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I'll put out the following possibly bold, possibly totally stupid conjectures. In any case, the statements below are not intended as mathematically rigorous statements, but may have some truth in them. Cheers!

Conjecture 1 Any naturally defined coherent sheaf on all smooth projective varieties is related to the sheaf of differentials via tensor operations and up to torsion.

A more concrete version is

Conjecture 2 Let $\mathscr L_X$ be a naturally defined line bundle on all smooth projective varieties $X$. Then some tensor power (possibly negative or zero) of $\mathscr L_X$ agrees with some tensor power of the canonical bundle.

Remark Jason pointed out that there are naturally defined torsion sheaves line bundles and Arend's idea of defining sheaves as push-forwards would also produce thatsheaves supported on proper closed subvarieties. This is the main motivation for the "up to torsion" part . If this "conjecture" is found sort of true than one could ask what the first conjecture and for taking powers of the natural torsion sheaves are...Also line bundle and the canonical bundle. Also notice that David Ben-Zvi's construction also produces sheaves that satisfy this these conjectures and the extensions in David Speyer's answer produce sheaves whose determinants are powers of the canonical sheaf.

A more concrete version is

Conjecture 2 The only naturally defined line bundles on all smooth projective varieties are the canonical bundle and the structure sheaf.

To get to Conjecture 1 from Conjecture 2 one could argue the following way. Since the claim is "up to torsion" we can mod out by the torsion and assume that our sheaf is torsion-free. Now since we are on a smooth variety this implies that it is locally free in codimension $2$ and actually, again by the "up to torsion" principle, we may assume that it is reflexive, that is, take the reflexive hull, or in other words, the push-forward of the restriction to the open set where it is a locally free sheaf. In other words, we may perform all tensor operations as if we had locally free sheaves and in particular, the (reflexive hull of the) determinant will be a line bundle. In other words, up to torsion, we obtained a natural line bundle. If that is either the structure sheaf or the canonical bundle, then we're in business.

The reasoning I can offer for Conjecture 2 is the following: If there is a natural line bundle, then we can ask whether it is ample (or its inverse is) and for those varieties that it is we obtain a natural embedding (after taking some power). Once we have this we can look at the corresponding Hilbert schemes and try to construct moduli spaces. For those varieties on which this mysterious line bundle is not ample we can still define a corresponding Kodaira dimension and study Iitaka fibrations and eventually work toward a corresponding classification theory. I don't think any of this has happened except for the version using the canonical sheaf. I believe that suggests that there are no other non-trivial natural line bundles.

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I'll put out the following possibly bold, possibly totally stupid conjecture conjectures. In any case, the statements below are not intended as mathematically rigorous statements, but may have some truth in them. Cheers!

Conjecture 1 Any naturally defined coherent sheaf on all smooth projective varieties is related to the sheaf of differentials via tensor operations and up to torsion.

Remark Jason pointed out that there are no other naturally defined torsion sheaves and Arend's idea of defining sheaves as push-forwards would also produce thatappear . This is the main motivation for the "up to torsion" part. If this "conjecture" is found sort of true than one could ask what the natural torsion sheaves are...Also notice that David Ben-Zvi's construction also produces sheaves that satisfy this and the extensions in David Speyer's answer produce sheaves whose determinants are powers of the canonical sheaf.

A more concrete version is

Conjecture 2 The only naturally defined line bundles on everyall smooth projective varietyvarieties are the canonical bundle and the structure sheaf.

The reasoning I

To get to Conjecture 1 from Conjecture 2 one could argue the following way. Since the claim is "up to torsion" we can offer mod out by the torsion and assume that our sheaf is torsion-free. Now since we are on a smooth variety this : If there implies that it is locally free in codimension $2$ and actually, again by the "up to torsion" principle, we may assume that it is reflexive, that is, take the reflexive hull, or in other words, the push-forward of the restriction to the open set where it is a naturally appearing locally free sheaf. In other words, we may perform all tensor operations as if we had locally free sheaves and in particular, the (reflexive hull of the) determinant will be a line bundle. In other words, up to torsion, we obtained a natural line bundle. If that is significantly different from either the structure sheaf of differentials)or the canonical bundle, then we we're in business.

The reasoning I can construct offer for Conjecture 2 is the following: If there is a natural line bundleand , then we can ask whether it is ample (or its inverse is) and for those varieties that it is we obtain a natural embedding (after taking some power). Once we have this we can look at the corresponding Hilbert schemes and try to construct moduli spaces. For those varieties on which this mysterious line bundle is not ample we can still define a corresponding Kodaira dimension and study Iitaka fibrations and eventually work toward a corresponding classification theory. I don't think any of this has happened except for the version using the canonical sheaf. I believe that suggests that there is are no other non-trivial natural line bundle.

One could complain that this only covers locally free sheaves, but I would further claim that a natural sheaf would have to be locally free at least on a general variety. In fact, if we allow non-smooth ones, as we actually do in canonical-sheaf-based-classification-theory, then even the differentials do not form a locally free sheaf. And of course, if it is coherent, then there will be a dense open set where it is locally free, so on smooth (or mildly singular) varieties we can still try to carry out the usual classification program.

Notice that if one allows extensions as in David's answer, then the obtained line bundlesare not new, they will be powers of the canonical sheaf. I guess that suggests that I should say instead that there are no naturally appearing (locally free) sheaves whose determinant is a power (possibly negative) of the canonical sheaf.

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I'll put out the possibly bold, possibly totally stupid conjecture that there are no other sheaves that appear naturally on every smooth projective variety.

The reasoning I can offer is this: If there is a naturally appearing locally free sheaf (that is significantly different from the sheaf of differentials), then we can construct a natural line bundle and we can ask whether it is ample (or its inverse is) and for those that it is we obtain a natural embedding. Once we have this we can look at the corresponding Hilbert schemes and try to construct moduli spaces. For those varieties on which this mysterious line bundle is not ample we can still define a corresponding Kodaira dimension and study Iitaka fibrations and eventually work toward a corresponding classification theory. I don't think any of this has happened except for the version using the canonical sheaf. I believe that suggests that there is no other natural line bundle.

One could complain that this only covers locally free sheaves, but I would further claim that a natural sheaf would have to be locally free at least on a general variety. In fact, if we allow non-smooth ones, as we actually do in canonical-sheaf-based-classification-theory, then even the differentials do not form a locally free sheaf. And of course, if it is coherent, then there will be a dense open set where it is locally free, so on smooth (or mildly singular) varieties we can still try to carry out the usual classification program.

Notice that if one allows extensions as in David's answer, then the obtained line bundles are not new, they will be powers of the canonical sheaf. I guess that suggests that I should say instead that there are no naturally appearing (locally free) sheaves whose determinant is a power (possibly negative) of the canonical sheaf.

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