Vector bundles over $\mathbb{P}^1( \mathbb{F}_n)$

Question

Let $\mathbb{F}_n$ be a finite field and $n$ a prime and $m \in \mathbb{N}_{\geq 1}$. The following question is about understanding partition functions of susy gauge theories on a complex variety (let's say 4 real dimensional) via the Weil conjectures.

More specifically, my question is weather it is known how to construct explicitly vector bundles over the the projective space $\mathbb{P}^m$ over $\mathbb{F}_n$. I know that Yoshioka has done some relevant work where he uses the Weil conjectures in order to calculate Poincare polynomials of the corresponding, to the finite field, complex algebraic variety but I am not sure which work of his this is nor whether he constructs vector bundles in the "finite field side". The connection to physics arises from the fact that in this way it should be possible to construct the generating function of the Poincare polynomials and thus the generating function of the $\mathcal{N}=4$ SYM theory on a complex 4 dimensional variety.

To this end I would like to understand how to construct such vector bundles (I only care about simple cases) if they are constructuble and then e.g. how to count the finite sections of line bundles over $\mathbb{P}^1(\mathbb{F}_n)$ although it would be interesting to understand also the case of general dimension $\mathbb{P}^m(\mathbb{F}_n)$.

Any relevant references would be appreciated.

Comments

Could you please explain how complex varieties can be related to finite fields rather than the field of complex numbers?

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By reduction mod $n$. I think it follows this idea more or less.One starts with a complex projective algebraic variety defined over $\mathbb{Q}$ which is given as  $X=Z(\{f\})$ for some set of homogeneous polynomials (I care about projective varieties only) $f_i \in \mathbb{Q}[x_0, \ldots, x_N]$. Then one can consider the "elimination" of the denominators such that the coefficients are relatively prime, thus we end up with polynomials in the integers relatively prime and then we reduce mod $n$. Thus we have polynomials $\bar{f} \in \mathbb{Z}/n\mathbb{Z}$ which are a subset of the ones we started with. Then, we can define the reduction of $X$ mod $n$ as $\bar{X}(\mathbb{F}_n) = Z(\bar{f}) \subset \mathbb{P}^{N}(\mathbb{F}_n)$ which is the projective space over the finite field $\mathbb{F}_n$ which has finitely many points. Now what I ask is to construct a vector bundle over this. The thing is that using Weil conjectures, the generalized Riemann function will give us the Poincare polynomial of the original $X$ if I have understood this story well. Reference.

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So it is in fact a real algebraic variety? Otherwise you should get a quadratic number field in place of the integers, and hence a finite field of order prime-squared.

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It is an algebraic variety defined over the corresponding field which is $\mathbb{F}_n$. The order can be any power of the prime $n$ (although then there is not a ring homomorphism to $\mathbb{Z}/n^r \mathbb{Z}$. This is the standard construction used in the Weil conjectures.

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By the way is this a negative vote on my comment? Nothing above is mine, these are standard constructions.

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Possibly relevant:

Work by Totaro, e.g., Moving codimension-one subvarieties over finite field

Work by Langer, e.g., Nef Line Bundles Over Finite Fields or On positivity and semistability of vector bundles in finite and mixed characteristics

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I only see a positive vote on your comment.

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I see a "red" 1, I thought someone gave a bad vote. I will check your references. Thanks

Answer 1

Bundles with a purely algebraic definition make sense on projective spaces over any basis. For example, over $\mathbb{P}^n_{\mathbb{F}_q}$, we still have the line bundles $\mathcal{O(k)}$, for $k$ integers (in fact, they are the only line bundles), we have the tangent bundle $T$, the cotangent bundle $\Omega$... All classical facts about these objects (like space of sections, cohomology...) that you might know over the complex numbers remain true if you replace everywhere complex numbers by a finite field.

The key difference between the complex and finite field cases will come from the more complicated vector bundles. A typical way to construct complicated vector bundles is as extension of simpler ones and the possible ways to do that are controlled by spaces of cohomological nature called Ext^1. Over complex numbers, these spaces are finite dimensional complex vector spaces, in particular with a continuous infinite number of points if non-zero, which is why there are continuous families of vector bundles and so non-trivial moduli spaces over complex numbers. Over a finite field, these Ext^1 are finite dimensional vector spaces over the finite field and in particular they have only finitely many elements, corresponding to finitely vector bundles! Computing these Ext^1 spaces is usually the way to construct interesting vector bundles and to try to count them in the finite field case.

Comments

@40227 Do you know the paper of Yoshioka I refer to?