François Castella

Bruno Sericola

Emmanuelle Anceaume

Yves Mocquard

Continuous-time stochastic analysis of rumor spreading with multiple operations

Keywords: Rumor spreading time, k-pull protocol, Poisson Process, Markov chain, Asymptotic analysis

In this paper, we analyze a new asynchronous rumor spreading protocol to deliver a rumor to all the nodes of a large-scale distributed network. This protocol relies on successive pull operations involving k different nodes, with k ≥ 2, and called k-pull operations. Specifically during a k-pull operation, an uninformed node a contacts k -1 other nodes at random in the network, and if at least one of them knows the rumor, then node a learns it. We perform a detailed study in continuous-time of the total time Θ k,n needed for all the n nodes to learn the rumor. These results extend those obtained in a previous paper which dealt with the discrete-time case. We obtain the mean value, the variance and the distribution of Θ k,n together with their asymptotic behavior when the number of nodes n tends to infinity.

Introduction

Randomized rumor spreading or gossiping is an important communication mechanism that allows the dissemination of information in large-scale and open networks. A large-scale and open network comprises a collection of sequential computing entities (e.g., processes, processors, nodes, agents, peers) that join and leave the system at any time, and communicate with one another by exchanging messages. Randomized rumor spreading was initially proposed by Deemers et al. [START_REF] Demers | Epidemic algorithms for replicated database maintenance[END_REF] for the update of a database replicated at different sites, and has then been adopted in many applications due to its robustness and simplicity. In contrast to reliable communication broadcasts which must provide agreement on the broadcast value with possibly additional ordering guarantees on the delivery of updates from different sources, a randomized rumor spreading procedure provides reliability only with some probability. A randomized spreading rumor protocol describes the rules required for one or more pieces of information known by an arbitrary node in the network (we call such a node an informed node) to be spread to all the nodes of the network. The push and pull protocols are the basic operations used by the nodes to propagate an information over the entire network [START_REF] Demers | Epidemic algorithms for replicated database maintenance[END_REF][START_REF] Doerr | Randomized Rumor Spreading Revisited[END_REF]. With the push operation, an informed node contacts some randomly chosen node in the system, and gives it the rumor while with the pull operation, an uninformed node contacts some random node and asks for the rumor. Note that in both cases the contacted node may already know the rumor or not. The same node can perform both operations according to whether it knows or not the rumor, which corresponds to the push-pull protocol, or performs only one, either a pull or push operation, which corresponds to the pull or push protocols respectively. One of the important questions raised by these protocols is the spreading time, that is the time needed for the rumor to be known by all the nodes of the network.

To answer such a question, one first needs to specify how synchronized nodes are, or in other words whether we suppose that all the nodes of the system act in a synchronous way or not. In the former case, the system model is synchronous while in the latter case it is asynchronous. The most studied one is the synchronous model. This model assumes that all the nodes of the network act synchronously, which allows the algorithms designed in this model to divide time in synchronized rounds. During each synchronized round, each node a of the network selects at random one of its neighbors b and either sends the rumor to b if a knows it (push operation) or gets the rumor from b if b knows the rumor (pull operation). In this model, the spreading time of a rumor is defined as the number of synchronous rounds necessary for all the nodes to know the rumor. When the underlying graph is complete, it has been shown by Frieze [11] that the number of rounds divided by log 2 (n) converges in probability to 1 + ln(2) when the number n of nodes in the graph tends to infinity. Further results have been established (see for example [START_REF] Pittel | On spreading a rumor[END_REF][START_REF] Karp | Randomized rumor spreading[END_REF] and the references therein), the most recent ones resulting from the observation that the rumor spreading time is closely related to the conductance of the graph of the network, see [START_REF] Giakkoupis | Tight bounds for rumor spreading in graphs of a given conductance[END_REF]. Investigations have also been done in different topologies of the network as in [START_REF] Chierichetti | Rumor spreading in social networks[END_REF][START_REF] Daum | Rumor spreading with bounded indegree[END_REF][START_REF] Fountoulakis | Rumor spreading on random regular graphs and expanders[END_REF][START_REF] Panagiotou | Randomized rumor spreading: the effect of the network topology[END_REF], in the presence of link or nodes failures as in [START_REF] Feige | Randomized broadcast in networks[END_REF], in dynamic graphs as in [START_REF] Clementi | Rumor spreading in random evolving graphs[END_REF], and in general graphs in terms of vertex expansion [START_REF] Giakkoupis | Tight bounds for rumor spreading with vertex expansion[END_REF]. Another alternative consists in letting the nodes make more than one call during the push or pull operations [START_REF] Panagiotou | Faster rumor spreading with multiple calls[END_REF]. The authors show that the push-pull protocol takes O (log n/ log log n) rounds in expectation if the number of neighbors of a node is chosen independently according to a power law distribution with exponent β ∈ (2, 3).

In large scale networks, that is in networks involving several thousands of nodes, assuming that all nodes act synchronously is a very strong assumption. Thus several authors, including [START_REF] Giakkoupis | How asynchrony affects rumor spreading time[END_REF][START_REF] Mocquard | Analysis of the propagation time of a rumour in large-scale distributed systems[END_REF][START_REF] Acan | On the push & pull protocol for rumour spreading[END_REF][START_REF] Doerr | Experimental analysis of rumor spreading in social networks[END_REF][START_REF] Pourmiri | Brief announcement: Ultra-fast asynchronous randomized rumor spreading[END_REF], suppose an asynchronous model, that is a model in which nodes asynchronously trigger operations with randomly chosen nodes in the system, either to push, pull or push-pull information. The asynchronous gossip protocol is usually modeled by a time-continuous stochastic (Markovian) process [START_REF] Giakkoupis | How asynchrony affects rumor spreading time[END_REF][START_REF] Mocquard | Analysis of the propagation time of a rumour in large-scale distributed systems[END_REF][START_REF] Acan | On the push & pull protocol for rumour spreading[END_REF][START_REF] Doerr | Experimental analysis of rumor spreading in social networks[END_REF][START_REF] Pourmiri | Brief announcement: Ultra-fast asynchronous randomized rumor spreading[END_REF]. This type of stochastic processes belongs to the death process category, which has many applications in demography, queuing theory, performance engineering, epidemiology, biology and many other distributed applications. For instance, in [START_REF] Daley | Epidemic Modelling: An Introduction[END_REF], an analysis of the SI (Susceptible-Infected) model -corresponding to an asynchronous push-pull model-allows us in some cases to explicit the state probabilities by using the Laplace transform on the Kolmogorov forward equation. However, these techniques prove ineffective when the transition rate is non-linear (Laplace transform inversion becomes a tricky exercise). Most of the rumor spreading protocols studied in the asynchronous models rely either on the push/pull operations or on the push operations. Indeed, pushing the information allows us to initiate the rumor very quickly but then struggles to reach the few uninformed nodes. In contrast, the pull algorithm attracted very little attention because this operation was long considered inefficient to spread a rumor within a large scale network [START_REF] Sanghavi | Gossiping with multiple messages[END_REF]. It is actually very useful in systems fighting against message saturation (see for instance [START_REF] Yao | A pull model IPv6 duplicate address detection[END_REF]). The ineffectiveness of the pull protocol stems from the fact that it takes some time before the rumor reaches a phase of exponential growth.

The objective of this paper is to further develop this line of inquiry by studying the k-pull protocol in the continuous-time case. This protocol counterbalances the slow initiation of pull-based rumour spreading protocols by increasing the chances of learning the rumor with each operation. A local clock following an exponential distribution with rate λ is associated with each uninformed node of the system. Each time the clock of an uniformed node rings, this node contacts k -1, with k ≥ 2 distinct nodes, chosen at random uniformly among the n -1 other nodes. If at least one of these contacted nodes knows the rumor, the initiator of the k-pull operation learns the rumor and clears its clock.

The remainder of the paper is organized as follows. In Section 2, we present the asynchronous k-pull protocol and introduce the random variable Θ k,n which represents the total amount of time needed for all the nodes to know the rumor. We prove in Section 3 that the mean number of k-pull operations needed to inform all the n nodes of the system, assuming that a single node initially knows the rumor, that is E(Θ k,n ), is equivalent to k ln(n)/(k -1)λ when the number of nodes n in the system tends to infinity. We also show that the limiting variance of Θ k,n is equal to (1 + 1/(k -1) 2 )π 2 /6λ 2 when n tends infinity. The distribution of the rumor spreading time Θ k,n is analyzed in Section 4. We provide explicit limiting distributions of Θ k,n -E(Θ k,n ) and Θ k,n -k ln(n)/(k -1)λ when n tends to infinity.

The model

We recall in the next subsection the discrete-time model of the k-pull rumor spreading. This model, which has been analyzed in [START_REF] Robin | Stochastic analysis of rumor spreading with multiple pull operations[END_REF], is important because the continuous-time model of the k-pull rumor spreading is based on the discrete-time model, even if the analysis is, as we will see, much more complicated and needs more refined results in continuous-time.

The discrete-time model

We consider a complete network of size n in which each node may be asked for a piece of information (pull event). The algorithm starts with a single node informed of the rumor. At each discrete time m ≥ 1, a single uninformed node s contacts k -1 distinct nodes, chosen at random uniformly among the n -1 other nodes. If at least one of these k -1 contacted nodes knows the rumor then node s learns it. Otherwise nothing happens. This is a k-pull operation.

