A connected compact complex manifold has
holomorphically trivial tangent bundle if and only if there is a connected complex Lie
group and a discrete subgroup such that is biholomorphic to the quotient manifold. In particular is homogeneous. If is K¨ahler, must be commutative and the quotient manifold is a compact complex torus.
The purpose of this note is to generalize this result to the non-compact K¨ahler case. Evidently, for arbitrary non-compact complex manifold such a result can not hold. For instance, every domain over C has trivial tangent bundle, but many domains have no automorphisms. So we consider the “open case” in the sense we consider manifolds which can be compactified by adding a divisor.
Following a suggestion of the referee, instead of only considering K¨ahler manifolds we consider manifolds in class C as introduced earlier. A compact complex manifold is said to be class in C if there is a surjective holomorphic map from a compact K¨ahler manifold onto . Equivalently, is bimeromorphic to a K¨ahler manifold
Ngojea threads za malaya kama hii ndio opening remarks. Nyinyi ni wale mmezoea vitu ziko na picha na sentense moja hapo chini explaining the photo. Hata as a kid you would throw away new books after perusing ukapata hazina picha ndani.
Hio sokwe at one point nilidhani he was guy i could engage in intellectual discourses but later realized ni idler judging by the way yeye hutafuta threads zimerudiwa ndio apake hio umeffi yake kwa threads za watu