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

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?

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.

• Thank you @Denis Nardin!... Oct 12 '21 at 22:26