By Daniel Huybrechts

Simply available comprises fresh advancements Assumes little or no wisdom of differentiable manifolds and sensible research specific emphasis on themes relating to reflect symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)

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**Extra resources for Complex Geometry: An Introduction**

**Example text**

1 . 2 . 9 (Wirtinger inequality) Let (V, ( , }) be an euclidian vector space endowed with a compatible almost complex structure / and the associated fundamental form LO. Let W C V be an oriented subspace of dimension 2m. The induced scalar product on W together with the chosen orientation define a natural volume form volw 6 f\2mW*. e. I(W) = W, and the orientation is the one induced by the almost complex structure. (The inequality is meant with respect to the isomorphism / \ m W* = K, voW H-> 1.

A (xm A ym). Note that I(xl) = —yl and I(yl) = x1. We tacitly use the natural isomorphism f\k V* Q* (Afc V)* given by (a! A . . A a fc )(«i A . . A vfe) = det (^(u,-))^.. e. The operator Hk does not depend on the almost complex structure /, but the operators I and IIp'q certainly do. Note that I is the multiplicative extension of the almost complex structure I on Vc, but I is not an almost complex structure. Since / is defined on the real vector space V, also I is an endomorphism of the real exterior algebra A*^We denote the corresponding operators on the dual space A* ^c a l s o by k II , np'q, respectively I.

Since B and 8 share the usual properties of the exterior differential d and reflect the holomorphicity of functions, it seems natural to build up a holomorphic analogue of the de Rham complex. As we work here exclusively in the local context, only the local aspects will be discussed. Of course, locally the de Rham complex is exact (see Appendix A) due to the standard Poincare lemma. We will show that this still holds true for B (and d). 7 (9-Poincare lemma in one variable) Consider an open neighbourhood of the closure of a bounded one-dimensional disc Be C BecU C C.