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From this Temkin's paper (at the end of section 1.1.3), I know that one may define Berkovich spaces that include both archimedean and non-archimedean worlds. This looks very interesting.

Temkin mentions an example: the affine line over $\mathbb Z$. But I was wondering if there are more examples to include both worlds(especially in higher dimensions).

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The definition of analytic space over $\mathbf{Z}$ was given by Berkovich in his foundational book "Spectral theory and analytic geometry over non-Archimedean fields" (see the beginning of section 1.4 and section 1.5). It is quite general, and anyway enough so that analytifications of schemes locally of finite type over $\mathbf{Z}$ make sense. (Let me mention that similar constructions exists over $\mathbb{Z}[1/N]$ or ring of integers of arbitrary number fields.)

The construction is actually quite simple. As a set, the affine analytic space of dimension $n$ over $\mathbf{Z}$ is just the set of multiplicative seminorms on $\mathbf{Z}[T_1,\dotsc,T_n]$. Of course, it also comes with a topology (which makes it a Hausdorff locally compact space) and a sheaf of functions. Once you have the affine space, you can take open subsets, closed analytic subsets, etc. and glue to get arbitrary spaces.

To understand better what is going on, let me first describe the space of dimension 0, usually denoted by $\mathcal{M}(\mathbf{Z})$. As I said before, it is the set of multiplicative seminars on $\mathbf{Z}$, so Ostrowski's theorem allows us to describe it explicitly: it contains the trivial absolute value (sending 0 to 0 and everything else to 1), the usual absolute value possibly raised to some power between 0 and 1 (the power 0 giving back the trivial absolute value) and for each prime number $p$, the $p$-adic absolute value raised to some power between 0 and $+\infty$ (the power 0 giving back the trivial absolute value, the power $+\infty$ giving the seminorm induced by the trivial absolue value on $\mathbf{Z}/p\mathbf{Z}$). I will not try to draw a picture here, but it looks like a spider with the trivial absolute value in the middle and a leg for each place.

Now, if you want to describe what the affine analytic space of dimension $n$ over $\mathbf{Z}$, you can look at its projection onto $\mathcal{M}(\mathbf{Z})$ and describe the fibers. As you could guess, the fiber over a $p$-adic absolute value (power not 0 nor $+\infty$) will be a honest Berkovich affine analytic space of dimension $n$ over $\mathbf{Q}_p$. (If the power is 0, i.e. the absolute valued is trivial, you get a space over $\mathbf{Q}$ trivially valued and if the power is $+\infty$, you get a space over $\mathbf{F}_p$ trivially valued.) Interestingly enough, over the usual absolute value (or a non-zero power of it), you get something quite close to $\mathbf{C}^n$, namely its quotient by the complex conjugation (with diagonal action). Moreover, the topology and the structure sheaf on this fiber are the usual ones.

These spaces have not been studied so much but some basic properties are known. In the references mentioned in Xarles' answer, I proved that they have nice local properties (first reference for the affine line, second for arbitrary spaces): the stalks of the structure sheaf are noetherian local rings (even excellent), the structure sheaf is coherent, etc. In his thesis, Thibaud Lemanissier proved that they are locally path-connected. In on-going joint work, we prove some cohomological properties (vanishing of coherent cohomology on disks, annuli, etc., GAGA-like theorems).

Since you asked about examples, let me add one to finish. It illustrates another nice feature of Berkovich spaces over $\mathbf{Z}$: they provide nice parameter spaces. Let me first consider the relative open punctured unit disk $\mathbf{D}^*$ over $\mathbf{Z}$, i.e. the subset of the affine line (with coordinate say $q$) defined by $\{0 < |q| < 1\}$. You can now take a relative $\mathbf{G}_m$ (i.e. the affine line minus 0) over $\mathbf{D}^*$. So over each point $x$ of $\mathbf{D}^*$, you have an analytic $\mathbf{G}_m$ that is defined over the residue field $\mathcal{H}(x)$ of the point (as for schemes, each point has an associated residue field that is a complete valued field).

