 So, first I would like to thank the organizer, Bertrand and Pepe. It's a pleasure to be here, to be back here, so I enjoy very much this meeting. This is a talk about holomorphic geometric structures on compact complex manifolds. And first I will state a result, which was an example for what I wanted to do. So it's a result of G's, about classification of holomorphic co-dimension one-foliations on complex tori. And I want to think about this result as having a generic situation, which is a first part of the statement, and an exceptional part. First of all, for complex tori with algebraic dimension zero, meaning they only admit constant function as meromorphic function, they do not admit other meromorphic function. The only foliations are given by the kernel of a closed holomorphic one form, since the holomorphic tangent bundle is trivial, those holomorphic forms are translation invariant. And this gives you all foliation of generic tori, so generic tori have algebraic dimension zero. And these are the only foliations on those tori, so here you have these complex torus. If you assume that the algebraic dimension is zero, then you can see easily that a holomorphic foliation of co-dimension one will be given by a map into the projecting space, the space of hyper planes in CN, which are tangent to the foliation. And since you assume that there are no non-constant meromorphic function, each such holomorphic function must be constant, so those foliation are translation invariant. Now there is an exceptional situation for those tori which admit vibrations over elliptic curves. If it is the case, you can pick up a non-constant meromorphic function on the elliptic curve and pull back this meromorphic closed one form, a UDZ, from the elliptic curve, and add any constant holomorphic form on the torus. Gives you a closed meromorphic one form, the kernel will be a holomorphic foliation, in fact it will be a foliation with maybe poles because this form has poles. The point is that if you assume that omega, small omega, vanishes on the fibers of the vibration, what you get is the vibration. So it's holomorphic everywhere. And the point is that in general, a local computation will show you that here you have a non-singular holomorphic foliation. So those are holomorphic foliations. And they are not translation invariant, but still they have a big group of symmetries because all translations in the kernel of pi give you translations which preserve the foliation. So in fact, all these foliation, either they are translation invariant in the generic case or they have a sub torus of co-dimension one which acts preserving the foliation. So you have lots of symmetries in any case. So the idea of this talk is to try to generalize this CRM for a broader class of geometric structure and for a broader class of complex compact manifolds. And let's see what kind of geometric structure I'm thinking of. The first one which are good to have in mind for the talk are holomorphic Riemannian metrics. They are complexified version of pseudo-Riemannian metrics. So they are holomorphic sections of the bundle of complex quadratic forms in the holomorphic tangent space. And you assume that they are of maximal rank. They are non-degenerate in each point. The flat case is the flat of the, say, complex Minkowski space of complex Euclidean space. So here you can think of the flat case as being a GX geometry where X is CN and G is the group of complex Euclidean motions. So a semi-direct product of the orthogonal group, complex orthogonal group, and CN acting on itself by translations. So this is like a flat Minkowski geometry. But of course in general there are local differential invariance as in the pseudo-Riemannian case. You will have a holomorphic Levy-Civita connection and a holomorphic tensor of curvature which shows you how far you are from the flat case. Another example of geometric structure is given by a fine connections. So a fine connections in the holomorphic tangent bundle so you can take derivatives of holomorphic vector fields with respect to other holomorphic vector fields. And this operator is completely determined by this Christopher coefficient. So in local coordinates you can write the derivatives of the coordinate vector fields and you have some holomorphic local functions gamma ajk which are Christopher coefficients of the connection. And for example when all of them are constant on CN that means that this connection is very particular it is translation invariant and it descends on complex tori on CN over any lattice so they admit this complex tori they admit holomorphic a fine connections. The point is that in general compact complex manifold will not admit this kind of holomorphic geometric structure and all the situation will be very particular very symmetric and one could try to classify all of them to classify say all compact complex manifold admitting holomorphic a fine connections or holomorphic Riemannian metrics. I will give you some however some interesting examples. So the first one was given by complex tori another interesting example are those manifold for which the holomorphic tangent bundle is parallelizable meaning you have n linearly independent holomorphic vector field in any moving frame which is holomorphic. And those manifolds are known to be exactly quotients of a complex group by a lattice so here P is a complex group and gamma is a lattice in P so those are palatable manifolds and here you have holomorphic flat holomorphic connections given by right invariant vector fields on P which descends on P over gamma and you decide that this globally defined holomorphic vector fields are parallel for your connection. Of course these connections are not torsion free because P is not a billion in general but these are flat holomorphic connection on these quotients and also those manifolds admit holomorphic Riemannian metrics coming from right invariant complex quadratic forms on P. There are other