# Affine Nomizu's algebras and compact complex surfaces with meromorphic affine connections.

Alexis Garcia

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Alexis Garcia. Affine Nomizu's algebras and compact complex surfaces with meromorphic affine connections. 2025. hal-05166996

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Affine Nomizu's algebras and compact complex surfaces with meromorphic affine connections.

Alexis Garcia¹,²*

¹*DIMAI, University of Florence, Viale Morgagni, Florence, 50134, FI, Italy.

Corresponding author(s). E-mail(s): alexispierre.garcia@unifi.it;

**Abstract**

We use the Cartan's description of holomorphic affine connections to give a tractable characterization of their Nomizu's algebra. We prove that in complex dimension two, either the affine connection is flat or the isotropy subalgebra stabilizes a line in the complex plane. Using the generalized Singer's theorem by Pecastaing, this implies the existence of a holomorphic foliation on the base manifold. We apply this result to classify compact Kähler surfaces of algebraic dimension zero admitting a meromorphic affine connection.

**Keywords:** Complex surfaces, Meromorphic affine connections, Nomizu's algebras, Cartan geometries

**MSC Classification:** 32J15, 32Q30, 53C05

# 1 Introduction

The classification of complex compact manifolds is certainly the central question in complex geometry, and is still open nowadays. The first answer was given for complex dimension one manifolds, by the Riemann's uniformization theorem.

This result can be explained as follows. Any orientable compact manifold can be endowed with a Riemannian metric, and in real dimension two, conformal classes of such metrics are in bijection with complex structures. Moreover, it was proved that on compact real manifolds of dimension two any such class admits a representative with constant Gaussian curvature. By the Killing-Hopf theorem, this implies that the manifold admits a $(G, X)$-structure where $X$ is a manifold endowed with a metric with the same constant curvature, and $G$ is a group of global isometries of $X$ acting
transitively. It turns out that $X$ is in any case an open subset of the complex projective line $\mathbb{P}^1$, and $G$ a subgroup of its complex automorphisms $\mathrm{PSL}_2(\mathbb{C})$.

Moreover, the $(G, X)$-structure thus obtained is *uniform*, in the sense that the developant map is an embedding. Therefore, compact Riemann surfaces are *uniformized* by the model $(G, G/P) = (\mathrm{PSL}_2(\mathbb{C}), \mathbb{P}^1)$, meaning that any such manifold is obtained by a quotient $q: U \to \Gamma\backslash U$ of an open subset $U \subset G/P$ by a discrete subgroup $\Gamma \subset G$, in such a way that $q$ is an etale cover.

We can ask whether we can split higher dimensional compact complex manifolds into classes which are uniformized by some model $(G, G/P)$, where $G$ is a complex Lie group and $P$ a closed complex subgroup. We can try to extend the Riemann's uniformization theorem by studying the existence of locally homogeneous Hermitian metrics on such manifolds. However, as shown by the Wall's classification [1], not any compact complex surface admits a $(G, G/P)$-structure where $G$ is a group of isometries acting transitively on a Hermitian space $G/P$. Therefore, we must study more general geometric structures. We could first restrict ourselves to *holomorphic geometric structures*, including holomorphic affine connections, holomorphic projective connections, and some holomorphic reduction of $k$-th order holomorphic frame bundle.

Most of these examples correspond to *holomorphic Cartan geometries* (see [5] for the definition in a singular setting, Theorem [4] for the equivalence with affine connections, and [2] for equivalence with other geometric structures including conformal ones). One hope for extending the Riemann's uniformization theorem using such geometric structures stands in the existence of *infinitesimally homogeneous holomorphic geometric structures* on any compact complex manifold $M$, meaning the infinitesimal automorphisms of the geometric structure span the tangent space at any $x \in M$. Indeed, in this case, the *generalized Singer's theorem* proved by Pecastaing in [3] implies that the manifold is endowed with a *flat* $(H, H_0)$-Cartan geometry induced by the infinitesimal symmetries (Theorem [12]). This is a generalization of $(G, G/P)$-structures, in the sense that when $H/H_0$ is a well-defined manifold, *flat* $(H, H_0)$-Cartan geometries coincide with $(H, H/H_0)$-structures. Here, $H$ is a complex Lie group whose Lie algebra is the *Nomizu's algebra* of the Cartan geometry at $x$, namely the (isomorphism class of) Lie algebra of germs of its infinitesimal automorphisms at $x$, and $H_0 = \exp_H(h_0)$ where $h_0$ is the subalgebra of germs vanishing at $x$. Therefore, a first question is: *Can we divide compact complex manifolds into classes endowed with infinitesimally homogeneous holomorphic geometric structures of a certain kind?*

Negative answers of the above question were given for simple geometric structures. Compact complex surfaces endowed with holomorphic affine connections were classified by Inoue and Kobayashi ([4]) and it turns out that these manifolds must be minimal, and do not cover all surfaces appearing in the Enriques-Kodaira classification. In general, the existence of a holomorphic geometric structure imposes restrictions on the geometry and topology of a compact complex manifolds (see [5] for the Kähler case).

In one other hand, holomorphic Cartan geometries admit a straightforward singular generalization called *meromorphic Cartan geometries* (5). These singular geometric structures are more flexible, as shows the 1. Moreover, the Pecastaing's theorem mentioned above applies to the complement of the singular locus, giving a flat $(H, H_0)$-Cartan geometry on this open dense subset of the manifold. Therefore, a new
question is: Does any complex compact manifold admit an infinitesimally homogeneous meromorphic Cartan geometry?

By the previous example, this question admits a positive answer for projective manifolds, and there is a generalization of the notion of uniformization given by finite maps from the manifold to a model geometry given by a manifold endowed with a model meromorphic geometric structure. In fact this also gives a positive answer for complex compact surfaces with algebraic dimension two, since these are bimeromorphic to projective ones. However, in [6], we classified complex compact surfaces of algebraic dimension one endowed with meromorphic affine connections, and it turns out that this does not cover all such surfaces (in particular such a surface must be an isotrivial elliptic fibration over a compact Riemann surface).

In this paper, we pursue the classification of compact complex surfaces endowed with meromorphic affine connection by the remaining case, namely algebraic dimension zero surfaces. In fact, we complete the classification in the Kähler case, which is mostly reduced to the case of K3 surfaces (since Kähler minimal surfaces of algebraic dimension zero are either tori or K3 surfaces, and tori obviously admits holomorphic affine connections).

The strategy is to study the possible models $(H, H_0)$ for the flat holomorphic Cartan geometry appearing in the Pecastaing's theorem, and prove that this geometric structure extends to a meromorphic geometric structure on the whole manifold. Motivated by the above considerations, we use the language of Cartan's geometries to study meromorphic affine connections, and work at the level of Lie algebras, giving a characterization of their *Nomizu's algebra* which holds for any Cartan geometry. Then we give a more tractable characterization in the affine case (i.e. for affine connections) and prove that, in the non-flat case, the isotropy subgroup $H_0$ stabilizes a complex line in $\mathbb{C}^2$. This implies the existence of a non-trivial holomorphic foliation on the manifold (Theorem 13). In fact, in the Kähler case, we are able to prove that there exists a holomorphic foliation even in the flat case. The classification in the Kähler case is then achieved (Theorem 15) using the result by Brunella classifying minimal Kähler surfaces endowed with non-trivial holomorphic foliations (Theorem 2).

# 2 Meromorphic affine connections and affine Cartan geometries

## 2.1 Analytic definition and properties

**Definition 1** Let $M$ be a complex manifold. A meromorphic affine connection on $M$ is a $\mathbb{C}$-linear morphism of sheaves $\nabla : TM|_{M\setminus D} \to \Omega^1|_{M\setminus D} \otimes TM|_{M\setminus D}$ where $D$ is a subset of empty interior, which satisfies the Leibniz identity:

$$ (1) \quad \forall U \subset M \setminus D, \forall f \in \mathcal{O}_U, X \in TM(U), \nabla(fX) = df \otimes X + f\nabla(X) $$

and the meromorphicity condition:

$$ (2) \quad \forall U \subset M, \forall X \in TM(U), \exists f \in \mathcal{O}_M(U), f\nabla(X) \in \Omega_M^1 \otimes TM(U) $$

The subset $D$ is called the pole of $\nabla$ whenever it is minimal among the subsets such that (2) holds for any $U \subset M$.
We can give a local characterization of meromorphic affine connections on M with pole D, among holomorphic affine connections on M \ D, implying in particular that D is an analytic subset of codimension on in M. Let ∇ be a holomorphic affine connection on M \ D, U ⊂ M and assume the existence of a local holomorphic trivialization TM|_U = O_U X_1 ⊕ ... ⊕ O_U X_n. Then there exists unique holomorphic one forms (A_j^k)_{j,k=1,...,n} such that:

$$ \nabla(X_j) = \sum_{j=1}^{n} A_j^k \otimes X_k $$

The matrix of one forms Mat(X_j)∇ := (A_j^k)_{j,k} is called the *matrix form* of ∇ in the basis (X_j)_{j=1,...,n}. If (Y_j)_{j=1,...,n} is another holomorphic trivialization of TM on U, and Q is the matrix of (Y_j)_{j=1,...,n} in (X_i)_{i=1,...,n}, then the gauge change formula implies:

$$ (3) \qquad Mat_{(Y_j)} \nabla = dQQ^{-1} + Q(Mat_{(X_i)} \nabla)Q^{-1} $$

Moreover, we can find sufficiently small neighborhoods U of any x ∈ M such that X_j = ∂/∂z_j for local coordinates (z_i)_{i=1,...,n} on U. Then A_j^k = ∑_{i=1}^n Γ_{i,j}^k dz_i for some unique holomorphic functions Γ_{i,j}^k called the *Christoffel symbols* of ∇. Then ∇ is meromorphic if and only if the local Christoffels symbols are meromorphic functions on any subset U endowed with holomorphic coordinates. The pole D is the union of the poles of the Christoffel symbols, proving our claim.

