2.1 Picard group and multipliers

Let us start with a quick overview of holomorphic line bundles, sufficient for our purpose (see [Reference Griffiths and Harris11, Reference Huybrechts15]). For an open subset $U \subset M$ , denote by $\mathcal {O}(U)$ the space of holomorphic functions on U, and by $\mathcal {O}^\times (U)$ the subspace of nonvanishing functions. The corresponding sheaves on M are denoted by $\mathcal {O}$ and $\mathcal {O}^\times $ , respectively. Let $\pi \colon L \to M$ be a line bundle and consider an open cover $\mathcal {U} = \{U_\alpha \}$ of M that trivializes $\pi $ (i.e., $\pi ^{-1}(U_\alpha ) \simeq U_\alpha \times \mathbb C$ ). The line bundle is uniquely determined by its transition functions $g_{\alpha \beta } \in \mathcal {O}^\times (U_\alpha \cap U_\beta )$ , which satisfy $ g_{\alpha \beta } g_{\beta \gamma } = g_{\alpha \gamma }$ . Two collections of transition functions $\{g_{\alpha \beta }\}$ , $\{\tilde g_{\alpha \beta }\}$ define the same bundle if and only if $\tilde g_{\alpha \beta } = \frac {f_\alpha }{f_\beta } g_{\alpha \beta }$ for some functions $f_\alpha \in \mathcal {O}^\times (U_\alpha )$ .

This leads to a description of line bundles in terms of Čech cohomology. Define the complex

$$\begin{align*}0 \xrightarrow{} \prod_{\alpha} \mathcal{O}^\times (U_\alpha) \xrightarrow{\delta^0} \prod_{\alpha \neq \beta} \mathcal{O}^\times(U_\alpha \cap U_\beta) \xrightarrow{\delta^1} \prod_{\alpha \neq \beta \neq \gamma \neq \alpha}\mathcal{O}^\times(U_\alpha \cap U_\beta \cap U_\gamma) \xrightarrow{} \cdots, \end{align*}$$

with differentials given by

$$ \begin{align*} (\delta^q \mu)_{\alpha_0 \cdots \alpha_{q+1}} = \prod_{i=0}^{q+1} \mu^{(-1)^{i+1}}_{\alpha_0 \cdots \hat \alpha_i \cdots \alpha_{q+1}} \Big|_{U_{\alpha_0} \cap \cdots \cap U_{\alpha_{q+1}}}, \qquad \mu = \{\mu_{\alpha_0 \cdots \alpha_q}\} \in \prod_{\alpha_0,\dots, \alpha_q} \mathcal{O}^\times(U_{\alpha_0} \cap \cdots \cap U_{\alpha_q}). \end{align*} $$

In particular, $\delta ^0$ and $\delta ^1$ are defined in the following way:

$$ \begin{align*} (\delta^0\mu)_{\alpha \beta} &= \mu_{\alpha}/\mu_{\beta}, \qquad && \mu = \{\mu_\alpha\} \in \prod_{\alpha} \mathcal{O}^\times (U_\alpha),\\ (\delta^1\nu)_{\alpha \beta \gamma} &=\frac{ \nu_{\alpha \gamma}}{ \nu_{\alpha \beta} \nu_{\beta \gamma}} , \qquad && \nu = \{\nu_{\alpha \beta}\} \in \prod_{\alpha \neq \beta} \mathcal{O}^\times(U_\alpha \cap U_\beta). \end{align*} $$

The first Čech-cohomology with respect to the fixed open cover $\mathcal {U}$ , defined by ${\check {\mathrm {H}}}^1(\mathcal {U}, \mathcal {O}^\times ) = \ker (\delta ^1)/\mathrm {im}(\delta ^0)$ , is the group of transition functions on $\mathcal {U}$ modulo the above equivalence relation.

The Picard group $\mathrm {Pic}(M)$ of equivalence classes of holomorphic line bundles over M can be described in terms of this cohomology group as follows:

• • If all line bundles are trivializable on the open charts in $\mathcal {U}$ (for instance, each $U_\alpha $ is biholomorphic to a polydisc with a possible factor $\mathbb C^\times $ ), then $\mathrm {Pic}(M) \simeq {\check {\mathrm {H}}}^1(\mathcal {U},\mathcal {O}^\times )$ .

• • In general, $\mathrm {Pic}(M) \simeq {\check {\mathrm {H}}}^1(M,\mathcal {O}^\times ):= \varinjlim {\check {\mathrm {H}}}^1(\mathcal {U}, \mathcal {O}^\times )$ is the direct limit as $\mathcal {U}$ becomes finer.

In both cases, the identification is a group isomorphism. In particular, if the conditions of Leray’s theorem hold, the first description is applicable (see [Reference Griffiths and Harris11, p.40] or the simpler Theorem 12.8 of [Reference Forster8], which will usually be sufficient for us).

For instance, the canonical line bundle $K_M = \Lambda ^{\dim M} T^*M$ (see [Reference Huybrechts15, Ch. 2.2]) always admits a canonical lift of ${\mathfrak g} \subset \mathcal {D}(M)$ . Thus, it is an (often nontrivial) ${\mathfrak g}$ -equivariant line bundle.

In general, the lift of a vector field $X\in {\mathfrak g}$ can be defined on $\pi ^{-1}(U_\alpha )\simeq U_\alpha \times {\mathbb C}$ by

$$\begin{align*}\hat X|_{U_\alpha} = X|_{U_\alpha} + \lambda_\alpha(X) u\partial_u,\quad \lambda_\alpha \in {\mathfrak g}^* \otimes \mathcal{O}(U_\alpha), \end{align*}$$

similar to formula (4.1) in [Reference Fels and Olver7]. To simplify notation, we will write X instead of $X|_{U_\alpha }$ when there is no room for confusion. The condition $[\hat X, \hat Y] = \widehat {[X,Y]}$ for each $X,Y\in {\mathfrak g}$ implies that $\lambda _\alpha $ satisfies

(2.1) $$ \begin{align} X(\lambda_\alpha(Y))-Y(\lambda_\alpha(X))= \lambda_{\alpha}([X,Y]), \qquad \forall X \in \mathfrak g. \end{align} $$

Changing the coordinate function on the fiber, $v=e^{\mu _\alpha }u$ for some function $\mu _\alpha \in \mathcal {O}(U_\alpha )$ gives

$$ \begin{align*} X+\lambda_\alpha(X)\,u \partial_u = X+(\lambda_\alpha(X)+X(\mu_\alpha))\,v \partial_v. \end{align*} $$

In this sense, two lifts $\lambda _\alpha , \tilde \lambda _\alpha $ on $U_\alpha $ are equivalent if and only if there exists a $\mu _\alpha $ satisfying

(2.2) $$ \begin{align} \tilde \lambda_\alpha(X) = \lambda_\alpha(X) + X( \mu_\alpha), \qquad \forall X \in \mathfrak g. \end{align} $$

The conditions (2.1) and (2.2) can be interpreted in terms of Lie algebra cohomology of $\mathfrak g$ with coefficients in the $\mathfrak g$ -module $\mathcal {O}(U_\alpha )$ . Consider the Chevalley-Eilenberg complex

$$\begin{align*}0 \xrightarrow{} \mathcal{O}(U_\alpha) \xrightarrow{d^0} {\mathfrak g}^*\otimes \mathcal{O}(U_\alpha) \xrightarrow{d^1} \Lambda^2 {\mathfrak g}^*\otimes \mathcal{O}(U_\alpha) \xrightarrow{} \cdots \end{align*}$$

where the maps $d^0$ and $d^1$ are given by

$$\begin{align*} (d^0 \mu_\alpha)(X) &= X(\mu_\alpha), \qquad &&\mu_\alpha \in \mathcal{O}(U_\alpha), \\ (d^1 \lambda_\alpha)(X,Y) &= X(\lambda_\alpha(Y))-Y(\lambda_\alpha(X))-\lambda_\alpha([X,Y]), \qquad &&\lambda_\alpha \in {\mathfrak g}^* \otimes \mathcal{O}(U_\alpha), \end{align*}$$

for $X,Y \in {\mathfrak g}$ (see [Reference Chevalley and Eilenberg6]). Notice that $\mathrm {Hom}({\mathfrak g},F)={\mathfrak g}^* \otimes F$ when one of the factors is finite-dimensional. If both factors are infinte-dimensional, a completion of the tensor product is required. We omit this from the notation, understanding by default that $\Lambda ^i {\mathfrak g}^*\otimes F$ may stand for $\mathrm {Hom}(\Lambda ^i {\mathfrak g}^*,F)$ here and below.

Define the cohomology groups

$$\begin{align*}{\mathrm{H}}^0(\mathfrak g,\mathcal{O}(U_\alpha)) = \ker(d^0), \qquad {\mathrm{H}}^i(\mathfrak g, \mathcal{O}(U_\alpha)) = \ker(d^i)/\mathrm{im}(d^{i-1}), \quad i>0. \end{align*}$$

It is clear that $\lambda _\alpha \in \mathfrak g^* \otimes \mathcal {O}(U_\alpha )$ defines a lift of $\mathfrak g$ to $U_\alpha \times \mathbb C$ if and only if $d^1 \lambda _\alpha =0$ . Furthermore, two cocycles $\lambda _\alpha , \tilde \lambda _\alpha $ define equivalent lifts if and only if $\tilde \lambda _\alpha = \lambda _\alpha + d^0 \mu _\alpha $ for some $\mu _\alpha \in \mathcal {O}(U_\alpha )$ . Thus, equivalence classes of lifts of $\mathfrak g|_{U_\alpha }$ to $U_\alpha \times \mathbb C$ are in one-to-one correspondence with elements in ${\mathrm {H}}^1(\mathfrak g, \mathcal {O}(U_\alpha ))$ . (Note also that ${\mathrm {H}}^0({\mathfrak g}, \mathcal {O}(U_\alpha ))=\mathcal {O}(U_\alpha )^{\mathfrak g}$ consists of ${\mathfrak g}$ -invariants.)

Remark 2.2. If $U_\alpha $ is a polydisc for each $\alpha $ , then any function in $\mathcal {O}^\times (U_\alpha )$ is of the form $e^\mu $ , and the argument above works. If $U_\alpha $ is a general open set, one replaces $e^{\mu _\alpha } f_\alpha $ with $\mu _\alpha f_\alpha $ , where $\mu _\alpha \in \mathcal {O}^\times (U_\alpha )$ . Then the local lifts are in one-to-one correspondence with elements in the cohomology group of the complex

(2.3) $$ \begin{align} 0 \xrightarrow{} \mathcal{O}^\times (U_\alpha) \xrightarrow{d^0 \log} {\mathfrak g}^*\otimes \mathcal{O}(U_\alpha) \xrightarrow{d^1} \Lambda^2 {\mathfrak g}^*\otimes \mathcal{O}(U_\alpha) \xrightarrow{} \cdots. \end{align} $$

We will use this slightly modified complex below with the notation

$$\begin{align*}{\tilde{\mathrm{H}}}^1({\mathfrak g},\mathcal{O}(U_\alpha)) = \frac{\ker(d^1)}{\mathrm{im}(d^0 \log)}. \end{align*}$$

Elements in ${\tilde {\mathrm {H}}}^1({\mathfrak g},\mathcal {O}(U_\alpha ))$ yield local lifts of ${\mathfrak g}$ to $\pi ^{-1}(U_\alpha )$ that may not glue together to a global lift on L. On $U_\alpha \cap U_\beta $ , a lift is given by both $X+\lambda _\alpha (X)\,u_\alpha \partial _{u_\alpha }$ and $X+\lambda _\beta (X)\,u_\beta \partial _{u_\beta }$ . The fiber coordinates relate on overlaps by $u_\alpha = g_{\alpha \beta } u_\beta $ , where the transition functions $\{g_{\alpha \beta }\}$ represent an element of ${\check {\mathrm {H}}}^1(\mathcal {U},\mathcal {O}^\times (M))$ . Thus, $X+\lambda _\alpha (X)\,u_\alpha \partial _{u_\alpha }$ becomes $X+(\lambda _\alpha (X)-X(g_{\alpha \beta })/g_{\alpha \beta })\,u_\beta \partial _{u_\beta }$ , resulting in the following compatibility condition on $U_\alpha \cap U_\beta $ :

(2.4) $$ \begin{align} \lambda_\alpha(X)-\lambda_\beta(X)=\frac{X(g_{\alpha \beta})}{g_{\alpha \beta}} = X(\log g_{\alpha \beta}), \qquad \forall X \in \mathfrak g. \end{align} $$

