Sanskar is a BSMS graduate from IISER Tirupati, currently doing his PhD in Leiden, Netherlands, in some intersection of algebraic/complex geometry and number theory.
talking about o-minimality
Quote
You can prove quasi-projectivity of analytically defined things. Like period images.
Deligne-Mumford compactification of
to show quasi-projectivity was classical difficult result, and also for [you have the] Baily–Borel compactification. This Tsimmermann Bakket Brunebarbe gang did all this o-minimal business and, in particular, proved such things for much more general class of objects, i.e., images of period mappings.
Which you could think of as the map
, for example, which sends a curve to its Jacobian, or equivalently, its (polarised) Hodge structure. This case you have Torelli, so if you prove quasi-projectivity for the image, then you also have it for Mg.
But then this stuff you can generalise a lot, using Hodge theory. That still works fine, but generalising Jacobians and all is difficult business.
So you define period domains and period mappings for general things using Hodge theory. And it’s analytic construction, so quasi-projectivity is far from clear. But o-minimal stuff comes to rescue.
The ideas are apparently initially from model theory, but it’s readable stuff even without knowing any model theory. Atleast the part about what is definable, etc. Some results:
https://arxiv.org/abs/1811.12230
https://arxiv.org/abs/2508.19215v2
Some analytic space being projective or quasi-projective adds a lot of algebraic restrictions. And sometimes one can show that the algebraic restrictions also imply the projectivity, etc. In any case, being algebraically defined gives you more control over the situation, while these analytic spaces can be however wild a priori. These periods mappings are defined using Hodge theory.
One silly lil example of this phenomenon: if you have a compact complex manifold which comes from a projective variety, then you know that it is Kähler, plus its function field will have transcendence degree equal to its dimension, coz the algebraic functions will be analytic too.
Now Siegel tells you that the function field of a (compact?) complex manifold has tr deg at most n. If it’s strictly less than n or not Kähler, it can’t be algebraic.
But what’s surprising is that the converse is also true. So if your cpct cplx mfld is Kähler and has enough analytic functions, then it must have come from an algebraic variety.
In particular, Riemann existence tells you that compact Riemann surfaces always have nontrivial meromorphic functions on them. Plus, they’re anyways all Kähler, so they all come from algebraic curvss.
So somehow this stupid criterion characterises the algebraicity of compact complex mflds.
- Sanskar, on 18 Febuary, ‘26