Let ϕ : M → M' be a map of complex manifolds, such that there exists S ⊂ M and S' ⊂ M' proper analytic subsets such that ϕ restricts as a locally biholomorphic map between M \ S and M' \ S'. Suppose the existence of a meromorphic affine connection ∇' on M'. Then there exists (see [6], Lemma 2) a unique meromorphic affine connection ϕ*∇' on M satisfying:

$$ \forall X, Y \in TM(U), T\varphi(Y \rfloor \varphi^* \nabla'(X)) = T\varphi(Y) \rfloor \nabla'(T\varphi(X)) $$

**Definition 2** Let ϕ : M → M' and ∇' as above. The meromorphic affine connection ϕ*∇' is the *pullback* of ∇' by ϕ.

This enables to construct many examples of meromorphic affine connections:

**Example 1** Let M ⊂ P^N be a projective complex manifold of complex dimension n ≥ 1. Then there exists ([7], Proposition ) a finite ramified cover q : M → P^n. Now, on P^n = SL_{n+1}(C)/P (here P is the stabilizer of C_1 ⊂ C^{n+1}), we have the family (X_i)_{i=1,...,n} of global holomorphic vector fields defined by:

$$ X_i(x) := \frac{d}{dt} |_{t=0} \exp_{SL_{n+1}} (tC_{1,i+1}) \cdot x $$

spanning T_x P^n for any x ∈ P^n \ D_0 where C_{1,i+1} = e_1^* ⊗ e_{i+1}, and D_0 is the hypersurface obtained as the quotient of {z_1 = 0} ⊂ C^{n+1}. Then, given any x ∈ P^n and local holomorphic coordinates (z_1, ..., z_n) on a neighborhood U of x, the matrix of (X_j)_{j=1,...,n} is
in $(\frac{\partial}{\partial z_i})_{i=1,...,n+1}$ is holomorphic. Its inverse $Q^{-1}$ is a matrix of meromorphic functions on $U$, so that the meromorphic affine connection $\nabla_U$ on $U$ with matrix form $dQQ^{-1}$ in $(\frac{\partial}{\partial z_i})_{i=1,...,n}$ is a meromorphic affine connection. By the gauge change formula (3), $\nabla_U$ is the unique meromorphic affine connection such that $\nabla_U(X_j) = 0$ for any $j = 1,...,n$. These meromorphic affine connections therefore glue to an unique meromorphic affine connection $\nabla_0$ on $\mathbb{P}^n$. By the above considerations, its pullback $q^*\nabla_0$ is a meromorphic affine connection on $M$.

**Definition 3** Let $M$ be a complex manifold endowed with a meromorphic affine connection $\nabla$.

1. A local automorphism of $(M, \nabla)$ is local biholomorphism $\varphi : U \to \varphi(U) \subset M$ such that $\varphi^*\nabla = \nabla$.

2. An infinitesimal automorphism of $(M, \nabla)$ is a local holomorphic vector field $X$ on some open subset $U \subset M$ such that the flow $\phi_X^t$ is a local automorphism of $(M, \nabla)$ whenever defined. For $x \in M$, denote:

$$ (4) \qquad \mathfrak{till}_{M,\nabla}(x) = \{\text{germs of infinitesimal automorphism at } x\} $$

which is a Lie subalgebra of $((TM)_x, [\cdot, \cdot]_{TM})$.

3. The meromorphic affine connection $\nabla$ is infinitesimally homogeneous if there exists $S \subset M$ with $\mathring{S} = \emptyset$ such that for any $x \in M \setminus S$, $T_x M$ is spanned by the evaluations $X(x)$ of elements $X \in \mathfrak{till}_{M,\nabla}(x)$. In this case, (4) is called an *affine Nomizu algebra*.

Recall the following result on compact complex surfaces ([8], Theorem 4.5). Let $M$ be a complex compact surface, then there exists a surjective holomorphic map $\pi : M \to M_0$ to a minimal complex compact surface $M_0$ ([8], Definition p.79) and a codimension one analytic subset $S \subset M$ such that:

(i) $\pi|_{M \setminus S} : M \setminus S \to M_0 \setminus \{\text{points}\}$ is a biholomorphism.

(ii) Any local meromorphic function on $M$ is the pullback of a meromorphic function on $M_0$.

Then we have a bijection between meromorphic affine connections on $M$ and $M_0$ constructed as follow. Given a meromorphic affine connection $\nabla$ on $M$, $\nabla_0 = (\pi_{M \setminus S}^{-1})^*\nabla$ is a meromorphic affine connection on $M_0 \setminus \{\text{points}\}$. Then the Hartogs theorem proves that the local Christoffel symbols of $\nabla_0$ extends as meromorphic affine functions across the points. Hence $\nabla_0$ is a well-defined meromorphic affine connection on $M_0$. The inverse is given by the pullback by $\pi$. Therefore, the classification of compact complex surfaces endowed with meromorphic affine connections is reduced to minimal surfaces.

**Definition 4** Let $\nabla$ be a meromorphic affine connection on $M$. The torion and curvature of $\nabla$ are the meromorphic sections $T_\nabla$ and $R_\nabla$ of $\wedge^2 \Omega_M^1 \otimes TM$ and $\wedge^2 \Omega_M^1 \otimes \text{End}(TM)$ defined by:

$$ T_\nabla(X, Y) = \nabla_X(Y) - \nabla_Y(X) - [X, Y]_{TM} $$
and:

$$R_{\nabla}(X, Y)(Z) = \nabla_X(\nabla_Y(Z)) - \nabla_Y(\nabla_X(Z)) - \nabla_{[X, Y]_{TM}}(Z)$$

## 2.2 Previous classification results

The compact complex surfaces endowed with holomorphic affine connections were classified by Inoue and Kobayashi in [4].

Let $M$ be a minimal compact surface of algebraic dimension two. This means that there exists a biholomorphism $\pi : M \setminus S \to M' \setminus S'$ where $S, S'$ are codimension one analytic subsets, $M'$ is a projective surface, and local meromorphic functions on $M$ are pullbacks of local meromorphic functions on $M'$. As explained in the previous section, the last property implies a bijection between meromorphic affine connections on $M$ and $M'$. By 1, and the existence of a minimal model for any complex compact surface, we get that any complex compact surface of algebraic dimension two is endowed with a meromorphic affine connection.

The case of algebraic dimension one was classified in [6], where we proved that minimal surfaces of algebraic dimension one endowed with meromorphic affine connections are the isotrivial ones with exception of Class $IV_0$ in Theorem 22 of [9].

To complete the classification of compact surfaces endowed with meromorphic affine connections, it remains to treat the case of algebraic dimension zero. In this case, we have the following powerful:

**Theorem 1** [Dumitrescu, 10] Let $M$ be a compact complex manifold of algebraic dimension zero. Any meromorphic affine connection on $M$ is infinitesimally homogeneous in the sense of 3.

## 2.3 The example of Kummer surfaces and K3 surfaces admitting a holomorphic foliation

Let $M' = \mathbb{C}^2/\Lambda$ be a complex torus of dimension two ($\Lambda$ being a lattice of rank four in $\mathbb{R}^4$), and with algebraic dimension $a(M') = 0$. Denote by $\tilde{q} : \mathbb{C}^2 \to M'$ its universal covering, and pick global holomorphic coordinates $(z_1, z_2)$ on its total space. Then, given any $Q_1, Q_2 \in \mathfrak{gl}_2(\mathbb{C})$, the holomorphic affine connection $\tilde{\nabla}_{Q_1, Q_2}$ having matrix in $(\frac{\partial}{\partial z_1}, \frac{\partial}{\partial z_2})$ given by $dz_1 \otimes Q_1 + dz_2 \otimes Q_2$ is clearly invariant by the translations on $\mathbb{C}^2$. This defines an unique holomorphic affine connection $\nabla'_{Q_1, Q_2}$ on $M'$ by:

$$ (5) \qquad \tilde{q}^* \nabla'_{Q_1, Q_2} = \tilde{\nabla}_{Q_1, Q_2} $$

Concerning K3 surfaces, we have the following result by Brunella ([?]):

**Theorem 2** (Brunella, [?]) Let $M$ be a compact Kahler surface with $a(M) = 0$. Suppose it admits a (non-trivial) holomorphic foliation $\mathcal{F} \subset TM$. Then either $M$ is a two torus, or
there exists $M'$ as above, and a diagram:

$$ (6) \qquad \begin{array}{ccc} & M' & \\ M \xrightarrow{\pi_0} & \downarrow \pi & \\ & M_0 & \end{array} $$

where:

• $\pi: M' \to M_0$ is a map of analytic spaces obtained as the quotient map by a finite cyclic group $\Gamma$ of automorphisms of $M'$.

• $\pi_0: M \to M_0$ is a map of analytic spaces

• There exists codimension two analytic subspaces $S_0 \subset M_0$ and $S' \subset M'$ such that $\pi'|_{M'\setminus S'}: M' \setminus S' \to M_0 \setminus S_0$ is a finite unramified cover of complex manifolds, and $\pi_0|_{M\setminus\pi_0^{-1}(S_0)}: M \setminus \pi_0^{-1}(S_0) \to M_0 \setminus S_0$ is a biholomorphism.

• $\mathcal{F} = \ker(\pi_0^*(\nu_0))$ where $\pi^*\nu_0 = dz_1$

In particular, *Kummer surfaces* corresponds to K3 surfaces as in Theorem 2 with $\pi$ given by the quotient of $M'$ by the automorphism lifting by $\tilde{q}$ to $(z_1, z_2) \to (-z_1, -z_2)$, and $\pi_0$ the blowing-up along the sixteen singular points of the quotient $M_0$.

Now, let $M$ be any K3 surface as in Theorem 2. We prove the existence of a (affine flat) meromorphic affine connection on $M$. For, let $\tilde{\Gamma}$ be the subgroup of automorphisms of $\mathbb{C}^2$ lifting an element of $\Gamma$ in Theorem 2. Then by the last property in the theorem, $\tilde{\Gamma}$ is a subgroup of automorphisms of the fibration $(z_1, z_2) \mapsto z_1$. Moreover, since $\mathcal{K}_M$ is trivial, $\tilde{\Gamma}$ must preserves the volume form $dz_1 \wedge dz_2$. Hence, any element $\tilde{\varphi} \in \tilde{\Gamma}$ is of the form:

$$ (7) \qquad \tilde{\varphi}(z_1, z_2) = (\alpha z_1 + b, \frac{1}{\alpha} z_2 + f(z_1)) $$

with $\alpha \in \mathbb{C}$ a root of unity, $b \in \mathbb{C}$ and $f$ an entire function on $\mathbb{C}$. Since $\tilde{\varphi}^k$ is a translation for $k$ big enough, the same arguments as in the proof of Theorem 39 in [11] implies that $f(z_1) = \gamma z_1 + \delta$ for some $\gamma, \delta \in \mathbb{C}$. As a consequence, the matrix of the differential of $\tilde{\varphi}$ in $(\frac{\partial}{\partial z_1}, \frac{\partial}{\partial z_2})$ is:

$$ \mathrm{Mat}_{\frac{\partial}{\partial z_1}, \frac{\partial}{\partial z_2}} T \tilde{\varphi} = \begin{pmatrix} \alpha & 0 \\ \gamma & \frac{1}{\alpha} \end{pmatrix} $$

and $\alpha, \gamma$ only depends on the choice of a generator of $\Gamma$. By the explicit formula for the pullback by an automorphism (see [12], ), the holomorphic affine connection $\nabla'_{Q_1, Q_2}$ on $M'$ is invariant by $\Gamma$ if and only if:

$$ (8) \qquad \frac{1}{\alpha} \begin{pmatrix} \frac{1}{\alpha} & 0 \\ -\gamma & \alpha \end{pmatrix} Q_1 \begin{pmatrix} \alpha & 0 \\ \gamma & \frac{1}{\alpha} \end{pmatrix} = Q_1 \quad \text{and} \quad \alpha \begin{pmatrix} \frac{1}{\alpha} & 0 \\ -\gamma & \alpha \end{pmatrix} Q_2 \begin{pmatrix} \alpha & 0 \\ \gamma & \frac{1}{\alpha} \end{pmatrix} = Q_2 $$
As a consequence (see for example Lemma in [12]), given any $Q_1, Q_2$ satisfying (8), there exists a unique holomorphic affine connection $\nabla_{0,Q_1,Q_2}$ on $M_0 \setminus S_0$ such that $\pi'^*\nabla_{0,Q_1,Q_2} = \nabla'_{Q_1,Q_2}$. We obtain a meromorphic affine connection:

$$ (9) \qquad \nabla_{Q_1,Q_2} = \pi_0^* \nabla_0 $$

on $M$.

