Reflexive sheaf

In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a reflexive sheaf is a locally free sheaf and, in practice, a reflexive sheaf is thought of as a kind of a vector bundle modulo some singularity. The notion is important both in scheme theory and complex algebraic geometry.

For the theory of reflexive sheaves, one works over an integral noetherian scheme.

A reflexive sheaf is torsion-free. The dual of a coherent sheaf is reflexive.[1] Usually, the product of reflexive sheaves is defined as the reflexive hull of their tensor products (so the result is reflexive.)

A coherent sheaf F is said to be "normal" in the sense of Barth if the restriction is bijective for every open subset U and a closed subset Y of U of codimension at least 2. With this terminology, a coherent sheaf on an integral normal scheme is reflexive if and only if it is torsion-free and normal in the sense of Barth.[2] A reflexive sheaf of rank one on an integral locally factorial scheme is invertible.[3]

A divisorial sheaf on a scheme X is a rank-one reflexive sheaf that is locally free at the generic points of the conductor DX of X.[4] For example, a canonical sheaf (dualizing sheaf) on a normal projective variety is a divisorial sheaf.

See also

Notes
  1. Hartshorne 1980, Corollary 1.2.
  2. Hartshorne 1980, Proposition 1.6.
  3. Hartshorne 1980, Proposition 1.9.
  4. Kollár, Ch. 3, § 1.

References
  • Hartshorne, R.: Stable reflexive sheaves. Math. Ann.254 (1980), 121–176
  • Hartshorne, R.: Stable reflexive sheaves. II, Invent. Math. 66 (1982), 165–190
  • Kollár, János, "Chapter 3", Book on Moduli of Surfaces

Further reading
  • Greb, Daniel; Kebekus, Stefan; Kovacs, Sandor J.; Peternell, Thomas (2011). "Differential Forms on Log Canonical Spaces". Publications mathématiques de l'IHÉS. 114: 87–169. arXiv:1003.2913. doi:10.1007/s10240-011-0036-0.


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