Ivancevic_Applied-Diff-Geom

440 Applied Differential Geometry: A Modern Introduction(p 1 , p 2 )−forms with p 1 +p 2 = p. Since the harmonic forms represent the cohomologyclasses in a 1–1 way, we find the important result that for Kählermanifolds,H p (M) = H p,0 (M) ⊕ H p−1,1 (M) ⊕ · · · ⊕ H 0,p (M).That is, the Dolbeault cohomology can be viewed as a refinement of the deRham cohomology. In particular, we haveb p = h p,0 + h p−1,1 + . . . + h 0,p ,where h p,q = dim H p,q (M) are called the Hodge numbers of M.The Hodge numbers of a Kähler manifold give us several topologicalinvariants, but not all of them are independent. In particular, the followingtwo relations hold:h p,q = h q,p , h p,q = h m−p,m−q . (3.243)The first relation immediately follows if we realize that ω ↦→ ω maps∂−harmonic (p, q)−forms to ¯∂−harmonic (q, p)−forms, and hence can beviewed as an invertible map between the two respective cohomologies. Aswe have seen, the ∂−cohomology and the ¯∂−cohomology coincide on aKähler manifold, so the first of the above two equations follows.The second relation can be proved using the map∫(α, ω) ↦→ α ∧ ωfrom H p,q × H m−p,m−q to C. It can be shown that this map is nondegenerate,and hence that H p,q and H m−p,m−q can be viewed as dual vectorspaces. In particular, it follows that these vector spaces have the samedimension, which is the statement in the second line of (3.243).Note that the last argument also holds for de Rham cohomology, inwhich case we find the relation b p = b n−p between the Betti numbers.We also know that H n−p (M) is dual to H n−p (M), so combining thesestatements we find an identification between the vector spaces H p (M)and H n−p (M). Recall that this identification between p−form cohomologyclasses and (n − p)−cycle homology classes represents the Poincaréduality. Intuitively, take a certain (n − p)−cycle Σ representing a homologyclass in H n−p . One can now try to define a ‘delta function’ δ(Σ) whichis localized on this cycle. Locally, Σ can be parameterized by setting pcoordinates equal to zero, so δ(Σ) is a ‘pD delta function’ – that is, it is anobject which is naturally integrated over pD submanifolds: a p−form. ThisM