**Proposition 3** Let $M$ be a K3 surface with $a(M) = 0$, admitting a non trivial holomorphic foliation. Let $\alpha, \gamma$ as in (7). Then the meromorphic affine connections on $M$ are the meromorphic affine connection $\nabla_{Q_1,Q_2}$ constructed as in (9), with $Q_1, Q_2$ given by:

$$ (10) \qquad Q_1 = \begin{pmatrix} \frac{\gamma\alpha}{1-\alpha^2}b_1 & b_1 \\ \frac{(3\alpha^2-1)\gamma^2}{1-\alpha^2}b_1 & -\frac{\gamma\alpha}{1-\alpha^2}b_1 \end{pmatrix} \quad \text{and} \quad Q_2 = \begin{pmatrix} 0 & 0 \\ d_2 & 0 \end{pmatrix} $$

Moreover, if $\gamma = 0$ and $\alpha^3 \neq 1$, then $Q_1 = Q_2 = 0$ that is the only meromorphic affine connection on $M$ is obtained from the standard affine structure on $M'$.

*Proof* Let $\nabla'$ be any meromorphic affine connection on $M$. Then the pullback by $(\pi_0|_{M\setminus\pi^{-1}(S_0)})^{-1}$ is a meromorphic affine connection $\nabla'_0$ on $M_0\setminus S_0$. Its pullback $\nabla' = \pi^*\nabla'_0$ is a meromorphic affine connection on $M' \setminus S'$. Since $S'$ is of codimension two, the Hartogs theorem implies that the Christoffel symbols of $\nabla'$ are the restrictions of meromorphic functions defined on $M'$. Hence $\nabla'$ is a meromorphic affine connection on $M'$. Since $a(M') = 0$, the Christoffel are in fact constant, so that $\nabla' = \nabla_{Q_1,Q_2}$ as in (5), for suitable $Q_1, Q_2 \in \mathfrak{gl}_2(\mathbb{C})$. The pullback $\tilde{\nabla}_{Q_1,Q_2}$ to $\mathbb{C}^2$ must be invariant by the lifted automorphism group $\tilde{\Gamma}$. Now we prove $\mathrm{Tr}(Q_1) = \mathrm{Tr}(Q_2) = 0$. Indeed, let $\eta$ be any trivialization of $\mathcal{K}_M$. Then there exists an unique $\eta_0 \in \mathcal{K}_{M_0\setminus S_0}$ whose pullback by $\pi_0$ is $\eta$. Then, since $a(M') = 0$, the pullback $(\pi')^*\eta_0$ is a constant multiple of the trivialization of $\mathcal{K}_{M'}$ induced by $dz_1 \wedge dz_2$. But then we obtain:

$$ \nabla_{0,Q_1,Q_2}(\eta_0) = \alpha_0 \otimes \eta_0 $$

for some holomorphic one form $\alpha_0$ on $M_0 \setminus S_0$. Hence:

$$ \nabla_{Q_1,Q_2}(\eta) = \pi_0^* \nabla_{0,Q_1,Q_2}(\pi_0^*\eta_0) = \pi_0^*\alpha_0 \otimes \pi_0^*\eta_0 = \pi_0^*\alpha_0 \otimes \eta $$

where $\pi_0^*\alpha_0$ is a holomorphic one form on $M$. Since $M$ is a K3 surface, $\pi_0^*\alpha_0 = 0$. This exactly means that $\tilde{\nabla}_{Q_1,Q_2}(dz_1 \wedge dz_2) = 0$, that is $\mathrm{Tr}(Q_1) = \mathrm{Tr}(Q_2) = 0$. Explicit computations show that (8) and the previous property are equivalent to (10). $\square$

## 2.4 Description as meromorphic Cartan geometries

In order to obtain a tractable characterization of affine Nomizu's algebras, we will use the equivalence between (meromorphic) affine connections on complex manifolds of dimension $n$, and the following objects modeled on the homogeneous space $G/P = \mathbb{C}^n$ where $G = \mathrm{Aff}(\mathbb{C}^n)$ is the affine Lie group, and $P = \mathrm{GL}_n(\mathbb{C})$ the stabilizer of the origin, that is the linear subgroup.

**Definition 5** Let $G$ be a complex Lie group and $P$ a complex Lie subgroup, and denote $\mathfrak{g} = \mathrm{Lie}(G)$ for any complex Lie group.
1. A meromorphic $(G, P)$-Cartan geometry on a complex manifold $M$ is a pair $(E, \omega)$ where $p: E \to M$ is a holomorphic $P$-principal bundle and $\omega$ is a global meromorphic section of $\Omega_E^1 \otimes g$ (whose poles will be denoted by $\tilde{D} \subset E$) satisfying:

(i) For $b \in GL_n(\mathbb{C})$, $b^*\omega = Ad(b^{-1}) \circ \omega$.

(ii) For $e \in E \setminus \tilde{D}$, $B \in \mathfrak{p}$, $\omega(\frac{d}{dt}|_{t=0}e \cdot \exp_G(tB)) = B$

(iii) For $e \in E \setminus \tilde{D}$, $\omega(e)$ is an isomorphism of complex vector spaces between $T_e E$ and $g$

2. A local automorphism of $(E, \omega)$ is a local automorphism of holomorphic principal bundle $\Psi: p^{-1}(U) \to p^{-1}(\varphi(U))$ ($\varphi$ is the local automorphism of $M$ induced by $\Psi$) such that $\Psi^*\omega = \omega$. An infinitesimal automorphism of $(E, \omega)$ is a local holomorphic vector field $\tilde{X}$ on $E$ whose flow is a local automorphism whenever defined. We denote:

$$ \text{kill}_{E,\omega}(e) = \{\text{germs of infinitesimal automorphisms of } (E, \omega) \text{ at } e\} $$

3. A meromorphic $(G, P)$-Cartan geometry $(E, \omega)$ on $M$ is infinitesimally homogeneous if and only if there exists an open dense subset $M \setminus S$ such that for any $x \in M \setminus S$ and $e \in p^{-1}(e)$, $T_x M$ is spanned by $Tp(\tilde{X}(x))$ for $\tilde{X} \in \text{kill}_{E,\omega}(e)$.

To explain this equivalence, first consider the holomorphic frame bundle $p: R \to M$ of $M$, which is a holomorphic $P$-principal bundle. Recall a theorem by Ehresman ([13]) which implies a bijection between smooth principal connections on the smooth principal bundle underlying $R$ and smooth connections on the smooth vector bundle underlying $TM$ (compaptible with the complex structure). Starting with a meromorphic affine connection $\nabla$ on $M$, with pole $D$, we can extend the corresponding maps $\nabla(U \setminus D)$ over open subsets $U \subset M$ as smooth affine connections on $U \setminus D$, and check that the corresponding smooth principal connection on $p^{-1}(U \setminus D)$ is in fact a meromorphic one form on $p^{-1}(U)$. By unicity, these glue to a unique meromorphic principal connection on $R$, namely a holomorphic principal connection on $R \setminus p^{-1}(D)$ extending as a meromorphic one form on $R$.

This can be straightforwardly to a bijection between meromorphic principal connections on $R$, which are holomorphic principal connections on an open dense subset of $R$ extending as meromorphic one forms, and meromorphic affine connections.

Cartan observed ([14]) that, just as the canonical parallel transport on $G/P$ can be recovered from the Maurer-Cartan form $\omega_G$ on $G$ (the translations are spanned by the left invariant vector fields projecting onto $\mathbb{C}^n$ by $\omega_G$), affine connections on a complex manifold $M$ correspond in general to parallelisms on the frame bundle of $M$ with the same properties as $\omega_G$, with the exception that the parallelism is not given by left invariant vector fields through a local Lie group action.

Indeed, recall that there exists on the holomorphic frame bundle $p: R \to M$ a holomorphic one form with values in $\mathbb{C}^n$, the canonical form, which is up to automorphism of $R$ the unique one form which is equivariant and whose kernel is $ker(dp)$. This characterizes the isomorphism class of $R$ among $P$-principal bundles on $M$. On the model, we have the $P$-invariant decomposition:

$$ (11) \qquad g = \mathbb{C}^n \oplus \mathfrak{p} $$

so that $\omega_G = \theta_G \oplus \tilde{\omega}_G$, and $\theta_G$ is in fact a canonical form on $G$, which therefore is isomorphic to $R(G/P)$. Now, conversely, given a meromorphic affine connection $\nabla$ on
$M$, with corresponding meromorphic principal connection $\tilde{\omega}$ on $R$,

$$ \omega := \theta \oplus \tilde{\omega} $$

is a meromorphic $(G, P)$-Cartan connection on $R$ (5). Morover, since the Ehresmann theorem also states that local automorphisms $\varphi$ of $\nabla$ are in bijection with local automorphisms $\Psi$ of the holomorphic principal bundle $R$ such that $\Psi^*\omega = \omega$ (namely $\Psi(e) := T\varphi(e)$), local automorphisms of $\nabla$ correspond bijectively to local automorphisms of $(R, \omega)$. We obtained:

**Theorem 4** [Meromorphic Affine Cartan's equivalence] Let $M$ be a complex manifold, $(R, \theta)$ its holomorphic frame bundle with its canonical one form. Then:

1. There is a bijective map between the set of isomorphism class of meromorphic $(G, P)$-Cartan geometries on $M$ and the set of isomorphism classes meromorphic affine connections on $M$

2. Any element of the first set in 1. admits a unique representant of the form $(R, \omega)$ with $\mathrm{proj}_{\mathbb{C}^n} \circ \omega = \theta$.

3. Let $\nabla$ be a meromorphic affine connection on $M$ and $\omega$ the associated Cartan connection as in 2. Then local automorphisms of $(M, \nabla)$ are in bijection with local automorphisms of $(R, \omega)$.