To analyze this k-pull model, we introduce the discrete-time stochastic process Y = {Y m , m ≥ 0} where Y m represents the number of informed nodes at time m. The stochastic process Y is a discrete-time homogeneous Markov chain with n states where states 1, . . . , n -1 are transient and state n is absorbing. From the description of the protocol, we deduce that when the Markov chain Y is in state i at time m, then at time m + 1, either it remains in state i if none of the k -1 chosen nodes were informed of the rumor, or it transits to state i + 1 if at least one of the k -1 chosen nodes were informed of the rumor. We denote by P the transition probability matrix of Markov chain Y . The non zero entries of matrix P are thus P i,i and P i,i+1 , for any i = 1, . . . , n -1. Obviously, we get, for any i = 1, . . . , n -1, P i,i = 1 -P i,i+1 , where

P i,i =                n -1 -i k -1 n -1 k -1 if i ≤ n -k 0 otherwise.
Indeed, given that Y t = i, i.e., when i nodes are informed of the rumor at time t, we have Y t+1 = i if and only if, at time t + 1, the set of k -1 chosen nodes (i.e., k -1 among n -1) must be chosen among the n -1 -i non informed nodes. We denoted in [START_REF] Robin | Stochastic analysis of rumor spreading with multiple pull operations[END_REF] by T k,n the random variable defined by

T k,n = inf{t ≥ 0 | Y m = n}
which represents the spreading time, that is the total number of k-pull operations needed for all the nodes in the network to know the rumor. The spreading time distribution can thus be expressed as a sum of independent random variables S k,n (i), where

S k,n (i) is the sojourn time of Markov chain Y in state i. For all i = 1, . . . , n -k, S k,n (i) follows a geometric distribution with parameter p k,n (i) = 1 -P i,i and S k,n (i) = 1, for i = n -k + 1, . . . , n -1. Thus T k,n verifies T k,n = n-1 i=1 S k,n (i) = k -1 + n-k i=1 S k,n (i). Observe that p k,n (i) =          1 - k-1 h=1 1 - i n -h if 1 ≤ i ≤ n -k 1 if n -k + 1 ≤ i ≤ n -1.
(1)

The continuous-time model

Based on this discrete-time model, a local clock following an exponential distribution with rate λ is associated with each uniformed node of the system. Each time the clock of an uniformed node s rings, this node contacts k -1, with k ≥ 2, distinct nodes, chosen at random uniformly among the n -1 other nodes. If at least one of these contacted nodes knows the rumor, node s learns it and clears its clock (i.e., s remains contactable but does not contact other nodes). As in the discrete-time case, we suppose that the k-pull operation, i.e., the time for a node to contact k -1 other nodes and to receive their response, is instantaneous. Since the clock of an uninformed node rings after a time that is exponentially distributed with rate λ, we naturally introduce the continuous-time Markov chain Z = {Z t , t ≥ 0}, where Z t represents the number of informed nodes at continuous-time t ≥ 0. Specifically the transitions of Z occur at successive instants τ 0 = 0, τ 1 , . . ., where the τ i -τ i-1 , i ≥ 1, are independent and exponentially distributed with rate (n -i)λp k,n (i), and the p k,n (i) are given by [START_REF] Acan | On the push & pull protocol for rumour spreading[END_REF]. Hence the global clock of the process rings according to an exponential distribution whose rate is proportional to the amount of uniformed nodes. Note that a jump of process Z corresponds to a state change of process Y .

Observe also that the continuous-time model of the rumor spreading corresponds to a physical time, that is the total amount of time needed for all the n nodes to learn the rumor, while the discrete-time model stands for the total number of k-pull operations needed for all the n nodes to learn the rumor.

We denote by Θ k,n the random variable defined by

Θ k,n = inf{t ≥ 0 | Z t = n}
which represents the continuous-time model i.e., the total amount of time needed for all the nodes to know the rumor. The spreading time Θ k,n can thus be expressed as a sum of independent and exponentially distributed random variables. Specifically, introducing the notation

U k,n (i) = τ i -τ i-1 , for i ≥ 1 with τ 0 = 0, we have Θ k,n = n-1 i=1 U k,n (i), (2)
where U k,n (1), . . . , U k,n (n -1) are independent and U k,n (i) is exponentially distributed with rate (ni)λp k,n (i). The authors of [START_REF] Robin | Stochastic analysis of rumor spreading with multiple pull operations[END_REF] used two technical lemmas to analyze the moments and the distribution of the rumor spreading time in the discrete-time case. These lemmas allowed them to provide lower and upper bounds for the probabilities p k,n (i). These bounds are sufficiently precise to deal with the sum of geometric random variables with parameters p k,n (i), but they are not precise enough to deal with the sum of exponential random variables with rates (n -i)p k,n (i) as in the continuous-time case. We thus consider here a different method to analyze the problem.

Moments of the rumor spreading time

We analyze in this section the first two moments of the rumor spreading time by using appropriate lower and upper bounds. Observe that the bounds obtained for p k,n (i) in [START_REF] Robin | Stochastic analysis of rumor spreading with multiple pull operations[END_REF] do not allow us to obtain the limiting behavior of these first two moments in the continuous-time case. The following technical lemma is used to obtain their asymptotic behavior.

Lemma 3.1 Let g be a C ∞ (infinitely differentiable) function on interval [0, 1]. If g(0) = 0 then n i=1 1 i g i n ∼ n-→∞ g(0) ln(n). If g(1) = 0 then n i=1 1 i g 1 - i n ∼ n-→∞ g(1) ln(n).
Proof. Function g being C ∞ on interval [0, 1], it is Lipschitz and so, for all x, y ∈ [0, 1], we have

|g(x) -g(y)| ≤ C|x -y|, where C = max z∈[0,1] |g (z)|
and g is the derivative of function g. We then have, by taking x = i/n and y = 0,

n i=1 1 i g i n -g(0) n i=1 1 i ≤ n i=1 1 i g i n -g(0) ≤ C n i=1 1 i i n = C.
This, together with the fact that

n i=1 1/i ∼ n-→∞ ln(n), implies that n i=1 1 i g i n ∼ n-→∞ g(0) n i=1 1 i ∼ n-→∞ g(0) ln(n),
which completes the proof of the first equivalence. The same steps with x = 1 -i/n and y = 1 lead to the second equivalent.

Expected rumor spreading time

Using ( 1) and ( 2), the expected rumor spreading time writes

E(Θ k,n ) = 1 λ n-1 i=1 1 (n -i)p k,n (i) = 1 λ n-k i=1 1 (n -i)p k,n (i) + 1 λ k-1 i=1 1 i . (3)
Using the fact that 0 ≤ h ≤ k in Relation (1), we easily get

1 -1 - i n k-1 ≤ p k,n (i) ≤ 1 -1 - i n -k k-1 . ( 4
)
Introducing the notation

α n = n-k i=1 1 (n -i)p k,n (i) , these inequalities lead to γ n ≤ α n ≤ β n , (5)
where

β n = n-k i=1 1 (n -i) 1 -1 - i n k-1 and γ n = n-k i=1 1 (n -i) 1 -1 - i n -k k-1 .
In the following two lemmas, we obtain an equivalent for both β n and γ n when n tends to infinity. Lemma 3.2 For all k ≥ 2, we have

β n ∼ n-→∞ k ln(n) k -1 .
Proof. For all k ≥ 2 and y ∈ R, using the identity

1 -y k-1 = (1 -y) 1 + y + • • • + y k-2 ,
and introducing the function f k is defined, for all x ∈ [0, 1] by

f k (x) = 1 1 + (1 -x) + (1 -x) 2 + • • • + (1 -x) k-2 ,
we obtain

β n = n-k i=1 f k i n (n -i) 1 -1 - i n = n-k i=1 n i(n -i) f k i n = n-k i=1 1 i + 1 n -i f k i n (6) = n-k i=1 1 i f k i n + n-1 i=k 1 i f k 1 - i n = n i=1 1 i f k i n + n i=1 1 i f k 1 - i n - n i=n-k+1 1 i f k i n - k-1 i=1 1 i f k 1 - i n - 1 n f k (0). Observing that lim n-→∞ n i=n-k+1 1 i f k i n + k-1 i=1 1 i f k 1 - i n + 1 n f k (0) = f k (1) k-1 i=1 1 i ,
we obtain

β n = n i=1 1 i f k i n + n i=1 1 i f k 1 - i n -f k (1) k-1 i=1 1 i + ε(n), (7)
where ε(n) is such that lim n-→∞ ε(n) = 0. Since the function f k is C ∞ on interval [0, 1] and verifies f k (0) = 1/(k -1) = 0 and f k (1) = 1 = 0, k being fixed, we obtain, using Lemma 3.1,

β n ∼ n-→∞ (f k (0) + f k (1)) ln(n) = k ln(n) k -1 ,
which completes the proof.

We consider now the term γ n .

Lemma 3.3 For all k ≥ 2, we have

γ n ∼ n-→∞ k ln(n) k -1 .
Proof. Using the function f k defined in the proof of Lemma 3.2, we obtain easily

γ n = n-k i=1 f k i n -k (n -i) 1 -1 - i n -k = (n -k) n-k i=1 1 i(n -i) f k i n -k = n -k n n-k i=1 1 i + 1 n -i f k i n -k (8)
The function

f k being C ∞ on interval [0, 1], defining C k = max z∈[0,1] |f k (z)
| where f k is the derivative of f k , we get, using Relations ( 6) and ( 8),

γ n - n -k n β n ≤ n -k n n-k i=1 1 i + 1 n -i f k i n -k -f k i n ≤ C k n -k n n-k i=1 1 i + 1 n -i ik n(n -k) ≤ C k k n n-k i=1 1 i + n-1 i=k 1 i .
This bound tends to 0 when n tends to infinity. We thus have, from Lemma 3.1,

lim n-→∞ γ n - n -k n β n = 0 and n -k n β n ∼ n-→∞ β n ∼ n-→∞ k ln(n) k -1 ,
which completes the proof.

Theorem 3.4 For all k ≥ 2, we have

E(Θ k,n ) ∼ n-→∞ k ln(n) (k -1)λ .
Proof. Relations (3) and ( 5) give

γ n λ + 1 λ k-1 i=1 1 i ≤ E(Θ k,n ) ≤ β n λ + 1 λ k-1 i=1 1 i .
The use of Lemma 3.2 and Lemma 3.3 leads to the desired result.

We now show that the following refined analysis of the term β n leads to a precise description of the asymptotic behavior of the expected rumor spreading time E(Θ k,n ). We need the following lemma in which γ is the Euler-Mascheroni constant given by γ ≈ 0.5772156649.

