Remark 2.3. Semiabelian varieties are commutative groups (see [Reference Iitaka25, Lemma 4]), so we will denote their Chevalley decompositions with additive notation ( $0$ instead of $1$ ).

Remark 2.7 (Universal cover of a semiabelian variety).

As complex manifolds, every semiabelian variety G is isomorphic to ${\mathbb C}^g/{\mathbb Z}^r$ for $g = \dim G$ and some $r\in {\mathbb Z}_{\ge 0}$ , where ${\mathbb Z}^r$ is embedded into ${\mathbb C}^g$ as a discrete subgroup. In particular, $r\le 2g$ . Also, ${\mathbb Z}^r$ must generate ${\mathbb C}^g$ as a ${\mathbb C}$ -vector space: otherwise, G would have a nontrivial holomorphic homomorphism to ${\mathbb C}$ , contradicting Proposition 2.6. In particular, $g\le r$ . The universal cover of G is given by the exponential map of Lie groups $\exp :TG\to G$ , where is the tangent space of G at the identity $e\in G$ . Note that, since G is an abelian group, $\exp $ is a group homomorphism, where $TG$ is seen as a group under addition. Note that $TG\cong {\mathbb C}^g$ , and $\exp ^{-1}(e)$ is identified with the lattice ${\mathbb Z}^r$ through this identification, so $r=\mathrm {rank}\ \pi _1(G)$ .

Remark 2.8. Let $f:G_1\to G_2$ be an algebraic morphism between two semiabelian varieties. Up to translation, f is also a group homomorphism. Indeed, up to translation in $G_2$ , we can assume that f takes the identity to the identity. Such f induces a linear map (the differential of f at the identity) between the universal covers given by the exponential map, which implies that f is also a group homomorphism.

Remark 2.10. Since U has the homotopy type of a finite CW complex, $H_i(U^f,{\mathbb K})$ is a finitely generated R-module.

Remark 2.13. There is a natural R-module isomorphism $H_i(U^f,{\mathbb K})\cong H_i(U,{\mathcal L})$ . This follows from the definition of the right-hand side, since the chain complex that computes it is the same chain complex that computes the homology of $U^f$ (see [Reference Dimca11, Section 2.5]).

Remark 2.14. If $V\subset U$ is a simply connected open set, $\pi ^{-1}(V)\cong \pi _1(G)\times V$ . For any $\gamma \in \pi _1(U)$ , the action of $\gamma $ on the stalk ${\mathcal L}_x$ is given by multiplication by $f_*(\gamma )\in \pi _1(G)$ .

Remark 2.15. Let ${\mathcal S}$ be the sheaf (of sets) of lifts of f to $TG$ , that is, $\Gamma ({\mathcal S},V)= \{\iota \colon V\to TG \mid \exp \circ \iota =f \}$ for any open set $V\subseteq U$ . For every $x\in U$ , the stalk ${\mathcal S}_x$ is canonically isomorphic to $\exp ^{-1}(f(x))$ , and it carries a $\pi _1(G)$ -action coming from the action on $TG$ .

On the other hand, a basis of the stalk of ${\mathcal L}_G$ at $f(x)$ is given by $\exp ^{-1}(f(x))$ , where each point $z'$ on the fiber corresponds to the locally constant function that is 1 around $z'$ and $0$ elsewhere on the fiber, and this bijection is also compatible with the $\pi _1(G)$ action. The same can be said of the stalk of ${\mathcal L}$ at x, since $f^{-1}$ preserves stalks.

This provides us with a map of sheaves ${\mathcal S}\to {\mathcal L}$ that sends ${\mathcal S}$ to a basis of ${\mathcal L}$ on each open set, and it is compatible with the action of $\pi _1(G)$ . Thus, a (locally defined) function $ \iota \colon U\to TG$ such that $\exp \circ \iota = f$ can be seen as a (local) section of ${\mathcal L}$ , and these locally form a ${\mathbb K}$ -basis. For $\gamma \in \pi _1(G)$ , we will denote .