2.2 A double complex

To better understand the compatibility condition, consider the double complex

where $C^{p,q}$ are given by

$$ \begin{align*} C^{0,q} &= \prod_{\alpha_0, \cdots, \alpha_q} \mathcal{O}^\times (U_{\alpha_0} \cap \cdots\cap U_{\alpha_q}),\\ C^{p,q} &= \prod_{\alpha_0, \cdots, \alpha_q} \Lambda^p \mathfrak g^* \otimes \mathcal{O}(U_{\alpha_0} \cap \cdots\cap U_{\alpha_q}), \quad p \geq 1, \end{align*} $$

and the differentials $\delta ^{p,q} \colon C^{p,q} \to C^{p,q+1}$ and $d^{p,q} \colon C^{p,q} \to C^{p+1,q}$ are defined for $p=0$ by

$$ \begin{align*} (\delta^{0,q} \mu)_{\alpha_0 \cdots \alpha_{q+1}} &= \prod_{i=0}^{q+1} \mu_{\alpha_0 \cdots \hat \alpha_i \cdots \alpha_{q+1}}^{(-1)^{i+1}} \Big|_{U_{\alpha_0} \cap \cdots \cap U_{\alpha_{q+1}}}, \\ (d^{0,q}\mu_{\alpha_0 \cdots \alpha_q})(X) &= X(\log \mu_{\alpha_0 \cdots \alpha_q}) = \frac{X(\mu_{\alpha_0 \cdots \alpha_q})}{\mu_{\alpha_0 \cdots \alpha_q}}, \end{align*} $$

and for $p> 0$ by

$$ \begin{align*} (\delta^{p,q} \mu)_{\alpha_0 \cdots \alpha_{q+1}} &= \sum_{i=0}^{q+1} (-1)^{i+1} \mu_{\alpha_0 \cdots \hat \alpha_i \cdots \alpha_{q+1}} \Big|_{U_{\alpha_0} \cap \cdots \cap U_{\alpha_{q+1}}}, \\ (d^{p,q}\mu_{\alpha_0 \cdots \alpha_q})(X_0, \dots, X_p) &=\sum_{i=0}^{p} (-1)^i X_i(\mu_{\alpha_0 \cdots \alpha_q}(X_0,\dots, X_{i-1}, X_{i+1}, \dots, X_p))\\ +\sum_{i<j}(-1)^{i+j}&(\mu_{\alpha_0 \cdots \alpha_q}([X_i,X_j],X_0,\dots, X_{i-1}, X_{i+1}, \dots, X_{j-1}, X_{j+1}, \dots, X_p)). \end{align*} $$

We will sometimes write $C^{p,q}({\mathfrak g},\ \mathcal {U})$ for precision when there would otherwise be ambiguity. The horizontal lines (q fixed) are nearly Chevalley-Eilenberg complexes of $\mathfrak g$ with coefficients in the $\mathfrak g$ -modules $\mathcal {O}(U_\alpha ), \mathcal {O}(U_{\alpha } \cap U_{\beta })$ , etc; however, $(C^{0,q},d^{0,q})$ are adjusted in accordance with Remark 2.2. The vertical lines (p fixed) are Čech complexes with respect to the open cover $\mathcal {U}$ .

The total complex corresponding to the double complex $C^{\bullet , \bullet }$ is defined as follows:

$$\begin{align*}\mathrm{Tot}^r(C) = \prod_{p+q=r} C^{p,q}, \qquad \partial^r=\sum_{p+q=r} (d^{p,q}+(-1)^p\delta^{p,q})\colon \mathrm{Tot}^r(C) \to \mathrm{Tot}^{r+1}(C).\end{align*}$$

The identity $\partial ^{i+1} \circ \partial ^i =0$ expresses the fact that the double complex is a commutative diagram. The cohomology groups of the total complex are defined in the usual way:

$$\begin{align*}{\mathrm{H}}^0(\mathrm{Tot}^\bullet(C))= \ker(\partial^0), \qquad {\mathrm{H}}^i(\mathrm{Tot}^\bullet(C))= \frac{\ker(\partial^i)}{\mathrm{im}(\partial^{i-1})}.\end{align*}$$

The double complex also gives us several complexes of cohomology groups. The cohomology groups with respect to $d^{i,j}$ (with j fixed) make up the following complexes:

Simultaneously, the cohomology groups with respect to $\delta ^{i,j}$ (with i fixed) also give complexes:

Thus, we get the induced cohomology groups

$$\begin{align*}{\mathrm{H}}_\delta^i({\mathrm{H}}_d^j(C^{\bullet, \bullet})) = \frac{\ker(\delta^{j,i}_*)}{\mathrm{im}(\delta^{j,i-1}_*)}, \qquad {\mathrm{H}}_d^i({\mathrm{H}}_\delta^j(C^{\bullet,\bullet})) = \frac{\ker(d^{i,j}_*)}{\mathrm{im}(d^{i-1,j}_*)}, \end{align*}$$

for $i \geq 1$ and

$$\begin{align*}{\mathrm{H}}_\delta^0({\mathrm{H}}_d^j(C^{\bullet, \bullet})) = \ker(\delta_*^{j,0}), \qquad {\mathrm{H}}_d^0({\mathrm{H}}_\delta^j(C^{\bullet,\bullet})) = \ker(d_*^{0,j}). \end{align*}$$

In the general setting of the total complex, we have the two projections (homomorphisms)

defined by

$$\begin{align*}\Phi_1([(g,\lambda)]) = [g], \qquad \Phi_2([(g,\lambda)]) = [\lambda], \end{align*}$$

where $[(g,\lambda )]$ denotes the equivalence class of $(g,\lambda ) \in \ker (\partial ^1)$ . We use the notation $g = \{g_{\alpha \beta }\}$ for an element in $C^{0,1}$ and $\lambda =\{\lambda _\alpha \}$ for an element in $C^{1,0}$ , and note that

$$\begin{align*}{\mathrm{H}}_{\delta}^1(C^{0,\bullet})={\check{\mathrm{H}}}^1(\mathcal{U},\mathcal{O}^\times), \qquad {\mathrm{H}}_{d}^1(C^{\bullet,0})= \prod_\alpha {\tilde{\mathrm{H}}}^1(\mathfrak g,\mathcal{O}(U_\alpha)). \end{align*}$$

Lemma 2.4. The maps $\Phi _1, \Phi _2$ have the following kernels:

$$\begin{align*}\ker(\Phi_1) \simeq {\mathrm{H}}_d^1({\mathrm{H}}_\delta^0(C^{\bullet,\bullet})), \qquad \ker(\Phi_2) \simeq {\mathrm{H}}_\delta^1({\mathrm{H}}_d^0(C^{\bullet,\bullet})). \end{align*}$$

Lemma 2.4 is equivalent to exactness of the two sequences

(2.5) $$ \begin{align} &0 \longrightarrow {\mathrm{H}}_d^1({\mathrm{H}}_\delta^0(C^{\bullet,\bullet})) \longrightarrow {\mathrm{H}}^1(\mathrm{Tot}^\bullet(C)) \longrightarrow \mathrm{im}(\Phi_1) \longrightarrow 0 , \end{align} $$

(2.6) $$ \begin{align} &0 \longrightarrow {\mathrm{H}}_\delta^1({\mathrm{H}}_d^0(C^{\bullet,\bullet})) \longrightarrow {\mathrm{H}}^1(\mathrm{Tot}^\bullet(C)) \longrightarrow \mathrm{im}(\Phi_2) \longrightarrow 0 , \end{align} $$

and furthermore, we have

$$\begin{align*}\varinjlim {\mathrm{H}}_\delta^1({\mathrm{H}}_d^0(C^{\bullet,\bullet})) = {\check{\mathrm{H}}}^1(M, (\mathcal{O}^\times)^{\mathfrak g}), \qquad \varinjlim {\mathrm{H}}_d^1({\mathrm{H}}_\delta^0(C^{\bullet,\bullet})) = {\tilde{\mathrm{H}}}^1({\mathfrak g}, \mathcal{O}(M)), \end{align*}$$

where $(\mathcal {O}^\times )^{\mathfrak g} \subset \mathcal {O}^\times $ is the subsheaf of ${\mathfrak g}$ -invariants.

Lemma 2.6. The images of $\Phi _1,\Phi _2$ are given by the following exact sequences:

$$ \begin{align*} 0 \longrightarrow \mathrm{im}(\Phi_1) \longrightarrow {\mathrm{H}}_d^0({\mathrm{H}}_\delta^1(C^{\bullet, \bullet})) \longrightarrow {\mathrm{H}}_d^2({\mathrm{H}}_\delta^0(C^{\bullet,\bullet})), \\ 0 \longrightarrow \mathrm{im}(\Phi_2) \longrightarrow {\mathrm{H}}_\delta^0({\mathrm{H}}_d^1(C^{\bullet,\bullet})) \longrightarrow {\mathrm{H}}_\delta^2({\mathrm{H}}_d^0(C^{\bullet,\bullet})). \end{align*} $$

While $\mathrm {Pic}(M) = \varinjlim {\mathrm {H}}_\delta ^1(C^{0,\bullet })$ plays an important role, the group ${\mathrm {H}}_d^1(C^{\bullet ,0})$ will in general grow without bound as the cover $\mathcal {U}$ becomes finer. Therefore, as a counterpart to ${\check {\mathrm {H}}}^1(\mathcal {U},\mathcal {O}^\times )= {\mathrm {H}}_\delta ^1(C^{0,\bullet })$ , we define $\mathfrak {M}_{\mathfrak g}(\mathcal {U}) := {\mathrm {H}}_\delta ^0({\mathrm {H}}_d^1(C^{\bullet , \bullet }))$ which can be interpreted as the collection of local (infinitesimal) multipliers of ${\mathfrak g}$ with respect to the cover $\mathcal {U}$ that are equivalent on overlaps. Note that this is also a reasonable definition in this context due to Lemma 2.6.

The group $\mathfrak {M}_{\mathfrak g}(M)$ should not be confused with the group of global multipliers ${\tilde {\mathrm {H}}}^1({\mathfrak g},\mathcal {O}(M))$ , which is often trivial as the algebra of global functions $\mathcal {O}(M)$ may be small (for instance, ${\mathbb C}$ for compact M). To see the difference, consider an element $\lambda = \{\lambda _\alpha \} \in \ker (d^{1,0})$ . We have $[\lambda ] \in {\mathrm {H}}_d^1({\mathrm {H}}_\delta ^0(C^{\bullet ,\bullet }))$ if and only if $\lambda _\alpha = \lambda _\beta $ on $U_\alpha \cap U_\beta $ , and $[\lambda ] \in \mathfrak {M}_{\mathfrak g}(\{U_\alpha \})={\mathrm {H}}_\delta ^0({\mathrm {H}}_d^1(C^{\bullet ,\bullet }))$ if and only if $\lambda _\alpha = \lambda _\beta + d \log \mu _{\alpha \beta }$ for some $\mu =\{\mu _{\alpha \beta }\} \in \mathcal {O}^\times (U_\alpha \times U_\beta )$ .

2.3 The equivariant Picard group

From the description of lifts at the end of Section 2.1, we see that the pair $(g,\lambda )\in C^{0,1}\times C^{1,0}$ defines a ${\mathfrak g}$ -equivariant line bundle if and only if

$$\begin{align*}\delta^{0,1} g=0,\quad d^{1,0}\lambda=0,\quad d^{0,1}g=\delta^{1,0}\lambda \quad\Leftrightarrow\quad (g,\lambda)\in\ker(\partial^1). \end{align*}$$

The three conditions correspond to the cocycle condition for transition functions, the cocycle condition for the local lift (2.1) and the compatibility condition (2.4), respectively. Rescaling the fiber coordinates $u_\alpha $ in the line bundle corresponds exactly to changing the cocycle $(g,\lambda )$ by a coboundary in $\mathrm {im}(\partial ^0)$ .

Denoting by $\mathfrak {C}_{\mathfrak g}$ the modified Chevalley-Eilenberg sheaf complex (2.3), $\mathrm {Pic}_{\mathfrak g}(M)$ may be identified with the first hypercohomology $\mathbb {H}^1(M,\mathfrak {C}_{\mathfrak g})$ , cf. [Reference Griffiths and Harris11, Ch. 3.5] for a discussion of hypercohomology $\mathbb {H}^q$ .

The maps $\Phi _1 \colon {\mathrm {H}}^1(\mathrm {Tot}^\bullet (C)) \to {\mathrm {H}}_\delta ^1(C^{0,\bullet })$ and $\Phi _2 \colon {\mathrm {H}}^1(\mathrm {Tot}^\bullet (C)) \to {\mathrm {H}}_d^1(C^{\bullet ,0})$ induce maps

$$\begin{align*}\Phi_1 \colon \mathrm{Pic}_{\mathfrak g}(M) \to \mathrm{Pic}(M), \qquad \Phi_2 \colon \mathrm{Pic}_{\mathfrak g}(M) \to \mathfrak{M}_{\mathfrak g}(M),\end{align*}$$

denoted by the same letters (Lemma 2.6 justifies the choice of codomain for the second map). We define $\mathrm {Pic}^{\mathrm {red}}_{\mathfrak g}(M):=\mathrm {im}(\Phi _1\times \Phi _2)\subset \mathrm {Pic}(M)\times \mathfrak {M}_{\mathfrak g}(M)$ , which we call the reduced ${\mathfrak g}$ -equivariant Picard group, and denote by $\Psi _1,\Psi _2$ the projections

$$\begin{align*}\Psi_1 \colon \mathrm{Pic}^{\mathrm{red}}_{\mathfrak g}(M) \to \mathrm{Pic}(M), \qquad \Psi_2 \colon \mathrm{Pic}_{\mathfrak g}(M) \to \mathfrak{M}_{\mathfrak g}(M).\end{align*}$$

Then $\varpi :=\Phi _1\times \Phi _2$ epimorphically maps $\mathrm {Pic}_{\mathfrak g}(M)$ to $\mathrm {Pic}^{\mathrm {red}}_{\mathfrak g}(M)$ .

