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Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures? I get from Wikipedia how to calculate for a punctured and pinched curve (http://en.wikipedia.org/wiki/Hodge_structure#Mixed_Hodge_structures), but I want more! I want tables and numbers and everything explicit and spoonfed. Thanks!

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Within our department, have you asked Sam? He's spent a lot of time learning Hodge theory. –  Mike Skirvin Dec 1 '10 at 3:35
    
even I am looking for some thing like that. –  J Verma Dec 1 '10 at 3:57
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Thanks for the responses so far. I will dig in before accepting an answer. (Hint: active links most likely to get the vote. When I say spoonfed, I mean spoonfed. [Yeah, I'm talkin' to you, Emerton!]) –  Eric Zaslow Dec 1 '10 at 15:48
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7 Answers

up vote 11 down vote accepted

I can't believe nobody has yet mentioned the book Period mappings and period domains by Carlson, Müller-Stach and Peters. Chapter 1, the introduction, is written almost as a story, starting with the pure Hodge structure on the cohomology of a Riemann surface and introducing mixed Hodge structures by looking at degenerations. Further niceties await the reader in subsequent chapters.

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This was the easiest reference for the basic ideas, gently spoonfed -- thanks! It's probably enough for my purposes -- i.e., constructing a filtration on the cohomology of some complex which should be the cohomology of a singular algebraic variety qua central fiber, hence have a mixed Hodge structure. Thanks to the others, too. –  Eric Zaslow Dec 4 '10 at 2:11
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Dear Zaz,

If you look at Deligne's "Hodge I" article (actually an ICM talk, maybe from 1970 or thereabouts, which should be available at the IMU's electronic database of ICM talks), you will find a table giving at least some info about weights and Hodge numbers in various contexts (smooth but open, proper but possibly singular, etc.); more precisely, he states the ranges that the various numbers can lie in.

Otherwise, the basic principle is that (say in the case of a smooth variety first) you compactify your variety, write down the various long exact sequences that come to mind relating the cohomology of the open variety to that of its compactification, and then use the fact the maps of MHS are strict for weight and Hodge filtrations, so that you can read of the numbers of the MHS you care about (the cohomology of the open variety) from the MHS of other things appearing (which are compact, and so in some sense known: more precisely, the compactification is compact and smooth, so is known --- in principle --- by usual Hodge theory, while the boundary will be a normal crossings divisor, so is compact but possibly singular --- but in the mildest possible way --- and is also of lower dimension, so you can imagine that you know it by induction on dimension and/or because it's compact and very close to being smooth).

This is not the same as giving you a table, unfortunately; you wanted fish and I am giving you (at best) some kind of fishing implement (or maybe a spoon). Sorry about that.

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Matt, thanks for all the fish(y things). 42. –  Sándor Kovács Dec 1 '10 at 6:22
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Sometimes you just want your fish on a spoon! I don't need to know what the carburetor does just to drive to the mall. –  Eric Zaslow Dec 1 '10 at 15:50
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I just noticed the question. The references mentioned so far are good. So I'll just do the example that I normally do on a blackboard when anyone asks me. Take a smooth complex projective variety $X$ with a smooth divisor $D$. Set $U=X-D$. There's a long exact sequence $$ \ldots H^i(X)\to H^i(U)\to H^{i-1}(D)\to H^{i+1}(X)\ldots$$ say with rational or complex coefficients. What are the maps? The first is restriction, the second using $\mathbb{C}$ coefficients is a residue map, and the third is the Gysin map which is of type $(1,1)$ (or you want to want to get fancy you need a Tate twist here). The mixed Hodge theory of $U$ can be read off from this.

For example, for the Hodge numbers $$\dim H^{pq}(U)= \dim im[H^{p-1,q-1}(D)\to H^{pq}(X)]+\dim ker[H^{pq}(D)\to H^{p+1,q+1}(X)]$$ and so on. (By the way, $H^{pq}$ is taken to be the $(p,q)$ part of the $p+q$ weight graded quotient.)

OK, let me make it more concrete. Let $X$ be a surface with irregularity $q=0$, perhaps $\mathbb{P}^2$, then $D \subset X$ is a curve of say genus $g$. Then from above, the interesting Hodge numbers are $$h^{20}(U)=h^{02}(U)=h^{20}(X)$$ $$h^{11}(U)=h^{11}(X)-1$$ $$h^{12}(U)=h^{21}(U)=g$$

Maybe that's enough for now.

I can't resist squeezing in one more example. Suppose $D$ on the above surface $X$ can be contracted to a point in a normal surface $Y$. So for example, $Y$ might be a cone over a plane curve, and $X$ the blow up of the vertex. Using duality and the standard exact sequence for compactly supported cohomology, we can conclude $$H^i(Y)=H_c^i(U)= H^{4-i}(U)^*(-2),\quad i>0$$ As far as Hodge numbers are concerned, the dual means $(p,q)\mapsto (-p,-q)$ and $(-2)$ means shift by $(2,2)$.

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This lecture by Sam Payne is a very nice introduction:

http://archive.msri.org/communications/vmath/VMathVideos/VideoInfo/4162/LV/LaunchVideo?videoid=13602

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I'll vouch for that; it was a great talk. The video links at MSRI have changed; I think the video Jim is referring to is now at msri.org/web/msri/online-videos/-/video/showVideo/3674 –  Ravi Vakil Jun 10 '11 at 18:08
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Clare Voisin's book "Hodge Theory and Complex Algebraic Geometry I" goes over the case of a smooth (noncompact) variety in quite an elementary way. I also found Peters and Steenbrink's book "Mixed Hodge Structures" to be useful - it has a lot of detail and takes you all the way from Hodge theory of compact Kahler manifolds to mixed Hodge modules.

I think these are written in a slightly more elementary/textbook style than Deligne's papers (though I also recommend looking at these).

I second Matt's remark on calculating mixed Hodge numbers. If you can find long exact sequences relating the cohomology of your variety to that of other varieties whose Hodge structure you know, then you can use the fact that the maps in the LES are maps of mixed Hodge structures to try to read off mixed Hodge numbers.

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thanks for the references. –  J Verma Dec 1 '10 at 7:22
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Deligne's long "Fundamental group of the projective line minus three points" paper has several examples of mixed motives right near the beginning (in section 2, "examples") whose associated mixed Hodge structures are described very explicitly. That's where I learned what a mixed Hodge structure was, at any rate.

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

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A.H. Durfee, ``A naive guide to mixed Hodge theory,'' Singularities, Part 1 (Arcata, Calif., 1981), 313-320, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983.

is quite nice to read, but probably doesn't handle any more cases than you already know.

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