Lemma 3.5 Let g be a C ∞ function on interval [0, 1]. If g(0) = 0 then n i=1 1 i g i n = g(0) ln(n) + γg(0) + 1 0 g(x) -g(0) x dx + ε(n). If g(1) = 0 then n i=1 1 i g 1 - i n = g(1) ln(n) + γg(1) + 1 0 g(1 -x) -g(1) x dx + ε(n), where ε(n) is such that lim n-→∞ ε(n) = 0.
Proof. Using the integral form of the remainder for the Taylor series of function g, we get

g(x) -g(0) = xG(x), where G(x) = 1 0 g (tx)dt,
function g being the derivative of g. We then have

n i=1 1 i g i n -g(0) n i=1 1 i = n i=1 1 i i n G i n = 1 n n i=1 G i n -→ n→∞ 1 0 G(x)dx,
since the last term is a Riemann sum. We then use the following well-known development of the harmonic sum

n i=1 1 i = ln(n) + γ + ε(n),
where ε(n) is such that lim n-→∞ ε(n) = 0. In the same way, we have

g(1 -x) -g(1) = -xG(x), where G(x) = 1 0 g (1 -tx)dt
and we obtain

n i=1 1 i g 1 - i n -g(1) n i=1 1 i = - n i=1 1 i i n G i n = - 1 n n i=1 G i n -→ n→∞ - 1 0 G(x)dx.
Again the development of the harmonic sum completes the proof.

The following theorem gives a more precise description of the asymptotic behavior of β n .

Lemma 3.6 For all k ≥ 2, we have

β n = k ln(n) k -1 + γk k -1 - k-1 i=1 1 i - ln(k -1) k -1 + ε(n), where ε(n) is such that lim n-→∞ ε(n) = 0.
Proof. Since f k (0) = 1/(k -1) and f k (1) = 1, Relation (7) and Lemma 3.5 give

β n = k ln(n) k -1 + γk k -1 - k-1 i=1 1 i + 1 0 f k (1 -x) + f k (x) x - k (k -1)x dx + ε(n).
We need to compute the quantity

I := 1 0 f k (1 -x) + f k (x) x - k (k -1)x dx.
Coming back to the definition of f k , we introduce the polynomial

q k (x) = 1 + x + • • • + x k , so that f k (x) = 1/q k-2 (1 -x) and f k (1 -x) = 1/q k-2 (x). On top of that, we observe that k/(k -1) = 1 + 1/(k -1) = f k (0) + f k (1)
. These two observations lead to

I = 1 0 1 x 1 q k-2 (x) - 1 q k-2 (0) + 1 q k-2 (1 -x) - 1 q k-2 (1)
dx, and the variable change x := 1 -x to deal with the second difference leads to

I = 1 0 1 x 1 q k-2 (x) - 1 q k-2 (0) + 1 1 -x 1 q k-2 (x) - 1 q k-2 (1)
dx.

Now the whole point in order to compute this integral is to factorize x in the first difference 1/q k-2 (x) -1/q k-2 (0), and to factorize 1 -x in the second difference 1/q k-2 (x) -1/q k-2 (1), so as to remove the apparent singularities and to recover computable quantities.

Concerning the first difference, we observe that

q k-2 (0) -q k-2 (x) = -x -x 2 -• • • -x k-2 = -xq k-3 (x), which gives 1 x 1 q k-2 (x) - 1 q k-2 (0) = - q k-3 (x) q k-2 (0)q k-2 (x) = - q k-3 (x) q k-2 (x) .
The other difference requires slightly more attention. We claim that the following formula holds

q k-2 (1) -q k-2 (x) = (1 -x) (k -2)q k-3 (x) -xq k-3 (x) . ( 9
)
Admitting the result for the time being, we obtain

1 1 -x 1 q k-2 (x) - 1 q k-2 (1) = (k -2)q k-3 (x) -xq k-3 (x) q k-2 (1)q k-2 (x) = (k -2)q k-3 (x) -xq k-3 (x) (k -1)q k-2 (x) .
Eventually, this provides

I = 1 0 - q k-3 (x) q k-2 (x) + (k -2)q k-3 (x) -xq k-3 (x) (k -1)q k-2 (x) dx = - 1 k -1 1 0 q k-3 (x) + xq k-3 (x) q k-2 (x) dx = - 1 k -1 1 0 q k-3 (x) + xq k-3 (x) xq k-3 (x) + 1 dx,
where the last equality comes from the observation q k-2 (x) = xq k-3 (x) + 1. As a final result, we recover, since (xq k-3 (x) + 1) = xq k-3 (x) + q k-3 (x), the value

I = - ln (p k-3 (1)) k -1 = - ln (k -1) k -1 .
There remains to prove formula [START_REF] Feige | Randomized broadcast in networks[END_REF]. The formula can easily be proved using a recursion procedure. Alternatively, one may write, using the fact that 1 -

x j = (1 -x) j-1 =0 x , the relations q k-2 (1) -q k-2 (x) = k-2 j=1 1 -x j = (1 -x) k-2 j=1 j-1 =0 x = (1 -x) k-3 =0 k-2 j= +1 x = (1 -x) k-3 =0 (k -2 -)x = (1 -x) (k -2)q k-3 (x) -xq k-3 (x) ,
which completes the proof.

The following theorem will be used in Corollary 4.4 to get an asymptotic behavior of the distribution of Θ k,n . Theorem 3.7 For all k ≥ 2, we have

E(Θ k,n ) = 1 λ k ln(n) k -1 + γk k -1 - ln(k -1) k -1 + ε(n), where ε(n) is such that lim n-→∞ ε(n) = 0.
Proof. Relations (3) and ( 5) give

γ n λ + 1 λ k-1 i=1 1 i ≤ E(Θ k,n ) ≤ β n λ + 1 λ k-1 i=1 1 i .
Using Lemma 3.6 and the fact that lim n-→∞ (γ n -β n ) = 0 (see the end of the proof of Lemma 3.3), we easily get the desired result.

Variance of the rumor spreading time

The following lemma is needed to obtain the limiting value of the variance of Θ k,n when n tends to infinity.

Lemma 3.8 Let g be a C ∞ function on interval [0, 1]. If g(0) = 0 then lim n-→∞ n i=1 1 i 2 g i n = g(0) π 2 6 . If g(1) = 0 then lim n-→∞ n i=1 1 i 2 g 1 - i n = g(1) π 2 6 .
Proof. Function g being C ∞ on interval [0, 1], it is Lipschitz and so, for all x, y ∈ [0, 1], we have

|g(x) -g(y)| ≤ C|x -y|, where C = max z∈[0,1] |g (z)|
and g is the derivative of function g. We then have, by taking x = i/n and y = 0,

n i=1 1 i 2 g i n -g(0) n i=1 1 i 2 ≤ n i=1 1 i 2 g i n -g(0) ≤ C 1 n n i=1 1 i ∼ n-→∞ C ln(n) n .
This means that lim

n-→∞ n i=1 1 i 2 g i n = g(0) ∞ i=1 1 i 2 = g(0) π 2 6 .
The same steps with x = 1 -i/n and y = 1 lead to the second result.

Using (1) and ( 2), the variance of the rumor spreading time writes

V(Θ k,n ) = 1 λ 2 n-1 i=1 1 (n -i) 2 p 2 k,n (i) = 1 λ 2 n-k i=1 1 (n -i)p 2 k,n (i) + 1 λ 2 k-1 i=1 1 i 2 . (10)
As we did for the expected rumor spreading time and introducing the notation

δ n = n-k i=1 1 (n -i) 2 p 2 k,n (i)
,

we obtain η n ≤ δ n ≤ ζ n , (11)
where

ζ n = n-k i=1 1 (n -i) 2 1 -1 - i n k-1 2 , η n = n-k i=1 1 (n -i) 2 1 -1 - i n -k k-1 2 .
In the following two lemmas, we obtain the limiting value of both ζ n and η n when n tends to infinity. Lemma 3.9 For all k ≥ 2, we have

lim n-→∞ ζ n = 1 + 1 (k -1) 2 π 2 6 - k-1 i=1 1 i 2 .
Proof. Using again the function f k introduced in the proof of Lemma 3.2, we obtain

ζ n = n-k i=1 f 2 k i n (n -i) 2 1 -1 - i n 2 = n-k i=1 n 2 i 2 (n -i) 2 f 2 k i n . (12)
Observing that

n 2 i 2 (n -i) 2 = 1 (n -i) 2 + 1 i 2 + 2 n 1 i + 1 n -i , (13)
we have

ζ n = n-k i=1 1 (n -i) 2 f 2 k i n + n-k i=1 1 i 2 f 2 k i n + 2 n n-k i=1 1 i + 1 n -i f 2 k i n .
We denote respectively by ζ n,1 , ζ n,2 and ζ n,3 these three sums. Concerning ζ n,3 , we have

n-k i=1 1 i + 1 n -i f 2 k i n = n-k i=1 1 i f 2 k i n + n-1 i=k 1 i f 2 k 1 - i n .
Applying Lemma 3.1 with function g = f 2 k , we obtain

n-k i=1 1 i f 2 k i n + n-1 i=k 1 i f 2 k 1 - i n ∼ n-→∞ f 2 k (0) + f 2 k (1) ln(n) = 1 + 1 (k -1) 2 ln(n) which implies that lim n-→∞ ζ n,3 = lim n-→∞ 2 1 + 1 (k -1) 2 ln(n) n = 0.
Concerning ζ n,2 , since we have

ζ n,2 = n-k i=1 1 i 2 f 2 k i n , applying Lemma 3.8 with function g = f 2 k , we obtain lim n-→∞ ζ n,2 = f 2 k (0) π 2 6 = 1 (k -1) 2 π 2 6 .
Finally, for term ζ n,1 , we have

ζ n,1 = n-k i=1 1 (n -i) 2 f 2 k i n = n-1 i=k 1 i 2 f 2 k 1 - i n
Applying again Lemma 3.8 with function g = f 2 k , we obtain

lim n-→∞ ζ n,1 = f 2 k (1) ∞ i=k 1 i 2 = π 2 6 - k-1 i=1 1 i 2 .
Putting these three limits together leads to lim n-→∞

ζ n = 1 + 1 (k -1) 2 π 2 6 - k-1 i=1 1 i 2 ,
which completes the proof.

We analyze now the limiting value of η n when n tends to infinity.

