An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifolds

Abstract

feme/ (d β : C~(ζ 9 ) -> C°°(ζ Q+1 )) and B e = image (d 2 : C°°(ζ q -1 ) -• C°°(ζ)), 0 < ^ < n.TTien ί/iβ cohomology groups Z q /B q are canonically isomorphic to the sheaf theoretic cohomology groups H q (X, Ω(ξ)).We introduce hermitian metrices in the bundles ξ and T(X).Then there are canonical hermitian metrices in the bundles ζ q , 0 < q < n.Let d* 2 : C°°(ζ q+ι ) -+ C°°(ζ q ) be the adjoint of the differential operator d 2 : C°°(ζ q ) -+ C°°(ζ q+ι ) with respect to the hermitian metrices in the bundles ζ q , ζ q+ \ and let ζ e = 0 ζ 2q and ζ° = 0 ζ 2q+1 .Then the operator d 2 + d* 2 maps C°°(ζ e ) into C°°(ζ°) and is q easily seen to be an elliptic operator.The following proposition is an immediate consequence of Theorem 1.2 and the complex analogue of the Hodge decomposition theorem.Proposition 1.3.The analytic index of the operator is equal to the Euler-Poincare characteristic of X with coefficients in the sheaf Ω(ξ), that is, χ(X, Ω(ξ)) = dim (kernel of d 2 + d* 2 : C°°(ζ e ) -> C°°(ζ 0 )) -codim (image of d 2 + d* g : C°°(ζ e ) -> C TO (ζ 0 )) .The adjoint of the operator d 2 + d* 2 : C°°(ζ e ) -* C°°(ζ°) is the operator d 2 + d* 2 : C°°(ζ 0 ) -^ C°°(ζ e ) and we have (d 2 + rf* g )W, + d* g ) = d f d* f + dV.= -4 , J 2 being the complex analogue of the Laplace-Beltrame operator.The operator Δ 2 is a self-adjoint elliptic operator from C°°(ζ q ) -> C°°(ζ Q ), 0 < (? < n.Let ^ be a non-negative real number, and S q (λ) be the eigenspace of the operator Δ 2 \ C°°(ζ 9 ) -• C°°(ζ α ) corresponding to >ί.Then the following proposition is an immediate consequence of an argument due to Atiyah Bott; see [4, §3].

Locations

• Journal of Differential Geometry - View - PDF