator of $\pi_1\mathbb{D}^\times$).

In the algebraic case you can think of $\eta=\operatorname{Spec}A\smallsetminus\{s\}$ as the algebraic analogue of $\mathbb{D}^\times$, and the geometric point $\bar\eta$ as the algebraic analog of the universal cover $\mathbb{H}$. Then the Galois group of $\bar \eta/\eta$ is the one inducing the monodromy action.