Lemma 3.10 For all k ≥ 2, we have

lim n-→∞ η n = 1 + 1 (k -1) 2 π 2 6 - k-1 i=1 1 i 2 .
Proof. Using again the function f k introduced in the proof of Lemma 3.2, we obtain

η n = n-k i=1 f 2 k i n -k (n -i) 2 1 -1 - i n -k 2 = (n -k) 2 n 2 n-k i=1 n 2 i 2 (n -i) 2 f 2 k i n -k . ( 14
)
The function f k being Lipschitz on interval [0, 1], defining

C k = max z∈[0,1] |(f 2 k ) (z)| where (f 2 k ) is the derivative of f 2
k , we get, using Relations ( 12) and ( 14),

η n - (n -k) 2 n 2 ζ n ≤ (n -k) 2 n 2 n-k i=1 n 2 i 2 (n -i) 2 f 2 k i n -k -f k i n ≤ C k (n -k) 2 n 2 n-k i=1 n 2 i 2 (n -i) 2 ik n(n -k) ≤ C k k(n -k) n 2 n-k i=1 n 2 i 2 (n -i) 2 .
Using Relation [START_REF] Giakkoupis | Tight bounds for rumor spreading with vertex expansion[END_REF], we have

lim n-→∞ n-k i=1 n 2 i 2 (n -i) 2 = ∞ i=1 1 i 2 + ∞ i=k 1 i 2 = π 2 3 - k-1 i=1 1 i 2 .
It follows that lim

n-→∞ η n - (n -k) 2 n 2 ζ n = 0.
This result together with the result of Lemma 3.9 leads to lim n-→∞

η n = lim n-→∞ ζ n = 1 + 1 (k -1) 2 π 2 6 - k-1 i=1 1 i 2 ,
which completes the proof.

Theorem 3.11 For all k ≥ 2, we have

lim n-→∞ V(Θ k,n ) = 1 + 1 (k -1) 2 π 2 6λ 2 .
Proof. Relations [START_REF] Fountoulakis | Rumor spreading on random regular graphs and expanders[END_REF] and [START_REF] Frieze | The shortest-path problem for graphs with random arc-lengths[END_REF] give

η n λ 2 + 1 λ 2 k-1 i=1 1 i 2 ≤ V(Θ k,n ) ≤ ζ n λ 2 + 1 λ 2 k-1 i=1 1 i 2 .
The use of Lemma 3.9 and Lemma 3.10 leads to the desired result.

Distribution of the rumor spreading time

This section provides explicit limiting distributions such as Θ k,n -E(Θ k,n ) when n tends to infinity. We introduce the notation µ k,n (i) = λ(n -i)p k,n (i). Recall that U k,n (i) is exponentially distributed with rate µ k,n (i) and that

Θ k,n = n-1 i=1 U k,n (i).
The main result of this section is Theorem 4.2 whose proof needs the following lemma.

1 i 2 p 2 k,2n+1 (2n + 1 -i) = 0. Proof. Introducing the notation ∆ m,n (k) = n i=m 1 (2n + 1 -i) 2 1 -1 - i 2n + 1 k-1 2
we obtain, using inequality (4),

n i=m 1 (2n + 1 -i) 2 p 2 k,2n+1 (i) ≤ ∆ m,n (k).
Using the function f k defined in the proof of Lemma 3.2 we write

∆ n,m (k) = n i=m (2n + 1) 2 (2n + 1 -i) 2 i 2 f 2 k i 2n + 1 .
Observing that function f k is increasing on interval [0, 1] and that f k (1) = 1, we obtain using Relation [START_REF] Giakkoupis | Tight bounds for rumor spreading with vertex expansion[END_REF], with 2n + 1 instead n,

∆ n,m (k) ≤ n i=m (2n + 1) 2 (2n + 1 -i) 2 i 2 = n i=m 1 (2n + 1 -i) 2 + n i=m 1 i 2 + 2 2n + 1 n i=m 1 i + 1 2n + 1 -i = 2n+1-m i=n+1 1 i 2 + n i=m 1 i 2 + 2 2n + 1 n i=m 1 i + 2n+1-m i=n+1 1 i = 2n+1-m i=m 1 i 2 + 2 2n + 1 2n+1-m i=m 1 i ≤ 2n+1-m i=m 1 i 2 + 2(1 + ln(2n + 1 -m) 2n + 1 .
The lim m-→∞ lim sup n-→∞ of both terms is 0 because i 1/i 2 is a converging series. This proves the first relation.

Concerning the second relation, introducing the notation

Λ m,n (k) = n i=m 1 i 2 1 -1 - 2n + 1 -i 2n + 1 k-1 2
we obtain in the same way, using inequality (4),

n i=m 1 i 2 p 2 k,2n+1 (2n + 1 -i) ≤ Λ m,n (k)
and, using function

f k , Λ n,m (k) = n i=m (2n + 1) 2 (2n + 1 -i) 2 i 2 f 2 k 1 - i 2n + 1 .
As we did for term ∆ n,m (k), we have

Λ m,n (k) ≤ n i=m (2n + 1) 2 (2n + 1 -i) 2 i 2 ,
which in turn leads to the same result. We are now able to prove the following theorem. Theorem 4.2 Let (Z i ) i≥1 be a sequence of i.i.d. random variables exponentially distributed with rate 1 and let W be defined by

W = ∞ i=1 Z i -1 i .
We then have 2) as n -→ ∞ where W (1) and W (2) are i.i.d. with the same distribution as W .

Θ k,n -E(Θ k,n ) L --→ 1 (k -1)λ W (1) + 1 λ W (
Proof. Recalling that µ k,n (i) = λ(n -i)p k,n (i), we have, for all n ≥ 2, i ∈ {1, . . . , n -1} and x ≥ 0,

P{µ k,n (i)U k,n (i) > x} = P{U k,n (i) > x/µ k,n (i)} = e -x .
Thus if Z i is exponentially distributed with rate 1, we have µ k,n (i)U k,n (i) L = Z i . Moreover since the (U k,n (i)) i∈{1,...,n-1} are independent, the (Z i ) i≥1 are also independent.

Observing now that for each fixed i and k, we have, from Relation (1),

p k,n (i) = i k-1 h=1 1 n -h + o(1/n), we obtain lim n-→∞ µ k,n (i) = λ lim n-→∞ (n -i)p k,n (i) = (k -1)λi.
From Relation (1), we also trivially get

lim n-→∞ µ k,n (n -i) = λi lim n-→∞ p k,n (n -i) = λi. Defining R k,n (i) = U k,n (i) -E(U k,n (i)) we obtain, since E(U k,n (i)) = 1/µ k,n (i), R k,n (i) = U k,n (i) -E(U k,n (i)) = µ k,n (i)U k,n (i) -1 µ k,n (i) L --→ Z i -1 (k -1)λi as n -→ ∞. (15)
In the same way, we get

R k,n (n -i) = µ k,n (n -i)U k,n (n -i) -1 µ k,n (n -i) L --→ Z i -1 λi as n -→ ∞. (16)
Suppose that n is odd, i.e., that n = 2 + 1. Defining

V k, = i=1 R k,2 +1 (i) and V k, = i=1 R k,2 +1 (2 + 1 -i), we have Θ k,2 +1 -E(Θ k,2 +1 ) = 2 i=1 [U k,2 +1 (i) -E(U k,2 +1 (i))] = 2 i=1 R k,2 +1 (i) = V k, + V k, . (17)
Observe that the random variables V k, and V k, are independent. The rest of the proof consists in checking the hypothesis of the principle of accompanying laws of Theorem 3.1.14 of [START_REF] Stroock | Probability Theory: An Analytic View[END_REF]. We introduce the notation

V k, ,m = m-1 i=1 R k,2 +1 (i) and V k, ,m = m-1 i=1 R k,2 +1 (2 + 1 -i).
Using the fact that E(R k,n (i)) = 0 and that the R k,n (i) are independent, we have

E (V k, -V k, ,m ) 2 = E   i=m R k,2 +1 (i) 2   = V i=m R k,2 +1 (i) = i=m V(R k,2 +1 (i)) = i=m V(U k,2 +1 (i)) = i=m 1 µ 2 k,2 +1 (i) = 1 λ 2 i=m 1 (2 + 1 -i) 2 p 2 k,2 +1 (i)
and, in the same way, Using now the Markov inequality, we obtain, for all ε > 0,

E (V k, -V k, ,m ) 2 = i=m 1 µ 2 k,2 +1 (2 + 1 -i) = 1 λ 2 i=m 1 i 2 p 2
P{|V k, -V k, ,m | ≥ ε} = P{(V k, -V k, ,m ) 2 ≥ ε 2 } ≤ E((V k, -V k, ,m ) 2 ) ε 2
and

P{ V k, -V k, ,m ≥ ε} = P{(V k, -V k, ,m ) 2 ≥ ε 2 } ≤ E((V k, -V k, ,m ) 2 ) ε 2 .
Putting together these results, we deduce that for all ε > 0, we have Using ( 15) and ( 16) and the fact that the R k,n (i) are independent, we have

V k, ,m L --→ W k,m and V k, ,m L --→ W k,m as -→ ∞. ( 19
)
The hypothesis of the principle of accompanying laws of Theorem 3.1.14 of [START_REF] Stroock | Probability Theory: An Analytic View[END_REF] are properties [START_REF] Panagiotou | Randomized rumor spreading: the effect of the network topology[END_REF] and [START_REF] Panagiotou | Faster rumor spreading with multiple calls[END_REF].

We can thus conclude that

V k, L --→ 1 (k -1)λ W and V k, L --→ 1 λ W as -→ ∞.
This means, from Relation [START_REF] Mocquard | Analysis of the propagation time of a rumour in large-scale distributed systems[END_REF], that 2) as -→ ∞,

Θ k,2 +1 -E(Θ k,2 +1 ) L --→ 1 (k -1)λ W (1) + 1 λ W (
where W (1) and W (2) are independent and identically distributed as W . The same reasoning applies in the case where n = 2 . which completes the proof.

Lemma 4 . 1

 41 For all k ≥ 2, we have lim

  k, -V k, ,m ) 2 ) = lim k, -V k, ,m ) 2 ) = 0.

  k, -V k, ,m | ≥ ε} = lim m-→∞ lim sup -→∞ P{ V k, -V k, ,m ≥ ε} = 0.