Proposition 2.9. $\mathrm {Pic}^{\mathrm {red}}_{\mathfrak g}(M) \simeq \ker (\partial ^1)/\sim $ , where the equivalence relation is defined by $(g,\lambda )\sim (\tilde g, \tilde \lambda )$ if $\tilde g=g \cdot \delta ^{0,0} \nu $ and $\tilde \lambda = \lambda + d^{0,0} \mu $ , where $\mu ,\nu \in C^{0,0}$ satisfy

$$\begin{align*}\mu/\nu \in \ker (d^{0,1} \circ \delta^{0,0})=\ker(\delta^{1,0} \circ d^{0,0}). \end{align*}$$

Let us investigate the relationship between $\mathrm {Pic}_{\mathfrak g}(M)$ and $\mathrm {Pic}^{\mathrm {red}}_{\mathfrak g}(M)$ . By Proposition 2.9, the admissible pair $(\nu ,\mu )\in C^{0,0}\times C^{0,0}$ characterizing the freedom in choice of representatives $(g,\lambda ) \in \ker (\partial ^1)$ for the reduced group can be rewritten as $(\nu ,\mu )=(\nu /\mu ,1)\cdot (\mu ,\mu )\simeq (\nu /\mu ,\mu ) \in \ker (d^{0,1}\circ \delta ^{0,0})\times C^{0,0}$ . It follows that (for a good cover) $\mathrm {Pic}^{\mathrm {red}}_{\mathfrak g}(M)$ is equal to

$$\begin{align*}\frac{\ker(\partial^1)}{\mathrm{im}(\tilde\partial)\cdot\mathrm{im}(\partial^0)} = \frac{\ker(\partial^1)}{\frac{\mathrm{im}(\tilde \partial)}{\mathrm{im}(\tilde \partial)\cap\mathrm{im}(\partial^0)} \cdot\mathrm{im}(\partial^0)}= \frac{{\mathrm{H}}^1(\mathrm{Tot}^\bullet (C))}{\frac{\mathrm{im}(\tilde \partial)}{\mathrm{im}(\tilde \partial) \cap \mathrm{im}(\partial^0)}}, \end{align*}$$

where the map $\tilde \partial \colon \ker (d^{0,1}\circ \delta ^{0,0})\to C^{0,1}\times C^{1,0}$ is defined as $\delta ^{0,0}\times 0$ . We have

$$ \begin{align*} \mathrm{im}(\partial^0) &= \{(\delta^{0,0} \mu, d^{0,0} \mu)\in C^{0,1}\times C^{1,0} \mid\mu\in C^{0,0}\}, \\ \mathrm{im}(\tilde \partial) &= \{(\delta^{0,0} \kappa, 0) \in C^{0,1} \times C^{1,0} \mid \kappa \in \ker(d^{0,1} \circ \delta^{0,0})\} \simeq \ker(d^{0,1} \circ \delta^{0,0}) /\ker(\delta^{0,0}). \end{align*} $$

If $d^{0,0} \mu =0$ , then $\mu \in \ker (\delta ^{1,0} \circ d^{0,0}) = \ker (d^{0,1} \circ \delta ^{0,0})$ , and therefore,

$$ \begin{align*} \mathrm{im}(\tilde\partial)\cap\mathrm{im}(\partial^0) &= \{(\delta^{0,0} \mu,0) \in C^{0,1} \times C^{1,0} \mid \mu \in \ker (d^{0,0}) \} \\ &= \delta^{0,0}(\ker(d^{0,0})) \simeq \ker(d^{0,0})/(\ker(\delta^{0,0}) \cap \ker(d^{0,0})). \end{align*} $$

It follows that

$$ \begin{align*} \frac{\mathrm{im}(\tilde \partial)}{\mathrm{im}(\tilde \partial) \cap \mathrm{im}(\partial^0)} = \frac{\ker(d^{0,1} \circ \delta^{0,0})}{\ker(\delta^{0,0}) \cdot \ker(d^{0,0})}. \end{align*} $$

Defining

(2.7) $$ \begin{align} T_{\mathfrak g}(\mathcal{U}) :=\frac{\ker(d^{0,1} \circ \delta^{0,0})}{\ker(\delta^{0,0}) \cdot \ker(d^{0,0})}, \qquad T_{\mathfrak g}(M) := \varinjlim T_{\mathfrak g}(\mathcal{U}) \end{align} $$

gives us the relation between $\mathrm {Pic}_{\mathfrak g}(M)$ and $\mathrm {Pic}^{\mathrm {red}}_{\mathfrak g}(M)$ :

Proposition 2.10. The following sequence is exact:

(2.8) $$ \begin{align} 0\to T_{\mathfrak g}(M)\longrightarrow\mathrm{Pic}_{\mathfrak g}(M)\xrightarrow{\varpi}\mathrm{Pic}^{\mathrm{red}}_{\mathfrak g}(M)\to 0. \end{align} $$

The commutative diagram in Figure 1 gives relations between the groups we have considered, leading to vanishing conditions for $\mathrm {Pic}_{\mathfrak g}(M)$ and various isomorphisms.

Figure 1 Commutative diagram: The dotted and dashed long sequences as well as three straight line sequences are exact.

The diagram contains the short exact sequence (2.8), the (direct limit of) short exact sequences (2.5)–(2.6), and also two longer exact sequences. For instance, exactness at $\delta ^{0,0}_*$ and $d^{0,0}_*$ can be seen as follows. If $\mu \in \ker (d^{0,1} \circ \delta ^{0,0})$ , then $\delta ^{0,0} \mu \in \ker (d^{0,1}) \cap \ker (\delta ^{0,1})$ and $d^{0,0} \mu \in \ker (\delta ^{1,0}) \cap \ker (d^{1,0})$ , whence $[\delta ^{0,0} \mu ] \in {\mathrm {H}}_\delta ^1({\mathrm {H}}_d^0(C^{\bullet ,\bullet }))$ and $[d^{0,0} \mu ] \in {\mathrm {H}}_d^1({\mathrm {H}}_\delta ^0(C^{\bullet ,\bullet }))$ . We have $[\delta ^{0,0} \mu ]=0$ if and only if there exists an element $\mu _0 \in \ker (d^{0,0})$ such that $\delta ^{0,0} \mu = \delta ^{0,0} \mu _0$ . This happens if and only if $\mu =\frac {\mu }{\mu _0} \mu _0 \in \ker (\delta ^{0,0}) \cdot \ker (d^{0,0})$ . By a similar argument, $[d^{0,0} \mu ] = 0$ if and only if $\mu \in \ker (\delta ^{0,0}) \cdot \ker (d^{0,0})$ .

Note that the maps ${\check {\mathrm {H}}}^1(M,(\mathcal {O}^\times )^{\mathfrak g}) \to \mathrm {Pic}_{\mathfrak g}(M)$ and ${\tilde {\mathrm {H}}}^1({\mathfrak g},\mathcal {O}(M)) \to \mathrm {Pic}_{\mathfrak g}(M)$ in the commutative diagram are defined by $[g] \mapsto [(g,0)]$ and $[\lambda ] \mapsto [(1,-\lambda )]$ , respectively.

Notice $\mathrm {im}(\Psi _1)= \mathrm {im}(\Phi _1)$ and $\mathrm {im}(\Psi _2)= \mathrm {im}(\Phi _2)$ , which are described by Lemmata 2.4 and 2.6.

Propositions 2.10 and 2.13 combine into Theorem 1.1.

Example 2.14 (Rational curve).

Consider the projective space $\mathbb CP^1$ with charts $U_0\simeq \mathbb C^1(x)$ and $U_\infty \simeq \mathbb C^1(y)$ , with coordinates related by $y=1/x$ on $U_0 \cap U_\infty $ . The Lie algebra $\mathfrak {sl}(2,\mathbb C)$ acts naturally on this space with the basis $X,Y,Z$ given in local coordinates:

$$\begin{align*}X|_{U_0} = \partial_x, \;\; Y|_{U_0} = x \partial_x, \;\; Z|_{U_0} = x^2 \partial_x, \qquad\quad X|_{U_\infty} = -y^2 \partial_y, \;\; Y|_{U_\infty} = -y\partial_y, \;\; Z|_{U_\infty} = -\partial_y. \end{align*}$$

Let $\lambda _i$ be a representative of an element in $H^1(\mathfrak {sl}(2,\mathbb C),\mathcal {O}(U_i))$ for $i=0,\infty $ .

Taking $\mu _0= e^{\int \lambda _0(X_0) dx}$ (the integral sign denotes the anti-derivative on $\mathbb C$ ) gives

$$\begin{align*}(\lambda_0-d^{0,0} \mu_0)(X_0) =\lambda_0(X_0)-\partial_x (\log \mu_0) =0, \end{align*}$$

so we can without loss of generality assume that $\lambda _0(X)=0$ . The values $\lambda _0(Y)$ and $\lambda _0(Z)$ are now determined by the cocycle conditions

$$ \begin{align*} X(\lambda_0(Y))-Y(\lambda_0(X))&=\lambda_0 ([X,Y]) = \lambda_0(X) = 0, \\ X(\lambda_0(Z))-Z(\lambda_0(X))&=\lambda_0 ([X,Z]) = 2 \lambda_0(Y), \\ Y(\lambda_0(Z))-Z(\lambda_0(Y))&=\lambda_0 ([Y,Z]) = \lambda_0(Z). \end{align*} $$

This leads to

$$\begin{align*}\lambda_0(X)=0, \qquad \lambda_0(Y)=\tfrac12 A, \qquad \lambda_0(Z) = A x. \end{align*}$$

Analogous computations on $U_\infty $ give

$$\begin{align*}\lambda_\infty(X)=By, \qquad \lambda_\infty(Y)=\tfrac12 B, \qquad \lambda_\infty(Z) = 0. \end{align*}$$

Next, we require that $d^{0,1} g_{0 \infty } = (\delta ^{1,0} \lambda )_{0 \infty }$ for some $\delta ^{0,1}$ -cocycle $g_{0 \infty } \in \mathcal {O}^\times (U_0 \cap U_\infty )$ . Evaluating this on X, Y and Z leads to the following overdetermined system of ODEs:

$$\begin{align*}\frac{\partial_x (g_{0 \infty})}{g_{0 \infty}} = -\frac{B}{x}, \qquad \frac{x\partial_x (g_{0 \infty})}{g_{0 \infty}} = \frac{A-B}2 , \qquad \frac{x^2\partial_x (g_{0 \infty})}{g_{0 \infty}} = Ax. \end{align*}$$

The system has a solution if and only if $A=-B$ , in which case, $g_{0 \infty } = C x^A$ . This solution is holomorphic on $U_0 \cap U_\infty $ if and only if $A \in \mathbb Z$ . The constant C can be set equal to $1$ by multiplying $g_{0 \infty }$ with $(\delta ^{0,0} \mu )_{0 \infty }$ for $\mu = \{\mu _0=1, \mu _\infty =C\} \in \ker (d^{0,0})$ . Thus, the global lifts are given by $(\lambda _0, \lambda _\infty )$ with $A=-B \in \mathbb Z$ , and the corresponding $\mathfrak {sl}(2,\mathbb C)$ -equivariant line bundle has transition function $g_{0 \infty } = x^{A}$ . To sum up, the cover $\mathcal {U}=\{U_0,U_\infty \}$ is nice and we get

$$\begin{align*}\mathrm{Pic}_{\mathfrak{sl}(2,\mathbb C)}(\mathbb CP^1)\simeq \mathrm{Pic}^{\mathrm{red}}_{\mathfrak{sl}(2,\mathbb C)}(\mathbb CP^1)\simeq \mathbb Z. \end{align*}$$

The first isomorphism is a consequence of ${\check {\mathrm {H}}}^1(M, (\mathcal {O}^\times )^{\mathfrak g}) \simeq {\mathrm {H}}_\delta ^1({\mathrm {H}}_d^0(C^{\bullet ,\bullet })) =0$ and Corollary 2.11. Since $\mathbb CP^1$ is covered by two open charts, we have $C^{0,2}=0$ , implying ${\mathrm {H}}_d^0(C^{\bullet ,2})=0$ and ${\mathrm {H}}_\delta ^2({\mathrm {H}}_d^0(C^{\bullet ,\bullet }))=0$ . By Lemma 2.6, $\mathrm {im}(\Psi _2) \simeq {\mathrm {H}}_\delta ^0({\mathrm {H}}_d^1(C^{\bullet ,\bullet })) = \mathfrak {M}_{\mathfrak g}(\{U_0,U_\infty \})$ , which for this cover is isomorphic to $\mathbb Z$ .

A straightforward generalization of this computation gives $T_{\mathfrak g}({\mathbb C} P^n)=0$ , $\mathfrak {M}_{\mathfrak g}({\mathbb C} P^n)={\mathbb C}$ (for $n=1$ this is parametrized by the above $A=-B$ , but with a finer cover $\mathcal {U}$ it is unconstrained: $A\in {\mathbb C}$ ) and $\mathrm {Pic}_{\mathfrak g}({\mathbb C} P^n)=\mathbb Z$ for ${\mathfrak g}=\mathfrak {sl}(n+1,{\mathbb C})$ .

Remark 2.15. Recall that $\mathrm {Pic}({\mathbb C} P^n) = \{\mathcal {O}_{{\mathbb C} P^n}(k)\}_{k\in \mathbb Z}\simeq \mathbb Z$ , where $\mathcal {O}_{\mathbb CP^n}(0)$ is the trivial line bundle and $\mathcal {O}_{\mathbb CP^n}(-1)$ is the tautological line bundle and for $k>0$ :

$$\begin{align*}\mathcal{O}_{\mathbb CP^n}(-k) = \mathcal{O}_{{\mathbb C} P^n}(-1)^{\otimes k}, \qquad \mathcal{O}_{\mathbb CP^n}(k) =\big(\mathcal{O}_{{\mathbb C} P^n}(-1)^{\otimes k}\big)^*. \end{align*}$$

The canonical line bundle is $K_{\mathbb CP^n}=\Lambda ^n T^* \mathbb CP^n= \mathcal {O}_{\mathbb CP^n}(-n-1)$ , cf. [Reference Huybrechts15, Ch. 2.2].

