What is the interface between functional analysis and algebraic geometry?

Question

This is a very open ended curiosity of mine and I would be grateful to hear any comments in this direction. In particular I am interested in functional analysis/algebraic geometry books/papers references which show this bridge from functional analysis into algebraic geometry.

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I am not sure if it's related but what are the good references for "functional analysis on manifolds"? Like how do we characterize the function space based on the domain manifold properties or for specific manifolds like say spheres. (the related things I see are courses like, Pierre Albin's lecture notes or Yaiza Canzani's ones, but these seem more about understanding specific differential operators on manifolds rather than the space of functions on a manifold)

Like is there any meaning to wondering, "What is the Hilbert space of functions on a sphere?"

Answer 1

Have a look at how the Hirzebruch-Riemann-Roch can be deduced as a special case of the Atiyah-Singer index theorem. The idea is to consider the hodge operator $\overline{\partial} + \overline{\partial}^\ast$ on the bundle of differential forms with values in a given holomorphic vector bundle $V$ over a complex manifold $X$. This is an elliptic differential operator, and the index theorem says that its Fredholm index is the integral of the product of the Chern class of $V$ and the Todd class of $X$. On the other hand the Fredholm index is the holomorphic Euler characteristic of $V$ by Hodge theory, yielding HRR.

The proof of the index theorem invariably uses a lot of functional analysis - either analysis of heat kernels, pseudo-differential operator theory, or operator algebras, depending on your preferences. This provides a foundation for a number of other interactions between algebraic geometry and functional analysis, like generalizations to holomorphic non-commutative spaces or analytic counterparts of Grothendieck-Riemann-Roch.

Answer 2

One connection (which is perhaps more geometric analytic than functional analytic), is the relationship between $K$-stability, which is an algebro-geometric notion and the existence of constant scalar curvature Kahler metrics. In particular, the Yau–Tian–Donaldson conjecture states that there is a deep connection between these two notions.

Yau–Tian–Donaldson conjecture: A smooth polarised variety $(X,L)$ admits a constant scalar curvature Kähler metric in the class of $c_1(L)$ if and only if the pair $(X,L)$ is K-polystable.

Initially, this might seem like more of a connection between algebraic and differential geometry. However, the existence of a constant scalar curvature Kahler metric boils down to the existence for a particularly nasty PDE, so the work in this area is quite analytic (see, e.g., this paper of Chen and Cheng). I should note that one direction of this conjecture is known (existence of a cscK metric implies $K$-polystability for polarized varieties). At present, the other direction is still open, although it was proven for Fano manifolds in 2012.

Answer 3

As an example in which algebraic geometry and functional analysis (mildly) interact: in the book Several complex variables with connections to algebraic geometry and Lie groups by J.L.Taylor, if I remember correctly, there is a chapter on GAGA in which sheaves of Fréchet spaces are considered.

Answer 4

I do not know if this is what you are looking for but I heard people speaking of Banach Algebraic Geometry. I can also note the fascinating work of Semyon Alesker (using algebra-geometric tools on say space of convex bodies).

On a high level there is certain similarities between the ideas in two fields. But, I personally prefer more concrete connections better. The real algebraic objects coming out of optimization problems, or convex objects with a lot symmetry coming out of Quantum Information Theory seems to be forcing an interface between algebraic geometry and convex geometric analysis ( local theory of Banach spaces if you like). There is also new developments in random real geometry which seems to require a blend of func analysis with AG.