Remark 2.17. There is a canonical isomorphism ${\mathcal {H}\mathrm {om}}_R({\mathcal L},R)\cong \overline {{\mathcal L}}$ . On our local ${\mathbb K}$ -bases of ${\mathcal L}$ and $\overline {{\mathcal L}}$ , it is given by the pairing:

$$\begin{align*}\overline{{\mathcal L}} \times {\mathcal L} \longrightarrow \underline R_U \end{align*}$$

defined by $\langle \gamma _1 \cdot \overline {\iota },\gamma _2\cdot \iota \rangle = \gamma _1\gamma _2\in \pi _1(G) \subseteq R$ for every $\iota ,\gamma _1,\gamma _2$ . One readily verifies that this is well-defined. Extending it in a ${\mathbb K}$ -bilinear way makes it automatically R-bilinear and it induces the above isomorphism.

Remark 2.18. The observation from Remark 2.17 also holds if we replace R by ${\mathbb K}[H]$ , where H is a finite index subgroup of $\pi _1(G)$ . Indeed, let $H\gamma _1,\ldots ,H\gamma _n$ be the distinct elements of $\pi _1(G)/H$ , seen as right cosets. Then, $\{\gamma _1\cdot \iota ,\ldots ,\gamma _n\cdot \iota \}$ is a local ${\mathbb K}[H]$ -basis of ${\mathcal L}$ , and ${\mathcal L}$ is a rank n free ${\mathbb K}[H]$ -module. Similarly, since $\pi _1(G)$ is abelian, $\{\gamma _1^{-1}\cdot \overline {\iota },\ldots ,\gamma _n^{-1}\cdot \overline {\iota }\}$ is a local ${\mathbb K}[H]$ -basis of $\overline {{\mathcal L}}$ . We define the ${\mathbb K}[H]$ -bilinear pairing $\overline {{\mathcal L}}\times {\mathcal L}\to \underline {{\mathbb K}[H]}_U$ by the ${\mathbb K}$ -bilinear extension of the pairing given by

$$ \begin{align*}\langle \delta_1\gamma_i^{-1}\cdot\overline{\iota},\delta_2\gamma_j\cdot\iota\rangle=\left\{\begin{array}{lr} 0 &\text{if }i\neq j,\\ \delta_1\delta_2 &\text{if }i= j \end{array}\right.\end{align*} $$

for all $\delta _1,\delta _2\in H$ , $i,j\in \{1,\ldots ,n\}$ . One readily verifies that this is well-defined and induces the isomorphism of sheaves of ${\mathbb K}[H]$ -modules

$$ \begin{align*} {\mathcal{H}\mathrm{om}}_{{\mathbb K}[H]}({\mathcal L},{\mathbb K}[H])\cong\overline{{\mathcal L}}, \end{align*} $$

which is also an isomorphism of sheaves of R-modules.

Remark 2.23 ( $R_m$ and $R/\mathfrak m^m$ are isomorphic R-modules).

Let be the augmentation ideal of R, and let $m\geq 1$ . The image of $\mathfrak m$ by the ${\mathbb K}$ -linear ring monomorphism described in Definition 2.22 lies in $\prod _{j=1}^\infty \operatorname {\mathrm {Sym}}^j H_1(G,{\mathbb K})$ , so one gets an induced ring homomorphism $R/\mathfrak m^m\to R_m$ , which is also an R-module homomorphism. In fact, it is an R-module isomorphism. To see this, it suffices to see that $R/\mathfrak m^m\to R_m$ is an isomorphism of ${\mathbb K}$ -vector spaces. Let $\gamma _1,\ldots ,\gamma _r$ be a basis of generators of $\pi _1(G)$ , and let us consider the bases

$$ \begin{align*} \bigcup_{j=0}^{m-1}\{(\gamma_1-1)^{i_1}\cdots(\gamma_r-1)^{i_r}&\mid i_l\geq 0 \text{ for all }l=1,\ldots,r,\quad i_1+\cdots+i_r= j\},\\ \bigcup_{j=0}^{m-1}\{(\log\gamma_1)^{i_1}\cdots(\log\gamma_r)^{i_r}&\mid i_l\geq 0 \text{ for all }l=1,\ldots,r,\quad i_1+\cdots+i_r= j\} \end{align*} $$

of $R/\mathfrak m^m$ and $R_m$ , respectively, where both are ordered in the same way by increasing order of j, and amongst the ones with the same j, by lexicographical order. The square matrix representing the R module homomorphism between $R/\mathfrak m^m$ and $R_m$ (seen as a ${\mathbb K}$ -linear homomorphism) in these bases is triangular with ones along the diagonal.

