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# tangentBundle -- get the tangent bundle

- Usage:
    `tangentBundle X`
    `tangentBundle f`
- Inputs:
  - X, an abstract variety,
  - f, an abstract variety map,
- Outputs:
  - an abstract sheaf, the (absolute) tangent bundle of $X$ or the (relative) tangent bundle of $f$

## Description

    i1 : X = abstractProjectiveSpace' 2

    o1 = X

    o1 : a flag bundle with subquotient ranks {2, 1}

    i2 : Y = abstractProjectiveSpace'_2 X

    o2 = Y

    o2 : a flag bundle with subquotient ranks {2, 1}

    i3 : tangentBundle Y

    o3 = a sheaf

    o3 : an abstract sheaf of rank 4 on Y

    i4 : chern oo

                            2            2         2     2       2 2
    o4 = 1 + (3h + 3h) + (3h  + 9h*h + 3h ) + (9h*h  + 9h h) + 9h h

                  QQ[][H   ..H   , h]
                        1,1   1,2
         ------------------------------------[H   ..H   , h]
         (- H    - h, - H    - H   h, -H   h)  1,1   1,2
             1,1         1,2    1,1     1,2
    o4 : ---------------------------------------------------
                 (- H    - h, - H    - H   h, -H   h)
                     1,1         1,2    1,1     1,2

    i5 : tangentBundle Y.StructureMap

    o5 = a sheaf

    o5 : an abstract sheaf of rank 2 on Y

    i6 : chern oo

                    2
    o6 = 1 + 3h + 3h

                  QQ[][H   ..H   , h]
                        1,1   1,2
         ------------------------------------[H   ..H   , h]
         (- H    - h, - H    - H   h, -H   h)  1,1   1,2
             1,1         1,2    1,1     1,2
    o6 : ---------------------------------------------------
                 (- H    - h, - H    - H   h, -H   h)
                     1,1         1,2    1,1     1,2

## See also

- cotangentBundle -- get the cotangent bundle

## Ways to use tangentBundle:

- `tangentBundle(AbstractVariety)`
- `tangentBundle(AbstractVarietyMap)`

## For the programmer

The object tangentBundle is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Schubert2.m2:1857:0.