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I know three examples of interesting vector bundles (not taking into consideration what one gets from them using algebraic operations):

  • Tangent bundle of a smooth manifold

  • Tautological bundle over Grassmannian

  • Invertible sheaf associated with divisor on algebraic variety

Could you give me other interesting examples, better connected with algebraic geometry?

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  • $\begingroup$ Check out book "Vector Bundles and Projective Modules" by Richard Swan. $\endgroup$ – Vlad May 23 '15 at 14:42
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    $\begingroup$ There exists a one-to-one correspondence between locally free sheaves of finite rank and vector bundles on a scheme. Check out Hartshorne's Algebraic Geometry, Exercise II.5.18. $\endgroup$ – rla May 23 '15 at 22:03
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    $\begingroup$ One thing to point out is that on an arbitrary variety it's usually quite hard to produce vector bundles other than the canonical bundle, hence its importance. One thing that might interest you is the "Serre correspondence" between codimension 2 subvarieties that are complete intersections and certain vector bundles of rank 2. I think this is in Griffiths-Harris. In general that book is a goldmine for geometric constructions, although it was and is very hard going for me. $\endgroup$ – Hoot May 23 '15 at 23:30

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