Corollary 4 . 3 1 Z 2 0 P Z 1

 431201 For all x ∈ R and k ≥ 2, we havelim n-→∞ P {Θ k,n -E(Θ k,n ) ≤ x} = ∞ 0 exp -t -t -(k-1) e -(k-1)λx-kγ dt.Proof. L. Gordon has proved in[START_REF] Gordon | Bounds for the distribution of the generalized variance[END_REF] that i ) are i.i.d. exponential with rate 1. Thus, by definition of W in Theorem 4.2, we haveW L = -γ -ln(Z 1 ).Introducing W (1) L = -γ -ln(Z 1 ) and W (2) L = -γ -ln(Z 2 ), we obtain from Theorem 4.2, for all x ∈ R,lim n-→∞ P {Θ k,n -E(Θ k,n ) > x} = P ≥ e -λx-kγ/(k-1) = ∞ ≥ t -(k-1) e -(k-1)λx-kγ e -t dt = ∞ 0 exp -t -t -(k-1)e -(k-1)λx-kγ dt, which completes the proof.

Corollary 4 . 4

 44 For all x ∈ R and k ≥ 2, we havelim n-→∞ P Θ k,n -k ln(n) (k -1)λ ≤ x = ∞ 0 exp -t -t -(k-1) e -(k-1)λx k -1 dtProof. Observe first that we haveP Θ k,n -k ln(n) (k -1)λ ≤ x = P Θ k,n -E(Θ k,n ) ≤ x + k ln(n) (k -1)λ -E(Θ k,n ) .From Theorem 3.7, we have limn-→∞ k ln(n) (k -1)λ -E(Θ k,n ) = -γk -ln(k -1) (k -1)λ . It follows from Corollary 4.3 that lim n-→∞ P Θ k,n -k ln(n) (k -1)λ ≤ x = ∞ 0 exp -t -t -(k-1)e -(k-1)λx k -1 dt.


Indranil Biswas

Sorin Dumitrescu

FUJIKI CLASS C AND HOLOMORPHIC GEOMETRIC STRUCTURES

Keywords: 2010 Mathematics Subject Classification. 53B35, 53C55, 53A55 Holomorphic geometric structure, Fujiki class C, algebraic reduction, torus bundle

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class C, whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class C and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.

Introduction

The article deals, in particular, with holomorphic geometric structures, in the sense of Gromov [Gr, DG], on compact complex manifolds. The definition is very general (see Section 2), and interesting classical examples of such structures to keep in mind are holomorphic tensors, holomorphic affine connections, holomorphic projective connections and holomorphic conformal structures. Compact complex manifolds bearing any of these kind of geometric structures are rather special. Based on results obtained in [START_REF] Dumitrescu | Structures géométriques holomorphes sur les variétés complexes compactes[END_REF][START_REF] Biswas | Holomorphic Affine Connections on Non-Kähler manifolds[END_REF] we formulate the following:

Question 1.1. Is it true that any holomorphic geometric structure of affine type on any compact complex manifold with trivial canonical bundle is locally homogeneous? Question 1.1 is known to have a positive answer if the manifold is either Kähler (hence Calabi-Yau) [START_REF] Dumitrescu | Structures géométriques holomorphes sur les variétés complexes compactes[END_REF], or if it is in Fujiki class C with a polystable holomorphic tangent bundle (with respect to some Gauduchon metric) [BD]. The results of [START_REF] Dumitrescu | Structures géométriques holomorphes sur les variétés complexes compactes[END_REF] show that the answer is also yes for compact parallelizable manifolds and for Ghys's deformations of the complex structure on the parallelizable manifolds SL(2, C)/Γ with Γ being a uniform lattice [START_REF] Ghys | Déformations des structures complexes sur les espaces homogènes de SL(2, C)[END_REF].

Here we prove that Question 1.1 has a positive answer for compact complex torus principal bundles over compact Kähler Calabi-Yau manifolds (Theorem 1.2).

Since compact complex surfaces with trivial canonical bundle are either complex tori, or K3 surfaces, or primary Kodaira surfaces (elliptic principal bundles over elliptic curves) [START_REF] Barth | Compact complex surfaces[END_REF]Chapter 6], from Theorem 1.2 it follows that the answer to Question 1.1 is yes when the dimension of the manifold is two.

More precisely, our result in this direction is: Theorem 1.2. Let X be a compact complex torus holomorphic principal bundle over a compact Kähler manifold with trivial first Chern class in H 2 (X, R). Then any holomorphic geometric structure of affine type φ on X is locally homogeneous.

The previous result also stands for holomorphic projective connections and for holomorphic conformal structures even though these two geometric structures are not of affine type. This is because on manifolds with trivial canonical bundle these two geometric structures admit global representatives which are of affine type, namely, a holomorphic affine connection and a holomorphic Riemannian metric respectively.

On the other hand, the previous result does not work in general for non-affine geometric structures. To give an example, recall that compact complex tori T n = C n /Λ (with Λ a cocompact lattice) admit holomorphic foliations that are defined by nonconstant holomorphic maps into the complex projective space CP n-1 which are not translation invariant [Gh]. Together with the standard holomorphic parallelization of the holomorphic tangent bundle of T n they form holomorphic rigid geometric structures of non-affine type in Gromov's sense (see Definition 2.1 and Definition 2.2) which are not locally homogeneous.

We expect the answer for Question 1.1 to be yes for all holomorphic geometric structures of affine type on compact complex manifolds in the Fujiki class C that have trivial canonical bundle. We also expect holomorphic Riemannian metrics to be always locally homogeneous on compact complex manifolds (this was proved in complex dimension three [START_REF] Dumitrescu | Homogénéité locale pour les métriques riemanniennes holomorphes en dimension 3[END_REF]) and holomorphic affine connections to be always locally homogeneous on compact complex manifolds with trivial canonical bundle; it may be noted that contrary to the case of holomorphic Riemannian metrics, here the condition on the triviality of the canonical bundle is not automatically satisfied and is in fact even necessary: there exists non-locally homogeneous holomorphic affine connections on principal elliptic bundles with odd first Betti number (hence non-Kähler) over Riemann surfaces of genus g ≥ 2 [START_REF] Dumitrescu | Connexions affines et projectives sur les surfaces complexes compactes[END_REF].

In the case where the holomorphic geometric structure φ is rigid, Theorem 1.2 enables us to gather information about the fundamental group of the manifold X:

Corollary 1.3. Let X be a compact complex torus holomorphic principal bundle over a compact Kähler manifold with trivial first Chern class. If X is endowed with a holomorphic rigid geometric structure of affine type φ, then the fundamental group of X is infinite.

Corollary 1.4. Let X be a compact complex manifold in Fujiki class C bearing a holomorphic affine connection in T X. Then the fundamental group of X is infinite.

Notice that Theorem 1.2 applies to many complex torus principal bundles which are not Kähler. For example, one could consider the complex Heisenberg group of upper triangular (3 × 3) matrices with complex entries and the compact parallelizable manifold obtained by taking the quotient of this group by the lattice of matrices with Gaussian integers as entries. This quotient is biholomorphic to a (non-Kähler) principal elliptic bundle over a two-dimensional compact complex torus. Many other examples and results about complex torus principal bundles can be found in [Hof].

Recall that complex projective spaces admit the standard flat holomorphic projective connection. Also recall that the smooth quadric z 2 0 + z 2 1 + . . . + z 2 n+1 = 0 in CP n+1 is endowed with a canonical (flat) holomorphic conformal structure given by the quadratic form (this standard holomorphic conformal structure is invariant by the subgroup PO(n + 2, C) of the complex projective group PGL(n + 2, C)). More generally, if G is a complex semi-simple Lie group and P ⊂ G a parabolic subgroup, then the rational manifold G/P is equipped with the standard flat Cartan geometry with model (G, P ).

For holomorphic rigid geometric structures and for holomorphic Cartan geometries of algebraic type (definitions are given in Section 2) we prove the following: Theorem 4.2. Let X be a compact complex manifold in the Fujiki class C, of complex dimension n and of algebraic dimension n -d, with d > 0. If X admits a holomorphic rigid geometric structure φ, then the fundamental group of X is infinite.

Theorem 4.3. Let X be a compact complex simply connected manifold in the Fujiki class C. If X bears a holomorphic Cartan geometry of algebraic type, then X is projective.

Theorem 4.2 and Theorem 4.3 are in the same spirit as Borel-Remmert result asserting that compact simply connected homogeneous manifolds in Fujiki class C are projective. Indeed, it should be mentioned that it was conjectured that compact simply connected complex manifolds bearing holomorphic Cartan geometries are homogeneous manifolds. Borel-Remmert theorem states precisely that compact homogeneous manifolds in class C are biholomorphic to a product of a projective rational homogeneous manifold with a complex torus [Fu] (p. 255).

In the special case where the algebraic dimension of X is zero, Theorem 4.3 asserts that X does not bear holomorphic Cartan geometries of algebraic type. This was proved recently in [BDM] (Theorem 4.1) even for manifolds which are not necessarily in class C.

Theorem 4.3 implies the following: Corollary 1.5. Let X be a compact complex manifold in Fujiki class C with trivial first Chern class in H 2 (X, R). If X bears a holomorphic Cartan geometry of algebraic type, then the fundamental group of X is infinite.

Notice that Corollary 1.5 should be seen as a natural generalization of Corollary 1.4. Indeed, manifolds X in Fujiki class C bearing holomorphic affine connections on T X have vanishing Chern classes (see proof of Corollary 1.4 in Section 3).

Geometric structures and symmetries

Let X be a complex manifold of complex dimension n. For any integer k ≥ 1, we associate the principal bundle of k-frames R k (X) -→ X , which is the bundle of k-jets of local holomorphic coordinates on X. The corresponding structural group D k is the group of k-jets of local biholomorphisms of C n fixing the origin. We note that D k is a complex affine algebraic group.

Definition 2.1. A holomorphic geometric structure of order k on X is a holomorphic D kequivariant map φ from R k (X) to a complex algebraic manifold Z endowed with an algebraic action of D k . The geometric structure φ is said to be of affine type if Z is a complex affine manifold.

Holomorphic tensors are holomorphic geometric structures of affine type of order one. More precisely, a holomorphic tensor on X is a holomorphic GL(n, C)-equivariant map from the frame bundle R 1 (X) to a linear complex algebraic representation W of GL(n, C). Holomorphic affine connections are holomorphic geometric structures of affine type of order two [Gr, DG]. Holomorphic foliations and holomorphic projective connections are holomorphic geometric structure of non-affine type.

