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I'm trying to understand variations of Hodge structure. I understand that this is a very broad field, and that many of the concepts have been extended to algebraic geometry over fields other than $\mathbb{C}$, and so forth. In particular, there is a version of the monodromy theorem by Grothendieck, which is rather incomprehensible to me. I am asking this question in the context of variations of Hodge structure for Calabi-Yau manifolds and mirror symmetry, so I hope that the answer can be given with that in mind.

Given a family of projective Kähler manifolds $\pi:\mathcal{X}\to\Delta^*$, where $\Delta^*$ denotes the punctured disk, we get a monodromy operator $T\in\text{GL}(H^k(X_t,\mathbb{C}))$, for $t\in\Delta^*$. This operator is obtained from the local system associated to the family, which gives a representation $\rho:\pi_1(\Delta^*)\to\text{GL}(H^k(X_t,\mathbb{C}))$, and $T$ is the image of a generator of the fundamental group under this representation. One can then show that $T$ is quasi-unipotent, meaning that there are integers $m,N$ such that $(T^m-I)^N=0$. After pulling back along the map $z\mapsto z^m$, we may as well assume that $m=1$, so that $T$ is in fact unipotent. One then defines the nilpotent operator $$N=\log T=(T-I)+\dots+(-1)^{N+1}(T-I)^N/N!$$ Associated to this operator is a weight filtration of $H^k(X_t,\mathbb{Q})$, called the monodromy weight filtration. It is uniquely defined by $$N(W_i)\subseteq W_{i-2}\quad\quad N^k:W_{N+k}/W_{N+k-1}\xrightarrow{\cong}W_{N-k}/W_{N-k-1}$$ I have two questions about this.

  1. Why do we impose this filtration on the cohomology with rational coefficients, rather than with complex coefficients? Does this make a difference, if we restrict our attention to complex manifolds and their variations of Hodge structure?
  2. What does this filtration represent, geometrically? Up until this point, there has been a nice geometric interpretation for the concepts which were introduced (e.g. the Hodge bundles, Gauss-Manin connection, the period map, etc.), but I have no idea how to think about this filtration in geometric terms.
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Given a nilpotent endomorphism $N$ of a finite dimension vector space $V$, Jordan canonical form implies that we can decomponse $V$ into a sum of "blocks" on which we can find bases satisfying $Ne_1 = e_{2}, Ne_2=e_3,\ldots Ne_k=0$. To make it basis independent, pass to the filtration $$ \langle e_k\rangle\subset \langle e_k, e_{k-1}\rangle \subset \ldots$$ After suitable indexing, this is the monodromy weight filtraton $W_\bullet$.

The point I wanted to make is that $W$ may seem complicated, but it isn't that bad. Furthermore the construction works over $\mathbb{Q}$ (because $N$ has zero eigenvalues, and so has a Jordan form over this field). This is also useful further along in the story, when one gets to limit mixed Hodge structures: part of the definition of a mixed Hodge structure is that $W$ is rational.

Regarding your second question, I don't really have a good answer. It is clear that $W$ is quite natural from the point of linear algebra, and Schmid gave an analytic interpretation in terms of certain growth conditions. However, the geometric meaning seems much more elusive.

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  • $\begingroup$ Thank you prof. Arapura! Your book has been of great help to me on this journey. Regarding your answer: I understand that we can do this over $\mathbb{Q}$ but I was just wondering if one loses anything by doing the same construction over $\mathbb{C}$ instead, if the intent is to stay within the complex geometry world. I would assume not, because of the universal coefficients theorem. Would it be possible to elaborate on that slightly? Either way, thanks for your time. $\endgroup$ Jul 25, 2022 at 19:57
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    $\begingroup$ Your welcome. Universal coefficients tells you the inclusions $H^i(X,\mathbb{Z}), H^i(X,\mathbb{Q})\subset H^i(X,\mathbb(C))$ give "lattices". But these lattices are extra data that you may not want to throw away. $\endgroup$ Jul 25, 2022 at 21:18
  • $\begingroup$ It escaped my mind that the rational cohomology also gives us this lattice structure. That makes sense, then - we want to preserve this structure, but the integers don't allow us to define these matrices, so we must pass to rational cohomology. I will leave the question open in case someone has an insight about the geometric interpretation of this filtration, but your answer helps a great deal. $\endgroup$ Jul 25, 2022 at 21:31

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