Remark 2.24. Let ${\mathfrak m}\subset R$ be as in the previous remark. Taking inverse limits in the isomorphism between $\frac {R}{{\mathfrak m}^m}$ and $R_m$ from the previous remark, one obtains an R-module isomorphism

$$ \begin{align*} \varprojlim_m \frac{R}{{\mathfrak m}^m}\cong R_\infty. \end{align*} $$

Remark 2.26 (Relationship between homology and cohomology).

For all $m\neq 0$ , we have a chain of natural isomorphisms

$$ \begin{align*} {\mathcal{H}\mathrm{om}}_{\mathbb K}(R_m \otimes_R {\mathcal L},{\mathbb K}) &\cong {\mathcal{H}\mathrm{om}}_R({\mathcal L} ,\operatorname{\mathrm{Hom}}_{\mathbb K}(R_m,{\mathbb K})) & \text{(Tensor-hom adjunction)}\\ &= {\mathcal{H}\mathrm{om}}_R({\mathcal L} ,R_{-m})\\ &\cong {\mathcal{H}\mathrm{om}}_R({\mathcal L}, R) \otimes_R R_{-m} & \text{(Because }{\mathcal L}\text{ is locally free over }R) \\ &\cong\overline{{\mathcal L}}\otimes_R R_{-m}.& \text{Remark~2.17} \end{align*} $$

Since R is commutative, one can identify $\overline {{\mathcal L}}\otimes _R R_{-m}$ with $R_{-m}\otimes _R\overline {{\mathcal L}}$ . We apply Proposition 2.25 to $L = R_m \otimes _R {\mathcal L}$ , and the above to get R and $R_\infty $ -module isomorphisms for all $i\geq 0$ and $m\neq 0$ :

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\mathbb K}(H_i(U,R_m \otimes_R {\mathcal L}),{\mathbb K}) &\cong H^i(U,{\mathcal{H}\mathrm{om}}_{\mathbb K}(R_m \otimes_R {\mathcal L},{\mathbb K})). & \text{Proposition~2.25} \\ &\cong H^i(U,R_{-m}\otimes_R\overline{{\mathcal L}} ). \end{align*} $$

Remark 2.27. Let H be a finite index subgroup of $\pi _1(G)$ , and let $\pi _H:G_H\to G$ be the corresponding finite cover. $G_H$ is the quotient of $TG$ by H, where $H\leq \pi _1(G)$ acts by deck transformations. In particular, $G_H$ is a commutative algebraic group, which is in fact a semiabelian variety (see [Reference Conrad8, Section 3], for example), and $\pi _H$ is a morphism of algebraic groups. In that case, we may define , and $R^H_\infty $ , $R^H_m$ and $R^H_{-m}$ analogously as in Definition 2.20 using $H_1(G_H,{\mathbb K})$ instead of $H_1(G,{\mathbb K})$ . Note that ${\mathcal L}$ is locally free of finite rank as a sheaf of $R^H$ -modules. Hence, using Remark 2.17, the argument in Remark 2.26 can be replicated to obtain $R^H$ and $R^H_\infty $ -module isomorphisms

$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\mathbb K}(H_i(U,R^H_m\otimes_{R^H}{\mathcal L}),{\mathbb K})\cong H^i(U,R_{-m}^H\otimes_{R^H}\overline{{\mathcal L}}) \end{align*} $$

for all $m\neq 0$ and all $i\geq 0$ .

Remark 2.32 (Existence of the Albanese map).

The Albanese map $\alpha _U$ exists for any smooth connected complex algebraic variety (see [Reference Fujino21]), and hence the Albanese variety $G_U$ is well defined up to algebraic isomorphism, which, up to translation, will be a group homomorphism as well by Remark 2.8. Once $G_U$ is fixed, $\alpha _U$ is uniquely defined defined up to translation in $G_U$ and isomorphism of algebraic groups from $G_U$ to itself.

Remark 2.33. If U is a smooth connected complex projective variety, $G_U$ in Definition 2.31 is an abelian variety, and $\alpha _U$ is the usual Albanese map.

