Let \(X\) be a smooth algebraic variety (in the affine space \(\mathbb{A}_{\mathbb{K}}^{n}\) over algebraically closed field \(\mathbb{K}\) of characteristic zero). A spray over \(X\) is a triple \((E ,p ,s)\) consisting of a vector bundle \(p :E \rightarrow X\) of rank \(k\) and a morphism \(s :E \rightarrow X\) such that \(s\) and \(p\) are identical on the zero section of \(E .\) The variety \(X\) is called elliptic if it admits a spray \((E ,p ,s)\) which is dominating at each point \(x \in X\) i.e. the restriction of the differential \(ds\) of \(s\) to the tangent space to the fiber \(E_{x} : =p^{ -1}(x)\) at \(0_x\) is a surjection onto \(T_{x}X .\) The authors define also a local counterpart of this notion:
1. \(X\) is locally elliptic if for any \(x \in X\) there is a local spray in a neighbourhood of \(x\) dominating at \(x .\)
2. \(X\) is subelliptic if it admits a family of sprays \((E_{i} ,p_{i} ,s_{i})\) defined over the whole \(X\) which is dominating at each point \(x \in X\) which means \(T_{x}X\) is the sum of images of the tangent space to the fiber \(E_{x}\) at \(0_x\) of the differentials \(ds_{i}\).
The main theorem gives the equivalence of global and local ellipticity: For a smooth algebraic variety \(X\) the following are equivalent:
1. \(X\) is elliptic,
2. \(X\) is subelliptic,
3. \(X\) is locally elliptic.