A first interest in taking the Cartan's picture when dealing with affine connection is the following interpretation for the curvature and torsion:

**Definition 6** Let $(G, P)$ be any complex Klein geometry and $(E, \omega)$ a meromorphic $(G, P)$-Cartan curvature on a complex manifold $M$, with singular locus $D$. The Cartan curvature of $(E, \omega)$ is the $P$-equivariant map from $E$ to the $P$-representation $(\Lambda^2(\mathfrak{g}/\mathfrak{p})^* \otimes \mathfrak{g}) \subset (\Lambda^2\mathfrak{g}^*) \otimes \mathfrak{g}$ defined for any $e \in E \setminus p^{-1}(D)$ by:

$$ \forall A, B \in \mathfrak{g}, k_{\omega}(A, B) = \omega(e)[\omega^{-1}(A), \omega^{-1}(B)]_{TE} - [A, B]_{\mathfrak{g}} $$

**Lemma 5** Let $M$ be a complex manifold endowed with a meromorphic affine connection $\nabla$ with pole $D$. Let $\omega$ be the unique meromorphic Cartan connection on $R$ associated as in Theorem 4, and $k_{\omega}$ its Cartan curvature. and $e \in R \setminus p^{-1}(D)$. Then:

$$ k_{\omega} = e^{-1}T_{\nabla} \oplus e^{-1}R_{\nabla} $$

where we decomposed $(\Lambda^2\mathfrak{g}^*) \otimes \mathfrak{g}$ as in (23), $\mathfrak{p}$ is identified with $\mathrm{End}(\mathbb{C}^n)$ and $e^{-1}$ is the isomorphism between the tensors on $T_{p(e)}M$ and tensors on $\mathbb{C}^n$ induced by $e$.
# 3 Nomizu's algebras of affine connections

## 3.1 Nomizu's algebras of Cartan geometries

Let $(E, \omega) \xrightarrow{p} M$ be a meromorphic $(G, P)$-Cartan geometry on a complex manifold $M$ of complex dimension $n \ge 2$, with singular locus $D$, and $e \in \mathring{p}^{-1}(D)$. Denote:

$$ (12) \qquad \iota_e := \omega(e) \circ ev_e : (TE)_e \to \mathfrak{g} $$

where $ev_e$ stands for the evaluation of holomorphic vector fields at $e$. Then:

$$ (13) \qquad \mathfrak{h} := \iota_e(\mathfrak{kill}_{E,\omega}(e)) \subset \mathfrak{g} $$

is a vector subspace endowed with the Lie bracket $[\cdot, \cdot]_h$ defined by:

$$ (14) \qquad [\iota_e(\tilde{X}_1), \iota_e(\tilde{X}_2)]_h = \iota_e([\tilde{X}_1, \tilde{X}_2]_{TE}) $$

It contains the Lie subalgebra:

$$ (15) \qquad \mathfrak{h}_0 = \iota_e(\mathfrak{kill}_{E,\omega}(e)^0) $$

Now, assume that $(E, \omega)$ is infinitesimally homogeneous at $x := p(e) \in M$. Then by definition, there exists $\tilde{X}_1, \dots, \tilde{X}_n \in \mathfrak{kill}_{E,\omega}(e)$ such that $(Tp(\tilde{X}_1(e)), \dots, Tp(\tilde{X}_n(e)))$ is a basis of $T_x M$. Equivalently, $\tilde{X}_1, \dots, \tilde{X}_n$ is a complementary subspace of $(\mathfrak{kill}_{E,\omega}(e))^0$ in the Nomizu's algebra at $e$, i.e.

$$ (16) \qquad \exists \mathbb{C}^n \simeq \mathfrak{m} \subset \mathfrak{h}, \mathfrak{h} = \mathfrak{m} \oplus \mathfrak{h}_0 \text{ and } \mathfrak{m} \cap \mathfrak{p} = \{0\} $$

Moreover:

**Lemma 6** Let $(E, \omega)$ be an infinitesimally homogeneous holomorphic $(G, P)$-Cartan geometry, $e \in E$, $\iota_e$ and $\mathfrak{h} = \iota_e(\mathfrak{kill}_{E,\omega}(e))$ as in (12). Then:

$$ (17) \qquad [A, B]_h = -k_\omega(e)(A, B) - [A, B]_g $$

*Proof* Since both sides of (17) are $\mathbb{C}$-bilinear maps, it is sufficient to prove this equality for $A, B$ ranging in a basis $(\mathfrak{e}_i)_{i=1,...,m}$ of $\mathfrak{h}$. Complete this basis into a basis $(\mathfrak{e}_i)_{i=1,...,N}$ of $\mathfrak{g}$. Let $i \in \{1, ..., m\}$ and $\tilde{X}_i \in \mathfrak{kill}_{E,\omega}(e)$ such that $\iota_e(\tilde{X}_i) = \mathfrak{e}_i$. Then for any $j \in \{1, ..., N\}$ we have:

$$ (18) \qquad \begin{aligned} 0 &= \omega(e)([\tilde{X}_i, \omega^{-1}(\mathfrak{e}_j)]_{TE}) \\ &= k_\omega(e)(\mathfrak{e}_i, \mathfrak{e}_j) - \mathcal{L}_{\omega^{-1}(\mathfrak{e}_j)}\omega(\tilde{X}_i)(e) + [\mathfrak{e}_i, \mathfrak{e}_j]_g \end{aligned} $$
As a consequence:

$$
\begin{align*}
[\mathfrak{e}_i, \mathfrak{e}_j]_h &= \omega(e)([\tilde{X}_i, \tilde{X}_j]_T E) \\
&= -k_\omega(e)(\mathfrak{e}_i, \mathfrak{e}_j) - (\mathcal{L}_{\tilde{X}_i} \omega(\tilde{X}_j))(e) + (\mathcal{L}_{\tilde{X}_j} \omega(\tilde{X}_i))(e) - [\mathfrak{e}_i, \mathfrak{e}_j]_g \\
&= -k_\omega(e)(\mathfrak{e}_i, \mathfrak{e}_j) - (\mathcal{L}_{\omega^{-1}(\mathfrak{e}_i)}\omega(\tilde{X}_j))(e) + (\mathcal{L}_{\omega^{-1}(\mathfrak{e}_j)}\omega(\tilde{X}_i))(e) - [\mathfrak{e}_i, \mathfrak{e}_j]_g \\
&= -k_\omega(e)(\mathfrak{e}_i, \mathfrak{e}_j) - [\mathfrak{e}_i, \mathfrak{e}_j]_g
\end{align*}
$$

as stated.

□

In particular, since the *P*-equivariancy of a Cartan connection implies that $k_\omega$
vanishes when contracted with any element of $\mathfrak{p}$, the following holds:

$$
(19) \qquad \forall B \in \mathfrak{h}_0, [B, \cdot]_{\mathfrak{h}} = -[B, \cdot]_{\mathfrak{g}}|_{\mathfrak{h}}
$$

Conversely, let $\mathfrak{h} \subset \mathfrak{g}$ a vector subspace endowed with a Lie bracket $[,]_h$. Assume that there exists a Lie subalgebra $\mathfrak{h}_0 \subset \mathfrak{h}$ contained in $\mathfrak{p}$ such that (16) and (19) hold. Let $H$ be any simply connected Lie group with Lie algebra $(-\mathfrak{h}, -[,]_h)$, $H_0 \subset H$ the Lie subgroup with Lie algebra $\mathfrak{h}_0$, and $V \subset \mathfrak{m}$ a neighborhood of 0 such that $\exp|_V : V \to M \subset H$ is one to one. We define $E := M \times P$ and:

$$
\omega(x, b) = Ad(b^{-1})[\mathrm{proj}_1^* \omega_H(x, b)] + \mathrm{proj}_2^* \omega_P(x, b)
$$

which can be checked to be a (*G*, *P*)-Cartan connection on *E*. Moreover, *H*₀ is (iso-
morphic to) a Lie subgroup of *P*, so that, up to shrinking the neighborhood *V*,
*E*₀ := *M* · *H*₀ ⊂ *H* identifies with a *H*₀ reduction of the holomorphic *P*-principal bun-
dle *E* *p*=*proj*₁ *M* (compare Lemma 2.2.4 in [?]). Then the right invariant vector fields
of *H* restricts to *E*₀ as vector fields preserving *ω*<sub>*H*</sub>, that is *ω*|*E*₀. In other words, given
any (*h*, *h*₀) fulfilling (16) and (19), there exists a holomorphic Cartan geometry with
Nomizu's algebra (*h*, *h*₀).

Finally, consider two infinitesimally homogeneous meromorphic (*G*, *P*)-Cartan
geometries (*E*, *ω*) and (*E*', *ω*') over two complex manifolds *M* and *M*'. Suppose the
existence of an isomorphism Ψ : *E* → *E*′ of holomorphic principal bundles with
Ψ*ω*′ = *ω*. Let ϕ : *M* → *M*′ be the isomorphism induced by Ψ, *x* ∈ *M* and
*x*′ = ϕ(*x*). Then for any *e* ∈ *p*⁻¹(*x*) and *e*′ ∈ *p*⁻¹(*x*′), there exists *b* ∈ GL*ₙ*($\mathbb{C}$) such
that *e*′ = Ψ(*e*) · *b*. Moreover, by definition of the map ι*<sub>e</sub>*′*, and the property (i) in 5:

$$
\iota_{e'}(\mathfrak{K}ill_{E', \omega'}(e')) = \iota_{\Psi(e) \cdot b}(\mathrm{TR}_b(\mathfrak{K}ill_{E', \omega'}(\Psi(e))))) = \mathrm{Ad}(b^{-1})(\iota_{\Psi(e)}(\mathfrak{K}ill_{E', \omega'}(\Psi(e)))))
$$

Also, since $(\Psi^{-1})^*\omega = \omega'$, we get:

$$
\iota_{\Psi(e)}(\mathfrak{K}ill_{E', \omega'}(\Psi(e))) = \omega'(\Psi(e))(T\Psi(\mathfrak{K}ill_{E, \omega}(e))) = \iota_e(\mathfrak{K}ill_{E, \omega}(e))
$$

We have proved:

**Proposition 7** Let (G, P) be a complex Klein geometry.
1. The map:

$$
(20) \qquad \mathcal{F} : \{ \text{Pointed } (G, P)-\text{Nomizu's algebras} \} \to \{ \text{Lie algebras } (\mathfrak{h}, [\cdot]_{\mathfrak{h}}) \text{ with } \mathfrak{h} \subset \mathfrak{g}, (16) \text{ and } (19) \} \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mapsto (\iota_e(\text{till}_{E,\omega}(e)), [\cdot]_{\mathfrak{h}})
$$

is surjective.

2. The above map induces a bijection between isomorphism classes, defined as follow. An isomorphism between $\mathfrak{till}_{E,\omega}(e)$ and $\mathfrak{till}_{E',\omega'}(e')$ is a local isomorphism $\Psi: U \to U'$ between a neighborhood $U$ of $e$ in $E$ and $U'$ of $e'$ in $E'$ such that $p'(\Psi(e)) = p'(e')$ and $\Psi^*\omega' = \omega$. An isomorphism between $\mathfrak{h}$ and $\mathfrak{h}'$ is an element $b \in P$ such that $\mathrm{Ad}(b^{-1})(\mathfrak{h}) = \mathfrak{h}'$ and induces an isomorphism of Lie algebras.

