Why are the two kinds of algebraic geometry (complex manifolds/schemes) considered the same subject?

Algebraic geometry is, well, algebraic. If the theorem is formulated completely algebraically but you use analytic tools to prove it, you could argue that you're working in algebraic geometry using analytic methods. If you're just studying complex manifolds or whatever for their own sake, then very few people would consider that to be algebraic geometry.
The field of complex geometry uses both algebraic and analytic tools. I think most people approach the topic from one of these sides, at least initially. I am approaching it from the differential side, and although I’ve tried to pick up some algebraic geometry along the way, I am still far more comfortable with studying PDEs than the more algebraic topics.

Because of how mixed the subject is, there will be papers classified under algebraic, differential, or both headings simultaneously on the arxiv.
by
Certainly smooth/PDE methods apply to algebraic manifolds. So the question is when can algebraic methods apply the smooth side.

Chow's Theorem is the clearest demonstration of the equivalence that I know of: All complete complex submanifolds of projective space are algebraic.

Then there are a bunch of theorems in complex geometry which go like "Condition X implies a certain bundle has enough holomorphic sections to embed the manifold in projective space."

Combining these gives many conditions under which a complex manifold ends up being algebraic in the end.
> scheme theory instead covers a spectrum of topics

Nice pun.

> Is there something deep and fundamental that I'm missing? Is it merely a historical quirk?

Yes, it's certainly historical. Algebraic geometry was developed first over the complex numbers, with some people using analysis and others using algebra. The methods of analysis often preceded the methods of algebra.  For example, Riemann surfaces were first studied (by Riemann) using analysis.  Many decades later Dedekind and Weber showed how to develop the subject algebraically so that it would work over an arbitrary algebraically closed field (of characteristic 0), such as the field of algebraic numbers.  The analytically proved theorems over C often told the algebraists what how to formulate results in more generality, but proving such generalizations often required totally new methods relying on no analysis. Compare theorems about elliptic curves or Jacobian varieties (or more generally abelian varieties) over C and over other fields. It's not surprising that things were first done over C, where at least there is a good visual picture, before they were done over more abstract fields.

I think the main complaint you make has nothing to do with algebraic geometry at all: it could be made about any highly developed subject. Two specialists in some area (algebraic geometry, functional analysis, number theory, etc) might have great difficulty talking with each other because their subject has expanded so much that there can be different research trends in the subject with rather little overlap in terms of background needed to understand either the formulation of theorems or the methods of proof.  I am reminded of the first paragraph in the preface of Conway's *A Course in Functional Analysis*:

> Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the
work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear
space, or the linear operators on the space, or both.
by
Note that one of these papers was cross-posted on arXiv between different subject areas. The primary subject area is mathematical physics, with algebraic geometry and differential geometry considered secondary. So all they're saying is that some algebraic geometers may be interested in it.
I think the thing that you are missing is the distinction is not at all clear cut and that there are many instances where the two subjects are interwoven and blend more or less seamlessly, both historically (starting from the amazing fact that compact Riemann surfaces are algebraic and the Lefschetz principle showing that complex algebraic geometry gives insight into algebraic geometry over any field of characteristic 0) and in contemporary research.

A (stereo)typical complex algebraic geometer, with no number-theory/arithmetic geometry sideline,  is interested in understanding complex projective algebraic varieties, their geometry/topology, their birational classification, their algebraic cycles, their moduli, their vector/principal bundles, their enumerative geometry, their link with mathematical physics, symplectic geometry, representation theory, etc. For this, (s)he is willing to use all the tools at their disposal:

\- topology: singular cohomology and fundamental groups mostly, characteristic classes often, but also sometimes topological K-theory, complex cobordism...

\- analysis: basic differential geometry, hermitian metrics, L\^2-cohomology, Hodge theory, potential theory...

\- algebraic analysis: D-modules with regular and irregular singularities, the Riemann-Hilbert correspondence, microlocal sheaf theory...

\- reduction to positive characteristic in general, and finite fields in particular: point counting and the Weil conjectures to compute Betti numbers over C, Mori's proof via Bend-and-Break that Fano varieties are rationally connected...

\- pure Grothendieck style AG: defining moduli functors and studying their representability by schemes/stacks, , even using some derived algebraic geometry to correctly define enumerative counts...

and much much more besides.

Of course any one researcher or any one paper on the subject is not going to showcase all the techniques but there is a profound unity to the subject.
Maybe the real plan by the Algebraic Geometers to take over all of maths is to trick all the other mathematicians into renaming their subjects as Algebraic Geometry.
First, one could make the claim that a lot of math subcategories on the ArXiv are nearly as broad. math.CO (the combinatorics tag) for instance, has tons of papers that seem extremely tangentially related to each other. Math nowadays is so interconnected, after all.

Second, Algebraic Geometry is in some sense "universal", so techniques from AG can be applied to tons of fields (Commutative Algebra, Differential Geometry, Mathematical Physics, etc.), so sometimes when a Differential Geometry paper uses enough Algebraic Geometry machinery, it may just be listed under the math.AG umbrella.

&#x200B;

Finally, there's also historical context to consider. The AG of old was heavily concerned with "how many reducible plane cubics pass through 7 points over C" or something like that. In the 19th century, questions like these were often solved heuristically. In the early 20th century, varieties were conjured up to try and answer these questions in a more general setting. This, however, lead to more heavy machinery being introduced, to the point where a mathematician in 1890 reading a paper in their field written in 1930, solving a question they may have been thinking about,  would be completely and utterly confused with the language being casually thrown around. Then this process repeated itself when Grothendieck/Serre/Deligne came around (and the aforementioned types of questions were solved using the language of schemes/Chow groups/Chern classes in the 1960s through the 1980s). However, all these things are in some sense "Algebraic Geometry", so as a result, a wide array of things are now considered Algebraic Geometry, even though they may (initially) seem like completely disjoint fields of study. It just happens that these seemingly disjoint fields of study originated from the same (now somewhat defunct) field of study, and that defunct field of study and everything that branched from it (the two disjoint fields included) were consistently referred to as Algebraic Geometry.
I'm with you on this one. The non-schematic stuff should just be categorused as "complex geometry" in much the same way that everything to do with rigid analytic varieties and adic spaces (e.g. p-adic Hodge theory, perfectoid spaces, etc.) is not considered to strictly be algebraic geometry but p-adic geometry (or sometimes rigid analytic geometry), even though p-adic geometry is much more algebraic than complex geometry.

0 like 0 dislike