We will see later that $\Psi _1$ and $\Phi _1$ may be non-injective. The following shows it for $\Psi _2$ and $\Phi _2$ .

Example 2.16 (Elliptic curve).

Consider $\mathbb C^2$ with coordinates $(x,u)$ and two commuting maps

$$\begin{align*}h_1(x,u)= (x+1,u), \qquad h_2(x,u)= (x+\omega_1, \omega_2 u), \end{align*}$$

where $\omega _1\in {\mathbb C}\setminus {\mathbb R}$ , $\omega _2\neq 0$ . Both of these maps respect the projection $\mathbb C^2 \to \mathbb C$ given by $(x,u)\mapsto x$ , and the vector field $\partial _x$ . Thus, in the quotient by the $\mathbb Z^2$ action generated by $h_1,h_2$ , we get that the vector field $\partial _x$ on the elliptic curve $\Gamma = \mathbb C/\mathbb Z^2$ lifts to the vector field $\partial _x$ on the line bundle $\mathbb C^2/\mathbb Z^2$ over the elliptic curve. This line bundle $L_\omega $ is topologically trivial but holomorphically nontrivial for $\omega _2 \neq e^{2 \pi i \omega _1}$ , and all line bundles of this form lie in $\ker (\Psi _2)$ ; see Section 27 of [Reference Arnold2] for details. The general lift of ${\mathfrak g}$ is given by $\partial _x+c\,u\partial _u$ , $c\in {\mathbb C}$ .

For holomorphic curves, we have a short exact sequence (where $c_1$ is the first Chern class)

$$\begin{align*}0\to\mathrm{Pic}^0(\Gamma)\longrightarrow\mathrm{Pic}(\Gamma)\stackrel{c_1}\longrightarrow H^2(\Gamma,\mathbb Z)\to0, \end{align*}$$

and for elliptic curves $\mathrm {Pic}^0(\Gamma )=\mathrm {Div}^0(\Gamma )\simeq \Gamma $ , $x\mapsto x-x_0$ , whence $\mathrm {Pic}(\Gamma )\simeq \Gamma \oplus \mathbb Z$ .

The summand $\mathbb Z$ in $\mathrm {Pic}(\Gamma )$ corresponds to divisors $m\cdot x_0$ , $m\in \mathbb Z$ , $x_0\in \Gamma $ . However, for topologically nontrivial line bundles, $m=c_1(L)\neq 0$ , the algebra ${\mathfrak g}$ does not possess a lift to L. Indeed, such a lift would define a flat connection, at which point we can use the formula $c_1(L)=\Big [\frac {-1}{2\pi i}\mathrm {tr} R_\nabla \Big ]$ . Alternatively, denoting by $\pi :{\mathbb C}\to \Gamma $ the quotient-projection by the lattice $\langle 1,\omega _1\rangle $ , the pullback $\pi ^*L$ is trivial and can be identified with ${\mathbb C}^2(x,u)$ , on which $\mathbb Z^2$ acts through the above $h_1,h_2$ . Invariance of the lift $\partial _x+f(x)\,u\partial _u$ gives periodicity $f(x+1)=f(x)$ and the constraint $f(x+\omega _1)-f(x)=2\pi im$ , which are incompatible unless $m=0$ .

It is easy to see that $\mathfrak {M}_{\mathfrak g}(\Gamma )=0$ . Moreover, $T_{\mathfrak g}(\Gamma )={\mathbb C}$ as it corresponds to 0-cochains $c_\alpha e^{sx}\in \mathcal {O}(U_\alpha )$ modulo local constants $\{c_\alpha \}$ (so the quotient coordinate is s). This can be also identified with ${\tilde {\mathrm {H}}}^1({\mathfrak g},\mathcal {O}(M))={\mathbb C}$ generated by global 1-form $dx$ on $\Gamma $ .

We conclude:

$$\begin{align*}\mathrm{Pic}^{\mathrm{red}}_{\mathfrak g}(\Gamma)=\Gamma,\quad \mathrm{Pic}_{\mathfrak g}(\Gamma)={\mathbb C}^2/\mathbb Z^2. \end{align*}$$

Note that the equivariant Picard group can be identified with ${\check {\mathrm {H}}}^1(M,(\mathcal {O}^\times )^{\mathfrak g}) =({\mathbb C}^\times )^2$ , but simultaneously, it corresponds to trivial one-dimensional bundle, with fibers ${\mathbb C}(c)$ , over $\Gamma $ . This fits well the commutative diagram of Figure 1.

Corollary 2.12 gives a sufficient condition for $\Phi _1$ to be injective. For a connected algebraic group G, Mumford’s Proposition 1.4 in [Reference Mumford22] gives a sufficient condition for the map $\mathrm {Pic}_G(M)\to \mathrm {Pic(M)}$ to be injective, in terms of nonexistence of a homomorphism $G \to GL(1,\mathbb C)$ . (The group of G-equivariant line bundles will be discussed in Section 2.5.) This does not straightforwardly adapt to the infinitesimal analytic setting, yet below we obtain a result inspired by that of Mumford.

Let ${\mathfrak g}_p\subset {\mathfrak g}$ denote the isotropy algebra of the point $p\in M$ :

$$\begin{align*}{\mathfrak g}_p = \{X \in {\mathfrak g} \mid X_p = 0\}. \end{align*}$$

Let ${\mathrm {H}}^k_{dR}(M)$ denote the holomorphic de Rham cohomology of M. It is known that in the affine case (for Stein manifolds) as well as for the compact Kähler case, this coincides with the singular cohomology ${\mathrm {H}}^k(M,{\mathbb C})$ , see [Reference Grothendieck12, Reference Griffiths and Harris11]. In general, the holomorphic de Rham cohomology ${\mathrm {H}}^k_{dR}(M)$ is equal to the hypercohomology $\mathbb {H}^k(M,\Omega ^\bullet _M)$ of the sheaf of holomorphic forms on M.

Proof. Consider $[\lambda ] \in Z$ . Since its representative $\lambda $ is defined globally on M, the value $\lambda (X)_p \in \mathbb C$ is well defined for any $p \in M$ and any $X \in {\mathfrak g}$ . Transitivity of ${\mathfrak g}$ implies that for every $v \in T_p M$ , there exists $X\in {\mathfrak g}$ satisfying $X_p=v$ , which means that the following linear function on $T_p M$ is well defined:

$$\begin{align*}\alpha_p \colon X_p \mapsto \lambda(X)_p. \end{align*}$$

This gives a well-defined 1-form $\alpha $ on M. Choosing $X_i\in {\mathfrak g}$ for $v_i\in T_pM$ such that $v_i=(X_i)_p$ , we get for every $p\in M$ ,

$$ \begin{align*} (d \alpha)_p(v_1, v_2) &= d \alpha (X_1,X_2)_p = \left( X_1(\alpha(X_2)) -X_2(\alpha(X_1)) - \alpha([X_1,X_2])\right)_p = d^{1,0} \lambda(X_1,X_2)_p = 0. \end{align*} $$

Thus, $d\alpha =0$ , so the closed 1-forms in two cohomologies correspond; the same clearly concerns exact 1-forms. This yields the embedding $[\lambda ]\mapsto [\alpha ]$ .

If ${\mathrm {H}}^1_{dR}(M)=0$ , then $\alpha =d\log \mu $ for some $\mu \in \mathcal {O}^\times (M)$ , and therefore, $\lambda =d^{0,0}\mu $ .

Proof. Consider an element $[\lambda ]\in {\tilde {\mathrm {H}}}^1({\mathfrak g},\mathcal {O}(M)) \setminus Z$ (by our assumption and Lemma 2.17, this set is nonempty). Since $\delta ^{1,0} \lambda =0$ , $\lambda $ defines a global lift of ${\mathfrak g}$ to the trivial bundle $M \times \mathbb C$ ,

$$\begin{align*}{\mathfrak g}^\lambda = \{ X + \lambda(X) u \partial_u \mid X \in {\mathfrak g}\}.\end{align*}$$

For any point $p \in M$ , all vector fields of the Lie subalgebra

$$\begin{align*}\{Y + \lambda(Y) u \partial_u \mid Y \in {\mathfrak g}_p\} \subset {\mathfrak g}^\lambda \end{align*}$$

are tangent to the fiber $\pi ^{-1}(p)$ . Therefore, the restriction to the fiber results in a Lie algebra homomorphism:

(2.9) $$ \begin{align} {\mathfrak g}_p\ni Y \mapsto \lambda(Y)_p u \partial_u \in \mathfrak{gl}(1,{\mathbb C}). \end{align} $$

By definition of Z, $\lambda (Y)_p\not \equiv 0$ for a generic point p. Thus, at this point, (2.9) is surjective. Since, for a transitive ${\mathfrak g}$ , the isotropies ${\mathfrak g}_p$ at different points p are conjugate, the claim follows.

The existence of a surjective Lie algebra homomorphism to $\mathfrak {gl}(1,\mathbb C)$ implies that ${\mathfrak g}_p$ has an ideal of codimension 1, the kernel of (2.9). Corollary 2.12 then leads to the following statement.

As a consequence, $\Phi _1$ is injective if ${\mathfrak g}_p$ is perfect $[{\mathfrak g}_p,{\mathfrak g}_p]={\mathfrak g}_p$ (this also applies to infinite-dimensional Lie algebras ${\mathfrak g}$ ) and, in particular, if ${\mathfrak g}_p$ is semisimple (for finite-dimensional ${\mathfrak g}$ ).

Example 2.20 (Special affine algebra on the plane).

Consider the Lie algebra

$$\begin{align*}{\mathfrak g} = \langle \partial_x, \partial_y, y \partial_x, x\partial_y, x \partial_x-y\partial_y\rangle \subset \mathcal{D}(\mathbb C^2). \end{align*}$$

The Lie algebra is transitive with simple isotropy ${\mathfrak g}_p$ . By Proposition 2.18, we have ${\tilde {\mathrm {H}}}^1({\mathfrak g},\mathcal {O}(M))=0$ , implying that $\mathrm {Pic}_{\mathfrak g}(\mathbb C^2)\to \mathrm {Pic}(\mathbb C^2)$ is injective by Corollary 2.12. Since $\mathrm {Pic}(\mathbb C^2) =0$ , it follows that $\mathrm {Pic}_{\mathfrak g}(\mathbb C^2) = 0$ .

Next, for the Lie algebra

$$\begin{align*}\mathfrak{h}=\langle y \partial_x, -xy \partial_x-y^2 \partial_y, \partial_x,-x^2 \partial_x-xy\partial_y,2x\partial_x+y\partial_y \rangle \subset \mathcal{D}(\mathbb C^2), \end{align*}$$

the isotropy of the point $p=0$

$$\begin{align*}\mathfrak{h}_0 = \langle y \partial_x, -xy \partial_x-y^2 \partial_y,-x^2 \partial_x-xy\partial_y,2x\partial_x+y\partial_y \rangle \end{align*}$$

is 4-dimensional and solvable, and it has a 3-dimensional ideal. In this case, $\mathrm {Pic}_{\mathfrak {h}}({\mathbb C}^2)={\mathbb C}$ .

Note that ${\mathfrak g}$ and $\mathfrak {h}$ can be viewed as the same Lie subalgebra $\mathfrak {sl}(2,{\mathbb C})\ltimes {\mathbb C}^2\subset \mathfrak {sl}(3,{\mathbb C})\subset \mathcal {D}(\mathbb CP^2)$ restricted to two different open charts of $\mathbb CP^2$ .

2.4 Line bundles admitting a transversal lift

We start by recalling some basic information about divisors; cf. [Reference Griffiths and Harris11, Reference Huybrechts15]. Let $\mathcal {O}^\times $ denote the multiplicative sheaf of nonvanishing holomorphic functions on a complex manifold M, and $\mathcal {M}^\times $ the sheaf of meromorphic functions that are not identically zero on M. A divisor D is a global section of $\mathcal {M}^\times /\mathcal {O}^\times $ . It is defined by a collection of functions $f_\alpha \in \mathcal {M}^\times (U_\alpha )$ for an open cover $\{U_\alpha \}$ of M, such that $f_\alpha /f_\beta \in \mathcal {O}^\times (U_\alpha \cap U_\beta )$ . Any divisor D gives rise to a line bundle, denoted by $[D]$ , whose transition functions are given by $g_{\alpha \beta } = f_\alpha /f_\beta \in \mathcal {O}^\times (U_\alpha \cap U_\beta )$ , and the long exact sequence (see [Reference Griffiths and Harris11]) relates the group of divisors $\mathrm {Div}(M) := {\check {\mathrm {H}}}^0 (M,\mathcal {M}^\times /\mathcal {O}^\times )$ to the Picard group on M:

(2.10) $$ \begin{align} \cdots\to{\check{\mathrm{H}}}^0(M,\mathcal{M}^\times)\to\mathrm{Div}(M)\to\mathrm{Pic}(M)\to{\check{\mathrm{H}}}^1(M,\mathcal{M}^\times)\to\cdots \end{align} $$

Here, ${\check {\mathrm {H}}}^0(M, \mathcal {M}^\times )$ is the group of global meromorphic functions on M, and $\mathrm {Div}(M)/{\check {\mathrm {H}}}^0(M,\mathcal {M}^\times )$ is the group of equivalence classes of divisors (equivalent divisors give equivalent line bundles).

