By Mark Andrea A De Cataldo
This ebook is a written-up and extended model of 8 lectures at the Hodge thought of projective manifolds. It assumes little or no historical past and goals at describing how the speculation turns into steadily richer and extra appealing as one specializes from Riemannian, to Kähler, to advanced projective manifolds. even though the facts of the Hodge Theorem is passed over, its outcomes - topological, geometrical and algebraic - are mentioned at a few size. The particular houses of complicated projective manifolds represent a big physique of data and readers are guided via it with the aid of chosen routines. regardless of beginning with only a few must haves, the concluding bankruptcy works out, within the significant distinct case of surfaces, the evidence of a different estate of maps among advanced projective manifolds, which used to be found in basic terms relatively lately.
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Additional info for Hodge Theory of Projective Manifolds
Show that iff u is harmonic. ∆ = ∆ . e. it coincides with its adjoint and is an elliptic second order linear partial differential operator. 35, Exercise 9, 16, 18 (the wave equation is not elliptic and there is no regularity theorem for it), 21 (elliptic operators for vector bundles). 3. Verify that if (M, g) is the Euclidean space Rn with the standard metric and u = |I|=p uI dxI , then ∆(u) = − I ∂ 2 uI j ∂x2j dxI . 12 and [Griffiths and Harris, 1978], p. 83. 3 Lectures on the Hodge Theory of Projective Manifolds Harmonic forms and the Hodge Isomorphism Theorem Let (M, g) be a compact oriented Riemannian manifold.
As the reader should check, P = P in the space with conjugation HomC (VC , VC ) = HomR (V, V )⊗R C. • Let W be a complex vector space. The conjugate W of W is the complex vector space such that W = W as real vector spaces, but such that the scalar multiplication in W is defined as γ · w := γw 27 January 29, 2007 28 22:22 World Scientific Book - 9in x 6in test Lectures on the Hodge Theory of Projective Manifolds where, on the right-hand side we are using the given complex scalar multiplication on W.
G. 13) etc. can be seen using J via the eigenspace decomposition of TX (C) with respect to J ⊗ IdC . 5): VR ⊆ VR ⊗R C = V ⊕ V where VR is the real vector space underlying V, V and VR ⊗R C has the conjugation operation. 6 and starts with a V as above and considers V ⊕ V instead. We mention this equivalence in view of the use of Hermitean metrics on TX . These metrics are defined as special tensors in TX∗ ⊗C TX∗ . 16) we can view the ∗ ∗ tensor h as an element of TX∗ ⊗C TX∗ ⊆ TX (C) ⊗C TX (C). This is also convenient in view of the use of the real alternating form associated with a Hermitean metric which can then be viewed as a real ∗ element of Λ2C (TX (C)).
Hodge Theory of Projective Manifolds by Mark Andrea A De Cataldo