3. Given any infinitesimally homogeneous holomorphic $(G, P)$-Cartan geometry $(E, \omega) \xrightarrow{p} M$ and $e \in E$, denoting $(\mathfrak{h}, [\cdot]_{\mathfrak{h}}) = \mathcal{F}(\mathfrak{till}_{E,\omega}(e))$ we have:

$$
(21) \qquad [\cdot]_{\mathfrak{h}} = -[\cdot]_{\mathfrak{g}} - k_{\omega}(e)
$$

3.2 Alternative description for affine connections

Now we work with $G = \mathrm{Aff}(\mathbb{C}^n)$ and $P = GL_n(\mathbb{C})$. Fix a subalgebra $\mathfrak{h}_0 \subset \mathfrak{p}$ and
$w \in (\bigwedge^2 \mathfrak{g}^*) \otimes \mathfrak{g}$ vanishing when contracted by any $B \in \mathfrak{p}$. Decompose:

$$
(22) \qquad w = \tau \oplus \kappa
$$

with respect to the *P*-invariant decomposition:

$$
(23) \qquad \left( \bigwedge^2 \mathfrak{g}^* \right) \otimes \mathfrak{g} = \left( \bigwedge^2 \mathfrak{g}^* \right) \otimes \mathbb{C}^n \oplus \left( \bigwedge^2 \mathfrak{g}^* \right) \otimes \mathfrak{p}
$$

Clearly, any $(\mathfrak{h}_0, \tau \oplus \kappa)$ as above arises as $\mathfrak{h}_0 = \mathfrak{h} \cap \mathfrak{p}$ and $\tau \oplus \kappa = [\cdot, \cdot]_{\mathfrak{h}} - (\cdot, \cdot]_{\mathfrak{g}}$ for some element in the image of the map in Theorem 7.

We now give a more tractable description of the subset corresponding to a fixed
pair $(\mathfrak{h}_0, \tau \oplus \kappa)$ as above. By definition, it is in bijection with the set of vector subspaces
$\mathfrak{h} \subset \mathfrak{g}$ endowed with a bilinear map $[\cdot, \cdot]_{\mathfrak{h}} : \mathfrak{h} \times \mathfrak{h} \to \mathfrak{g}$ satisfying (16) and (19), with
the additional property that $[\cdot, \cdot]_{\mathfrak{h}}$ is a Lie bracket on $\mathfrak{h}$ (i.e. has values in $\mathfrak{h}$ and satisfies
the Jacobi identity). We define:

$$
(24) \quad \mathcal{N} := \{ \text{Subspaces } \mathfrak{h} \subset \mathfrak{g}, \text{ with a bilinear } [\cdot, \cdot]_{\mathfrak{h}} : \mathfrak{h} \times \mathfrak{h} \to \mathfrak{g} \text{ satisfying (16) and (19)} \}
$$

Now, pick any inner product on $\mathfrak{h}$ and denote by $\mathfrak{h}_0^{\perp}$ the orthogonal complement of $\mathfrak{h}_0$ in $\mathfrak{p}$, and $\mathrm{proj}_{\mathfrak{h}_0^{\perp}}$ the projection onto $\mathfrak{h}_0^{\perp}$ with respect to $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{h}_0^{\perp}$. there is the
following bijection:

(25)

$$ \mathcal{N} \quad \longleftrightarrow \quad \left\{ \begin{array}{c} (\mathfrak{h}_0, T, \tau \oplus \kappa) \mid \begin{array}{l} \mathfrak{h}_0 \hookrightarrow \mathfrak{p}, \\ T \in \operatorname{Hom}_{\mathbb{C}}(\mathbb{C}^n, \mathfrak{h}_0^{\perp}), \\ \tau \oplus \kappa \in (\wedge^2(\mathfrak{g}/\mathfrak{p})^*) \otimes \mathfrak{g} \end{array} \\ (\mathfrak{h}_T := \mathfrak{h}_0 \oplus \{A+T(A) \mid A \in \mathbb{C}^n\}, -[,\mathfrak{g}] - (\tau \oplus \kappa) \end{array} \right\} $$

$$ (\mathfrak{h}, [,]_{\mathfrak{h}}) $$

$$ \rightarrow (\mathfrak{h} \cap \mathfrak{p}, T_{\mathfrak{h}}(A) := \operatorname{proj}_{\mathfrak{h}_0^{\perp}}(A), [\cdot, \cdot]_{\mathfrak{h}} - (\cdot, \cdot]_{\mathfrak{g}}) $$

$$ (\mathfrak{h}_T := \mathfrak{h}_0 \oplus \{A+T(A) \mid A \in \mathbb{C}^n\}, -[,\mathfrak{g}] - (\tau \oplus \kappa) \leftarrow \mathfrak{h}_0, T, \tau \oplus \kappa) $$

$$ (\mathfrak{h}_0, T, \tau \oplus \kappa) $$

Let $(\mathfrak{h}, [\cdot, \cdot]_{\mathfrak{h}})$ be an element of $\mathcal{N}$ and $(\mathfrak{h}_0, T, \tau \oplus \kappa)$ its image by the above bijection. Let us interpret the condition that $[\cdot, \cdot]_{\mathfrak{h}}$ is a Lie bracket on $\mathfrak{h}$ in terms of $T, \tau \oplus \kappa$. Denote by $\mathfrak{m} = \{A+T(A) \mid A \in \mathbb{C}^n\}$ so that $\mathfrak{h} = \mathfrak{h}_0 \oplus \mathfrak{m}$. Since $\mathfrak{h}_0$ is already a subalgebra for $[\cdot, \cdot]_{\mathfrak{h}}$, the condition is equivalent to the properties $[\mathfrak{h}_0, \mathfrak{m}]_{\mathfrak{h}} \subset \mathfrak{h}$, $[\mathfrak{m}, \mathfrak{m}]_{\mathfrak{h}} \subset \mathfrak{h}$ and the Jacobi identity.

The first property is:

(26) $$ [A + T(A), B]_{\mathfrak{h}} = [A + T(A), B]_{\mathfrak{g}} = -T(B(A)) - [B, T(A)]_{\mathfrak{g}} \in \mathfrak{h} $$

or again:

(27) $$ \forall B \in \mathfrak{h}_0, (B \bullet T)(\cdot) = [B, T(\cdot)]_{\mathfrak{p}} - T(B(\cdot)) = 0 $$

where $B \bullet T = \frac{d}{dt}|_{t=0}\rho(exp_P(tB))(T)$ is the derivative of the representation of $P$ on $\operatorname{Hom}_{\mathbb{C}}(\mathbb{C}^2, \mathfrak{h}_0^{\perp}) \subset \mathfrak{g}^* \otimes \mathfrak{g}$. The second property is that for any $A, B \in \mathbb{C}^n$:

$$ -[A + T(A), B + T(B)]_{\mathfrak{g}} - \tau(A, B) - \kappa(A, B) \in \mathfrak{h} $$

(28) $$ \iff \underbrace{-\tau(A, B) - (T(A)(B) - T(B)(A))}_{\in \mathbb{C}^n} \oplus \underbrace{-\kappa(A, B) - [T(A), T(B)]_{\mathfrak{g}}}_{\in \mathfrak{p}} \in \mathfrak{h} $$

$$ \iff T(\tau(A, B) + \partial T(A, B)) = \kappa(A, B) + [T(A), T(B)]_{\mathfrak{g}} \mod \mathfrak{h}_0 $$

Note that we used the property (11) which is specific to the affine Klein geometry $(G, P)$. The Jacobi identity is equivalent to the properties:

(29)

$$ \forall A, B, C \in \mathbb{C}^n, \mathfrak{C} = \sum_{\text{cycl}(A,B,C)} [[A+T(A), B+T(B)]_{\mathfrak{g}} + \tau(A,B) + \kappa(A,B), C+T(C)]_{\mathfrak{g}} \\ + \tau(\tau(A,B) + \partial(A,B), C) + \kappa(\tau(A,B) + \partial T(A,B), C) $$
and for $B \in \mathfrak{h}_0$, and $A, C \in \mathbb{C}^n$:

$$ 
\begin{align*}
& [B, \tau(A, C)]_{\mathfrak{g}} + [B, \partial T(A, C)]_{\mathfrak{g}} \\
\oplus [B, \kappa(A, C)]_{\mathfrak{g}} + [B, [T(A), T(C)]_{\mathfrak{g}}]_{\mathfrak{g}} &= [B, [A + T(A), C + T(C)]_{\mathfrak{h}}]_{\mathfrak{h}} \\
&= [A + T(A), [B, C + T(c)]_{\mathfrak{g}}]_{\mathfrak{h}} + [[B, A + T(A)]_{\mathfrak{g}}, C]_{\mathfrak{h}} \\
&= \tau(A, [B, C]_{\mathfrak{g}}) + \tau([B, A]_{\mathfrak{g}}, C) \\
&\quad + \partial T([B, A]_{\mathfrak{g}}, C) + \partial T(A, [B, C]_{\mathfrak{g}}) \\
&\quad + \oplus \kappa(A, [B, C]_{\mathfrak{g}}) + \kappa([B, A]_{\mathfrak{g}}, C) \\
&\quad + [T(A), [B, T(C)]_{\mathfrak{g}}]_{\mathfrak{g}} + [[B, T(A)]_{\mathfrak{g}}, T(C)]_{\mathfrak{g}}
\end{align*}
$$

or again, using the Jacobi identity on $[\cdot, \cdot]_{\mathfrak{g}}$ and (27):

$$ \forall B \in \mathfrak{h}_0, [B, \tau \oplus \kappa(A, C)]_{\mathfrak{g}} - \tau \oplus \kappa(A, [B, C]_{\mathfrak{g}}) - \tau \oplus \kappa([B, A]_{\mathfrak{g}}, C) = 0 $$

i.e.:

$$ (30) \qquad \mathfrak{h}_0 \subset \mathrm{Lie}(\mathrm{Stab}_P(\tau \oplus \kappa)) $$

We will not expand the property (29), since it is trivial in dimension two. In view of Theorem 4, this directly implies:

**Proposition 8** Let $G = Aff(\mathbb{C}^2)$ and $P = GL_2(\mathbb{C})$. Denote by $\mathcal{P}$ the set of triples $(\mathfrak{h}_0, T, \tau \oplus \kappa)$ with $\mathfrak{h}_0$ a Lie subalgebra of $\mathfrak{p}$, $T \in \mathrm{Hom}_{\mathbb{C}}(\mathbb{C}^n, \mathfrak{h}_0)$ and $\tau \oplus \kappa \in (\Lambda(\mathfrak{g}/\mathfrak{p})^*) \otimes \mathfrak{g}$ such that (28),(27),(29) and (30) hold. There is an onto map:

$$ (31) \qquad \begin{gathered} \mathcal{F}_1 : \{ \text{Pointed affine Nomizu's algebras} \} \to \mathcal{P} \\ (\mathrm{till}_{M, \nabla}(x), e \in \mathrm{R}^1(M)_x) \mapsto (\mathfrak{h}_0, T) := (\iota_e(\mathrm{till}_{\mathrm{R}^1(M), \omega}(e)) \cap \mathfrak{p}, T_{\mathfrak{h}}, \tau \oplus \kappa) \end{gathered} $$

with $\omega$ the Cartan connection on $\mathrm{R}^1(M)$ associated to the affine connection $\nabla$. Moreover $\mathcal{F}_1$ defined as in (12), is an onto map inducing a bijection between isomorphism classes.