Definition 2.21. Let $\mathfrak g \subset \mathcal {D}(M)$ be a Lie algebra of vector fields on M. The divisor $D=\{f_\alpha \}$ defined on the open cover $\{U_\alpha \}$ of M is a ${\mathfrak g}$ -invariant divisor if for each $\alpha $

$$\begin{align*}X(f_\alpha)=\lambda_\alpha(X) f_\alpha, \qquad \forall X \in {\mathfrak g}, \end{align*}$$

where $\lambda _\alpha \in {\mathfrak g}^* \otimes \mathcal O(U_\alpha )$ . The group of ${\mathfrak g}$ -invariant divisors is denoted by $\mathrm {Div}_{\mathfrak g}(M)$ . The collection $\lambda =\{\lambda _\alpha \}$ is called the weight of D.

It follows that $\mathfrak g$ is tangent to the set of zeros of D, and also to the set of poles. In this way, D defines a (possibly reducible) invariant hypersurface in M.

Proof. To show that the pair defines a ${\mathfrak g}$ -equivariant line bundle, we must verify that $\partial ^1(g,\lambda )=0$ . It is clear that $\delta ^{0,1} g=0$ since $g_{\alpha \beta }$ are transition functions of $[D]$ . Next, the condition $d^{1,0} \lambda _\alpha =0$ holds for each $\alpha $ since $f_\alpha \not \equiv 0$ , and for arbitrary vector fields $X,Y\in {\mathfrak g}$ , we have

$$\begin{align*}\lambda_\alpha([X,Y]) f_\alpha = [X,Y](f_\alpha) = X(Y(f_\alpha))-Y(X(f_\alpha)) = (X(\lambda_\alpha(Y))-Y(\lambda_\alpha(X)))f_\alpha. \end{align*}$$

What remains is to verify that the weights $\lambda _\alpha $ are compatible with the transition functions $g_{\alpha \beta }= f_\alpha /f_\beta $ . On $U_\alpha \cap U_\beta $ , we have

$$\begin{align*}\lambda_\alpha(X) f_\alpha = X(f_\alpha) = X(g_{\alpha \beta} f_\beta) = X(g_{\alpha \beta}) f_\beta + g_{\alpha \beta} X(f_\beta) = \left(\frac{X(g_{\alpha \beta})}{g_{\alpha \beta}}+\lambda_\beta(X)\right) f_\alpha, \end{align*}$$

which is equivalent to $\delta ^{1,0}\lambda = d^{0,1} g$ .

Next, to show that the ${\mathfrak g}$ -equivariant bundle is independent of representative functions $f_\alpha $ of D, take another representative $\tilde f_\alpha = \mu _\alpha f_\alpha $ with $\mu _\alpha \in \mathcal {O}^\times (U_\alpha )$ . This results in an equivalent ${\mathfrak g}$ -equivariant line bundle $(\{\tilde g_{\alpha \beta }\}, \{\tilde \lambda _\alpha \})$ : $\tilde g_{\alpha \beta } = g_{\alpha \beta } \mu _\alpha /\mu _\beta $ and $\tilde \lambda _\alpha (X) = \lambda _\alpha (X) + X(\mu _\alpha )/\mu _\alpha $ for all $X \in {\mathfrak g}$ .

The lifted Lie algebra ${\mathfrak g}^{\lambda } \subset \mathcal {D}([D])$ is given locally on $\pi ^{-1}(U_\alpha ) \simeq U_\alpha \times \mathbb C$ by

$$\begin{align*}{\mathfrak g}^{\lambda}|_{\pi^{-1}(U_\alpha)} = \{\hat X|_{\pi^{-1}(U_\alpha)}=X|_{U_\alpha}+\lambda_\alpha(X) u \partial_u \mid X \in {\mathfrak g}\}. \end{align*}$$

The lift ${\mathfrak g}^\lambda $ has exactly the same form as the lifts in Section 2.1, the only difference being that $\lambda $ is now specifically determined by a ${\mathfrak g}$ -invariant divisor. Thus, we have a map

$$\begin{align*}j_{\mathfrak g} \colon \mathrm{Div}_{\mathfrak g}(M)\to\mathrm{Pic}_{\mathfrak g}(M),\qquad D\mapsto([D], {\mathfrak g}^\lambda). \end{align*}$$

In general, this map is neither injective nor surjective. For instance, the kernel of this map contains all global ${\mathfrak g}$ -invariant functions in $\mathcal {M}^\times (M)$ . If ${\mathfrak g}$ is transitive, then $\mathrm {Div}_{\mathfrak g}(M)=0$ , but $\mathrm {Pic}_{\mathfrak g}(M)$ may be nontrivial as in Example 2.14. The next example exhibits nontrivial $\mathrm {Div}_{\mathfrak g}(M)$ .

Example 2.23 ( $\mathfrak {sl}(2,\mathbb C) \subset \mathcal {D}(\mathbb CP^2)$ ).

The manifold $\mathbb CP^2$ is covered by the three charts

$$ \begin{align*} U_3 &= \{[x:y:z] \in \mathbb CP^2 \mid z \neq 0\}, \\ U_2 &= \{[x:y:z] \in \mathbb CP^2 \mid y \neq 0\} , \\ U_1 &= \{[x:y:z] \in \mathbb CP^2 \mid x \neq 0\}, \end{align*} $$

on which coordinates are given respectively by

$$\begin{align*}(x_1,x_2)=(x/z,y/z), \qquad (y_1,y_3) = (x/y,z/y), \qquad (z_2,z_3) = (y/x,z/x). \end{align*}$$

Consider the Lie algebra $\mathfrak {sl}(2,\mathbb C) \subset \mathfrak {sl}(3,\mathbb C)$ given in the respective charts by

$$ \begin{gather*} \langle x_2\partial_{x_1}, x_1\partial_{x_2}, x_1\partial_{x_1}-x_2\partial_{x_2} \rangle, \\ \langle\partial_{y_1}, -y_1^2\partial_{y_1}-y_1 y_3\partial_{y_3}, 2y_1\partial_{y_1}+y_3\partial_{y_3} \rangle, \\ \langle -z_2^2\partial_{z_2}-z_2 z_3\partial_{z_3},\partial_{z_2}, -2z_2\partial_{z_2}-z_3\partial_{z_3} \rangle. \end{gather*} $$

A computation shows that the Chevalley-Eilenberg cohomology groups are

$$\begin{align*}{\mathrm{H}}^1 (\mathfrak{sl}(2,{\mathbb C}),\mathcal{O}(U_3)) = 0, \qquad {\mathrm{H}}^1 (\mathfrak{sl}(2,{\mathbb C}),\mathcal{O}(U_2)) = {\mathbb C}^2, \qquad {\mathrm{H}}^1 (\mathfrak{sl}(2,{\mathbb C}),\mathcal{O}(U_1)) = {\mathbb C}^2 \end{align*}$$

with representative cocycles

$$\begin{align*}\lambda_3 = (0,0,0), \qquad \lambda_2= (0, B_1 y_1+B_2 y_3^2,-B_1), \qquad \lambda_1 = (C_1 z_2+C_2 z_3^2,0,C_1). \end{align*}$$

The holomorphic transition functions, compatible via overdetermined system (2.4), exist only for $B_2=C_2=0$ , $C_1=B_1=b$ and are given by formulae

$$\begin{align*}g_{32} = A_1 y_3^{b}=A_1 x_2^{-b}, \qquad g_{31} = A_2 z_3^{b}=A_2 x_1^{-b}, \qquad g_{21} = A_3 z_2^{b} = A_3 y_1^{-b}. \end{align*}$$

Requiring $g_{\alpha \beta }$ to be holomorphic gives the further restriction $b\in \mathbb Z$ . The constants $A_1, A_2, A_3$ can be set equal to $1$ by multiplying with an $\mathfrak {sl}(2,\mathbb C)$ -invariant $\delta ^{0,0}$ -coboundary. We conclude:

$$\begin{align*}\mathrm{Pic}_{\mathfrak{sl}(2,{\mathbb C})}(\mathbb CP^2)\simeq{\mathrm{H}}^1(\mathrm{Tot}^\bullet(C))=\mathbb Z. \end{align*}$$

In this case, $\mathrm {Div}_{\mathfrak {sl}(2,{\mathbb C})}({\mathbb C} P^2)\simeq \mathrm {Pic}_{\mathfrak {sl}(2,{\mathbb C})} ({\mathbb C} P^2)$ . The unique divisor $D=\{f_1,f_2,f_3\}$ corresponding to $b\in \mathbb Z$ is given by

$$\begin{align*}f_1 = z_3^{-b}, \qquad f_2 = y_3^{-b}, \qquad f_3=1. \end{align*}$$

We will now describe an obstruction for the existence of invariant divisors, elaborating upon [Reference Fels and Olver7]. The following definition is adapted from [Reference Anderson and Fels1] where it was used for group actions.

Note that singular orbits of $\hat {\mathfrak g}$ may project non-injectively (but indeed surjectively) to ${\mathfrak g}$ -orbits on M (see Example 2.27 below). The following is a reformulation of Theorem 1.3.

Proof. Let $D=\{f_\alpha \}$ be a $\mathfrak g$ -invariant divisor with weight $\lambda =\{\lambda _\alpha \}$ , and $[D]$ the corresponding line bundle defined by transition functions $g_{\alpha \beta } = f_\alpha /f_\beta $ . Any element $\hat X \in {\mathfrak g}^\lambda $ takes on $U_\alpha $ the form $\hat X|_{U_\alpha } = X|_{U_\alpha } + \lambda _\alpha (X) u \partial _u$ for some $X \in {\mathfrak g}$ . A straightforward computation shows that

$$\begin{align*}\hat X(u/f_\alpha) = \frac{\hat{X}(u) f_\alpha - u \hat{X}(f_\alpha)}{f_\alpha^2} = \frac{u \lambda_\alpha(X) f_\alpha-u X(f_\alpha)}{f_\alpha^2} =0. \end{align*}$$

Thus, the function $u/f_\alpha $ on $U_\alpha \times \mathbb C$ is a meromorphic absolute invariant (constant on ${\mathfrak g}^\lambda $ -orbits). It follows that the dimension of generic $\mathfrak g$ -orbits on $U_\alpha $ is equal to the dimension of generic ${\mathfrak g}^\lambda $ -orbits on $U_\alpha \times \mathbb C$ . This holds simultaneously on each $U_\alpha $ , and therefore globally on M.

Returning to Example 2.14 on $\mathfrak {sl}(2,{\mathbb C})\subset \mathcal {D}({\mathbb C} P^1)$ , we observe that generic orbits of any nontrivial lift are 2-dimensional. Thus, there are no nontrivial invariant divisors, which also follows from the fact that $\mathfrak {sl}(2,\mathbb C)$ is transitive on $\mathbb CP^1$ . Here is another demonstration of Proposition 2.25.

Example 2.27 ( $\mathfrak {aff}(1,\mathbb C) \subset \mathcal {D}(\mathbb CP^1)$ ).

Consider again coordinate charts $U_0 \simeq \mathbb C^1(x)$ and $U_\infty \simeq \mathbb C^1(y)$ of $\mathbb CP^1$ , with the Lie subalgebra ${\mathfrak g}=\mathfrak {aff}(1,{\mathbb C})=\langle X,Y\rangle \subset \mathfrak {sl}(2,{\mathbb C})$ given by

$$\begin{align*}X|_{U_0}=\partial_x,\ Y|_{U_0}=x\partial_x, \qquad X|_{U_\infty}=-y^2\partial_y,\ Y|_{U_\infty}=-y\partial_y. \end{align*}$$

General representatives $\lambda _s$ of elements in ${\mathrm {H}}^1(\mathfrak {aff}(1,{\mathbb C}),\mathcal {O}(U_s))$ , for $s=0,\infty $ , in basis $(X,Y)$ are given by

$$\begin{align*}\lambda_0=(0,A), \qquad \lambda_\infty=(B_2 y,B_1),\qquad A,B_1,B_2\in{\mathbb C}. \end{align*}$$

A general compatible transition function exists only when $A=B_1-B_2$ , in which case it is cohomologous to $g_{0 \infty }=y^{B_2}=x^{-B_2}$ . Requiring $g_{0 \infty }$ to be holomorphic results in $B_2\in \mathbb Z$ . The local lifts corresponding to $\lambda _0$ and $\lambda _\infty $ are given by

$$\begin{align*}{\mathfrak g}^{\lambda_0} = \langle \partial_x, x\partial_x + (B_1-B_2) u \partial_u \rangle, \qquad {\mathfrak g}^{\lambda_\infty} = \langle -y^2 \partial_y + B_2 y u \partial_u, -y \partial_y + B_1 u \partial _u\rangle. \end{align*}$$

It is clear that the generic orbit dimension is 1 if and only if $B_2=B_1$ . In this case, we get the invariant divisor D given by $f_0=1$ and $f_\infty =y^{-B_1}$ . Thus,

$$\begin{align*}\mathrm{Div}_{\mathfrak g}({\mathbb C} P^1)=\mathbb{Z}\subsetneqq{\mathbb C}\times\mathbb{Z}=\mathrm{Pic}_{\mathfrak g}({\mathbb C} P^1). \end{align*}$$

Note that the map $\Psi _1\colon \mathrm {Pic}_{\mathfrak g}({\mathbb C} P^1)\to \mathrm {Pic}({\mathbb C} P^1)$ is not injective: $\ker (\Psi _1) \simeq {\mathrm {H}}_d^1({\mathrm {H}}_\delta ^0(C^{\bullet ,\bullet }))={\mathbb C}$ . Similar to Example 2.14, we have $\mathfrak {M}_{\mathfrak g}(M) = \mathbb C$ even though $\mathfrak {M}_{\mathfrak g}(\{U_0, U_\infty \}) = \mathbb Z$ .