nice examples of geometric manifolds given by the deformations of the complex structure on SL2 of C over gamma if you take this palatable manifold SL2 of C over gamma G's show that this manifold this complex manifold is not rigid in fact you can see easily that this is a Clifford Klein form of some very geometric homogeneous space. You can think as X, X as being SL2 of C and at G as being SL2 times SL2 meaning here you have a holomorphic geometry which is like the complexification of ADS3 so the G action on X preserve the holomorphic Riemannian metric coming from the killing quadratic form on SL2 of C which is like the complexified version of the anti-desitter space in dimension 3. So one can try to take the formation of SL2 of C over gamma by considering this GX structure and deforming the holonomy morphism from gamma into SL2 times SL2 and this gives you other holomorphic GX structures on the real manifold SL2 of C over gamma and the point is that G's proved that the underlying complex structure of these GX geometries are different so as soon as the GX geometries are not isomorphic the complex structures are different in fact the deformation space of the complex structure is exactly given by the deformation space of this GX structure meaning that you have interesting complex structures near this one SL2 of C over gamma which are exotic because the generic ones do not admit any non-trivial holomorphic vector field so they are not palisable manifold they do not admit any non-trivial holomorphic vector field but they still admit a holomorphic Riemannian metric which is locally isomorphic to this complexified anti-desitter space in dimension 3 so these are interesting geometric manifolds but however all the examples we have are very symmetric so one could try to say something about the symmetries of holomorphic geometric structure on compact complex manifold so I will give you a first example in this direction which is a theorem saying that on any complex manifold you don't need compactness of algebraic dimension zero meaning the only meromorphic functions on those manifolds are the constant ones all these geometric structure which are called rigid in Gromov sense should be locally modeled on some GX structure at least on an open dense set away from a analytic subset in the manifold so let's say that this theorem comes is inspired by a previous result of Bogomolov which proved the same theorem for tensors and also of course by the work of Gromov about rigid geometric structure so the examples I gave you before holomorphic Riemannian metrics and holomorphic connections and palizations are all rigid rigid means that local automorphism for the geometric structure is completely determined by a finite jet and for connection this comes because of the fact that a local automorphism will send complex complex geodesics on complex geodesics preserving parametrization so local automorphism for connections are locally determined by the one jet by the differential so this works for any rigid geometric structure the idea would be that there exists an open dense set you such that you is locally modeled on some complex homogeneous space for G complex the group and age a closed subgroup so maybe away from an analytic subset you have a GX structure on your manifold and the geometric structure is locally isomorphic to some G invariant geometric structure on see on this homogeneous space so the idea would be to extend this open dense set to all of the manifold to get to get this GX structure on all of the manifold and try to classify compact complex manifold locally modeled on homogeneous basis so this is what I will show you here it works if we put some extra conditions on the manifold or on the geometric structure so let's see how this is a direct generalization of at least of the generic part in G's theorem so let's see why holomorphic foliation on complex tori with algebraic dimension zero must be translation invariant in fact this gives more that gives the fact that any holomorphic geometric object on a complex torus of algebraic dimension zero is translation invariant so foliations are not rigid there are too many local automorphism preserving a foliation local automorphism are infinite dimension but the point is that if you take a foliation on a complex torus of algebraic dimension zero on the complex torus you already have a rigid geometric structure which is the translation structure and this theory of Gromov enables one to put together the foliation or any geometric structure with this parallelization so you put together in some extra rigid geometric structure all these translation fields on the torus and the foliation give you a geometric structure and by the first year and since you are of algebraic dimension zero this must this must be locally homogeneous on a dense open set in the torus locally homogeneous means that you have lots of local symmetries because all vector fields in the algebra of G will be well defined on the open set you as local vector fields preserving the geometric structure so you have lots of vector fields locally defined preserving the geometric structure and they are transitive on an open then set so let's look to a vector field X which is a local symmetry for G in particular that means that the flow of X preserves the foliation and the flow of X commute with all these translations but commuting with translation on the torus means you are a translation so X is in fact a linear combination of translations so it's a translation so in fact local symmetries are globally defined they are translations and they must act with an open then so be so you must have all of them in the symmetry group so in fact here local symmetries are all translations in particular the foliation is translation invariant and it works in any dimension and for any holomorphic geometric structure on the torus in particular any holomorphic geometric structure on the torus come through a G X structure through