### 3.3 Computations in dimension two and a first application to the classification

We now use the characterization (8) to describe a subset, depending on the orbit of $\tau \oplus \kappa$, of admissible Lie algebras arising as Nomizu's algebras of holomorphic affine connections on complex surfaces.

**Proposition 9** Let $(\mathrm{R}^1(M), \omega)$ be the $(G, P)$-Cartan geometry corresponding to an infinitesimally homogeneous holomorphic affine connection $\nabla$ on a complex surface $M$, and $e \in$
R¹(M). Let $\mathfrak{h} = \mathcal{F}(\mathfrak{h}\mathfrak{I}\mathfrak{I}_{E,\omega}(e))$ be the image by the map in (7) and the isotropy $\mathfrak{h}_0 = \mathfrak{h} \cap \mathfrak{p}$
Then:

• Either $\nabla$ is affine flat, in which case $\langle = (\mathfrak{g}, [\cdot, \cdot]_{\mathfrak{g}})$,

• or, up to replacing $e$ by $e \cdot b$ for some $b \in \mathrm{GL}_2(\mathbb{C})$, $\mathfrak{h}_0 \subset \mathrm{Lie}(\mathrm{Stab}_P(\mathbb{C}\mathfrak{e}_1))$.

Proof First, it is immediate that the $P$-orbits of elements $\tau \oplus \kappa$ in $(\bigwedge^2 \mathfrak{g}^*) \otimes \mathfrak{g}$ vanishing when contracted with any $B \in \mathfrak{p}$ are represented by one of the elements in the following tables:

<table>
  <thead>
    <tr>
      <th>Case</th>
      <th>a) affine flat</th>
      <th>b)flat with torsion</th>
      <th>c)generic curvature</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>τ ⊕ κ</td>
      <td>τ = 0, κ = 0</td>
      <td>τ = e<sub>1</sub><sup>*</sup> ∧ e<sub>2</sub><sup>*</sup> ⊗ e<sub>1</sub>,<br>κ = 0</td>
      <td>κ = e<sub>1</sub><sup>*</sup> ∧ e<sub>2</sub><sup>*</sup> ⊗ (<table style="display: inline-block; vertical-align: middle; text-align: center;"><tbody><tr><td style="border-bottom: 1px solid black; padding: 0 0.2em;">λ<sub>1</sub></td></tr><tr><td style="padding: 0 0.2em;">0</td></tr></tbody></table> <table style="display: inline-block; vertical-align: middle; text-align: center;"><tbody><tr><td style="border-bottom: 1px solid black; padding: 0 0.2em;">ε</td></tr><tr><td style="padding: 0 0.2em;">λ<sub>2</sub></td></tr></tbody></table>),<br>λ<sub>1</sub> ≠ λ<sub>2</sub></td>
    </tr>
  </tbody>
</table>

and:

<table>
  <thead>
    <tr>
      <th>Case</th>
      <th>d1)Nilpotent curvature</th>
      <th>d2)Special curvature with torsion</th>
      <th>d3) Special curvature, torsionfree</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>τ ⊕ κ</td>
      <td>κ = e<sub>1</sub><sup>*</sup> ∧ e<sub>2</sub><sup>*</sup> ⊗ (<table style="display: inline-block; vertical-align: middle; text-align: center;"><tbody><tr><td style="border-bottom: 1px solid black; padding: 0 0.2em;">0</td></tr><tr><td style="padding: 0 0.2em;">0</td></tr></tbody></table> ε)</td>
      <td>κ = e<sub>1</sub><sup>*</sup> ∧ e<sub>2</sub><sup>*</sup> ⊗ λId,<br>λ ≠ 0<br>τ ≠ 0</td>
      <td>κ = e<sub>1</sub><sup>*</sup> ∧ e<sub>2</sub><sup>*</sup> ⊗ λId,<br>λ ≠ 0<br>τ = 0</td>
    </tr>
  </tbody>
</table>

In case a), the holomorphic affine connection $\nabla$ has vanishing torsion and curvature. As a consequence, there are local biholomorphisms $\varphi: U \to V \subset \mathbb{C}^2$ such that $\nabla = \varphi^*\nabla_0$ where $\nabla_0$ is the standard affine connection on $\mathbb{C}^2$. It follows that $\mathfrak{h}$ is the Nomizu's algebra of $(\mathbb{C}^2, \nabla_0)$.

In case b), we remark that $\mathfrak{h}_0 \subset \mathrm{Lie}(\mathrm{Stab}_P(\tau \oplus \kappa))$ must stabilize $\mathbb{C}\mathfrak{e}_1$. Let $B := \begin{pmatrix} \alpha & \beta \\ 0 & \gamma \end{pmatrix} \in \mathfrak{h}_0$. Then:

$$0 = B \cdot \tau = (-\alpha \mathfrak{e}_1^*) \wedge \mathfrak{e}_2^* \otimes \mathfrak{e}_1 + \mathfrak{e}_1^* \wedge (-\beta \mathfrak{e}_2^*) \otimes \mathfrak{e}_1 + \mathfrak{e}_1^* \wedge \mathfrak{e}_2^* \otimes (\alpha \mathfrak{e}_1) = \beta \mathfrak{e}_1^* \wedge \mathfrak{e}_2^* \otimes \mathfrak{e}_1$$

i.e. $\beta = 0$. This proves the inclusion in the third column.

Now we treat the cases of the second table. That is, up to replacing $e$ by $e \cdot b$ for some $b \in \mathrm{GL}_2(\mathbb{C})$:

$$\kappa = \mathfrak{e}_1^* \wedge \mathfrak{e}_2^* \otimes \begin{pmatrix} \lambda_1 & \epsilon \\ 0 & \lambda_2 \end{pmatrix}$$

with $(\lambda_1, \lambda_2, \epsilon) \in \mathbb{C}^3 \setminus \{(0,0,0)\}$. Let $B = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \in \mathfrak{h}_0 \subset \mathrm{Lie}(\mathrm{Stab}_P(\kappa))$. If $\lambda_1 \neq 0$ or $\lambda_2 \neq 0$, then $B \in \mathfrak{sl}_2$, so that $\delta = -\alpha$ and:

$$(32) \qquad 0 = \begin{bmatrix} \lambda_1 & \epsilon \\ 0 & \lambda_2 \end{bmatrix} = \begin{pmatrix} \gamma\epsilon & 2\alpha\epsilon + \beta(\lambda_2 - \lambda_1) \\ \gamma(\lambda_1 - \lambda_2) & -\gamma\epsilon \end{pmatrix}$$

In particular, if $\lambda_1 \neq \lambda_2$ then $\gamma = 0$, so that $B \in \mathrm{Lie}(\mathrm{Borel}(\mathrm{SL}_2))$ and we obtain case c). If $\lambda_1 = \lambda_2 = 0$ but $\epsilon \neq 0$, then:

$$(33) \qquad 0 = B \cdot \kappa = \mathfrak{e}_1^* \wedge \mathfrak{e}_2^* \otimes \begin{pmatrix} \gamma\epsilon & -2\delta\epsilon \\ 0 & -\gamma\epsilon \end{pmatrix}$$

so that $\gamma = \delta = 0$ i.e. $B \in \mathrm{Lie}(\mathrm{Borel}(\mathrm{GL}_2))$, and we obtain case d1).

Now, assume $\lambda_1 = \lambda_2 \neq 0$ and $\epsilon = 0$. First assume $\tau \neq 0$. Since $\mathrm{SL}_2(\mathbb{C})$ stabilizes $\kappa$, we can assume (up to replacing $e$ by $e \cdot b$, $b \in \mathrm{SL}_2(\mathbb{C})$) that $\tau = \mathfrak{e}_1^* \wedge \mathfrak{e}_2^* \otimes \mathfrak{e}_1$. We have already observed that this implies $\mathfrak{h}_0 \subset \begin{Bmatrix} 0 & \beta \\ 0 & \delta \end{Bmatrix} |_{\beta, \delta \in \mathbb{C}}$.
Assume now that $\tau = 0$. Then $\mathrm{Stab}_P(\tau \oplus \kappa) = \mathrm{SL}_2(\mathbb{C})$ (the direct inclusion is obtained by considering the action on the trace of $\kappa$, i.e. the first Chern form). We first prove that the Lie algebra $\mathfrak{t}$ of the Cartan algebra of diagonal matrices in $\mathfrak{sl}_2(\mathbb{C})$ intersects $\mathfrak{h}_0$ trivially. Indeed, if it is not the case, this means that $\mathfrak{h}_0$ contains $\mathfrak{t}$. But then, either $\mathfrak{h}_0 = \mathfrak{sl}_2$ or $\mathfrak{h}_0$ is already contained in the Lie algebra of the Borel subgroup, up to conjugation, proving case d3). Indeed, suppose that $\mathfrak{h}_0$ is not contained in any conjugated subgroup of the Lie algebra of the Borel subgroup, but contains $\mathfrak{t}$. Then it contains an element

$$B = \begin{pmatrix} \alpha & \beta \\ \gamma & -\alpha \end{pmatrix}$$

with $\gamma \neq 0$. Then:

$$\begin{pmatrix} \alpha & 2\beta \\ 0 & -\alpha \end{pmatrix} = B - \left[ \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & -\frac{1}{2} \end{pmatrix}, B \right]_{\mathfrak{p}} \in \mathfrak{h}_0$$

If $\beta = 0$, then $\mathfrak{h}_0$ was already contained in a conjugated subgroup of the Borel subgroup.

If $\beta \neq 0$, then $\mathfrak{h}_0$ contains $\mathfrak{t}$ and $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, so it contains the standard Borel subgroup. Since it contains $B$ as above, it contains also $\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, so finally $\mathfrak{h}_0 = \mathfrak{sl}_2$. Thus, we can assume for now on that $\mathfrak{h}_0 = \mathfrak{sl}_2$. In this case:

$$T(\mathfrak{e}_i) = \begin{pmatrix} \mu_i & 0 \\ 0 & \mu_i \end{pmatrix}$$

for some complex numbers $\mu_1, \mu_2 \in \mathbb{C}$. Then condition (27) implies that for any $i \in \{1, 2\}$ and $\alpha \in \mathbb{C}$:

$$0 = \left[ \begin{pmatrix} \alpha & 0 \\ 0 & -\alpha \end{pmatrix}, \begin{pmatrix} \mu_i & * \\ * & \mu_i \end{pmatrix} \right] \pm \begin{pmatrix} \alpha\mu_i & 0 \\ 0 & \alpha\mu_i \end{pmatrix} = \begin{pmatrix} \alpha\mu_i & * \\ * & \alpha\mu_i \end{pmatrix} \mod \mathfrak{sl}_2$$

so that $\mu_i = 0$, i.e $T = 0$. But then, condition (28) rewrites as:

$$0 = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} \mod \mathfrak{sl}_2$$

a contradiction since $\lambda \neq 0$. Hence $\mathfrak{t} \cap \mathfrak{h}_0 = \{0\}$.