Proposition 2.25 can be viewed as a global version of [Reference Fels and Olver7, Th. 5.4]. According to it, locally, in smooth regular case, the statement allows a converse, giving a criterion for the (local) existence of relative invariants. Globally, in general analytic context, there is no converse to Proposition 2.25, due to other reasons for nonexistence of meromorphic invariant divisors/relative invariants. This is shown in the following simple example and also in a more complicated example of Section 3.4. Yet, in the following section, we will give a converse statement in the algebraic context.

Example 2.28. Consider the Lie algebra ${\mathfrak g}=\langle x^2\partial _x\rangle \subset \mathcal {D}({\mathbb C})$ . All line bundles over ${\mathbb C}$ are trivial, $\mathrm {Pic}_{\mathfrak g}({\mathbb C})=0$ , while we have $\mathrm {Pic}_{\mathfrak g}({\mathbb C})={\mathbb C}^2$ . A general representative cocycle of ${\tilde {\mathrm {H}}}^1({\mathfrak g},\mathcal {O}(\mathbb C))$ has the form $\lambda (x^2\partial _x)=A+Bx$ with $A,B\in {\mathbb C}$ , and the corresponding lifted Lie algebra is

$$\begin{align*}{\mathfrak g}^{\lambda} = \langle x^2\partial_x+(A+Bx)u\partial_u\rangle\subset\mathcal{D}({\mathbb C}\times{\mathbb C}). \end{align*}$$

Generic orbits of both ${\mathfrak g}$ and ${\mathfrak g}^\lambda $ are 1-dimensional for any choice of A and B; thus, ${\mathfrak g}^\lambda $ is transversal. However, the general solution of the system $x^2\partial _x(f(x)) = (A+Bx) f(x)$ is

$$\begin{align*}f(x)=x^B e^{-A/x}. \end{align*}$$

This is a (meromorphic) ${\mathfrak g}$ -invariant divisor on $\mathbb C$ only when $A=0$ and $B \in \mathbb Z$ (i.e., $\mathrm {Div}_{\mathfrak g}({\mathbb C} P^1)=\mathbb {Z}$ and not all equivariant line bundles come from invariant divisors).

2.5 Lie group vs Lie algebra approach

Let G be a Lie group acting on M. We consider the group $\mathrm {Pic}_G(M)$ of G-equivariant line bundles over M. In the setting of algebraic schemes, it was studied in [Reference Mumford22, Ch. 1.3]. Here, we give a different description of $\mathrm {Pic}_G(M)$ , emphasizing its relation to $\mathrm {Pic}_{\mathfrak g}(M)$ when ${\mathfrak g}$ is the Lie algebra of vector fields corresponding to the Lie group action, but demonstrate that, in general, $\mathrm {Pic}_G(M)$ is not isomorphic to $\mathrm {Pic}_{\mathfrak g}(M)$ .

Definition 2.29. A lift $\hat \rho $ of a group action $\rho \colon G \times M \to M$ to a line bundle $\pi \colon L \to M$ is a map $\hat \rho \colon G \times L \to L$ such that $\rho _g \colon L \to L$ is a vector bundle automorphism for each $g \in G$ , and the following diagram commutes:

The pair $(\pi \colon L\to M,\hat \rho )$ is called a G-equivariant line bundle. The space of such bundles, modulo the natural equivalences, has the group structure with the operation of tensor product. The group of G-equivariant line bundles $\mathrm {Pic}_G(M)$ is called the G-equivariant Picard group.

We assume G acts by biholomorphisms on M. The general description of G-equivariant line bundles over M can be done in terms of a cohomology theory that generalizes both the Čech cohomology and the Lie group cohomology with coefficients in the G-module $\mathcal {O}^\times (M)$ .

Let $\pi \colon L \to M$ be a line bundle. Assume there exists a lift of the group action to L (i.e. for each $\varphi \in G$ , there exists a (holomorphic) vector bundle automorphism $\hat \varphi $ on L, satisfying $\pi (\hat \varphi (p)) = \varphi (\pi (p))$ for each $p \in L$ (to simplify formulas, we use the notation $\varphi = \rho _g$ and $\hat \varphi = \hat \rho _g$ ). Let $\mathcal {U}=\{U_\alpha \}$ be a trivializing chart for L, and $u_\alpha $ be a (linear) fiber coordinate on $\pi ^{-1}(U_\alpha ) \simeq U_\alpha \times \mathbb C$ . Then $\hat \varphi $ acts on $u_\alpha $ in the following way:

(2.11) $$ \begin{align} \hat \varphi^*(u_\alpha) = \Lambda_{\alpha \beta}(\varphi) u_\beta, \qquad \Lambda_{\alpha \beta}(\varphi) \in \mathcal{O}^\times\left(U_\beta \cap \varphi^{-1}(U_\alpha)\right). \end{align} $$

Composing with a second element in the Lie group gives

$$\begin{align*}\hat \psi^*(\hat \varphi^*(u_\alpha)) = \psi^*(\Lambda_{\alpha \beta}(\varphi)) \Lambda_{\beta \gamma}(\psi) u_\gamma \end{align*}$$

on $U_\gamma \cap \psi ^{-1}(U_\beta \cap \varphi ^{-1}(U_\alpha ))$ . Simultaneously, on $U_\gamma \cap \psi ^{-1}(\varphi ^{-1}(U_\alpha ))$ , we have

$$\begin{align*}(\hat \psi^* \circ \hat \varphi^*)(u_\alpha) = \Lambda_{\alpha \gamma} (\varphi \circ \psi) u_\gamma. \end{align*}$$

Thus, on $U_\gamma \cap \psi ^{-1}(U_\beta ) \cap \psi ^{-1}(\varphi ^{-1}(U_\alpha ))$ we get

(2.12) $$ \begin{align} \psi^*(\Lambda_{\alpha \beta}(\varphi)) \Lambda_{\beta \gamma}(\psi) = \Lambda_{\alpha \gamma} (\varphi \circ \psi). \end{align} $$

When $\varphi $ is equal to the identity transformation on M, equation (2.11) gives $\Lambda _{\alpha \beta }(\mathrm {id}) = g_{\alpha \beta }$ , where $g_{\alpha \beta }$ is the transition function of $\pi $ on $U_\alpha \cap U_\beta $ . Setting $\varphi =\mathrm {id}$ in (2.12) gives

$$\begin{align*}\Lambda_{\alpha \gamma}(\psi) = \psi^*(g_{\alpha \beta})\Lambda_{\beta \gamma}(\psi), \end{align*}$$

while setting $\psi =\mathrm {id}$ leads to

$$\begin{align*}\Lambda_{\alpha \gamma} (\varphi) = \Lambda_{\alpha \beta}(\varphi) g_{\beta \gamma}. \end{align*}$$

The last equality shows that if the transition functions are given, then $\Lambda _{\alpha \beta }(\varphi )$ on $U_\beta \cap U_\alpha \cap \varphi ^{-1}(U_\alpha )$ is uniquely determined by $\Lambda _{\alpha \alpha }(\varphi )$ .

Next, changing the fiber coordinates $v_\alpha = \mu _\alpha u_\alpha $ for $\mu _\alpha \in \mathcal {O}^\times (U_\alpha )$ ,

$$\begin{align*}\hat \varphi^*(\mu_\alpha) \Lambda_{\alpha \beta}(\varphi) u_\beta= \hat \varphi^*(\mu_\alpha u_\alpha)= \hat \varphi^*(v_\alpha)= \tilde \Lambda_{\alpha \beta}(\varphi) v_\beta= \tilde \Lambda_{\alpha \beta} (\varphi) \mu_\beta u_\beta \end{align*}$$

results in the equivalence relation

$$\begin{align*}\Lambda_{\alpha \beta}(\varphi) \sim \frac{\varphi^*(\mu_\alpha)}{\mu_\beta} \Lambda_{\alpha \beta}(\varphi). \end{align*}$$

Introducing the differentials

$$ \begin{align*} (D^0 \mu (\varphi))_{\alpha \beta} &= \frac{\varphi^*(\mu_\alpha)}{\mu_\beta}, \\ (D^1 \Lambda(\varphi, \psi))_{\alpha \beta \gamma} &= \frac{\Lambda_{\alpha \gamma}(\varphi \circ \psi)}{\psi^*(\Lambda_{\alpha \beta}(\varphi)) \Lambda_{\beta \gamma} (\psi)}, \end{align*} $$

we see that $\Lambda =\{\Lambda _{\alpha \beta }\}$ defines a G-equivariant line bundle over M if and only if $D^1 \Lambda =0$ . Moreover, the G-equivariant line bundles defined by $\Lambda $ and $\tilde \Lambda $ are equivalent if and only if $\tilde \Lambda = \Lambda \cdot D^0 \mu $ for some $\mu $ . We define the action group cohomology for a given cover $\mathcal {U}$

$$\begin{align*}{\mathrm{H}}^1_{\mathcal{U}}(G,\mathcal{O}^\times)= \frac{\ker(D^1)}{\mathrm{im}(D^0)}, \end{align*}$$

and in general, we use the direct limit of this cohomology, ${\mathrm {H}}^1(G,\mathcal {O}^\times ):=\varinjlim {\mathrm {H}}^1_{\mathcal {U}}(G,\mathcal {O}^\times )$ .

Let us note that we consider not abstract but rather continuous (van Est) group cohomology; cf. [Reference Fuks9]. In fact, the above specifies cochains to be holomorphic.

Any Lie group action gives rise to a Lie algebra $\mathfrak g$ of vector fields. Consider a one-parameter group $\varphi _t \subset G$ and the corresponding vector field X. Denote the vector field on L corresponding to $\hat \varphi _t$ by $\hat X$ . For small t, the set $U_\alpha \cap \varphi _t^{-1}(U_\alpha )$ is nonempty, and on this set, we have

$$\begin{align*}\hat \varphi_t^*(u_\alpha) = \Lambda_{\alpha \alpha}(\varphi_t) u_\alpha. \end{align*}$$

When t approaches $0$ , $U_\alpha \cap \varphi _t^{-1}(U_\alpha )$ approaches $U_\alpha $ , and the Lie derivative of $u_\alpha $ with respect to $\hat X$ on $U_\alpha $ is given by

$$\begin{align*}L_{\hat X}(u_\alpha) = \frac{d}{dt} \Big|_{t=0} \Lambda_{\alpha \alpha}(\varphi_t) u_\alpha. \end{align*}$$

Comparing this to the lifts $\hat X=X +\lambda _{\alpha }(X) u_\alpha \partial _{u_\alpha }$ discussed in Section 2.1 results in the relation

$$\begin{align*}\lambda_\alpha(X) = \frac{d}{dt} \Big|_{t=0} \Lambda_{\alpha \alpha}(\varphi_t). \end{align*}$$

Thus, a G-equivariant line bundle on M yields a ${\mathfrak g}$ -equivariant line bundle for ${\mathfrak g}=\mathrm {Lie}(G)$ . However, the map

$$\begin{align*}\mathrm{Pic}_G(M) \to \mathrm{Pic}_{\mathfrak g}(M) \end{align*}$$

in general is neither injective nor surjective. Non-injectivity is illustrated by an action of a discrete group, like $\mathbb Z_m:z\mapsto z^m$ on ${\mathbb C} P^1$ . Non-surjectivity is demonstrated as follows.

Example 2.32 (Projective action revisited).

The Lie groups $SL(2,{\mathbb C})$ and $PGL(2,{\mathbb C})$ act on ${\mathbb C} P^1$ by Möbius transformations. In the open cover given by charts $U_0\simeq {\mathbb C}(x)$ and $U_\infty \simeq {\mathbb C}(y)$ , the action is

$$\begin{align*}\varphi=\left(\begin{matrix} a & b \\ c & d \end{matrix}\right):\quad \varphi^*(x) = \frac{ax+b}{cx+d}, \qquad \varphi^*(y) = \frac{dy+c}{by+a}. \end{align*}$$

For $SL(2,\mathbb C)$ , the lifts are given by

$$\begin{align*}\hat\varphi^*(u_0)=\frac{u_0}{(cx+d)^A}, \qquad \hat\varphi^*(u_\infty) = \frac{u_\infty}{(by+a)^A}, \end{align*}$$

where $A\in \mathbb Z$ , as in Example 2.14. However, for $PGL(2,\mathbb C)$ , the lifts are given by

$$\begin{align*}\hat\varphi^*(u_0)=\frac{(ad-bc)^{A/2} u_0}{(cx+d)^A}, \qquad \hat\varphi^*(u_\infty) = \frac{(ad-bc)^{A/2} u_\infty}{(by+a)^A}, \end{align*}$$

which is well defined if and only if $A=2m\in 2\mathbb Z$ . In other words, the line bundle $\mathcal {O}_{\mathbb CP^1}(1)$ is not $PGL(2,\mathbb C)$ -equivariant, but $\mathcal {O}_{\mathbb CP^1}(2)$ is.

