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I am reading Complex Geometry: An Introduction by Daniel Huybrechts.

Section on complex structures and Hermitian structures deals with

  1. Lefschetz operator denoted by $L$.
  2. Hodge * -operator.
  3. The dual Lefschetz operator denoted by $\Lambda$.
  4. Counting operator denoted by $H$.

A theorem giving a relation between commutators $[H,L]=2L, [H,\Lambda]=-2\Lambda, [\Lambda,H]=H$.

And then concludes with Hodge-Riemann bilinear pairing and Hodge-Riemann bilinear relation.

I am having difficulty with notation and the presentation.

This deals with only vector spaces and almost complex structures and no manifolds are introduced. It would be great if some one can give some reference/notes/article regarding this.

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    Basically that is a lie algebra representation of $sl_2$ on $H^*(X, \mathbb C)$. – 2017-01-20
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    @JohnMa : Basically which is a lie algebra representation? – 2017-01-20
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    Now I take a look at that section, it is really linear algebra and has nothing to do with cohomology yet. I suggest you skip this part, if you are more interested in the geometry. You do not need that for 80% of the theory in this book. – 2017-01-20
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    @JohnMa : (Un)fortunately I am **supposed** to read that section as well.. – 2017-01-20

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