We now prove that this implies $\mathfrak{h}_0 \subset \mathrm{Lie}(\mathrm{Borel}(\mathrm{SL}_2))$, up to replacing $e$ by $e \cdot b$ for some $b \in \mathrm{SL}_2(\mathbb{C})$ (that is up to conjugation by $b \in \mathrm{SL}_2(\mathbb{C})$). Now, $\mathfrak{h}_0$ has either dimension 0, 1 or 2. In the two first cases, the result follows immediately. Hence, assume that $\mathfrak{h}_0 = \mathbb{C}B_1 \oplus \mathbb{C}B_2$. Up to conjugation by $b \in \mathrm{SL}_2(\mathbb{C})$, we can further assume:

$$B_1 = \begin{pmatrix} \mu & \nu \\ 0 & -\mu \end{pmatrix}$$

with $\nu \neq 0$. If $B_2$ stabilizes $\mathbb{C}\mathfrak{e}_1$, then $\mathfrak{h}_0$ is already contained in the Lie algebra of the Borel subgroup. Else, we have:

$$\begin{pmatrix} \alpha & \beta \\ \gamma & -\alpha \end{pmatrix}$$

for some $\alpha, \beta, \gamma \in \mathbb{C}$ with $\gamma \neq 0$. But then, there are two cases:

* If $\alpha = 0$, then $[B_1, B_2]_{\mathfrak{p}} = \begin{pmatrix} 2\nu\gamma & 0 \\ 0 & -2\nu\gamma \end{pmatrix} \in \mathfrak{h}_0$, contradicting $\mathfrak{t} \cap \mathfrak{h}_0 = \{0\}$.

* If $\alpha \neq 0$, then consider:

$$B'_2 = B_1 - \frac{\mu}{\alpha} B_1 = \begin{pmatrix} 0 & \beta - \frac{\mu}{\alpha} \\ \gamma & 0 \end{pmatrix} \in \mathfrak{h}_0$$
Then:

$$[B_1, B'_2]_p = \begin{pmatrix} 2(\nu\gamma - (\beta - \frac{\mu}{\alpha})) & 0 \\ 0 & -2(\nu\gamma - (\beta - \frac{\mu}{\alpha})) \end{pmatrix}$$

Now, up to conjugation of $B_1$ and $B_2$ by the same element $b \in SL_2(\mathbb{C})$, the above element lies in $t \setminus \{0\}$, so we get also a contradiction in this case.

Hence $\mathfrak{h}_0$ is contained in the Lie algebra of the Borel subgroup as claimed, proving case d3). □

# 4 Generalized Singer's theorem and application to the classification

## 4.1 Generalized Singer's theorem for Cartan geometries, after Pecastaing

We now recall the generalization of the Singer's theorem ([15]) to the context of Cartan geometries by Pecastaing ([3]). Let $(G, P)$ be a complex Klein geometry.

**Lemma 10** Let $(E, \omega) \xrightarrow{p} M$ be a holomorphic $(G, P)$-Cartan geometry and $U \subset M$ a non-empty open subset. Suppose that there exists a subspace $\mathfrak{h} \subset \mathfrak{g}$ and a submanifold $E_{\mathfrak{h}} \subset E$ projecting onto $U$ such that:

(i) The Cartan curvature $k_{\omega}$ is constant on $E_{\mathfrak{h}}$

(ii) $\omega(e)(T_e E_{\mathfrak{h}}) = \mathfrak{h}$ for any $e \in E_{\mathfrak{h}}$

Then $(E|_U, \omega)$ is infinitesimally homogeneous.

*Proof* By definition, we have to prove that for any $x \in U$, there exists a neighborhood $U_x$ and vector fields $\tilde{X}_1, \dots, \tilde{X}_{2n} \in TE(p^{-1}(U_x))$ such that:

(i') The evaluations of $Tp(X_1)(x), \dots, Tp(X_n)(x)$ span $T_x M$

(ii') $\mathcal{L}_{\tilde{X}_i} \omega = 0$ for any $i = 1, \dots, n$

By assumption (i), there exists a basis $(e_1, \dots, e_m)$ of $\mathfrak{h}$ such that $Tp(\omega^{-1}(e_1)), \dots, Tp(\omega^{-1}(e_n))$ span $T_x M$. By definition of $k_{\omega}$ and assumption (ii), $\left( \bigoplus_{i=1, \dots, n} \mathbb{C}\omega^{-1}(e_i), [\cdot]_{TE} \right)$ is a Lie subalgebra of the algebra of vector fields on $E_{\mathfrak{h}}$. In particular, $\mathfrak{h}$ is endowed with a Lie algebra structure such that $\omega$ restricts as a Lie homomorphism between both Lie algebras. By [16], Proposition, denoting by $H$ any simply connected Lie group with Lie algebra $\mathfrak{h}$, given any $e \in p^{-1}(x)$, there exists a neighborhood $\tilde{U}_x$ of $e$ and a biholomorphism $\varphi: \tilde{U}_x \to V \subset H$ such that $\omega = \varphi^*\omega_H$. For any $i = 1, \dots, m$, defined $\tilde{X}_i = T\varphi^{-1}(\tilde{X}_i)$ where $\tilde{X}_i$ is the right invariant vector field on $H$ with $\tilde{X}_i(\varphi(e)) = e_i$. By the choice of $(e_1, \dots, e_n)$, the vector fields $\tilde{X}_1, \dots, \tilde{X}_n$ satisfy (i'). Also, each $\tilde{X}_i$ satisfies:

$$\mathcal{L}_{\tilde{X}_i} \omega = \mathcal{L}_{\tilde{X}_i} \varphi^* \omega_H = \mathcal{L}_{\tilde{X}_i} \omega_H = 0$$

i.e. (ii') is satisfied. □
**Lemma 11** Let $(E, \omega) \xrightarrow{p} M$ be a holomorphic $(G, P)$-Cartan geometry. Let $\mathbb{V}$ be a $P$-representation and $f: E \to \mathbb{V}$ a $P$-equivariant map. Pick $e \in E$, and suppose that the level set $F = f^{-1}(\{f(e)\})$ is a complex submanifold. Suppose also that the covariant derivative $\mathcal{D}^1 f$ defined in ?? is constant on this level set. Then there exists a subspace $\mathfrak{h} \subset \mathfrak{g}$ so that

$$ \omega(e')(T_{e'}F) = \mathfrak{h} $$

for any $e' \in F$.

*Proof* Lemma 3.2 in [3]. □

Given a $P$-representation $\mathbb{V}$ recursively the $P$-representations $(\mathbb{V}^{(r)})_{r \ge 0}$ by:

$$ (34) \qquad \mathbb{V}^{(0)} = (\bigwedge^2 \mathfrak{g}^*) \otimes \mathfrak{g} \quad \text{and} \quad \mathbb{V}^{(r+1)} = \mathbb{V}^{(r)} \oplus \mathfrak{g}^* \otimes \mathbb{V}^{(r)} $$

**Definition 7** Let $(E, \omega) \xrightarrow{p} M$ be a holomorphic $(G, P)$-Cartan geometry and $f: E \to \mathbb{V}$ be a $P$-equivariant holomorphic map to a $P$-representation. Pick a basis $(e_1, \dots, e_N)$ for $\mathfrak{g}$. The iterated covariant derivatives of $f$ are the $P$-equivariant holomorphic maps $\mathcal{D}^r f: E \to \mathbb{W}^{(r)}$ defined recursively by:

$$ (35) \qquad \mathcal{D}^0 f = f \quad \text{and} \quad \mathcal{D}^{r+1} f = \mathcal{D}^r f \oplus \sum_{i=1}^N \mathfrak{e}_i^* \otimes \mathcal{L}_{\omega^{-1}(\mathfrak{e}_i)} f $$

**Theorem 12** [[3], Theorem 1] Let $(E, \omega) \xrightarrow{p} M$ be a meromorphic $(G, P)$-Cartan geometry with singular locus $D \subset M$. Then:

1. $(E, \omega)$ is infinitesimally homogeneous if and only for any $r \ge 0$, the iterated covariant derivative $\mathcal{D}^r k_\omega$ (7) takes values in a single $P$-orbit.

2. Under the assumption of 1., for any $e \in E \setminus p^{-1}(D)$, the level-set:

$$ (36) \qquad E_0 = \bigcap_{r \ge 0} (\mathcal{D}^r k_\omega)^{-1}(\{\mathcal{D}^r k_\omega(e)\}) \subset E \setminus p^{-1}(D) $$

has its tangent space at $e$ spanned by the evaluations of germs in $\text{kill}_{E,\omega}(e)$.

3. Under the setting of 2., denoting $\mathfrak{h} = \iota_e(\text{kill}_{E,\omega}(e))$ and $\mathfrak{h}_0 = \mathfrak{h} \cap \mathfrak{p}$, $E_0$ is a holomorphic $H_0$-reduction of $E \setminus p^{-1}(D)$.

*Proof* The direct implication in 1. is Lemma in [3]. We now prove the converse implication, as well as 2. and 3. For any $r \ge 0$ and $e \in E$, define:

$$ \mathfrak{p}_r(e) := \{B \in \mathfrak{p} \mid \tilde{B} \setminus d\mathcal{D}^r k_\omega(e) = 0\} \subset \mathfrak{p} $$

Clearly, $(\mathfrak{p}_r(e))_{r \ge 0}$ is a decreasing sequence of vector subspaces in $\mathfrak{p}$. As a consequence, there exists $s \le \dim(\mathfrak{p}) + 1$ such that $\mathfrak{p}_{s+1}(e) = \mathfrak{p}_s(e)$.

Pick $x \in M \setminus D$ and $e \in p^{-1}(x)$. By assumption $E_0$ contains a section $\sigma(U)$ of $E$ on a neighborhood $U$ of $x$. Up to shrinking $U$, this level set is a complex submanifold of $E$. Clearly, $\mathcal{D}^1(\mathcal{D}^s k_\omega)$ is constant along $\sigma$. Since $\mathfrak{p}_{s+1} = \mathfrak{p}_s$ is the Lie algebra of the structure group of $E_0$, it is also constant along the whole $E_0$. Hence, $\mathcal{D}^s k_\omega$ satisfies the assumptions of
**Theorem 11**, so there exists a complex vector subspace $\mathfrak{h} \subset \mathfrak{g}$ such that $\omega(e')(T_{e'}E_0) = \mathfrak{h}$ for any $e' \in E_0$. By definition of $f$ and the iterated covariant derivatives, $k_\omega$ is constant along $F$. Applying **Theorem 10**, we get that $(E|_U, \omega)$ is infinitesimally homogeneous. This proves 1.

