Locally Conformally Kahler Surfaces (Massimiliano Pontecorvo)

always taking so the surface would be a complex surface for example which in use as ramp was explore topics traction on the you can’t abandon a vice-president of course is a durable construction and then so this problem is a of course about funding the attributed metrics on so we take edge to the mission on earth and which you can always – and the axe so as a piano lesson yesterday Jay is an orthogonal complex structure on earth and the imaginary part of H is a to form in fact another generator phone this square is a volume of G 2 are constant and because they’re dimensional so this means that taking which product to beat them oh my god is always danger but because of the reanimation of S is for taking the dot product with Omega is actually an isomer oh my god is not 0 but it is exact so in

this case we say that G is a robbery confirm an econometric and finally because we are interested in is when and this is just closed so it’s not that the manometric is performant to color metric question we are interesting today was posed by man which complex complex surfaces SG amid and luckily of course on a compartment and even better members not mention this efficient is as lovely

simple and using the very useful notion of additional metric that can be proved that a luckily for medical surface with even be one so for surfaces compact effigies of course educate plus divine even implies globally conformal together that the metric is actually lovely complement together 17th nice good for in the case in this case D 1 equal to even vitamin question coincide with the very famous conjecture of Medina excellent dimension compensation two side pieces was proved by su and therefore this reduces Wiseman question to the case V one another reduction comes from that point so I thought we can reduce our question deconcini placement to minimal surfaces with the spectrum this is the interesting case one has to understand hey minimum

Oh any other chef yes sir so it means that it is no passion occurred yes whose sake position is minus one and we also assume that the first beta number is without loss of generality okay so the easiest example in this situation was already here of course is an observers the easiest top surface is a question of c2 – the origin divided by this simple action a nominal action for example ZW s 2 2 z 2w and in this case the fat metal can convert to – the origin and divided by these two factors nobody go formally an American in this sense to this covered surface answer which of course is smoothly just as one person did not need any arithmetic for the topological reason I was already remarked that is other examples this kind of surfaces object numbers were time to understand this question and for example the inter surface is produced by public services which were discovered in Tennessee independently by Eric in the seventies and they are some molecules which have to compact questions I will see cross the Opera plane but this action is a very confident and so this come in the three big families of a complex can be managed to

distract me from any kind of metrics all of them except for one dimensional remedy which is usually who passed last and and depends on our complex parameter as a matrix 99 my progress was done on this custom solution on they observers LCD so our surface is our surface of the universe at color is biomorphic to see to – the origin and the action in this case is very simple so you can construct but this action in the Jess situation can be is a more general gentleman and service are not actually possible to write an emetic explicitly the user the formation argument to show that a value of surfaces look on the corner mm true following important to see of course the other characteristic here is zero also in this case because actually D 2 is zero but more generally improved @sj with of course first patent Amin and assume also the director pistol easy as is Elsie the deserts are not going to formally accept so he actually do that this design was

sharp and he also proved that you see that this notion of Nagamani cannon is not stable under deformations it’s not an open public and complex deformations because I start with a locally colored surface with urea and then many neighbor at this point as as long as the parameter is complex not real cannot be very confident so this is different from the current situation where the caliper is preserved under small deformations so this actually takes an hour of many surfaces all the surfaces in classics pacification for example in all surfaces of plus seven with only characteristic zero so let’s see what left now so you can see that after the good work basement question as an answer which is not always positive except for surfaces with the one organs of the first baby number and other characteristics differences but actually so what’s like if you just apply the data conciliation complex surfaces then this means actually that p1 of s must be one actually one they could either dimensions of as whose van is infinity no positive power of the canonical bundle cannava in office sections and also no Erica Tristan is actually positive which in this case is agreeing to say that so this surface is these are perfect classic surfaces which is for the seminal plants in academic oscillation so 7 stands for very dimension managed infinity this stands for variable simulation extraneous and this plus stands for the fact that the truth cannot be zero so op surfaces are necessary no but yeah we look at Francis amino plus the second ready number is there and there is no classification aware yet of surfaces in this class there are many

open questions for example do they always omit terms there is no consideration there is some work done by Andre Perriman and he answered this question only for me to oversee necessarily what to do in that case it is possible to prove that this occurred but how about examples the only known examples in this class so good and it means a cutter surfaces are surface and they can be characterized as you see this surfaces always have an algebraic dimension equal to zero at the dimension instead these cancel the number of independent automatic functions and so they’re the only mode amorphic fractions are constant this implies that there can be almost silently many curves on the surface so the cutter surface is just in the best possible situation there is a maximal possible number of rational curves so it’s not that tiny many caves in fact almost the two of us Russian actors and now this is the limit situation so we have to look at this class and these are the known examples and we would like to address the question of whether there are many color matrix in the first examples take me from my crystal construction that actually more structure and Association so we found by the ocean and interceptor one so you know surfaces are a special tales

of campus if assistant the Tempe characterized by saying that this the deterioration curves cannot be arranged Vitali each of them as to belong to a cycle of so the configuration okay no it’s only everything it’s residual component of a secondary class so maybe I would like to say first that’s where I should say that the German genetic idea of turbulence while visiting adapting from maracas and so first of all as I said is that remember we had in dimension 4 so in this case the curvature and splits because as support is not so in this case the right density of the manometric splits in two parts the manometric can be seen as a and the mark is our two forms the two forms splits so the metric splits and absent well means that that Romanian matrix so you see of course the notional the Calico formally calorimetric is a conformal value and when this condition is also instead there is not only one photo a complex structure but actually to which we denote by G+ and again I was introduced by information flow it is a generalized carrier condition is a good like to having a permission matrix like this tablet in Germany support integrable and the differential equation is that the son of the leaf form should be zero see when we have an admission metric you can define before will the measure for and here because we have to have two reforms structure when the strategic vision device my question it

says that if valuation metric for dimensional manifold then which is happy so cool it was a condition of G condition then the matrix is automatically SJ and also satisfied this equation right so maybe look at the corollary of this search and using a recent result of our country were carried and our son welcome to the Halloween that there exist generalize cannot metric thank you it’s on every hyperbolic in Winona so we’re not use but for this nice current services these are important to produce examples in the first examples some of the first examples where by you know the packs of surfaces in this I think and we can even take care of the nonlinearities so using this medicine every hyperbolic you know surface wound up at point collection of points – K which is the anti canonical device analysis services so I prepare Aquino is a special custom yeah exactly the class in which you don’t up just one second but you have to cycle and this condition because this generalized color condition okay so we go back to last one question and should say that a lot of progress was made by backup together in a sequence of two papers recently he

showed first of all that acetate is a table is another property and the information Oh other services so you know this is not true in general by the room that can actually feel asleep together surface is a small deformation or a condo surfaces Dilek at the surface and it grew first of all that smooth definition about HCP car to surface is again at scale and then finally include another fitted in that every surface so I think this is a very strong result we like to think that you know our work somehow motivated matter to study this problem that this result actually surfaces of course because as I said the gasification of terminal placer stop complete but made for the moment we know that this is true only if be one of us we can say that although this answer is not always possible because it is this kids are embedded in our services with the work of Marco you know the answer to this emphasis