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Let $X$ be a scheme which is smooth over a noetherian base scheme $S$. Let $(\mathcal F,\nabla)$ be a flat vector bundle, i.e. $\mathcal F$ is a vector bundle of finite rank on $X$ and $\nabla$ is an integrable connection relative $S$.

Is there a notion in the literature of "unipotent" flat vector bundle and morphisms between them? And does this give an abelian category with objects isomorphism classes of such $(\mathcal F, \nabla)$? I haven't seen a clean formal approach to this in the literature.

I don't think it is abelian in the above wide context, but probably for $X$ being a smooth variety over a field or an abelian variety over a field.

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There's a general notion of G-bundle with flat connection for an algebraic group G, in particular for G unipotent. However, they form a groupoid and not an abelian category, so perhaps it's not what you want. Can you clarify what you're looking for? Does some author use this phrase? Or else can you give examples of what you want to axiomatize? –  Moosbrugger Dec 2 '11 at 23:07
    
Over $\mathbb{C}$, a vector bundle with flat connection can be viewed as a local system and hence as a representation of $\pi_1$. I think a vector bundle here is 'unipotent' if the corresponding representation of $\pi_1$ is unipotent. This seems to be an abelian category to me: it is the category of representations of the unipotent algebraic envelop of $\pi_1$. –  Keerthi Madapusi Pera Dec 3 '11 at 7:44
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In general, you might want to define a vector bundle with flat connection to be unipotent if it admits a filtration by sub-bundles, the associated gradeds of which are trivial bundles with connection. It is clear that the category of such objects is closed under extensions in the category of all vector bundles with flat connection. It is not immediately clear to me that this category will be abelian, though that does not seem unlikely. –  Keerthi Madapusi Pera Dec 3 '11 at 7:47
    
@Keerthi: thanks, that's a definition which makes good sense in my case. Two things: first, is the stability under extensions really clear? I don't immediately see how you get your filtration on the middle term. Second, where exactly do you see the problem of the category to be abelian? For example, if you denote by $Conn(X)$ the category of vector bundles (of finite rank) on a smooth variety over say $\mathbb C$, then one knows that this is abelian. Now in our case one would have to endow kernels and cokernels of unipotent bundles with a unipotent structure. But this is not clear, isn't it? –  Veen Dec 3 '11 at 8:33
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For abelian varieties you may want to look at Mukai's paper in Nagoya (1981) (especially example 2.9), also see 1.5 and 1.6 of 0802.1060. –  Hacon Dec 7 '11 at 18:59
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2 Answers

up vote 5 down vote accepted

As Keerthi Madapusi Pera points out in his comments, it is certainly reasonable to define a unipotent flat vector bundle as a flat vector bundle that is a successive extension of the trivial one $(\mathcal O_X,d)$. Over a general basis $S$ I don't know, but over a field there are plenty of references.

For instance let $k$ be a field of characteristic zero and $X$ a smooth geometrically connected scheme over $k$. Then according to

Deligne, P. Le groupe fondamental de la droite projective moins trois points.

http://www.math.ias.edu/files/deligne/GaloisGroups.pdf

$\S$ 10.26 the category of unipotent flat vector bundles over $X$ is even Tannakian (a fortiori abelian), giving rise to the "De Rham fundamental group" $\pi(X,x)_{DR}$.

If you want to work in positive characteristic, or with singularities, flat vector bundles are certainly not the right objects any longer (at least if you want to have some link with the fundamental group), one has to replace them with stratifications, see

Saavedra Rivano, Neantro Catégories Tannakiennes. Lecture Notes in Mathematics, Vol. 265.

http://www.springerlink.com/content/w141370u7g5u/

VI 1.2

or

dos Santos, João Pedro Pinto Fundamental group schemes for stratified sheaves J. Algebra 317 (2007), no. 2, 691–713.

http://www.sciencedirect.com/science/article/pii/S0021869307001640

One could also mention that over a complete scheme over a field, you don't need to consider connexions at all. For instance Nori in chapter IV of his phd

Nori, Madhav V. The fundamental group-scheme. Proc. Indian Acad. Sci. Math. Sci. 91 (1982), no. 2, 73–122.

http://www.springerlink.com/content/x246hl378t2737p5/

considers a scheme of finite type $X$ over a field $k$ such that $H^0(X,\mathcal O_X)=k$, and proceeds to show that the category of unipotent vector bundles is indeed Tannakian (the explanation is that one can deduce from this that unipotent vector bundles are in fact endowed with connections). Lemma 2 there may be of interest for you, because he shows exactly the abelianness in this similar context.

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Your last question "Furthermore, another question is if any extension of a flat vector bundle by unipotents is again unipotent." can be partially answered by the following two papers:

"A remark on "Connections and Higgs fields on a principal bundle"" http://arxiv.org/abs/1102.4216

"Unipotent Schottky bundles on Riemann surfaces and complex tori" http://arxiv.org/abs/1102.3006

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