Let X be a smooth projective variety of dimension d over an algebraically closed field k. Can you please list some examples of natural locally free or coherent sheaves that you can always construct on X, regardless whether X has some particular structure or is of general type?
I can only name the two obvious examples: the structural sheave and the sheaf of differentials (and obvious combinations of them, of course, which do not count).
I can believe that these are the only ones for curves, as the genus is the only discrete invariant available.
What about surfaces, for instance?
Best Answer
One formal approach to a problem like this is motivated by the "formal geometry" of Gelfand-Fuchs (which is the subject of other MO questions). We can define a "natural bundle" on a class of spaces to be one associated to a representation of a structure group of the given geometry. This is a common approach in differential geometry, where there's a lot known about natural differential operators etc. But in any case in the current context we can define a natural bundle on smooth n-dimensional varieties as one associated to a representation of the group of changes of coordinates on a formal n-dimensional disc. Every smooth variety has a canonical principal bundle for this group (with fiber over $x\in X$ the variety of isomorphisms between the completion of $X$ at $x$ and the formal disc), and so given a representation of the group we get a vector bundle on any variety of the given dimension.
All of the bundles discussed in the answers above are of this form (including jet bundles, sheaves of differential operators etc). If you take this as a definition of a natural bundle it's easy to prove the conjecture that they're all extensions of powers of bundles of forms: the group of changes of coordinates is an extension of $GL_n$ (the structure group of the tangent bundle) by a pro-unipotent group (changes of coordinate with derivative the identity). Hence all representations have filtrations with associated graded bundles associated to the (frame bundle of the) tangent bundle in the usual sense (ie all the Schur functors of the tangent bundle, like forms).
Edit: it's interesting to note that if we take representations of this group which extend to the Lie algebra of all derivations of formal power series in n variables (ie we have an action of $\partial_x$'s on the module as well) we get "all" natural flat bundles on smooth varieties --- such as the sheaves of all jets or all differential operators.