This example, borrowed from [Reference Mumford22, Ch. 1.3], works in any dimension n: the line bundle $\mathcal {O}_{{\mathbb C} P^n}(k)$ is $PGL(n+1,{\mathbb C})$ -equivariant iff $k\in (n+1)\mathbb Z$ (i.e., this group lifts only to the powers of the canonical bundle $K_{{\mathbb C} P^n}$ ). However, all bundles $\mathcal {O}_{{\mathbb C} P^n}(k)$ are $SL(n+1,{\mathbb C})$ -equivariant. (Note that the center of $SL(n+1,{\mathbb C})$ is $\mathbb Z_{n+1}$ and $PGL(n+1,{\mathbb C})=SL(n+1,{\mathbb C})/\mathbb Z_{n+1}$ .) This difference cannot be seen at the Lie algebra level, since the two Lie group actions give rise to the same Lie algebra of vector fields. Summarizing, we have

$$\begin{align*}\mathrm{Pic}_{SL(n+1,{\mathbb C})}({\mathbb C} P^n)= \mathrm{Pic}_{\mathfrak{sl}(n+1,{\mathbb C})}({\mathbb C} P^n)=\mathbb Z= \mathrm{Pic}(\mathbb CP^n) \supset (n+1)\mathbb Z=\mathrm{Pic}_{PGL(n+1,{\mathbb C})}({\mathbb C} P^n). \end{align*}$$

Note that in this example, both groups $PGL(n+1,{\mathbb C})$ and $SL(n+1,{\mathbb C})$ are algebraic, so this example illustrates a general result in [Reference Mumford22, Cor. 1.6] on G-linearization of high powers $L^m$ of an algebraic line bundle L. Next, we discuss a similar effect for invariant divisors.

Recall that an algebraic Lie algebra is ${\mathfrak g}=\mathrm {Lie}(G)$ for an algebraic Lie group G. If M is an algebraic variety, we call a Lie algebra ${\mathfrak g} \subset \mathcal {D}(M)$ algebraic if it is the Lie algebra of an algebraic action by an algebraic Lie group on M. The following is a converse to Proposition 2.25 in the algebraic context (there is a version of this statement for $\mathrm {Pic}_G(M)$ ).

Proof. Since ${\mathfrak g}$ is transversal, it admits on L an absolute invariant $I=I(x,u)$ , with x coordinate on M and u a fiber coordinate on L, such that $\partial _u(I)\not \equiv 0$ . This complements absolute invariants $J=J(x)$ obtained by pullback from M. By Rosenlicht’s theorem [Reference Rosenlicht26] the algebraicity of the action implies that the invariant I can be chosen rational in proper (local) variables $x,u$ (on $U_\alpha $ with algebraic overlaps). Decompose I into its Laurent series by the fiber variable u

(2.13) $$ \begin{align} I=\sum_{k=-N}^\infty h_k(x)u^k. \end{align} $$

Since $[u\partial _u,\hat {\mathfrak g}]=0$ , we get that $(u\partial _u)^r(I)$ is an absolute invariant for every r. The spectrum of the operator $u\partial _u$ on generators $u^k$ is simple, and due to rationality, the coefficients of I are determined by a finite number of base functions $h(x)$ . Thus, every term in the series (2.13) is an absolute invariant. Choose such invariant of the lowest (in absolute value) degree by u. This degree m does not depend on local coordinate chart $U_\alpha ,\alpha \in A$ , we are using, and we get

$$\begin{align*}I_\alpha=\frac{u_\alpha^m}{f_\alpha(x)}\quad\ \Longrightarrow\ \quad 1=\frac{I_\alpha}{I_\beta}=\frac{u_\alpha^m/f_\alpha(x)}{u_\beta^m/f_\beta(x)} =g^m_{\alpha\beta}\frac{f_\beta}{f_\alpha}\quad\text{ on }\quad U_\alpha\cap U_\beta. \end{align*}$$

The collection of functions $\{f_\alpha \in \mathcal {O}(U_\alpha ):\alpha \in A\}$ defines a ${\mathfrak g}$ -invariant divisor D with weight $\lambda _\alpha (X)=X(\log f_\alpha )$ , $X\in {\mathfrak g}$ , and the corresponding line bundle $[D]$ has transition functions

$$\begin{align*}\tilde{g}_{\alpha\beta}= \frac{f_\alpha}{f_\beta}=g_{\alpha\beta}^m. \end{align*}$$

Thus, $[D]=L^m$ , and the claim follows.

3.1 Lie algebra action on affine bundles

Let $\pi \colon E \to M$ be an affine bundle (of rank $r\ge 1$ ), and let $\hat {\mathfrak g} \subset \mathcal {D}_{\mathrm {proj}}(E)$ be a Lie algebra of projectable vector fields on E that preserves the affine structure in the fibers. In this section, we will focus on $\hat {\mathfrak g}$ -invariant divisors whose restriction to fibers are polynomials. We define for $U \subset M$

$$\begin{align*}\mathfrak{P}(U) = \{f \in \mathcal{O}(\pi^{-1}(U)) \mid f|_{\pi^{-1}(p)} \text{ is a polynomial for every } p \in U\}, \end{align*}$$

where polynomiality is checked in affine coordinates on E. This space of functions is preserved by automorphisms: If $\varphi \colon E \to E$ is an automorphism of affine bundles and $f \in \mathfrak {P}(U)$ , then $\varphi ^*f \in \mathfrak {P}(\varphi ^{-1}(U))$ .

Assume that $\mathcal {U}=\{U_\alpha \}$ is an open cover for M such that $\pi ^{-1}(U_\alpha )\simeq U_\alpha \times {\mathbb C}^r$ for any affine bundle $\pi $ . Then $\{\pi ^{-1}(U_\alpha )\}$ is open cover for the total space E.

In local coordinates $x^i, u^j$ on $\pi ^{-1}(U_\alpha ) \simeq U_\alpha \times \mathbb C^r$ , $f_\alpha $ takes the form

$$\begin{align*}f_\alpha = \sum_{|\sigma|\leq s} F_{\sigma}(x) u^{\sigma}, \qquad F_\sigma \in \mathcal{O}(U_\alpha). \end{align*}$$

Here, $u^\sigma =\prod (u^i)^{m_i}$ for the multi-index $\sigma =(m_1\dots m_r)$ . The defining functions of a polynomial divisor D satisfy $g_{\alpha \beta } = f_\alpha /f_\beta \in \mathcal {O}^\times (\pi ^{-1}(U_\alpha ) \cap \pi ^{-1}(U_\beta ))$ , where both the numerator and denominator are polynomials in $u^1, \dots , u^r$ . It follows that the polynomials must cancel each other out, which implies that the transition functions $g_{\alpha \beta } = f_\alpha /f_\beta $ are the pullback of functions $\tilde g_{\alpha \beta }\in \mathcal {O}^\times (U_\alpha \cap U_\beta )$ . Thus, we obtain the following proposition.

In the above argument, it is clear that the degree s can be taken to be the same for each $\alpha $ . We call the smallest such s the degree of the polynomial divisor D.

Next, we let $\hat {\mathfrak g} \subset \mathcal {D}_{\mathrm {proj}}(E)$ be a projectable Lie algebra of vector fields on E preserving the affine structure on fibers, and consider $\hat {\mathfrak g}$ -invariant polynomial divisors.

Proof. What remains to be proven is that the bundle $\pi \colon L \to M$ with transition functions $\tilde g_{\alpha \beta }$ admits a ${\mathfrak g}$ -lift. In local coordinates $x^i, u^j$ on $\pi ^{-1}(U_\alpha ) \simeq U_\alpha \times \mathbb C^r$ , each $X \in \hat {\mathfrak g}$ takes the form $X = a^i(x) \partial _{x^i} + (b_0^j(x)+b_l^j(x) u^l) \partial _{u^j}$ . Consider an invariant divisor D given by $f_\alpha = \sum _{|\sigma | \leq s} F_{\sigma }(x) u^\sigma $ , a polynomial in $u^1, \dots , u^r$ of degree s. We have

$$\begin{align*}X(f_\alpha) = \lambda_\alpha(X) f_\alpha. \end{align*}$$

Looking at the coordinate form of X, it is clear that $X(f_\alpha )$ is a polynomial in $u^1, \dots , u^r$ and, furthermore, that its degree is $\leq s$ . Thus, $\lambda _\alpha (X)=X(f_\alpha )/f_\alpha $ is a rational function in $u^1, \dots , u^r$ , and it is defined everywhere on $U_\alpha \times \mathbb C^r$ only if $\lambda _\alpha (X)$ is the pullback of a function $ \tilde \lambda _\alpha (X) \in \mathcal {O}(U_\alpha )$ . Thus, the $\hat {\mathfrak g}$ -equivariant line bundle over E defined by $(\{g_{\alpha \beta }\}, \{\lambda _\alpha \})$ is the pullback of the ${\mathfrak g}$ -equivariant line bundle over M defined by $(\{\tilde g_{\alpha \beta }\}, \{\tilde \lambda _\alpha \})$ .

Proposition 3.3 (which was reformulated in Theorem 1.4) is relevant, for instance, for investigation of relative invariants of tensor fields (and other geometric objects like affine connections) on a manifold M under the action of some Lie algebra ${\mathfrak g} \subset \mathcal {D}(M)$ .

Let us make a brief remark about invariant rational divisors. Each such is a ratio of two invariant polynomial divisors. The weights of invariant rational divisors form a lattice generated by weights of invariant polynomial divisors. In other words, we have the following relation:

(3.1) $$ \begin{align} \mathrm{Span}_{\mathbb Z} \left(\mathrm{Div}_{\mathfrak g}^{\mathrm{pol}}(M) \right) =\mathrm{Div}_{\mathfrak g}^{\mathrm{rat}}(M). \end{align} $$

3.2 Lie algebra action on jet bundles

Now we consider polynomial divisors on jet bundles. Most of the arguments here closely resemble those in Section 3.1, but some additional care must be taken. Our introduction to jets will be very brief, and we refer to [Reference Kruglikov, Lychagin, Krupka and Saunders19, Reference Vinogradov and Krasil’shchik21, Reference Olver23] for a more comprehensive treatment.

Let $J^k(E,m)$ denote the space of k-jets of codimension-m submanifolds of E, and $J^k \pi $ the space of k-jets of sections of the fiber bundle $\pi $ . In statements that are true for both $J^k(E,m)$ and $J^k \pi $ , we will use the notation $\mathbf {J}^k$ which can always be replaced with either of the two (an exception to this convention occurs only in Section 3.5). There are natural bundle structures $\pi _{k,l}\colon \mathbf {J}^k \to \mathbf {J}^l$ for $0 \leq l \leq k$ , and $\pi _k\colon J^k \pi \to M$ .

Coordinates on $J^k \pi $ and $J^k(E,m)$ are induced from coordinates on the total space of $\pi $ or E, respectively. Given a bundle $\pi \colon E \to M$ , and an open cover $\{U_\alpha \}$ of coordinate charts of E, the collection $\{\pi _{k,0}^{-1}(U_\alpha )\}$ is an open cover of $J^k \pi $ . The split coordinates $x^1,\dots x^n, u^1, \dots , u^m$ on $U_\alpha $ induce additional canonical coordinates $u_\sigma ^j$ , $|\sigma |\leq k$ where $\sigma $ is a multi-index, on $\pi _{k,0}^{-1}(U_\alpha )$ .

To get an open cover of $J^k(E,m)$ , we let $\{U_\alpha \}$ be an open cover of coordinate charts of E that trivializes the bundle $J^1(E,m) \to E$ . On each $U_\alpha $ , for a given set of $m+n$ coordinates, we choose a splitting $x^1,\dots , x^n, u^1, \dots , u^m$ , and we denote the corresponding coordinate chart on $J^1(E,m)$ by $U_\alpha ^{i_1 \cdots i_m}$ with $1 \leq i_1 < \cdots < i_m \leq \dim E$ . For each way of splitting, there is one chart. The split coordinates on $U_\alpha $ induce additional canonical coordinates $u_\sigma ^i$ ( $|\sigma |\leq k$ ) on $\pi _{k,1}^{-1}(U_\alpha ^{i_1 \cdots i_m})$ for $k \geq 1$ . The collection $\{\pi _{k,1}^{-1}(U_\alpha ^{i_1 \cdots i_m})\}$ is an open cover of $J^k(E,m)$ . We define for $U \subset \mathbf {J}^i$

$$\begin{align*}\mathfrak{P}_i(U) = \{f \in \mathcal{O}(\pi_{\infty,i}^{-1}(U)) \mid f|_{\pi_{\infty,i}^{-1}(p)} \text{ is a polynomial for every } p \in U\}. \end{align*}$$

Polynomiality is defined with respect to the canonical coordinates described above. For example, in the case of $J^k(E,m)$ , then $f \in \mathfrak {P}_j(U)$ if and only if we have, for each $\alpha $ ,

$$\begin{align*}f\Big|_{\pi_{\infty,j}^{-1}\left(U \right) \cap \pi_{\infty,1}^{-1}\left(U_\alpha^{i_1 \cdots im}\right)} = \sum_{j \leq |\sigma|\leq k, |\tau| \leq r} F^\sigma_\tau u_\sigma^\tau \end{align*}$$

for some $k,r \in \mathbb Z_{\geq 0}$ and a collection $\{F^\sigma _\tau \}$ of functions on $J^j(E,m)$ .