something which is locally homogeneous so in particular this G X structure is translation invariant so any holomorphic G X structure on the torus is translation invariant as soon as the torus is of algebraic dimension zero translation invariant means that your geometric structure your G X structure is very particular if you look to the holonomy morphis going from the fundamental group of the torus into G this will extend to CN as a complexity group homomorphism such that the image of this group CN in G acts with an open orbit meaning that in fact the universal cover of the torus covers an open set in the model and its G X structures come from this covering and is translation invariant and you take the G X structure on the caution by taking the caution by lambda so this translation invariant geometric structure G X are very particular and somehow you can decide now what G X structure can live on a torus with algebraic dimension zero it gives you an algebraic criteria to decide what are those G X structure it's algebraic you need to find in G a subgroup is homomorphic to CN which acts with an open orbit on X meaning you need to find a copy of CN in G which is transverse to the isotropy and then you do this construction and they are all G X structure on complex tori of algebraic dimension zero and the point is that we asked with Benjamin McKay this question is it true that all holomorphic G X structure on all complex tori are translation invariant so those which are of algebraic dimension zero are translation invariant but maybe not all of them and we do not know how to answer in general to the question we have some partial answers answers to this question yes for dimension two it's true for any complex torus yes if G is nil potent in general those which are translation invariant so those which are translation invariant are a closed and open set in the deformation space of just of G X structures but we don't know how to do this in general and no what I'm saying is maybe like this sorry you're the other way in fact you have the the sorry the holonomimorphism which is defined on the sorry yeah and in fact this picture if you look exactly what we did for for the complex torus exactly the same theorem is true the same theorem is true for palisable manifolds with algebraic dimension zero for example for for g for p sl2 of c or sln of c meaning all holomorphic geometric structure on palisable manifolds with algebraic dimension zero pulls back on p as right invariant geometric structures and all G X structures come from this kind of construction they are translation invariant so if you replace the fundamental group here by gamma you want to send it in G as a holonomy of the G X structure it will extend to a group morphism to P such that the image of P acts with an open orbit in X so gives you in general a condition for a palisable manifold P over gamma to admit G X structure as a corollary of this sln of c over gamma does not admit complex affine structures you cannot find such a group morphism from sln of c into the affine group of dimension n which acts with an open orbit with c on cn this come from the representation theory so gives you somehow nice result about the fact that those non-keller manifolds sln of c over gamma they do admit holomorphic affine connections they do admit flat holomorphic affine connections but they do they do not admit complex affine structures that do not admit flat torsion free holomorphic affine connection so this is for palisable manifolds you can find other kind of results of the same type if you assume something about the topology of the manifold let's assume here that the manifold is simply connected the point is that those manifolds which are compact simply connected of algebraic dimension zero they will not admit holomorphic affine connections the assumption about the topology is important because those sln of c over gamma they are of algebraic dimension zero they have could they have connections but of course they are not simply connected so the proof go like this first there is a very general argument about symmetries of those connections maybe it was wrong sorry okay so this was the first argument as soon as you have any rigid geometric structure on those manifolds you have lots of symmetries which are globally defined in fact toroidal means that you have an action so toroidal toroidal means action of c star to the n with a dense open orbit good example to have in mind as a toroidal manifold admitting affine connect affine structures but not simply connected of course are the following think about hopf manifold let's do it in dimension two so a surface take the quotient of z1 z2 by a non by linear construction with different eigenvalues give you a surface which is diffeomorphic to s1 times s3 there is a complex affine structure on the quotient the quotient is a smooth manifold as soon as you assume there is a contraction here you have a complex affine structure on the quotient and you also have an action of this toroidal group given by two by two diagonal matrices because this action commute with the group for which by which you take the quotient so this action descend on the quotient in an action which is uh which has an open dense orbit away from the projection of the axis on those hopf surfaces you have two elliptic curves coming from the axis and this action will be transitive away from these two elliptic curves and these two elliptic curves will be the only curves in the surface so the so the algebraic dimension is zero but of course this is not simply connected if you want to have a simply connected example you need to take an adaptation of this take what we call kalabi ekman manifolds it goes like this take c2 minus zero times c2 minus zero and take the quotient by one parameter group which acts properly and freely and the action is holomorphic so you will get a quotient which is a complex compact manifold of dimension three in fact one can see that