Now, by the proof of **Theorem 11**, any infinitesimal automorphism $\tilde{X} \in \text{kill}_{E,\omega}(e)$ is tangent to $E_0$. This implies that $\omega(e)(\text{ker}(dp) \cap TE_0) \supset \mathfrak{h}_0$ (remark that $\mathfrak{h}_0$ is independant of $e$). The converse inclusion is immediate: let $B \in \omega(e)(\text{ker}(dp) \cap TE_0)$. By the above remarks, $E_0$ satisfies the assumptions of **Theorem 11**. The proof of this lemma shows that there exists an infinitesimal automorphism $\tilde{X}_B$ near $e$ such that $\omega(e)(\tilde{X}_B) = \omega(e)(\omega^{-1}(B)) = B$, that is $B \in \mathfrak{h}_0$. $\square$

## 4.2 Infinitesimally homogeneous meromorphic affine connections on surfaces

We now compare **Theorem 12** and **Theorem 9** to prove:

**Proposition 13** Let $M$ be a complex surface. Suppose the existence of an infinitesimally homogeneous meromorphic affine connection $\nabla$ on $M$, with singular locus $D$. Then:

* Either $\nabla$ is affine flat,

* Or there exists a holomorphic foliation on $M$.

*Proof* Let $(G, P) = (\text{Aff}(\mathbb{C}^2), GL_2(\mathbb{C}))$ $\omega$ be the meromorphic $(G, P)$-Cartan connection on $R^1(M)$ associated to $\nabla$ as in ?? . Then $(R^1(M), \omega) \xrightarrow{p} M$ is infinitesimally homogeneous, so by **Theorem 12**, 1., there exists a $P$-equivariant meromorphic map $f: E \to \mathbb{V}$, with $\mathbb{V} = (\wedge^2 \mathfrak{g}^*) \otimes \mathfrak{g} \oplus \mathbb{V}'$ as a $P$-representation, and $f = k_\omega \oplus f'$ for some $P$-equivariant meromorphic map $f': E \to \mathbb{V}'$, taking values in a single $P$-orbit $v_0 = (\kappa \oplus \tau) \oplus v'_0 \in \mathbb{V}$ and such that:

$$E_0 := f^{-1}(f(e))$$

is a holomorphic $H_0$-structure on $M \setminus D$, where $H_0 = \text{Stab}_P(v_0) \subset \text{Stab}_P(\tau \oplus \kappa)$. Also, by **Theorem 12**, 2., $\mathfrak{h}_0 := \text{Lie}(H_0) = \iota_e(\text{kill}_{E,\omega}(e))$. Now, suppose $\nabla$ is not affine flat, i.e. $\tau \oplus \kappa \neq 0 \oplus 0$. By **Theorem 9**, up to replacing $e$ by $e \cdot b$ for some $b \in GL_2(\mathbb{C})$, we have:

$$\mathfrak{h}_0 \subset \text{Lie}(B)$$

where $B \subset GL_2(\mathbb{C})$ is the stabilizer of $\mathbb{C}\mathfrak{e}_1$. Consider:

$$(37) \qquad E_1 = E_0 \times_{H_0} B \subset E$$

By definition of $B$, this defines a holomorphic foliation $\mathcal{F}$ on $M \setminus D$ by:

$$\mathcal{F}(x) = \{e^{-1}(\mathbb{C}\mathfrak{e}_1) | e \in E_1 \cap p^{-1}(x)\}$$

The relative position of $E_1 \subset E \setminus p^{-1}(D)$ is defined by:

$$(38) \qquad \begin{array}{ccc} \tau : E \setminus p^{-1}(D) & \to & \mathbb{P}^1 = P/B \\ e & \mapsto & [b] \text{ s.t. } e \cdot b^{-1} \in E_1 \end{array}$$

As a subvariety of the affine variety $\mathbb{V}$, $P \cdot v_0 \simeq P/H_0$. Under this identification:

$$\tau = \pi_{B,H_0} \circ f$$
where $\pi_{B,H_0}: P/H_0 \to P/B$ is a well defined map of algebraic varieties (since $B$ stabilizes any of its complex Lie subgroup). This means that $\tau$ is the restriction of a meromorphic map.

Pick $x \in M$. Then there exists a holomorphic section $\sigma: U \to E$ on a neighborhood of $x$. Then $\tau \circ \sigma$ is a meromorphic map from $U \simeq V \subset \mathbb{C}^2$ to $\mathbb{P}^1$, that is:

$$ \tau \circ \sigma = \begin{bmatrix} f_1 \\ f_2 \end{bmatrix} $$

for some holomorphic functions $f_1, f_2$ on $M$. By definition of $R^1(M)$ and $\tau$, this means that $\mathcal{F}|_{U \setminus D}$ is spanned by the restriction to $U \setminus D$ of the holomorphic vector field $X = f_1\sigma^{-1}(\epsilon_1) + f_2\sigma^{-1}(\epsilon_2)$. This means that $\mathcal{F}$ is a holomorphic foliation of rank one on $M$. $\square$

## 4.3 Classification in the Kähler case

The classification of Kähler surfaces with $a(M) = 0$ admitting non affine flat meromorphic affine connections can be achieved by comparing Theorem 2 and Theorem 13. We now address the affine flat case:

**Proposition 14** Let $M$ be a $K3$ surface with $a(M) = 0$. If $M$ admits a meromorphic affine connection $\nabla$, then it admits a non-trivial holomorphic foliation.

*Proof* By Theorem 13, we can assume that $\nabla$ is affine flat. Also, we can assume that for any trivialization $\eta$ of $K_M$, $\nabla(\eta) = 0$. Indeed, in the contrary case, $\nabla(\eta) = \alpha \otimes \eta$ for some non-trivial global meromorphic one form one $M$. Then $\mathcal{F} = \ker(\alpha)$ is a non-trivial holomorphic foliation on $M$.

Denote by $D$ its singular locus. Then there exists a sequence of blow-ups at the isolated singularities of $D$, denoted by $\pi: M' \to M$, such that $D' := \pi^{-1}(D)$ is a normal crossing divisor ([17], 3.2.1). Then $M'$ is also Kähler and simply connected. In particular, the monodromy of the pullback $\nabla' = \pi^*\nabla$ at any $e' \in M' \setminus D'$ is an abelian group ([18], ). By a well-known result of Deligne ([17], see also [19], Theorem 8.35), we have:

$$ \tilde{H}^1(M \setminus D, \mathbb{C}) \simeq H^0(M, \Omega_M^1(\log(D))) \oplus H_M^{0,1} $$

where $\Omega_M^1(\log(D))$ is the sheaf of logarithmic one forms and $H_M^{0,1}$ the vector space from the Dolbeault cohomology. Since $M$ is simply connected, $H_M^{0,1} = \{0\}$. Also, $\eta' := \pi^*\eta$ defines a global meromorphic section of $K_{M'}$ with only poles supported at the exceptional divisors of $\pi$. We have therefore two possible cases:

* Either the above monodromy is unipotent, in which case there exists a non-trivial global holomorphic vector field $X$ on $M' \setminus D'$ which is parallel for $\nabla'$. To prove that this defines a holomorphic foliation on $M$, it is sufficient to prove that $X$ is the restriction of a meromorphic vector field on $M$. Pick $x'_0 \in D'$ which is not contained in an exceptional divisor. By the above remarks, there exists coordinates $(z_1, z_2)$ on a neighborhood of $x_0$ with $\eta' = \lambda dz_1 \wedge dz_2$ for some $\lambda \in \mathbb{C}^*$. Since $\nabla'$ is affine flat, $X = a \frac{\partial}{\partial z_1} + b \frac{\partial}{\partial z_2}$ is
satisfies $\nabla_X(Y) = [X, Y]$ for any vector field Y, so we obtain:

$$
\begin{align*}
0 &= (\nabla_X \eta')(\frac{\partial}{\partial z_1}, \frac{\partial}{\partial z_2}) \\
&= -\eta'(\nabla_X \frac{\partial}{\partial z_1}, \frac{\partial}{\partial z_2}) - \eta(\frac{\partial}{\partial z_1}, \nabla_X \frac{\partial}{\partial z_2}) \\
&= -\eta'([X, \frac{\partial}{\partial z_1}], \frac{\partial}{\partial z_2}) - \eta'(\frac{\partial}{\partial z_1}, [X, \frac{\partial}{\partial z_2}]) \\
&= -(q_{11}^1 + q_{21}^2)a + (q_{12}^1 + q_{22}^2)b
\end{align*}
$$

where $q_{ij}^k$ are the Christoffel symbols of $\nabla'$ in the coordinates $(z_1, z_2)$ If $q_{11}^1 + q_{21}^2 \neq 0$ or $q_{12}^1 + q_{22}^2 \neq 0$, then:

$$
\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} -(q_{11}^1 + q_{21}^2) \\ q_{12}^1 + q_{22}^2 \end{bmatrix}
$$

is a meromorphic map from $U$ to $\mathbb{P}^1$. This means that the holomorphic foliation on $M \setminus D$ spanned by $X$ extends as a holomorphic foliation on $M$, as required. Now, suppose that for any coordinates $(z_1, z_2)$ as above (in particular for any constant linear change of coordinates), we have $q_{11}^1 + q_{21}^2 = q_{12}^1 + q_{22}^2 = 0$. Then, considering new coordinates $(u_1, u_2) = (az_1 + bz_2, cz_1 + dz_2)$ with $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{GL}_2(\mathbb{C})$ and computing the new Christoffel symbols, we get that $q_{ij}^k = 0$ for any $i, j, k \in \{1, 2\}$. This means that $\nabla'$ is holomorphic on a neighborhood of any point which does not belong to the exceptional divisor of $\pi$. This implies that $\nabla$ is holomorphic. But then, since $M$ is simply connected, the monodromy of $\nabla$ is trivial. In particular, there would exist a non-trivial global holomorphic vector field on $M$, which does not hold on K3 surfaces, a contradiction.

* Or the monodromy at some $x \in M' \setminus D'$ of the previous local system defines a non-trivial element of $\check{H}^1(M' \setminus D', \mathbb{C}) \simeq H^0(M', \Omega_{M'}^1(\log(D')))$ (the representation associated with a stable line) in particular a non-trivial meromorphic one form $\alpha'$ on $M'$. Then there exists a non trivial meromorphic one form $\alpha$ on the complement of the isolated singularities of $D$ in $M$ such that $\alpha'|_{M\setminus E} = \pi^*\alpha$ (where we denoted by $E$ the exceptional divisor). By the Hartogs theorem, $\alpha$ extends to a non trivial meromorphic one form on $M$. This in turns defines a non-trivial holomorphic foliation on $M$. $\square$

We obtain:

**Theorem 15** Let $M$ be a minimal Kahler surface with $a(M) = 0$. Then $M$ admits a meromorphic affine connection if and only if $M$ is a two torus or a K3 surface admitting a holomorphic foliation. In any case, there exists an affine flat meromorphic affine connection on $M$.

**Acknowledgements.** We thank the Laboratoire de Mathématique d'Avignon, and the DIMAI of the University of Florence for their hospitality while writing this paper. This paper was written in the context of the project PRIN2022: "Real and Complex Manifolds: Geometry and Holomorphic Dynamics."
# Declarations

Not applicable

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