Semistable degeneration and limit mixed Hodge structure data

Let $\mathrm{\Delta }$ be a complex disk with coordinate $t$ and ${\mathrm{\Delta }}^{\ast }=\mathrm{\Delta }\setminus \{0\}$. A semistable degeneration datum consists of a proper holomorphic map $\bar{\pi }:\bar{\mathcal{X}}\to \mathrm{\Delta }$ with smooth total space, smooth over ${\mathrm{\Delta }}^{\ast }$, whose central fibre $D={\bar{\pi }}^{-1}(0)$ is a reduced simple-normal-crossing divisor; in the algebraic/projective case require $\bar{\pi }$ projective. Put ${\mathcal{X}}^{\ast }:=\bar{\mathcal{X}}\setminus D$ and define $\pi :=\bar{\pi }{\mathrm{\mid }}_{{\mathcal{X}}^{\ast }}:{\mathcal{X}}^{\ast }\to {\mathrm{\Delta }}^{\ast }$. The associated local system is ${\mathbb{V}}_{\mathbb{Z}}=R^k{\pi }_{\ast }\mathbb{Z}\mathrm{/}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}$, with monodromy $T$ around a positively oriented loop. If after an explicitly specified finite base change the monodromy is unipotent, define $N=\log T=(T-I)-\frac{1}{2}(T-I{)}^2+\frac{1}{3}(T-I{)}^3-\cdots ,$ which is a finite sum because $T-I$ is nilpotent.

The relative logarithmic de Rham complex is ${\mathrm{\Omega }}_{\bar{\mathcal{X}}\mathrm{/}\mathrm{\Delta }}^{\bullet }(\log D):={\mathrm{\Omega }}_{\bar{\mathcal{X}}}^{\bullet }(\log D)/({\bar{\pi }}^{\ast }{\mathrm{\Omega }}_{\mathrm{\Delta }}^1(\log 0)\wedge {\mathrm{\Omega }}_{\bar{\mathcal{X}}}^{\bullet -1}(\log D))\mathrm{.}$ After the specified base change, use the flat-marking convention $\mathrm{\Phi }(z+1)=T\mathrm{\Phi }(z)$ with $t=e^{2\pi iz}$. The canonical logarithmic extension is the holomorphic bundle on $\mathrm{\Delta }$ whose frame on the universal cover is $e^{zN}{e}_1,\dots ,e^{zN}{e}_m$ for a flat basis $e_1,\dots ,e_m$; its connection has logarithmic matrix $Ndt\mathrm{/}(2\pi it)$ in this frame. If the untwisted filtration $e^{-zN}\mathrm{\Phi }(z)$ extends holomorphically across $t=0$ by subbundles of this extension, call those subbundles the extended Hodge filtration and denote their fibre at $0$ by $F_{\mathrm{\infty }}$. Their extension is recorded only when it exists; this definition does not assert it.

For a finite-dimensional rational vector space carrying the nilpotent $N=\log T$, let $W(N)$ denote the unique finite increasing filtration satisfying $N{W}_j\subseteq W_{j-2}$ and such that the induced $N^{\mathrm{\ell }}:{\mathrm{Gr}}_{\mathrm{\ell }}^{W(N)}\to {\mathrm{Gr}}_{-\mathrm{\ell }}^{W(N)}$ is an isomorphism for every $\mathrm{\ell }\ge 0$. It is centred at zero; set $W(N)[-k{]}_j:=W(N{)}_{j-k}$, centred at $k$. A candidate limit mixed-Hodge structure consists of this shifted filtration and an extended limiting Hodge filtration when the latter exists. This definition asserts neither their existence for every degeneration nor a semistable-reduction theorem.

Polarized variation of Hodge structure and period map

Rational mixed Hodge structures

The nilpotent Lefschetz filtration is canonical and self-dual

Voisin, §§3.4.3–3.5

A proper algebraic family with reduced SNC central fibre is a geometric specialization with its own hypotheses. For a general polarized analytic variation, quasi-unipotence and the passage to unipotent monodromy must be proved separately; the word “degeneration” does not imply either conclusion.