Remark 2.35. Consider the pullback of (1.1) for the map $\alpha _U$ . If $H_1(U,{\mathbb Z})$ is torsion free, $U^{\alpha _U}$ is the universal abelian cover of U.

Remark 2.46. Our main example of a dgla is the following: suppose $(M^{\bullet },d,\cdot )$ is a differential graded associative algebra, that is, $(M^{\bullet },d)$ is a complex, and $\cdot $ is an associative product for which $d(a\cdot b)=(da)\cdot b + (-1)^{|a|} a\cdot (db)$ . Then, automatically $(M^{\bullet },d,[,])$ is a dgla with the bracket given by

$$\begin{align*}[a,b] = a\cdot b - (-1)^{|a||b|} b\cdot a. \end{align*}$$

Remark 2.48. Let $C^{\bullet }$ be a bounded complex of ${\mathbb K}$ -vector spaces. Then the vector space of ${\mathbb K}$ -linear endomorphisms $\operatorname {\mathrm {End}}^{\bullet }_{\mathbb K}(C^{\bullet })$ is a differential graded associative algebra, where the homogeneous elements of degree k are linear maps $\phi $ such that $\phi (C^j)\subseteq C^{j+k}$ . The product is composition, and the differential is the graded commutator with d, that is, if $\phi $ has degree $|\phi |$ ,

Note that with this differential, $H^j(\operatorname {\mathrm {End}}^{\bullet }_{\mathbb K}(C^{\bullet }))$ is the group of homotopy classes of morphisms of complexes $C^{\bullet }\to C^{\bullet }[j]$ (recall that by convention the differential on $C^{\bullet }[j]$ is $(-1)^jd$ ). By Remark 2.46, $\operatorname {\mathrm {End}}_{\mathbb K}^{\bullet }(C^{\bullet })$ is a dgla with the bracket given by the commutator.

Remark 2.49. Let $S_1\to S_2$ be a ring map. If $L^{\bullet }$ is a $S_1$ -dgla, $S_2\otimes _{S_1} L^{\bullet }$ becomes a dgla with the bracket . We are interested in the ring map ${\mathbb K}\to A$ and the ${\mathbb K}$ -dgla $L^{\bullet } = \operatorname {\mathrm {End}}^{\bullet }_{\mathbb K}(C^{\bullet })$ . In this case, $A\otimes _{\mathbb K} L^{\bullet } = \operatorname {\mathrm {End}}^{\bullet }_A(A\otimes _{\mathbb K} C^{\bullet })$ , and the Lie bracket extended from ${\mathbb K}$ coincides with the commutator of endomorphisms. Furthermore, is a sub-dgla.

Remark 2.51. In the notation of Lemma 2.50, if we let M be a (left) A-module, the analogous statements can be made for the complexes $M\otimes _{\mathbb K} C^{\bullet }$ , since these are simply obtained from $A\otimes _{\mathbb K} C^{\bullet }$ by tensoring over A with M.

Remark 2.52. Let ${\mathbb K}$ be a field, let A be a local Artinian ${\mathbb K}$ -algebra with maximal ideal ${\mathfrak m}$ . Let $(C^{\bullet },d,\cdot )$ be an A-cdga and let $(M^{\bullet },d)$ be a $C^{\bullet }$ -differential graded (left) module. In other words, multiplication induces an A-dga homomorphism $C^{\bullet }\to \operatorname {\mathrm {End}}_A(A\otimes _{\mathbb K} M^{\bullet })$ . Let us abuse notation and use the same letter for elements of $C^{\bullet }$ and their multiplication endomorphism.

1. 1. For any $\phi \in {\mathfrak m} C^1$ , $[\phi ,\phi ]=0$ , so the Maurer-Cartan equation is equivalent to $[d,\phi ]=0\in \operatorname {\mathrm {End}}_A(A\otimes _{\mathbb K} M^{\bullet })$ , and therefore the condition that $\phi $ is closed in $C^{\bullet }$ is sufficient for the Maurer-Cartan equation to hold.

2. 2. For any $\phi \in {\mathfrak m} C^1$ and $\rho \in {\mathfrak m} C^0$ , $[\rho ,d+\phi ]=-d \rho $ . Therefore, $[\rho ,[\rho ,d+\phi ]]=0$ . Applying Lemma 2.50, $e^\rho $ is an isomorphism between $(A\otimes _{\mathbb K} M^{\bullet },d+\phi )$ and $(A\otimes _{\mathbb K} M^{\bullet },d+\phi -d \rho )$ .