m is defo morphic to s3 times s3 if you put s3 as the unitary sphere in c2 so this is a complex structure on s3 times s3 if you take this one you can see that you have a vibration over p1 times p1 so it's not of algebraic dimension zero but you can take a deformation of the c action of this action as a one parameter subgroup in the linear space of c4 such that a is generic and a is semi-simple is a if a is generic this gives you the fact that the manifold is of algebraic dimension zero and if a is semi-simple gives you the fact that in gl4 of c you have a big stabilizer a big centralizer a centralizer of dimension four so on the quotient you will get you will get a three-dimensional a billionly group acting with a dense orbit so those are toroidal manifold of algebraic dimension zero which are simply connected in fact they are very easy examples of a broader construction given by Lopez de Medrano, Alberto Verzhovsky and Laurent Merzmann and recently there is a paper proving that generically these manifolds are of algebraic dimension zero published by Panof Verbitzky and Ustinovsky so these manifolds of Lopez de Medrano Verzhovsky and Merzmann are very nice construction of non-keller toroidal manifolds of algebraic dimension zero many of them admit complex affine structures some of them are simply connected but with no complex affine structure with no holomorphic affine connection so let's see how one can prove this proposition one the the starting point of the how one can get this toroidal action starting with the geometric structure the the first part of the proof will be to construct those globally defined vector fields on the manifold and this comes from a theorem of Nomizo which was first a theorem about killing fields for real analytic Riemannian matrix but this theorem was generalized by Gromov and Amores for any rigid geometric structure and the point is the following if you assume analyticity and you look to for local symmetries which exist here because we are in algebraic dimension zero and the previous theorem says that you have lots of local symmetries we are we want to show that they are global and Nomizo theorem says that you can extend those vector fields along any pass in the manifold such that at the end of the path the germ of the killing field only depend on the homotopy class of your path in particular if you are simply connected this local symmetry will extend to all of the manifold if moreover you are compact this globally defined vector field is complete so you have a one parameter group acting on the manifold and preserving the geometric structure so of course here we use this Nomizo result because our manifold was assumed to be simply connected so we have globally defined vector field I will denote by a all those globally defined vector field of the manifold such that their flow preserve our say geometric structure G so G was our rigid holomorphic geometric structure but you can think about the connection sorry yeah no no this this result is about extension of local symmetries so the point is of course in the real situation you cannot have this you take a flat metric here you have translation take a perturbation translation which are here will not extend so this this results say that it's about extension of symmetries so maybe what happens here is if you start with lots of vector fields in the open then set they will lack transitively then you will extend them and you come on this analytic on this analytic subset where they are where they do not act transitively they become in somehow culinary here but the point is they extend so I don't I don't say the action is transitive I think I say all these local vector fields extend to all of the manifold okay so this it's exactly like like happen in hope look here for for this group sorry this group the vector fields are z1 dd z1 and z2 dd z2 so they extend and they become collinear on the two elliptic curves which are the axis where z1 and z2 vanish okay so you have those globally defined vector fields whose flow preserve the geometric structure now like before it's a trick you put them together with the initial geometric structure in some extra geometric structure so you take another geometric structure in which you have your initial one and those vector fields gives you another more stringent geometric structure and you do the same for this new rigid geometric structure you look to a prime which are those vector fields whose flow preserve g prime meaning the flow preserve both g the initial geometric structure and commutes with all symmetries of g so by definition being in a prime means preserving both g and vector fields in a meaning that a prime is in the centralizer of a in particular a prime is a billion and since your manifold is still of algebraic dimension zero a prime must be transitive on an open dense set and since the manifold is compact a prime is in fact the Lie algebra of a group g prime which acts with an open dense orbit so there is an a billion group on the manifold preserving the geometric structure and acting with a dense open orbit and a little bit of work shows that it is toroidal it descends on c star to the end and now the second step of the proof it's local it's only differential geometry about symmetries of a connection so if you look to a billion Lie algebra acting by symmetry with a dense orbit and preserving a connection they will also preserve a flat torsion free a fine connection so this is local statement about symmetries of a connection and I will not prove here proposition two I will say something later on the point is that proposition two implies the theorem because for simply connecting manifolds the developing the developing map of a gx structure is directly defined on the manifold because the manifold is simply connected so if you have an affine structure the developing map of the affine structure will be defined directly on your compact manifold and there is a contradiction here