Let $E_{i}=F(L_{i},\rho_{i}),i=1,2$ be holomorphic vector bundles as above. There is an analog of the dg-category ${\cal C}(Y)$ in the case of complex numbers. We will denote it by ${\cal A}(Y)$. Objects of ${\cal A}(Y)$ are holomorphic vector bundles on $X$ of the type $E=F(L,\rho)$. Morphisms are sections of soft sheaves on $Y$. Namely, we define the sheaf $\underline{Hom}_{{\cal A}(Y)}(E_{1},E_{2})$ on $Y$ as the direct image $p_{\ast}(\underline{Hom}_{Dolb}(E_{1},E_{2}))$ (in the self-explained notation). Then $Hom_{{\cal A}(Y)}(E_{1},E_{2})$ are global sections of this sheaf. This sheaf corresponds to the sheaf
$\underline{Hom}_{{\cal C}(Y)}(E_{1},E_{2})$ in the non-archimedean geometry. Let us choose an open affine chart $U\subset Y,U\simeq{{\bf R}}^{n}$. Then $\Gamma(U,\underline{Hom}_{{\cal A}(Y)}(E_{1},E_{2}))$ contains a subsheaf of finite Fourier sums with respect to the natural action of the torus
$T^{n}$ on $\Gamma(U\times T^{n},\underline{Hom}_{Dolb}(E_{1},E_{2}))$. Thus we have the sheaf $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})$ which is an analog of the sheaf $\underline{Hom}_{{\cal C}(Y)}^{alg}(E_{1},E_{2})$ considered in the non-archimedean case. Notice that there exists a natural homomorphism of sheaves $j:p^{\ast}(\underline{\Omega}^{\ast}_{Y})\to\underline{\Omega}^{0,\ast}_{X}$. The image of $j$ consists of Dolbeault forms on $X$ which have coefficients locally constant along fibers of $p$. In local coordinates $j$ is given by the formula $f_{i_{1},...,i_{n}}(y)dy_{i_{1}}\wedge...\wedge dy_{i_{n}}\mapsto f_{i_{1},...,i_{n}}(y)d\overline{z}_{i_{1}}\wedge...\wedge d\overline{z}_{i_{n}}$, where $z_{k}=y_{k}-\sqrt{-1}x_{k},1\leq k\leq n$. It is easy to see that $j$ is compatible with the structure of dg-algebras on de Rham and Dolbeault forms. Thus for a pair of holomorphic vector bundles $E_{1}$ and $E_{2}$ on $X$ we have a canonical structure of dg-module over $\underline{\Omega}^{\ast}_{Y}$ on the sheaf $\underline{Hom}_{{\cal A}(Y)}(E_{1},E_{2})$. In the case when $E_{i}=F(L_{i},\rho_{i}),i=1,2$ the subsheaf $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})$ is also a sheaf of dg-modules over $\underline{\Omega}^{\ast}_{Y}$.

As in the non-archimedean case there is a canonical decomposition of the stalk $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})_{y},y\in Y$ into the direct sum of dg-modules of finite rank over $\underline{\Omega}^{\ast}_{Y}$. Summands are labeled by the homotopy classes $[\gamma]\in P(L_{1},L_{2},y)$ and called sectors. We will denote them by $\underline{Hom}_{{\cal A}(Y)}^{alg,[\gamma]}(E_{1},E_{2})_{y}$. Informally, sectors correspond to “Fourier components” of Dolbeault forms in $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})_{y}$ in the direction of torus fibers. Let us describe them more explicitly. For simplicity we will assume that $\rho_{i},i=1,2$ are rank one trivial local systems, and $L_{i},i=1,2$ intersect with each fiber of $p$ at exactly one point. Then near $p^{-1}(y)$ we can write $L_{i}=graph(df_{i})\,(mod(T_{Y}^{{\bf Z}})^{\vee}),i=1,2$, where $f_{i}$ are germs at $y$ of smooth functions on $Y$. ¿From the description of the Poincaré bundle $P$ we deduce that $\underline{Hom}_{{\cal A}(Y)}^{alg}(E_{1},E_{2})_{y}$ is canonically identified with the space of germs of $\overline{\partial}$-forms near $T^{n}=p^{-1}(y)$, endowed with the twisted differential $\overline{\partial}^{\prime}\alpha=\overline{\partial}\alpha+{i\over{\varepsilon}}\sum\partial f/\partial y_{i}d\overline{z}_{i}\wedge\alpha$, where $f=f_{1}-f_{2}$. Then the sector corresponding to a path $\gamma$ consists of Dolbeault forms $\alpha=\sum_{i_{1},...,i_{n}}exp(i\langle m,x\rangle/\varepsilon)f_{i_{1}...i_{n}}(y)d\overline{z}_{1}\wedge...\wedge d\overline{z}_{n}$. Here vector $m=m(\gamma)$ is the homotopy class of the loop in $T^{n}=p^{-1}(y)$ which is the composition of three paths:

1) the path $[0,1]\to T^{n},t\mapsto t(df_{1})_{y}\,mod\,(T_{Y}^{{\bf Z}})^{\vee}$;

2) the path $\gamma$;

3) the path $[0,1]\to T^{n},t\mapsto(1-t)(df_{2})_{y}\,mod\,(T_{Y}^{{\bf Z}})^{\vee}$.

A choice of sector corresponds to the choice of monomial $z_{1}^{i_{1}}...z_{n}^{i_{n}}$ in the non-archimedean case. Homotopy classes of paths in non-archimedean approach correspond the summands of Fourier series. Locally each sector can be identified with the de Rham complex on $Y$. Namely, to a form $\alpha=\sum_{i_{1},...,i_{n}}f_{i_{1}...i_{n}}(y)exp(\langle m,x\rangle)d\overline{z}_{i_{1}}\wedge...\wedge d\overline{z}_{i_{n}}$ we assign the form $\alpha_{m}=\sum_{i_{1},...,i_{n}}f_{i_{1}...i_{n}}(y)exp({1\over{\varepsilon}}\langle m,x\rangle)dy_{i_{1}}\wedge...dy_{i_{n}}$, where $m=m(\gamma)$ defines the sector. It is easy to see that the differential $\overline{\partial}^{\prime}$ on Dolbeault forms on $X$ corresponds to the de Rham differential $d$ on $\Omega^{\ast}(Y)$. In this way we obtain an isomorphism of complexes $\underline{Hom}_{{\cal A}(Y)}^{alg,[\gamma]}(E_{1},E_{2})_{y}\simeq\underline{\Omega}^{\ast}_{Y,y}\otimes{{\bf C}}$.

The set of exponents appearing in the expansion of $\alpha_{m}$ at $y$ corresponds to the spectrum $Sp_{y}(\alpha)$ considered in the non-archimedean case.