The same result also holds in the case of sheaves, as in Lemma 2.50.

Remark 2.54 (MHS in homology).

Let X be a complex algebraic variety. By work of Deligne [Reference Deligne9, Reference Deligne10], $H^i(X,{\mathbb K})$ carries a canonical and functorial MHS for all $i\geq 0$ . Since $H^i(X,{\mathbb K})$ is finite dimensional, its dual is canonically isomorphic to $H_i(X,{\mathbb K})$ . By Definition-Proposition 2.53, $H_i(X,{\mathbb K})$ also carries a canonical and functorial MHS.

Remark 2.55 (MHS on $R_m$ and $R_{-m}$ ).

Let $m>0$ , and let $R_m$ and $R_{-m}$ as in Definition 2.20. Since

$$ \begin{align*}R_{m}\cong\prod\limits_{j=0}^{m-1} \operatorname{\mathrm{Sym}}^j H_1(G,{\mathbb K})=\bigoplus\limits_{j=0}^{m-1}\operatorname{\mathrm{Sym}}^j H_1(G,{\mathbb K})\end{align*} $$

and the direct sum of MHSs is an MHS, the MHS on $H_1(G,{\mathbb K})$ endows both $R_m$ and $R_{-m}$ (its ${\mathbb K}$ -dual) with an MHS by Definition-Proposition 2.53.

Remark 2.56 (Pro-MHS).

Let $V=\varprojlim _m V_m$ , where each $V_m$ is a ${\mathbb K}$ -MHS for all $m\geq 1$ , and all the morphisms involved are MHS morphisms. This data can be regarded as a pro-MHS. There is a category of pro-MHS that can be constructed as the usual abstract nonsense pro-completion: one would simply have to replace the index set ${\mathbb Z}_{> 0}$ by a more general filtered set (or category) to define a pro-MHS in full generality. Morphisms are defined as follows. Suppose we are given two pro-MHS constructed in this way, $V=\varprojlim _m V_m$ and $W=\varprojlim _{m'} W_{m'}$ . Then,

Plainly, a morphism consists of: for every $m'$ one must choose an m and a morphism $V_m\to W_{m'}$ , and these must be all compatible in the obvious ways. In this paper, the only such morphisms that will appear will be constructed in the most naive way: for every $m'$ , we will take $m=m'$ . That is, given morphisms of MHS $V_m\to W_m$ for all $m\geq 1$ commuting with the linear maps $V_{m'}\to V_{m}$ and $W_{m'}\to W_{m}$ for all $m'\geq m\geq 1$ , we obtain a morphism of pro-MHSs $V\to W$ .

Remark 2.57. Inverse limits are left exact, and in the context of inverse limits of finite dimensional vector spaces, they are also right exact, since these inverse limits always satisfy the Mittag-Leffler condition. Hence, the category of pro-MHS has kernels, images and cokernels, and they coincide with the kernels, images and cokernels of the underlying vector spaces V.

Remark 2.61. The two definitions of Tate twist above are compatible in the following sense. Let $\mathbb {H}^i(X,{\mathcal K}^{\bullet }_{\mathbb K}(j))$ (resp. $\mathbb {H}^i(X,{\mathcal K}^{\bullet }_{\mathbb K})$ ) be the ${\mathbb K}$ -MHS induced in hypercohomology by the ${\mathbb K}$ -mixed Hodge complex of sheaves ${\mathcal K}^{\bullet }(j)$ (resp. ${\mathcal K}^{\bullet }$ ). By [Reference Peters and Steenbrink41, Theorem 3.18], $\mathbb {H}^i(X,{\mathcal K}^{\bullet }_{\mathbb K}(j))=\mathbb {H}^i(X,{\mathcal K}^{\bullet }_{\mathbb K})(j)$ .