in fact gives you the idea of why the theorem should be true the developing map defined on m into c n which is the space of the gx structure should be a local differomorphism for but on the other hand it's holomorphic so it must be constant so this is a contradiction at the end of the proof that the second step of of the theorem was proved in dimension two before by Adolfo and me in an attempt to understand those affine connections which are locally homogeneous on an open dense set but not on all of the manifold we try to understand if it could happen starting with this result about algebraic dimension zero can one have something which is quasi homogeneous meaning it's locally homogeneous on an open dense set but not on all of the manifold and this is an example which was found by Adolfo example is like this you assume it's a polynomial connection in c2 every crystal coefficients vanish except this one and you can easily find symmetries you will find k1 which are translations along y-axis along x-axis and k2 which is an affine vector field both of them preserve the connection and in fact not only both of them preserve the connection but they they span all the symmetry algebra of this connection the symmetry algebra is this one of dimension two which is non-commutative is the algebra of the affine group of the real line another way to see that this connection is locally homogeneous away from y equals zero is to compute curvature and to see that this tensor vanishes exactly on the geodesic y equals zero you can notice also that k1 and k2 are affine vector fields so they preserve the flat affine connection it's a subalgebra of a bigger one which preserve the flat affine connection which is exactly what happens in the previous example and after that with Adolfo we gave complete classification of germs of connections which are quasi homogeneous like this we have formulas but I will give here a qualitative statement statements I will say less so here is the classification of germs of connections in all situations so I don't give formulas here but in all situations the symmetry algebra is either the affine algebra of the real line so dimension two non-commutatively algebra dimension two or sl2 of c in the first situation the exceptional set where the clinic where the symmetry vector fields drop rank they collapse it will be on a complex geodesic going through zero in the second situation the exceptional point where they are not transitive is the origin and in all situation in adept coordinates those vector fields are affine so they preserve a flat torsion free affine connection the stardom connection so in some sense in this result there is previous step two in the theorem with mk four dimension two it's proved here and also since you have a this flat affine connection which is preserved you have this general statement about which is about differential geometry of many faults admitting this kind of kaza kaza homogeneous holomorphic affine connection they also admit a flat torsion free complex affine structure which is preserved by the same symmetry the important point here that is that here in the complex setting none none of those germs extend on a compact surface they do not extend and to check this one can use so this is a result about classification of local geometry of holomorphic affine connection on complex surfaces the point is that we know all compact complex surfaces admitting holomorphic affine connections there are hop surfaces inways surfaces torii codera surfaces vibrations elliptic vibrations there are classes i want to see the theorem as the equivalent of gcrm in somehow there is a generic case and an exceptional case the generic case said that the connection is locally homogeneous it's like the generic case in gcrm the exceptional case is the case of a vibration elliptic vibration of a remand surfaces of genius g greater than two and here on this vibration which is the exceptional case you have flat connection flat torsion free affine connection but as soon as the connection is not flat torsion free there is only one local symmetry which in fact is global extend to to to a globally defined vector field which is given by the fundamental vector field of the elliptic vibration it's more or less like for me like in formally it's like in gis classification they are the less symmetric connections on compact surfaces and still they have some symmetry so there's a natural question here in higher dimension what would be the less amount of symmetry for a holomorphic affine connection in dimension n on a compact manifold here we see that still we have some symmetry and one interesting corollary of this is that all these connections are projectively flat if you forget about parametrization if you look to the geodesics but you forget about the parametrization locally you can rectify geodesics online on p2 of c so and in fact it's more general not only those affine connections are projectively flat but if you take more generally holomorphic projective connections on surfaces meaning that you are able to cover a surface with open sets on each open set you have a holomorphic affine connection and on overlaps those two connections have the same geodesics if you forget about parametrization this gives you a new notion of geometric structure those holomorphic projective connections and all of them must be flat if they live on compact surfaces they are locally modeled on this projective geometry on dimension so once again strong symmetry condition constraint on a holomorphic projective connection to live on a compact manifold and the last example I will give is the example of holomorphic remanian matrix in dimension three and this in a recent work with Karin Melnik from Maryland we show that this kind of quasi homogeneous germ of holomorphic remanian metric which exists for connections in dimension two do not exist for holomorphic remanian metric in dimension three in dimension two this is clear