Edit about $\mathbf{G}_m$: Here I really mean the Berkovich analytic line minus 0. Still it behaves as you would expect from scheme theory: the $\mathcal{H}(x)$-rational points of $\mathbf{G}_m$ over $\mathcal{H}(x)$ are exactly $\mathcal{H}(x)^\times$ (and similarly $\mathbf{G}_m(K) = K^\times$ for any extension of $\mathcal{H}(x)$).

But you also have an element $q(x)$ in $\mathcal{H}(x)$ with absolute value between 0 and 1. So you can construct the quotient of the multiplication by $q(x)$ and end up with a Tate curve over $\mathcal{H}(x)$.

Edit: This is nothing but the Berkovich analogue of $\mathcal{H}(x)^\times/q(x)^\mathbf{Z}$. (Actually $\mathcal{H}(x)^\times/q(x)^\mathbf{Z}$ is exactly the set of $\mathcal{H}(x)$-rational points, but you there are points over arbitrary extensions too.)

Note that this works for all fields $\mathcal{H}(x)$, which could be $\mathbf{Q}_p$, $\mathbf{F}_p((t))$, $\mathbf{C}$, etc. The nice point is that this construction can be carried out globally (as opposed to fiber by fiber) in exactly the same way, giving rise to some sort of universal Tate curve (defined as an analytic space over $\mathbf{Z}$) with an analytic global uniformization map. (In joint work with Daniele Turchetti, we do similar constructions for Mumford curves of higher genus.)

This answer is already long enough I believe, so I will stop here, but if you have any questions, please ask. And sorry for the shameless self-promotion.

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  • $\begingroup$ This is really a great answer. I particularly love the last example. I have a few questions: (1) what if we replace $\mathbb Z$ by something else, say $\mathbb C[[t]]$ or $\mathbb C((t))$? I guess we may have similar things, right? For example, what should the possible $\mathcal H(x)$ in this case? (2) By $G_m$ do you simply mean $\mathbb Z^\times$, or $k^\times$ for some $k$? $\endgroup$ – Hang Aug 14 '18 at 14:47
  • $\begingroup$ In general, Berkovich spaces are defined over complete normed rings. If $A$ is such a ring, $\mathcal{M}(A)$ is the set of mult. seminorms bounded by the given norm. So you can define Berkovich spaces over $\mathbb{C}[[t]]$ or $\mathbb{C}((t))$ endowed with some $t$-adic norm. (For $\mathbb{Z}$, one can endow it with the usual abs. value and then all mult. seminorms are bounded by it.) But beware that you will not find usual complex spaces in this case since the usual absolute on $\mathbb{C}$ is not bounded by the trivial absolute value on $\mathbb{C}$ (induced by the $t$-adic absolute value). $\endgroup$ – Jérôme Poineau Aug 15 '18 at 7:35
  • $\begingroup$ Something else you can do it consider $\mathbb{C}$ endowed with the norm that is the maximum of the trivial absolute value and the usual absolute value. This way, you get usual complex spaces that "degenerate" onto a non-archimedean fiber (over $\mathbb{C}$ with trivial valuation). This has been used by Berkovich to show that (under some conditions) the weight 0 part of the cohomology of a family of complex spaces can be computed by the singular cohomology of a suitable Berkovich space. $\endgroup$ – Jérôme Poineau Aug 15 '18 at 7:41
  • $\begingroup$ I added a few words about $\mathbf{G}_m$ in the answer. Hope it helps. $\endgroup$ – Jérôme Poineau Aug 15 '18 at 7:51
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You should take a look at the paper by Jérôme Poineau:

La droite de Berkovich sur Z, Astérisque n° 334 (2010)

and other papers by the same autor, like

Espaces de Berkovich sur Z : étude locale, Inventiones mathematicae 194 (2013), n° 3, 535–590

that you can find in his web page.

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