The natural notion of symmetry of a holomorphic geometric structure is the following. A (local) biholomorphism of X preserves a holomorphic geometric structure φ if its canonical lift to R k (X) fixes each fiber of the map φ. Such a local biholomorphism is called a local isometry of φ.

A (local) holomorphic vector field on X is called a Killing vector field with respect to φ if its local flow acts on X by local isometries.

The connected component Aut 0 (X, φ), containing the identity element, of the automorphism group of (X, φ) is a complex Lie subgroup of the automorphism group of X. The corresponding Lie algebra is the vector space of globally defined (holomorphic) Killing vector fields for φ.

Definition 2.2. A holomorphic geometric structure φ is called rigid of order l in Gromov's sense if any local biholomorphism preserving φ is uniquely determined by its l-jet in any given point.

Holomorphic affine connections are rigid of order one in Gromov's sense (see [Gr] and the nice expository survey [DG]). The rigidity comes from the fact that local biholomorphisms fixing a point and preserving a connection linearize in exponential coordinates, so they are indeed completely determined by their differential at the fixed point.

Holomorphic Riemannian metrics, holomorphic projective connections and holomorphic conformal structures in dimension ≥ 3 are all rigid holomorphic geometric structures, while holomorphic symplectic structures and holomorphic foliations are non-rigid geometric structures [DG].

The sheaf of local Killing fields of a holomorphic rigid geometric structure φ is locally constant. Its fiber is a finite dimensional Lie algebra called the Killing algebra of φ [DG, Gr].

Gromov's study of local symmetries of analytic rigid geometric structure [DG, Gr] led, in the particular case of simply connected manifolds X, to the following description of Aut 0 (X, φ) (see Section 3.5 in [Gr]). The action of Aut 0 (X, φ) preserves each connected component of the fibers of a meromorphic map φ r : X -→ W r into an algebraic manifold W r (representing the r-jets of φ), and furthermore, the action of Aut 0 (X, φ) on each of these connected components is transitive. The two main ingredients of the proof are (1) the integrability result showing that, for any r large enough, local isometries are exactly the class of local biholomorphisms that preserve the r-jet of φ, and (2) the extendibility result proving that local Killing fields on simply connected manifolds extend to the entire manifold [Am, DG, Gr, No].

It now follows that Aut 0 (X, φ)-orbits in X are locally closed as they coincide with the connected components of the fibers of the meromorphic map φ r .

Inspiration of these results led to Theorem 2.1 in [START_REF] Dumitrescu | Meromorphic almost rigid geometric structures, Geometry, Rigidity and Group Actions[END_REF] which says that the Killing Lie algebra of φ act transitively on the connected components of the algebraic reduction of X (see also Theorem 3 in [START_REF] Dumitrescu | Structures géométriques holomorphes sur les variétés complexes compactes[END_REF]).

The algebraic dimension a(X) of X is the transcendence degree over C of the field of meromorphic functions M(X).

Let us recall the following classical result called the algebraic reduction theorem (see [START_REF] Ueno | Classification theory of algebraic varieties and compact complet spaces[END_REF]).

Theorem 2.3 (Algebraic Reduction, [START_REF] Ueno | Classification theory of algebraic varieties and compact complet spaces[END_REF]p. 25,Definition 3.3], [START_REF] Ueno | Classification theory of algebraic varieties and compact complet spaces[END_REF]p. 26,Proposition 3.4]). Let X be a compact connected complex manifold of dimension n and algebraic dimension a(X) = n -d. There exists a bi-meromorphic modification

Ψ : X -→ X and a holomorphic map t : X -→ V with connected fibers onto a (n -d)-dimensional projective manifold V such that t * (M(V )) = Ψ * (M(X)) .
Let π red : X -→ V be the meromorphic fibration given by t • Ψ -1 ; it is called the algebraic reduction of X.

This meromorphic fibration π red is called almost holomorphic if the Ψ-exceptional locus does not the intersect generic t-fibers.

Manifolds with maximal algebraic dimension are those for which the algebraic dimension coincides with the complex dimension. They are called Moishezon manifolds [START_REF] Ueno | Classification theory of algebraic varieties and compact complet spaces[END_REF]p. 26,Definition 3.5]. The algebraic reduction of a Moishezon manifold is a bimeromorphism with a smooth complex projective manifold [Mo], [START_REF] Ueno | Classification theory of algebraic varieties and compact complet spaces[END_REF]p. 26,Theorem 3.6].

More generally, a compact complex manifold is said to be in the Fujiki class C if it is the image of a compact Kähler space under a holomorphic map. A result of Varouchas says that a compact complex manifold belongs to Fujiki's class C if and only if it is bimeromorphic to a compact Kähler manifold (in other words, admits compact Kähler modifications) [START_REF] Varouchas | Kähler spaces and proper open morphisms[END_REF]Section IV.3]. Manifolds in Fujiki's class C share many common features with Kähler manifolds.

We will investigate in Section 4 the holomorphic geometric structures on compact complex manifolds in Fujiki class C.

The following theorem is proved using Theorem 2.1 in [START_REF] Dumitrescu | Meromorphic almost rigid geometric structures, Geometry, Rigidity and Group Actions[END_REF].

Theorem 2.4. Let X be a compact complex simply connected manifold of dimension n endowed with a holomorphic rigid geometric structure. Then there exists a connected complex abelian Lie subgroup L in the group of automorphisms of (X, φ) which preserves each fiber of the algebraic reduction π red , and which acts transitively on the generic fibers of π red . Moreover the following hold.

(i) If π red is almost holomorphic, then the L-orbits are compact. (ii) If X is in the Fujiki class C, then a(X) > 0.
Proof. It follows from Theorem 2.1 in [START_REF] Dumitrescu | Meromorphic almost rigid geometric structures, Geometry, Rigidity and Group Actions[END_REF] that the orbits of Aut 0 (X, φ) contain the generic orbits of π red (see also Theorem 3 in [START_REF] Dumitrescu | Structures géométriques holomorphes sur les variétés complexes compactes[END_REF]). Let {X 1 , • • • , X l } be a basis of the Lie algebra of Aut 0 (X, φ); these X i are globally defined holomorphic vector fields on X.

Consider the holomorphic rigid geometric structure φ which is the juxtaposition of φ with the family of holomorphic vector fields {X 1 , • • • , X l }. Denote by L the automorphism group Aut 0 (X, φ ) of φ . It can be shown that the Lie algebra of L = Aut 0 (X, φ ) is in fact the center of the Lie algebra of Aut 0 (X, φ). Indeed, an element of the family {X 1 , • • • , X l } preserves φ if and only if it commutes with all elements in this family (it belongs to the center of the Lie algebra). It follows that L is the maximal abelian connected subgroup in Aut 0 (X, φ).

Applying Theorem 2.1 in [START_REF] Dumitrescu | Meromorphic almost rigid geometric structures, Geometry, Rigidity and Group Actions[END_REF] to φ it follows that at generic points in X, the Killing Lie algebra of φ (and hence L ) acts transitively on the fibers of the algebraic reduction π red .

Let us consider the almost rigid meromorphic geometric structure (in the sense of Definition 1.2 in [START_REF] Dumitrescu | Meromorphic almost rigid geometric structures, Geometry, Rigidity and Group Actions[END_REF]) φ obtained by juxtaposing φ, the family of holomorphic vector fields {X 1 , • • • , X l } and the meromorphic map π red . When applied to φ , Theorem 2.1 in [START_REF] Dumitrescu | Meromorphic almost rigid geometric structures, Geometry, Rigidity and Group Actions[END_REF] says that, at generic points, the Killing Lie algebra of φ acts transitively on the fibers of the algebraic reduction π red . Notice that Killing vector fields of φ are exactly those Killing vector fields of φ lying in the center of the Killing Lie algebra of φ and preserving each fiber of π red . The connected component of identity in the automorphism group Aut(X, φ ) is the maximal connected complex abelian Lie subgroup L = Aut 0 (X, φ ) of Aut 0 (X, φ) (and hence of L ) preserving each fiber of the algebraic reduction.

(i) If π red is almost holomorphic, its generic fibers are compact. Therefore the generic (and hence all) L-orbits are compact.

(ii) If X is in the Fujiki class C, the connected component of identity Aut 0 (X) of the automorphism group of the (simply connected) manifold X is a complex linear algebraic group (see Corollary 5.8 in [Fu]). Then Aut 0 (X, φ) is the connected component of identity of the subgroup of Aut 0 (X) preserving each fiber of the meromorphic fibration φ r : X -→ W r .

In particular, Aut 0 (X, φ) is closed in Aut 0 (X). Moreover, L = Aut 0 (X, φ ) coincides with the connected component of identity of the maximal abelian subgroup of Aut 0 (X, φ) preserving each fiber of the algebraic reduction of X. It follows that L is closed in Aut 0 (X, φ). A priori L is not Zariski closed with respect to the complex linear algebraic group structure of Aut 0 (X). Its Zariski closure L * is an algebraic abelian subgroup of Aut 0 (X) in the terminology of Fujiki (see Definition 2.1 in [Fu]), so L * is a meromorphic subgroup of Aut 0 (X) (equivalently, the L * -action on X is compactifiable in Lieberman's terminology [START_REF] Lieberman | Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Fonctions de Plusierus Variables Complexes III[END_REF]Section 3]).

Since L * is a connected complex abelian algebraic group, a classical result of Rosenlicht [Ro] shows that L * is isomorphic to C p × C * q for some nonnegative integers p and q.

Assume, by contradiction, that a(X) = 0. Then L acts with an open dense orbit on X. Consequently, L * acts with an open dense orbit on X. Since L * is abelian, this open dense orbit is biholomorphic to L * and X is a compactification of L * . Since L * is algebraic, it can be seen as a Zariski open dense set in a complex rational manifold L * * [Ro]. The action of L * on X is meromorphic in Fujiki's sense, meaning the holomorphic action map L * × X -→ X extends to a meromorphic map L * * × X -→ X (see [Fu], Proposition 2.2 and Remark 2.3). Therefore, X is a bi-meromorphic image of L * * and hence it is a projective rational manifold (see Lemma 3.8 and Remark 4.1 in [Fu]). It follows that a(X) = n > 0: a contradiction.

