Splitting of the weight filtration

Question

All varieties are over $\mathbb{C}$. Notions related to weights etc. refer to mixed Hodge structures (say rational, but I would be grateful if the experts would point out any differences in the real setting).

I am trying to get some intuition for the geometric meaning of/when to expect the weight filtration on the cohomology groups $H^i(X)$ of a variety to split. By the weight filtration splitting, I mean that the Hodge structure on each $H^i(X)$ is a direct sum of pure Hodge structures.

The simplest situation in which this happens is the classical one of smooth projective varieties. The next simplest situation I think of is the "smooth" being weakened to "mild singularities" (for instance, rationally smooth). So at least in the projective case I think of the "mixed" as encoding singularities. This also gels well with the construction of these Hodge structures using resolution of singularities. Are there other helpful perspectives?

Instead of fiddling with the "smooth" one can make the variety non-compact (but still smooth say). Examples: affine $n$-space, tori. Here I don't know how to think of or when to expect the weight filtration to split. Any intuition would be appreciated. The only rough picture I have is that this encodes information about the complement in a good compactification. But I don't find this particularly illuminating.

Generalizing affine space and tori is the situation of toric varieties for which the weight filtration always splits (thanks to a lift of Frobenius to characteristic $0$). In general when should one expect a splitting of the weight filtration to be given "geometrically" by a morphism (or say correspondence in the context of Borel-Moore homology)?

Related is the following: when should one expect the Hodge structure on each $H^i(X)$ to be pure (not necessarily of weight $i$). Here I am again more interested in weakening the "projective" rather than the "smooth".

At the risk of being even more vague, let me add some motivation from left field. There are several situations in representation theory where one expects/knows that the weight filtration on some cohomology groups splits (and is even of Hodge-Tate type). For instance, the cohomology of intersections of Schubert cells with opposite Schubert cells. However, the reasoning/heuristic has, a priori, nothing to do with geometry but more with the philosophy of "graded representation theory" (ala Soergel, see for instance his ICM94 address) I would love to have a geometric reason/heuristic for this. Pertinent to this is also the question of when should one expect the canonical Hodge structure on extensions between perverse sheaves of geometric origin to be split Tate? The only examples I know come from representation theory (see Section 4 of Beilinson-Ginzburg-Soergel's "Koszul duality patterns in representation theory").

Answer 1

Here is a simple example to keep in mind. Let $\bar X$ be a smooth projective curve, and let $X=\bar X-\lbrace p_1, \ldots, p_n \rbrace$. The rational mixed Hodge structure on $H^1(X)$ is given by an extension $$0\to H^1(\bar X)\to H^1(X)\to \mathbb{Q}(-1)^{n-1}\to 0$$ This splits if and only if for each $i,j$ there exists logarithmic $1$-forms with singularities only at $p_i$ & $p_j$ and rational periods. By work of Carlson [Extensions of mixed Hodge structures] the obstructions are given by the classes $p_i-p_j\in Jac(\bar X)\otimes \mathbb{Q}$ in the Jacobian. In particular, these are usually nontrivial. By contrast $H^1(X,\mathbb{R})$ does split because such forms with real periods do exist. However, for more complicated examples, the real mixed Hodge structure need not split either. So in general, you shouldn't expect it. My intuition, however, is that when the varieties "come from linear algebra" this is more common, but I don't have a precise statement or explanation.

I'm not sure if this answers your question.

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Addendum Here is another perspective which may or may not make the issue clearer. The category of polarizable rational mixed Hodge structures is neutral Tannakian, so it is the category of representations of a pro-algebraic group, what I would call the universal Mumford-Tate group $MT$. The split ones constitute the subcategory of representations of the quotient of $MT$ by its pro-unipotent radical.

Answer 2

I don't have a general answer, but let me add some more examples.

1. For your second question, examples of smooth varieties with $H^i$ pure of the 'wrong' weight, a good example is the complement of an affine arrangement of hyperplanes in $\mathbf C^n$. In this case, the Hodge structure on $H^i$ is pure of type $(i,i)$. One way to see this is that the cohomology is generated as an algebra by $H^1$, and $H^1$ is spanned by logarithmic forms $$ \omega_H = \frac 1 {2\pi i} \mathrm d \log H,$$ where $H=0$ is the defining equation of one of the hyperplanes; the class of $\omega_H$ is of type $(1,1)$. A reference for this is Brieskorn's "Sur les groupes des tresses [d'après V.I. Arnol'd]". See also the simple and "motivic" proof in "Weights in cohomology groups arising from hyperplane arrangements" by Minhyong Kim, as well as near-simultaneous papers by Boris Shapiro and Gus Lehrer which present basically the same result.

2. Another example is when $X$ is an abelian variety, and $F(X,n)$ is the configuration space of $n$ points in $X$, i.e. the complement of the "big diagonal" in $X^n$. Then the mixed Hodge structure on $H^i(F(X,n))$ is always a direct sum of pure Hodge structures. A reference for this is Gorinov's preprint "Rational cohomology of the moduli spaces of pointed genus 1 curves" (on his webpage), Section 3.

3. Let $Y(N)$ be the open modular curve parametrizing full level $N$ structures on elliptic curves. Then $H^1(Y(N))$ is a sum of pure Hodge structures, by the criterion in Donu Arapura's answer and the theorem of Drinfel'd and Manin.