# Complex varieties as real affine varieties: how to recover complex structure?

If $$A$$ is a finitely generated $$\mathbb C$$-algebra without nilpotents, then $$A = \mathbb C[V]$$ is the $$\mathbb C$$-algebra of polynomial functions on $$V := \mathrm{maxSpec}(A)$$ (this is precisely the content of Hilbert's Nullstellensatz). We can define an associated $$\mathbb R$$-algebra, $$\mathbb R[V]$$, to be generated by the real parts of the the complex polynomial functions on $$V$$. This association induces a faithful forgetful functor $$\left\{\text{finitely generated nilpotent-free } \mathbb C\text{-algs}\right\} \to \left\{\mathbb R\text{-algs}\right\},$$ (where the functoriality comes from the fact that the $$\mathbb R$$-algebra is an algebra of functions on a set) or in schemes, $$\left\{\text{finite type reduced affine } \mathbb C\text{-schemes}\right\} \to \{\mathbb R\text{-schemes}\},$$ where closed points on the LHS are in one-to-one correspondence with real points on the RHS.

The questions that arise here are endless, and I'm almost certainly reinventing the wheel here, so a few informal questions: to what extent can the domain of this functor be extended? Does this lead anywhere interesting? I'm pretty sure this functor respects gluing at least, so it should extend to non-affine varieties; does, for example, $$\mathbb CP^n$$ become an affine real variety in this picture? (Note that the closed points of $$\mathbb CP^n$$ are in 1-1 correspondence with the real points of the real variety consisting of unitary projection matrices of trace/rank 1.)

For the purpose of this posting, a slightly imprecise question which I suspect has a very concrete answer that I just don't know: what structure on an $$\mathbb R$$-algebra or $$\mathbb R$$-scheme allows us to go in the reverse direction? I suspect the answer has to do with a $$\mathbb C$$-action on the tangent bundle, and I know the $$(\mathfrak m/\mathfrak m^2)^*$$ chracterization of the tangent space at a point, but I don't know how to characterize when a $$\mathbb C$$-action on tangent spaces is "regular", in the sense of being induced by polynomial functions somehow.

• The analogous question in differential geometry (when does a smooth manifold [of necessarily even dimension] admit a complex structure?) is well understood. There is a notion of "almost complex structure" which is an action $J$ on tangent spaces such that $J^2 = J \circ J = -Id$, i.e. $J$ is some kind of globalization of the imaginary unit $i$. An almost complex manifold is complex when $J$ can be "integrated" to a complex structure (complex manifolds have a natural almost complex structure, of course).... May 31, 2020 at 4:17
• ...It is a theorem (Newlander-Nirenberg) that integrability is equivalent to vanishing of a certain tensor field associated to $J$ called the Nijenhuis tensor, so one could hope for an algebraic analogue of this story. The Nijenhuis tensor is a section of $T_M \otimes (T^*_M)^{\otimes 2}$ (where we are considering the ordinary smooth tangent and cotangent bundles), so perhaps the analogous sections of the analogous algebraic bundles might tell you something. I suppose my use of the words "algebraic bundle" betrays that I've been assuming that these varieties are smooth; I have no idea... May 31, 2020 at 4:17
• ... if a similar construction on singular varieties could ever be expected to work, but perhaps one could reduce to the smooth case by invoking resolution of singularities. May 31, 2020 at 4:17
• @TabesBridges Your comment correctly suggests that the answer is actually very complicated. :) May 31, 2020 at 18:16

Congratulations, you've rediscovered Weil restriction! Here it is in the most general form I know:

Let $$S'\to S$$ be a morphism of schemes. Given any $$S'$$-scheme $$X'$$, we can consider the contravariant functor $$R_{S'/S}(X'):(\text{Sch}/S)^{op}\to \text{Set}$$ given by $$T\mapsto X'(T\times_S S').$$ If this functor is representable by an $$S$$-scheme $$X$$, then we say that $$X$$ is the Weil restriction of $$X'$$ along $$S'\to S$$, and we write $$X=R_{S'/S}(X')$$.

This is rather broad! Let us try and get a little better handle on it in the situation we care about.

Let $$S'\to S$$ be a finite locally free morphism. Let $$X'$$ be an $$S'$$-scheme so that for any $$s\in S$$ and any finite set $$P\subset X'\times_S\operatorname{Spec}\kappa(s)$$, there exists an affine open subscheme $$U'\subset X'$$ containing $$P$$. Then the functor $$R_{S'/S}(X')$$ is representable by an $$S$$-scheme. (For proof, see Neron Models by Bosch, Lutkebohmert, and Raynaud, section 7.6. This is actually a really good reference to have for everything I'm talking about in this post.)

In particular, this means that if $$X'$$ is quasiprojective over $$S'$$ (and $$S'\to S$$ is finite locally free) then the Weil restriction exists. Now let's get even more specific: if $$K\subset L$$ is a finite extension of fields of degree $$d$$ so that $$L/K$$ has basis $$e_1,\cdots,e_d$$ and $$X'$$ is affine over $$L$$, say $$\operatorname{Spec} L[x_1,\cdots,x_n]/(f_1,\cdots,f_r)$$, then we can write the Weil restriction as $$\operatorname{Spec} k[y_{ij}]/(g_{st})$$ where we take $$1\leq i\leq n$$, $$1\leq j\leq d$$, $$1\leq s\leq r$$, $$1\leq t\leq d$$, and set $$x_i=\sum e_jy_{ij}$$ as well as $$f_s=\sum e_tg_{st}$$. This exactly recovers what you've written down in terms of real parts.

• Is it interesting? I'd say yes! Among other places, it gets used a fair bit when dealing with abelian varieties and algebraic groups. (Don't ask me for details, because I don't know!) Trying to verify certain properties can get pretty hairy, which means it's not trivial! For instance, if we have a Zariski cover of $$X'$$, then the Weil restrictions of this cover don't necessarily cover $$X$$ even in the case when $$S'\to S$$ is a finite separable extension of fields, and lots of other things like this can go wrong!
• Does $$\Bbb CP^n$$ become a real affine variety under this? No, though there are more tricks under the sun in real algebraic geometry than just Weil restriction.