Remark 2.64. Suppose ${\mathcal K}^{\bullet }$ is a ${\mathbb K}$ -mixed Hodge complex of sheaves on X. By [Reference Peters and Steenbrink41, Theorem 3.18.II] the hypercohomology vector spaces $\mathbb {H}^*(X, {\mathcal K}^{\bullet }_{\mathbb K})$ inherit ${\mathbb K}$ -mixed Hodge structures. Furthermore, it can be easily shown that

$$\begin{align*}\mathbb{H}^{*}(X, {\mathcal K}^{\bullet}_{\mathbb K}[r]) \cong \mathbb{H}^{*+r}(X, {\mathcal K}^{\bullet}_{\mathbb K}),\end{align*}$$

where the ${\mathbb K}$ -mixed Hodge structure on the left-hand side has been induced by the translated ${\mathbb K}$ -mixed Hodge complex ${\mathcal K}^{\bullet }[r]$ .

Remark 2.67. ${\mathcal A}^{\bullet }_{U,{\mathbb C}}$ (and thus also ${\mathcal A}^{\bullet }_{U,{\mathbb R}}$ ) is a complex of $j_*$ -acyclic and $\Gamma $ -acyclic sheaves. The $\Gamma $ -acyclicity is well known (see [Reference Kashiwara and Schapira27, p. 127], for example). For the $j_*$ -acyclicity, it suffices to show that ${\mathbb H}^i(V_x\cap U,{\mathcal A}^k_{U,{\mathbb C}})=0$ for all $i>0$ , $k\geq 0$ , $x\in X$ and certain $V_x$ neighborhoods of x in X forming a basis. Let $Y=V_x\cap U$ , which is a complex (and thus real) analytic manifold. As a real analytic manifold, Y can be embedded into $Y\times \overline Y$ as the diagonal, where $\overline Y$ is the complex conjugate of Y, and the restriction of $\mathcal {O}_{Y\times \overline Y}$ (the sheaf of complex analytic functions on $Y\times \overline Y$ ) to Y is ${\mathcal A}^0_{U,{\mathbb C}}$ . By [Reference Cieliebak and Eliashberg6, Proposition 5.42], Y possesses arbitrarily small neighborhoods in $Y\times \overline Y$ which are Stein. Let W be one such Stein neighborhood. By Oka’s coherence theorem and Cartan’s theorem B, one gets ${\mathbb H}^i(W,{\mathcal O}_{Y\times \overline Y})=0$ for all $i>0$ , so $R\Gamma (W,{\mathcal O}_{Y\times \overline Y})\cong \Gamma (W,{\mathcal O}_{Y\times \overline Y})$ . Since Y is closed in the paracompact space W, taking direct limits on W approaching Y yields $R\Gamma (Y,{\mathcal A}^0_{U,{\mathbb C}})=\varinjlim\limits_{ W}\Gamma (W,{\mathcal O}_{Y\times\overline Y})=\Gamma(Y,{\mathcal A}^0_{U,{\mathbb C}})$ , obtaining the $j_*$ -acyclicity of ${\mathcal A}^0_{U,{\mathbb C}}$ . For $k\geq 1$ we follow the same argument, replacing ${\mathcal O}_{Y\times \overline Y}$ by $\Omega _{Y\times \overline Y}^k$ , which is locally free as an ${\mathcal O}_{Y\times \overline Y}$ -module and thus it is also coherent.

Remark 2.74. The isomorphism $Rj_* \underline {{\mathbb R}}_U\to j_*{\mathcal E}^{\bullet }_{U,{\mathbb R}}$ described in the proof of Corollary 2.72 is the one used to endow $H^*(U,{\mathbb R})$ with Deligne’s canonical mixed Hodge structure, using Deligne’s mixed Hodge complex of sheaves described at the beginning of this section.

The mixed Hodge complex of Navarro Aznar (Definition 2.69) also induces a mixed Hodge structure on $H^*(U,{\mathbb R})$ via the composition of $Rj_* \underline {{\mathbb R}}_U\to j_*{\mathcal E}^{\bullet }_{U,{\mathbb R}}$ above with ${\mathcal A}^{\bullet }_{X,{\mathbb C}}(\log D) \hookrightarrow j_*{\mathcal E}^{\bullet }_{U,{\mathbb C}}$ . By Corollary 2.72, both of these mixed Hodge structures on $H^*(U,{\mathbb R})$ coincide. Consequently, this MHS coincides with the one induced by the modified complex ${\mathcal N}_{X,D,n}^{\bullet }$ from Definition-Proposition 2.73 for all $n\geq \max \{2,\dim _{\mathbb R} U\}$ .