Notice that statement (ii) in Theorem 2.4 will be improved in Theorem 4.2 which asserts that simply connected manifolds in Fujiki class C bearing holomorphic rigid geometric structures are Moishezon.

The following classical result will be useful in the proof of Proposition 3.3. Lemma 2.5. Let X be a compact complex manifold with trivial canonical line bundle K X , and let L be a connected complex Lie group acting holomorphically on X. Then L preserves any nonzero holomorphic section ω of K X and, consequently, the smooth finite measure on X defined by the section ω ∧ ω.

Proof. Let K be a holomorphic fundamental vector field of the L-action on X. Consider the corresponding 1-parameter family of automorphisms of X. The automorphism for any t ∈ C will be denoted by Φ t . The Lie derivative L K ω of ω with respect to K is another holomorphic section of K X which must be of the form cω, for some c ∈ C (also called the divergence of K with respect to ω), because K X is trivial. This implies that (Φ t ) * ω = exp(ct) • ω, for all t ∈ C. Since the total volume X ω ∧ ω of X is invariant by automorphisms, it follows that the modulus of exp(ct) equals one. By Liouville Theorem the entire holomorphic map t -→ exp(ct) must be constant, equal to 1 (the value in t = 0). Consequently, we have c = 0 and ω is K-invariant. Since L is connected, it is generated by the flows of its fundamental vector fields. Since all of them preserve ω, we get that ω is L-invariant.

In the last section we will deal also with another concept of geometric structure: the Cartan geometry. Cartan geometries are geometric structures which are infinitesimally modeled on homogeneous spaces G/H, where G is a complex Lie group and H ⊂ G is a closed subgroup.

Denote by g and h the Lie algebras of the Lie groups G and H respectively. Then we have the following definition (see [Sh]).

Definition 2.6. A holomorphic Cartan geometry (P, ω) on X with model (G, H) is a holomorphic principal (right) H-bundle π : P -→ X endowed with a holomorphic g-valued 1-form ω satisfying:

(1) ω p : T p P -→ g is a complex linear isomorphism for all p ∈ P , (2) the restriction of ω to every fiber of π coincides with the left invariant Maurer-Cartan form of H, and (3) (R h ) * ω = Ad(h) -1 ω, for all h ∈ H, where R h is the right action of h on P and Ad is the adjoint representation of G on g.

The Cartan geometry is said to be of algebraic type if the image of H through the adjoint representation of G is a complex algebraic subgroup of GL(g).

Usual geometric structures such as holomorphic affine connections, holomorphic projective connections, or holomorphic conformal are examples of holomorphic Cartan geometry of algebraic type [Sh, Pe].

A (local) holomorphic vector field on X is a Killing field for the Cartan geometry (P, ω) if it admits a lift to P whose (local) flow commutes with the action of H and also preserves ω. The sheaf of local Killing fields of a holomorphic Cartan geometry is locally constant with fiber a finite dimensional Lie algebra [Me, Pe]: the Killing algebra of (P, ω).

Gromov's results describing the decomposition in orbits of the Killing algebra of a rigid geometric structure [Gr] was adapted to the context of Cartan geometries of algebraic type by Melnick and Pecastaing [Me, Pe]. Using their results, Theorem 1.2 in [START_REF] Dumitrescu | Killing fields of holomorphic Cartan geometries[END_REF] adapts Theorem 2.1 in [START_REF] Dumitrescu | Meromorphic almost rigid geometric structures, Geometry, Rigidity and Group Actions[END_REF] to Cartan geometries of algebraic type. Therefore, Theorem 2.4 stands also for holomorphic Cartan geometries of algebraic type.

Trivial canonical bundle and geometric structures

In this section we prove Theorem 1.2, and we deduce Corollary 1.3 and Corollary 1.4. We also prove Proposition 3.2 which is a particular case of Theorem 3.3 (the case where the canonical bundle is trivial).

Let us first prove Theorem 1.2.

Proof of Theorem 1.2. Let T be a complex torus of complex dimension d > 0, and let π : X -→ Y be a principal T -fibration over a compact Kähler manifold Y . Since the torus action trivializes the relative tangent bundle of the fibration, the exact sequence induced by dπ yields

K X π * K Y .
We also assume that

• the real first Chern class of K X vanishes (or equivalently, the real first Chern class of K Y vanishes), and • X is endowed with a holomorphic geometric structure φ of order k which is of affine type.

We need to consider only the case where the complex dimension n -d of Y is positive.

Otherwise, X coincides with the complex torus T and, since R k (T ) is isomorphic to D k × T , the geometric structure φ is completely determined by a holomorphic map from T to a complex affine manifold. This map must be constant, and consequently, φ is invariant under the action of T on X.

Since Y is Kähler with vanishing first Chern class, its canonical bundle K Y is of finite order [Be]. Replacing Y by a suitable finite unramified cover of it we will assume that K Y is trivial (equivalently, K X is trivial).

Denote by X 1 , • • • , X d a basis of the fundamental vector fields of the T -action on X; they are holomorphic global vector fields on X which span (and trivialize) the kernel of dπ (the vertical subbundle V ⊂ T X).

Assume, by contradiction, that φ is not locally homogeneous on X. Then Lemma 3.2 in [START_REF] Dumitrescu | Structures géométriques holomorphes sur les variétés complexes compactes[END_REF] proves that, for some integers a, b ≥ 0, there exists a nontrivial holomorphic section of (T X) ⊗a ⊗ ((T X) * ) ⊗b which vanishes at some point in X. Since K X is trivial, there is a canonical contraction isomorphism between T X and Λ n-1 (T * X). Using this isomorphism, we get a nontrivial holomorphic section section t of (T * X) ⊗m , with m = (n -1)a + b, such that t vanishes at some point in x 0 ∈ X.

We will get a contradiction using an induction on the power m.

First consider the case of m = 1. In this case t is a holomorphic one form. Note that t(X i ) is a holomorphic function on X vanishing at x 0 and therefore, it vanishes identically. This proves that V is in the kernel of t. The fibers of π being compact and connected, the form t is the pull-back of a holomorphic one form on Y . This leads to a holomorphic one form on Y vanishing at π(x 0 ) ∈ Y . Since Y is a Kähler Calabi-Yau manifold, holomorphic tensors on Y are parallel with respect to any Ricci flat Kähler metric on it [Be,p. 760,Principe de Bochner]. Hence they cannot vanish at a point without being trivial: a contradiction.

Let us consider now the case of m > 1. The induction hypothesis is that the vanishing holds if degree of the tensor is less than m.

We will show that for all 0 < k ≤ m, the contraction of t with any ordered family of k vector fields (they need not be distinct), chosen from the vector fields {X 1 , • • • , X d }, vanishes identically.

Let us consider first the case of k = m. When contracted with any ordered family of m vector fields (need not be distinct) chosen from {X 1 , • • • , X d }, the tensor t produces a holomorphic function on X which vanishes at x 0 . Therefore all those contractions vanish identically on X. Now assume that 0 < k < m. When contracted with any k vector fields among {X 1 , • • • , X d }, the tensor t produces a holomorphic section of (T * X) ⊗m-k vanishing at x 0 . Since m -k < m, the induction hypothesis on m holds. So this section of (T * X) ⊗m-k vanishes identically.

Since all those contractions vanishing identically, our tensor t is a holomorphic section of ((T X/V ) * ) ⊗m .

The fibers of π being compact and connected, the tensor t is a pull-back from Y . We get a nontrivial holomorphic section of ((T Y ) * ) ⊗m which vanishes at π(x 0 ) ∈ Y . But, as before, this holomorphic tensor should be parallel with respect to any Ricci flat metric on Y [Be]: a contradiction.

Remark 3.1. Assume that Y is a compact complex manifold such that any holomorphic geometric structure of affine type on it is locally homogeneous. The proof of Theorem 1.2 shows that any compact complex torus holomorphic principal bundle X over Y shares the same property (i.e. any holomorphic geometric structure of affine type on X is locally homogeneous as well).

Corollary 1.3 is a direct consequence of Theorem 1.2 because of the following proposition.

Proposition 3.2. Let X be a compact complex manifold whose canonical bundle is of finite order such that X admits a holomorphic rigid geometric structure φ. If φ is locally homogeneous, then the fundamental group of X is infinite.

Proof. Assume by contradiction that the fundamental group of X is finite. Replacing X by its universal cover and φ by its pull-back on the universal cover we assume that X is simply connected. Notice that K X has now become trivial.

Since φ is locally homogeneous, the Killing Lie algebra of φ is transitive on X. The extendibility result of local Killing fields on simply connected manifolds [Am, No, DG, Gr] implies that the connected component of identity of Aut(X, φ) acts transitively on X. It now follows that X is a compact complex homogeneous manifold. Since K X is trivial, this implies that X is a parallelizable manifold biholomorphic to a quotient of a connected complex Lie group by a co-compact lattice in it [Wa]. In particular, the fundamental group of X is infinite: a contradiction. Proposition 3.3. Let X be a compact complex manifold in the Fujiki class C, of complex dimension n and of algebraic dimension n -d, with d > 0. Assume that the canonical bundle K X of X is trivial and that X admits a holomorphic rigid geometric structure φ. Then the fundamental group of X is infinite.

Proof. We assume that X have complex dimension n and algebraic dimension n -d, with d > 0. Assume, by contradiction, that X have finite fundamental group. Replacing X by its universal cover and pulling-back φ, we can assume that X is compact simply connected. We have seen in the proof of Theorem 2.4 that the maximal connected abelian complex Lie subgroup L of the group of automorphisms of (X, φ) acts transitively on the generic fibers of the algebraic reduction (in particular L has positive dimension) and coincides with the automorphism group of the new holomorphic rigid geometric structure φ constructed by juxtaposing φ with a basis {X 1 , • • • , X l } of the Lie algebra of Aut(X, φ) (this construction was used earlier in the proof of Theorem 2.4). Moreover, since L also preserves the smooth finite volume defined by a nontrivial holomorphic section of K X (Lemma 2.5), Section 3.7 in [Gr] shows that the orbits of L are all compact (see also Section 3.5.4 in [DG]). The orbits must be compact complex tori covered by L .