If $\varphi \colon E \to E$ is a point transformation and $f \in \mathfrak {P}_i(U)$ , then $(\varphi ^{(\infty )})^* f \in \mathfrak {P}_i(\varphi ^{-1}(U))$ for $i \geq 1$ . Similarly, if $\pi \colon E \to M$ is a fiber bundle, $\varphi \colon E \to E$ is a fiber-preserving biholomorphism, and $f \in \mathfrak {P}_i(U)$ , then $(\varphi ^{(\infty )})^* f \in \mathfrak {P}_i(\varphi ^{-1}(U))$ for $i \geq 0$ . In particular, point transformations preserve $\mathfrak {P}_1$ , while fiber-preserving transformations do the same for $\mathfrak {P}_0$ . Based on this, we introduce the following notion.

From the proof, it follows that the order and degree of the polynomials $f_\alpha ^{i_1 \cdots i_m}$ and $f_\beta ^{j_1 \cdots j_m}$ agree. Therefore, the order and degree are also well-defined notions for $D = \{f_\alpha ^{i_1 \cdots i_m}\}$ , and this is true also for divisors on $J^k \pi $ . We define the weighted degree of the monomial $c(x,y,y_i) y_{\sigma _1}^{j_1} \cdots y_{\sigma _s}^{j_s} \in \mathfrak {P}_1(U_\alpha ^{i_1 \cdots i_m})$ (with $|\sigma _l| \geq 2$ for each l) to be $\sum _{l=1}^s |\sigma _l|$ , and the weighted degree of a sum of such to be the maximal weighted degree of its monomial parts. The weighted degree can be defined for a divisor in the same way that order and degree were defined above. (In the case when E is a bundle, the weighted degree also counts the first-order jet variables $y_i^j$ .)

Next, consider a Lie algebra of vector fields ${\mathfrak g}\subset \mathcal {D}(\mathbf {J}^0)$ ; in the case $\mathbf {J}^0=J^0\pi $ , assume also that ${\mathfrak g}$ is $\pi $ -projectable. The Lie algebra prolongs to a unique Lie algebra ${\mathfrak g}^{(k)} \subset \mathcal {D}(\mathbf {J}^k)$ ; see, for instance, [Reference Kruglikov, Lychagin, Krupka and Saunders19, Sec. 1.5]. We are interested in polynomial ${\mathfrak g}^{(k)}$ -invariant divisors on $\mathbf {J}^k$ .

Proof. On $J^k(E,m)$ we will use the split coordinate charts $\pi _{k,1}^{-1}(U_\alpha ^{i_1 \cdots i_m})$ . If $\pi \colon E\to M$ is a bundle, then the splitting is canonical, and we use the charts $\pi ^{-1}_{k,0}(U_\alpha )$ on $J^k \pi $ .

The prolongation of a vector field $X=a^i \partial _{x^i} + b^j \partial _{y^j} \in \mathfrak g \subset \mathcal {D}(E)$ is given by

$$ \begin{align*} X^{(k)} =a^i \partial_{x^i} + \sum_{0 \leq |\sigma|\leq k} b_\sigma^j \partial_{y_\sigma^j}, \end{align*} $$

where $b_{\sigma }^j$ are given recursively by (see [Reference Vinogradov and Krasil’shchik21, Th. 3.4])

(3.2) $$ \begin{align} b_{\sigma i}^j = D_{x^i}(b_\sigma^j)-y_{\sigma l}^j D_{x^i}(a^l). \end{align} $$

When $|\sigma |=d$ , it is clear that $b^j_{\sigma }$ is a sum of monomials of the form $c(x,y) y^{j_1}_{\sigma _1} \cdots y^{j_s}_{\sigma _s}$ with $|\sigma _l| \leq d$ for each l and $|\sigma _1|+\cdots + |\sigma _s| \leq d+1$ . Thus, the weighted degree of $b_\sigma ^j$ is $\leq d+1$ .

If $D=\{f_\alpha ^{i_1 \cdots i_m}\}$ is a polynomial invariant divisor on $J^k(E,m)$ of weighted degree d, then for any $X \in {\mathfrak g}$ , the function $X^{(k)}(f_\alpha ^{i_1 \cdots i_m})$ has weighted degree $\leq d+1$ . The equality $X^{(k)}(f_\alpha ^{i_1 \cdots i_m}) = \lambda _\alpha ^{i_1 \cdots i_m}(X) f_\alpha ^{i_1 \cdots i_m}$ implies that $\lambda _\alpha ^{i_1 \cdots i_m}(X) = X^{(k)}(f_\alpha ^{i_1 \cdots i_m})/f_\alpha ^{i_1 \cdots i_m}$ is holomorphic on $\pi _{k,1}^{-1}(U_\alpha ^{i_1 \cdots i_m})$ if and only if the polynomial parts of the denominator is canceled out by the numerator. In this case, $\lambda _\alpha ^{i_1 \cdots i_m}(X)$ is polynomial of weighted degree $\leq 1$ , meaning that it is the pullback of a function $\tilde {\lambda }_\alpha ^{i_1 \cdots i_m}(X) \in \mathcal {O}(U_\alpha ^{i_1 \cdots i_m})$ . Thus, we see that the ${\mathfrak g}^{(k)}$ -equivariant line bundle over $J^k(E,m)$ defined by the pair $(\{f_\alpha ^{i_1 \cdots i_m}/f_\beta ^{j_1 \cdots j_m}\}, \{\lambda _\alpha ^{i_1 \cdots i_m}\})$ is the pullback of a ${\mathfrak g}^{(1)}$ -equivariant line bundle over $J^1(E,m)$ .

If $E \to M$ is a fiber bundle, then we get a similar argument, but now $\lambda _\alpha (X)$ has weighted degree $0$ and is therefore the pullback of a function in $\mathcal {O}(U_\alpha )$ .

This proposition, whose second part was reformulated in Theorem 1.5, tells us that invariant polynomial divisors on $\mathbf {J}^k$ are sections of pullbacks of equivariant line bundles over $\mathbf {J}^1$ or $\mathbf {J}^0$ . In particular, they are controlled by ${\mathrm {H}}^1(\mathrm {Tot}^\bullet (C))$ , where $C^{p,q}=C^{p,q}({\mathfrak g}^{(1)},\{U_\alpha ^{i_1\cdots i_m}\})$ or $C^{p,q}=C^{p,q}({\mathfrak g},\{U_\alpha \})$ or, more precisely, by $\mathrm {Pic}_{{\mathfrak g}^{(r)}}(\mathbf {J}^r)$ for $r=1,0$ , respectively. It is remarkable that this fact is independent of the order k (one should compare to the statement of the Lie-Bäcklund theorem [Reference Vinogradov and Krasil’shchik21, Reference Olver23], although the proofs are different).

If the bundle $\pi $ has, in addition, an affine structure, then we can consider divisors with local defining functions $f_\alpha $ in

$$\begin{align*}\mathfrak{P}_{-1}(U_\alpha) = \{f \in \mathcal{O}(\pi_{\infty}^{-1}(U_\alpha)) \mid f|_{\pi_{\infty}^{-1}(p)} \text{ is a polynomial for every } p \in U_\alpha\} \end{align*}$$

(i.e., divisors that are polynomial on fibers of $\pi _k \colon J^k \pi \to M$ ). These are preserved under ( $k^{\text {th}}$ -prolongation of) morphisms of affine bundles, and we will refer to them as ‘polynomial divisors’ in this context. For such a divisor D, we have $[D]=\pi _k^* L$ for some line bundle $L \to M$ . We can apply the same ideas as above to obtain the following result, which we leave without proof.

Computations of invariants in jets often result in rational relative differential invariants, which are related to polynomial differential invariants via a jet analogue of formula (3.1). This will be demonstrated in the following examples.

3.3 Example A: Three-dimensional Heisenberg algebra on the plane

Consider the following Lie algebra of vector fields on the plane:

$$\begin{align*}\mathfrak g=\langle \partial_x,\partial_y, y\partial_x\rangle \subset \mathcal{D}(\mathbb C^2). \end{align*}$$

It has the structure relations of the Heisenberg algebra and it prolongs naturally to the Lie algebra ${\mathfrak g}^{(1)}$ of vector fields on $J^1(\mathbb C^2,1)$ . Choosing y as the dependent variable gives

$$\begin{align*}\mathfrak g^{(1)}|_{U_1} = \langle \partial_x, \partial_y, y \partial_x-y_1^2 \partial_{y_1} \rangle, \end{align*}$$

where $U_1 \subset J^1(\mathbb C^2,1)$ denotes the open chart determined by our choice of dependent variable on $\mathbb C^2$ . Taking instead x as the dependent variable results in a different chart $U_2 \subset J^1(\mathbb C^2,1)$ , where the prolongation of $\mathfrak g$ takes the form

$$\begin{align*}\mathfrak g^{(1)}|_{U_2} = \langle \partial_x, \partial_y, y\partial_x+\partial_{x_1} \rangle. \end{align*}$$

These two charts cover $J^1(\mathbb C^2,1)=U_1 \cup U_2$ . On overlap $U_1 \cap U_2$ we get $(x,y,y_1)\equiv (x,y,1/x_1)$ . In each of the two charts, we compute the Chevalley-Eilenberg cohomology:

$$\begin{align*}{\mathrm{H}}^1( \mathfrak g^{(1)}, \mathcal{O}(U_1)) = \mathbb C^2, \qquad {\mathrm{H}}^1( \mathfrak g^{(1)}, \mathcal{O}(U_2)) = 0. \end{align*}$$

A representative $\lambda _1$ of a general element in ${\mathrm {H}}^1({\mathfrak g}^{(1)},\mathcal {O}(U_1))$ takes the form

$$\begin{align*}\lambda_1(\partial_x)=0, \qquad \lambda_1(\partial_y)=0, \qquad \lambda_1 (y \partial_x-y_1^2 \partial_{y_1}) = A + B y_1, \qquad A,B \in \mathbb C. \end{align*}$$

The compatibility condition $\lambda _1(X) - \lambda _2(X) = X(g_{12})/g_{12}$ , $\forall X \in {\mathfrak g}^{(1)}$ gives the general transition function $g_{12} = C y_1^{-B} e^{A/y_1}$ . This function is holomorphic on $U_1 \cap U_2$ if and only if $B \in \mathbb Z$ . Changing the representative $(g,\lambda )\in C^{0,1}\times C^{1,0}$ by the coboundary $\partial ^0\mu $ where $\mu _1=1, \mu _2=C e^{A x_1}$ , we get $g_{12} = y_1^{-B}$ and

$$\begin{align*}\lambda_1 = (0,0,A+By_1), \qquad \lambda_2 = (0,0,A). \end{align*}$$

Thus, $\mathrm {Pic}_{{\mathfrak g}^{(1)}}(\mathbf {J}^1)={\mathbb C}\times \mathbb Z\to \mathrm {Pic}(\mathbf {J}^1)=\mathbb Z$ is epimorphic.

We identify a generating set $(I,\nabla )$ of absolute differential invariants in charts as follows:

$$\begin{align*}\Big(-\frac{y_2}{y_1^3},\frac{1}{y_1}D_x\Big)\ \text{ on }\ U_1\quad\longleftrightarrow\quad \big(x_2,D_y\big)\ \text{ on }\ U_2. \end{align*}$$

The invariant divisors on $J^1(\mathbb C^2,1)$ are generated by $f_1=y_1, f_2=1$ of weight $(A,B)=(0,-1)$ . Note that the invariant ODE $y_1=0$ is not visible from the local computations on $U_2$ . Indeed, its solutions are $y=\mathrm {const}$ for the independent variable y, which are not graphs $x=h(y)$ .

General ${\mathfrak g}^{(2)}$ -invariant divisors on $J^2(\mathbb C^2,1)$ are generated by $f=\{f_1,f_2\}$ and the absolute invariant I. In particular, the irreducible invariant submanifolds of codimension 1 in $\mathbf {J}^2$ are given by the divisors $\tilde f=\{\tilde {f}_1,\tilde {f}_2\}=\{y_2-C y_1^3,x_2+C\}$ of weight $(A,B)=(0,-3)$ , parametrized by $C \in \mathbb C$ .

Note that the nonzero parameter A above is not realizable by an invariant divisor (on $\mathbf {J}^1$ such are $y_1^{-B}$ ). Higher prolongations give no new weights of polynomial divisors, and we conclude, with the help of Proposition 3.7,

$$\begin{align*}\mathbb Z= j_{{\mathfrak g}^{(\infty)}}\mathrm{Div}_{{\mathfrak g}^{(\infty)}}^{\mathrm{rat}}(\mathbf{J}^\infty)\subset \mathrm{Pic}_{{\mathfrak g}^{(1)}}(\mathbf{J}^1)={\mathbb C}\times\mathbb Z. \end{align*}$$

3.4 Example B: Invariant divisors of curves in the projective plane

Consider the Lie algebra $\mathfrak {sl}(3,{\mathbb C})\subset \mathcal {D}({\mathbb C} P^2)$ of projective vector fields. Differential invariants of curves in the projective plane were studied already in 1878 by Halphen in his PhD thesis [Reference Huybrechts15] (see also the recent treatment [Reference Konovenko and Lychagin16] in the real case). In this section, we demonstrate how the framework developed in this paper sheds new light on those classical invariants.