because you have sectional curvature which will be constant so you'll be locally homogeneous everywhere in dimension three you need some work to show that this kind of degeneracy when you have local symmetries you extend them they will not drop rank they will stay linearly independent but even in this proof the important step is nomizo theorem that's why we do not know what happens for smooth germs of Lorentz matrix in dimension three we do not know that if there exist quasi homogeneous germs of Lorentz matrix in dimension three so our proof uses real analyticity so with this one can see that the theorem about the algebraic dimension zero will say that any holomorphic remanian metric on a compact complex manifold of dimension three with algebraic dimension zero is locally homogeneous on an open dense set but the result with Karin Melnik say that the open dense set is all of the manifold the more interesting point is that you can also prove that for algebraic dimension one two or three the metric is locally homogeneous so at the end we don't need the assumption of about the algebraic dimension of the complex structure as soon as a holomorphic remanian metric lives on a compact complex manifold it needs to come from some gx structure exactly as in these gis examples of deformations so of course now one can try to classify all examples and to understand all these homogeneous spaces g over h which are local models for the holomorphic remanian metric in a way like in the certain list of all locally homogeneous remanian metrics in real dimension three here we have locally homogeneous holomorphic remanian metrics in dimension complex dimension three and we try to understand these homogeneous spaces which are certain lists for the complex case and in fact there are four models here the first two models are in the first part of the theorem part i those are of constant sectional curvature the the complex Minkowski space and the complex anti-disseter space those geometries have compact manifolds locally modeled on them the flat geometry can live on a torus and the sl2 times sl2 geometry can live on sl2 of c over gamma or on the deformation space so these two geometries are nice constant sectional curvature geometries which lives on compact manifold there are compact manifolds locally modeled on them and there are still two other geometries given in part two of the theorem for which the symmetry group is solvable and of dimension four and they come from holomorphic remanian metric which are left invariant on the complex heisenberg group of dimension three and on the complex sol group of dimension three the point is that here in the second part of the theorem we prove some biber back rigidity all compact manifolds locally modeled on the heisenberg geometry or on the complex sol geometry they are rigid in their realization the olonomy group up to a finite cover will stay in a three-dimensional subgroup acting properly on the model so it will stay in the heisenberg group or in the sol group and in fact our theorem says that in the second case the manifold up to a finite cover is very rigid is a quotient is a palatable manifold is a quotient of the heisenberg group or of the sol group since heisenberg sol admit flat invariant holomorphic remanian metric in any case we are left with some constant sectional curvature holomorphic remanian metric on the manifold which comes naturally in the first situation because comes from the gx structure and comes from this biber back kind of rigidity in the second case look at the end we are left to understand completeness for those holomorphic remanian metric or constant sectional curvature the point is that the result of career at kringer which gives you completeness for lorenzian metrics in this setting they are not known and open questions in the setting of holomorphic remanian metrics but we have a recent result of nicolato lozon saying something interesting about those geometries it's a larger it more more more general result by i state here like this if you assume that those gx structure in the flat case and also in sl2 times sl2 case if you assume that the geometric structure is uniformizable or clinian meaning that you take an open set in the model and a quotient of the discrete subgroup in g which is compact then nicola is able to show he proved it and publish it that u is all of the manifold meaning that you don't have other uniformizable example than the complete ones and this is interesting also because gives you the fact that all those gx structure which are complete are closed in the deformation space because if you take a limit of complete of complete gx structure at the limit you will have a uniformizable one and for the second case sl2 times sl2 or l times l with l semisimply group of real rank one there is a result about the fact that those structures which are complete is an open subset in the deformation space this is a result of fanny casel and françois guériteau and olivier guichard and anna vina one of their result is the fact that if you have this kind of gx geometries in the second case those which are complete are a open subset in the deformation space so if you put together nicola's result with this one you have the fact that in the deformation space those which are sl2 times sl2 which are complete is a union of connected components so maybe there are exotic connected components we cannot see but we can you cannot find incomplete ones by deforming a complete one so anyway all of these results go toward the direction that if you have a holomorphic geometric structure like holomorphic magnetic holomorphic connection on a compact complex manifold either it is locally homogeneous or there is another one which is locally homogeneous and locally homogeneous means locally modeled on some homogeneous space and you hope to use complex rigidity to classify all examples so this would be very nice symmetric examples not generic thank you for your attention