The manifold X being in class C, the main theorem in [GW] asserts that the action of L factors through a holomorphic action of a compact complex torus T . In particular, generic orbits of the T -action have trivial stabilizer, while all orbits have discrete stabilizers [GW] (Lemma 2.1), and X is a holomorphic principal T -Seifert bundle. Now notice that X being simply connected, its Albanese map is trivial [Ue], and the proof of Proposition 6.9 in [Fu] asserts that any solvable Lie subalgebra of the Lie algebra of holomorphic vector fields on X have a common zero on any invariant closed analytic set (see also Proposition 1.3 in [GW]). In particular, (holomorphic) fundamental vector fields of the T -action should vanish identically on any compact complex torus embedded in X. This means that they should vanish identically on all T -orbits: a contradiction.

We now deduce Corollary 1.4 from Proposition 3.3.

Proof of Corollary 1.4. Let us consider a holomorphic affine connection in T X. We apply a method of [At] (see also [IKO]) to show that all Chern classes of T X of positive degree must vanish. By Chern-Weil theory we compute a representative of the Chern class c k (X, R) using a Hermitian metric on T X and the associated Levi-Civita connection. We get a representative of c k (X, R) which is a closed form on X of type (k, k). We perform the same computations using the holomorphic affine connection in T X and get another representative of c k (X, R) which is a holomorphic form on X. But on manifolds of type C, forms of different types (here (k, k) and (2k, 0)) are never cohomologous, unless they represent zero in cohomology. This implies the vanishing of c k (X, R), for all k > 0.

In particular, c 1 (X, R) = 0 and Theorem 1.5 in [To] implies that there exists a finite integer l such that K l X is holomorphically trivial. Assume, by contradiction, that the fundamental group of X is finite. We replace X by its universal cover which is a compact complex manifold in class C with trivial canonical bundle bearing a holomorphic affine connection. Proposition 3.3 implies that X is a Moishezon manifold. By Corollary 2 in [START_REF] Biswas | Holomorphic Cartan geometries and rational curves[END_REF], a Moishezon manifold X admitting a holomorphic Cartan geometry (in particular a holomorphic affine connection) must be a smooth complex projective manifolds. But a compact complex projective manifold bearing a holomorphic affine connection (and hence having trivial real Chern classes of positive degree) is covered by a compact complex torus [IKO]: a contradiction. Proposition 4.1. Let X be a compact simply connected Kähler manifold, of complex dimension n and of algebraic dimension n -1. Then X does not admit any holomorphic rigid geometric structure.

Proof. Assume, by contradiction, that X as in the proposition bears a holomorphic rigid geometric structure φ. Then Theorem 2.4 implies that there exists a holomorphic vector field K on X preserving the fibers of the algebraic reduction π red of X. The algebraic reduction π red of any compact Kähler manifold of algebraic dimension n -1 is known to be an almost holomorphic fibration. Then Theorem 2.4 implies that all K-orbits in X are compact.

By a Theorem of Holman, [Ho], the K-action factors through the action of a compact complex torus T of dimension one, and X is a holomorphic Seifert T -principal bundle. It can be shown that the T -action must have trivial stabilizers on all of X. Indeed, if an element g ∈ T fixes x ∈ X, then its differential at x preserves K and also acts trivially on the quotient T X/R • K (since the action fixes all fibers of the algebraic reduction). On the other hand, since T is compact, the action of g must be linearizable in the neighborhood of x. This implies that the action of g is trivial in the neighborhood of x and hence on X. It follows that g is the identity element in T , and hence the T -action is free. This implies that X is a T -principal bundle over a simply connected projective manifold (the basis V of the algebraic reduction). By a result of Blanchard [Bl] (see also Theorem 1.6 in [Hof]) those manifolds are Kähler if and only if the principal bundle is trivial. In particular, they are not simply connected if they are not Kähler: a contradiction. Notice that the above proof of Proposition 4.1 works for Kähler manifolds X for which the algebraic reduction π red is a nontrivial almost holomorphic map. Another point regarding the previous proof is that it adapts to holomorphic Cartan geometries of algebraic type. Indeed, one needs to apply Theorem 1.2 in [START_REF] Dumitrescu | Killing fields of holomorphic Cartan geometries[END_REF] in order to show that there exists a nontrivial holomorphic vector field K preserving the Cartan geometry as well as the fibers of the algebraic reduction. Therefore, the proof of Proposition 4.1 remains valid for holomorphic Cartan geometries of algebraic type.

Theorem 4.2. Let X be a compact complex manifold in the Fujiki class C, of complex dimension n and of algebraic dimension n -d, with d > 0. If X admits a holomorphic rigid geometric structure φ, then the fundamental group of X is infinite.

Proof. Assume, by contradiction, that X has finite fundamental group. We replace X by its universal cover and the rigid geometric structure by the pull-back of φ on the universal cover. In this way we may assume that X is simply connected. By Theorem 2.4, the action of the connected complex abelian Lie group L preserves the algebraic reduction π red : X -→ V . Since X is in class C, this L is a subgroup of the complex linear algebraic group Aut 0 (X) (Corollary 5.8 in [Fu]).

It follows that in Theorem 2.3 the model X may be chosen, using Hironaka's equivariant resolution with respect to Aut 0 (X) (see Lemma 2.4 point (4), Remark 2.4 point (2) and Lemma 2.5 in [Fu]), such that Aut 0 ( X) acts transitively, at the generic point in X, on the fibers of the map t : X -→ V .

Equivalently, , for any generic point x 0 ∈ X, there exist holomorphic vector fields {X

1 , • • • , X d } on X such that {X 1 (x 0 ), • • • , X d (x 0 )}
span the tangent space at x 0 to the corresponding fiber t -1 (t(x 0 )). Since the fiber t -1 (t(x 0 )) is compact and connected, by Blanchard Theorem, {X 1 , • • • , X d } span the tangent space to t -1 (t(x 0 )) on an open dense set in t -1 (t(x 0 )). It now follows that the Lie algebra of the stabilizer of t -1 (t(x 0 )) in Aut 0 ( X) contains {X 1 , • • • , X d }. Therefore, this stabilizer of t -1 (t(x 0 )) acts with an open dense orbit in t -1 (t(x 0 )). Consequently, t -1 (t(x 0 )) is an almost homogeneous space (the group is the stabilizer in Aut 0 ( X)).

The closure of any Aut 0 ( X)-orbit is a compact analytic subset in X ( [Fu], Lemma 2.4 point (2)). Recall that X being simply connected and in Fujiki class C, this Aut 0 ( X) is a complex linear algebraic group acting meromorphically on X [Fu] (Lemma 1 and Corollary 5.8). This means -as in the proof of Theorem 2.3 (ii) -that the action of Aut 0 ( X) on X extends to a meromorphic map defined on a rational manifold. In particular, the closure of any Aut 0 ( X)-orbit is the meromorphic image of a rational manifold and hence it is Moishezon.

The smooth fiber t -1 (t(x 0 )) of t is a closed analytic subset in the closure of the Aut 0 ( X)orbit of x 0 . Consequently, t -1 (t(x 0 )) is also Moishezon. It can be shown that the smooth fiber t -1 (t(x 0 )) does not admit any nontrivial holomorphic one-form. Indeed, any holomorphic one-form β on it would furnish a constant function when specialized on any of the vector fields {X 1 , • • • , X d }. Since all those vector fields vanish at some point in t -1 (t(x 0 )) (Proposition 1.3 in [GW] and the proof of Proposition 6.9 in [Fu]), it follows that β = 0. Since t -1 (t(x 0 )) does not admit any nontrivial holomorphic one-form, the first Betti number of the smooth fiber of t is trivial. Then the results of Campana [START_REF] Campana | Reduction d'Albanese d'un morphisme propre et faiblement Kählerien. II[END_REF]Corollary 2] and Fujiki (Proposition 2.5 in [START_REF] Fujiki | On the structure of compact manifolds in C[END_REF], see also [START_REF] Fujiki | Relative algebraic reduction and relative albanese map for fiber space in C[END_REF]) imply that X is Moishezon, and hence X is Moishezon: a contradiction. Theorem 4.3. Let X be a compact complex simply connected manifold in the Fujiki class C. If X bears a holomorphic Cartan geometry of algebraic type, then X is a complex projective variety.

Proof. By Corollary 2 in [START_REF] Biswas | Holomorphic Cartan geometries and rational curves[END_REF], a Moishezon manifold X admitting a holomorphic Cartan geometry must be a smooth complex projective manifolds.

We deal now with the case where the algebraic dimension of X is strictly less than the complex dimension of X. In view of Theorem 1.2 in [START_REF] Dumitrescu | Killing fields of holomorphic Cartan geometries[END_REF], the method used in the proof of Theorem 4.2 works for holomorphic Cartan geometries of algebraic type, showing that compact simply connected manifolds in class C which are not of maximal algebraic dimension (meaning not Moishezon) do not admit any holomorphic Cartan geometry of algebraic type.

Let us deduce Corollary 1.5 from Theorem 4.3.

Proof of Corollary 1.5. Assume, by contradiction, that the fundamental group of X is finite. Replacing X by its universal cover endowed with the pull-back of the Cartan geometry, enables one to assume that X is simply connected. Theorem 4.3 implies than that X is a projective manifold. With our assumption on the first Chern class, X is a projective Calabi-Yau manifold. But a projective Calabi-Yau manifold bearing a holomorphic Cartan geometry of algebraic type is covered by a compact complex torus [START_REF] Biswas | Holomorphic Cartan geometries and Calabi-Yau manifolds[END_REF][START_REF] Dumitrescu | Killing fields of holomorphic Cartan geometries[END_REF]: a contradiction.

4.

  Fujiki class C and geometric structures This section deals with holomorphic geometric structures on Fujiki class C manifolds with finite fundamental group. The main results are Theorem 4.2, Theorem 4.3 and Corollary 1.5. Let us start by proving the following weak version of Theorem 4.2:

Acknowledgements

We are very grateful to the referee for pointing out an error in a previous version; that part is now removed. We are grateful to Frédéric Campana for very helpful comments. We also thank Henri Guenancia for very valuable discussions. The first-named author is supported by a J. C. Bose Fellowship. The second-named author wishes to thank T.I.F.R. Mumbai for its hospitality.