One motivation of the present article is mirror symmetry of smooth projective Fano varieties \(F\) (always over \(\mathbb{C}\)). Unlike for compact Calabi-Yau threefolds, the mirror of \(F\) is not a variety but a Landau-Ginzburg model (with additional properties). This is a pair \((Y,\mathsf{w})\) consisting of a smooth variety \(Y\) and a regular function \(\mathsf{w}\colon Y \to \mathbb{A}^1\). Moreover, we assume that \((Y,\mathsf{w})\) ‘compactifies’ to \((X,f)\) where \(Y\subset X\) is a smooth, projective normal-crossing compactification of \(Y\) and \(f\colon X\to \mathbb{P}^1\) is a flat projective morphism with \(f_{|Y}=\mathsf{w}\). By abuse of notation, we call \((X,f)\) a Landau-Ginzburg model as well. In mirror symmetry, \((X,f)\) satisfies additional conditions leading to tame compactified Landau-Ginzburg models. However, the present article does not require these conditions.
Any Landau-Ginzburg model \((Y,\mathrm{w})\) determines the following cohomology groups \(H^k(Y,\mathsf{w})\), \(H^k(Y,Y_b)\) and Hodge numbers \(f^{p,q}(Y,\mathsf{w})\), \(h^{p,q}(Y,\mathsf{w})\) respectively:
(1) let \((\Omega_X^\bullet(\ast D),d+df\wedge)\) be the complex determined by the reduced normal-crossing divisor \(D=f^{-1}(\infty)\subset X\) and \(f\). Here \(\Omega_X^\bullet(\ast D)\) are the sheaves of differential forms with at most poles of along \(D\) of arbitrary order. It admits a quasi-isomorphic subcomplex \((\Omega_f^\bullet,d+df\wedge )\) and we define \(H^k(Y,\mathsf{w}):=\mathbb{H}^k(\Omega_X^\bullet(\ast D), d+df\wedge )\) as well as \[ f^{p,q}(Y,\mathsf{w}):=\dim H^q(X, \Omega_f^p), \quad p+q=k. \] The \(E_1\)-degeneration of the irregular Hodge filtration (due to Esnault-Sabbah-Yu, Kontesvich and M. Saito) implies \(\dim H^k(Y,\mathsf{w})=\sum_{p+q=k} f^{p,q}(Y,\mathsf{w})\).
(2) Let \(b\in \mathbb{P}^1\) be sufficiently close to \(\infty\) so that \(Y_b:=f^{-1}(b)\) is a smooth fiber. The relative cohomology groups \(H^k(Y,Y_b)\) (with \(\mathbb{C}\)-coefficients) carries the weight filtration \(^k W\) determined by the local monodromy around \(\infty\). These define the Hodge numbers \[ h^{p,q}(Y,\mathsf{w}):= \dim Gr_{2p}^{^k W}H^k(Y,Y_b),\quad p+q=k. \] In their investigation of homological mirror symmetry for smooth projective Fano varieties, L. Katzarkov et al. [Proc. Symp. Pure Math. 78, 87–174 (2008; Zbl 1206.14009)] conjecture that \[ f^{p,q}(Y,\mathsf{w})=h^{p,q}(Y,\mathsf{w}),\quad (*). \] (Note that the conventions differ slightly from Katzarkov-Kontsevich-Pantev.) There are various examples and counter-examples to this conjecture. Hence it is desirable to have sufficient conditions when the conjecture does hold.
This is achieved in the present article in an abstract setting by introducing rescaling structures of Hodge-Tate type. A rescaling structure is a triple \(\mathcal{H}=(\mathcal{H},\nabla,\chi)\). Here \(\mathcal{H}\) is a \(\mathbb{Z}\)-graded locally free \(\mathcal{O}_S(\ast(\lambda)_\infty)\)-module where \(S:=\mathbb{P}^1_\lambda\times \mathbb{C}_\tau\) with (affine) coordinates \((\lambda,\tau)\). Moreover, \[ \nabla\colon \mathcal{H}\to \mathcal{H}\otimes \Omega_S^1(\log \lambda \tau)((\lambda_0)). \] is a grade-preserving meromorphic connection. Note that the last factor is locally generated (as a \(\mathcal{O}_S\)-module) by \(\lambda^{-1}\tau^{-1}d\tau\) and \(\lambda^{-2}d\lambda\).
Finally, let \(\sigma \colon \mathbb{C}^*\times S \to S\) be the \(\mathbb{C}^*\)-action \(\sigma(\theta,\lambda,\tau)=(\theta\lambda, \theta \tau)\) and \(p_2\colon \mathbb{C}^*\times S\to S\) the projection. Then \(\chi\colon p_2^*\mathcal{H}\to \sigma^*\mathcal{H}\) is an isomorphism with additional conditions.
Clearly, the last item explains the name rescaling structure. It is motivated by the fact that \(\dim \mathbb{H}^\bullet(X;(\Omega_f^\bullet, \lambda d+ \tau\, df\wedge))\) is independent of \((\lambda,\tau)\in \mathbb{C}_\lambda\times \mathbb{C}_\tau\) for any Landau–Ginzburg model \((X,f)\). Using this observation, the author constructs a recaling structure \(\mathcal{H}_f=(\mathcal{H}_f,\nabla_f,\chi_f)\) associated to \((X,f)\).
The previous discussion on Hodge numbers generalizes to any rescaling structure \(\mathcal{H}\). More precisely, let \(V=\mathcal{H}_{(1,0)}\) be the fiber over \((1,0)\in S\). The author constructs two filtration \(F\) and \(W\) on \(V\), the Hodge and weight filtration of \(\mathcal{H}\). If \((F,W)\) are a mixed Hodge filtration on \(V\) of Hodge-Tate type (i.e. \(Gr_{2k}^W\) is of Hodge type \((k,k)\) for any \(k\in \mathbb{Z}\) and zero otherwise), then \(\mathcal{H}\) is defined to be of Hodge-Tate type.
For a general rescaling structure \(\mathcal{H}\), the author defines Hodge numbers \(f^{p,q}(\mathcal{H})\) and \(h^{p,q}(\mathcal{H})\) associated to \(\mathcal{H}\). Then the main result gives a sufficient condition for the equality f\(_-\)equals\(_-\)h:
Let \((X,f)\) be a Landau-Ginzburg model and \(\mathcal{H}_f\) be its rescaling structure. Then \(\mathcal{H}_f\) is of Hodge-Tate type iff the mixed Hodge structures \((H^k(Y,Y_\infty;\mathbb{Q}),F,W)\) are of Hodge–Tate type for all \(k\in \mathbb{Z}\). The latter are defined via the nearby cycles of \(f\) at \(\infty\).
If this is true, then \(f^{p,q}(\mathcal{H}_f)=f^{p,q}(Y,\mathsf{w})\), \(h^{p,q}(\mathcal{H}_f)=h^{p,q}(Y,\mathsf{w})\) and the equality \((*)\) holds.
Subsequent to the proof of this theorem, the author gives examples of Landau-Ginzburg models \((X,f)\), for \(\dim X=2,\) \(3\), such that \(\mathcal{H}_f\) is of Hodge-Tate type.
Even though the article is technical, it is clearly written and well structured.