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 its 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, http://www.math.uiuc.edu/~palbin/Math524.Spring2012/LectureNotesMay1.pdf or http://www.math.harvard.edu/~canzani/math253.html 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

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 3

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 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.