Applications of the theory of L2 estimates and positive by Demailly J.-P.

By Demailly J.-P.

Show description

Read Online or Download Applications of the theory of L2 estimates and positive currents in algebraic geometry PDF

Similar geometry and topology books

The Geometry of Time (Physics Textbook)

An outline of the geometry of space-time with the entire questions and matters defined with out the necessity for formulation. As such, the writer exhibits that this is often certainly geometry, with genuine buildings regularly occurring from Euclidean geometry, and which permit special demonstrations and proofs. The formal arithmetic at the back of those buildings is supplied within the appendices.

The Hilton Symposium, 1993: Topics in Topology and Group Theory (Crm Proceedings and Lecture Notes)

This quantity offers a cross-section of recent advancements in algebraic topology. the most element includes survey articles compatible for complex graduate scholars and execs pursuing learn during this region. a superb number of issues are coated, a lot of that are of curiosity to researchers operating in different parts of arithmetic.

Additional info for Applications of the theory of L2 estimates and positive currents in algebraic geometry

Example text

19) Lemma. In Cn , we have [d′′⋆ , L] = i d′ . Proof. 17, we get [d′′⋆ , L]u = d′′⋆ (ω ∧ u) − ω ∧ d′′⋆ u ∂ ∂ (ω ∧ u) + ω ∧ =− ∂z k ∂zk k Since ω has constant coefficients, we have [d′′⋆ , L] u = − k =− k by the derivation property of ∂ ∂z k ω= ∂ ∂z k k ∂u . ∂zk ∂ ∂u (ω ∧ u) = ω ∧ and therefore ∂zk ∂zk ∂ ∂z k ω∧ ∂u ∂zk ∂ ∂z k ω ∧ ∂u ∂zk −ω∧ ∂ ∂z k ∂u ∂zk . Clearly ∂ ∂z k dzj ∧ dz j = −i dzk , i 1≤j≤n hence [d′′⋆ , L] u = i dzk ∧ k ∂u = i d′ u. ∂zk The final step is to extend these results to an arbitrary K¨ahler manifold (X, ω) and an arbitrary hermitian vector bundle (F, h).

E. is also a symplectic 2-form. 15) Examples. n • C equipped with its canonical metric ω = i 1≤j≤n dzj ∧ dz j (or any other hermitian metric ω = i 1≤j,k≤n ωjk dzj ∧ dz k with constant coefficients) is K¨ahler. n • A complex torus is a quotient X = C /Γ by a lattice (closed discrete subgroup) Γ of rank 2n. Then X is a compact complex manifold. Any positive definite hermitian form ω = i ωjk dzj ∧ dz k with constant coefficients defines a K¨ahler metric on X. n • The complex projective space P is K¨ ahler.

24) now follows from the Hodge isomorphisms for De Rham and Dolbeault groups. The decomposition is canonical since H p,q (X) coincides with the set of classes in H k (X, C) which can be represented by d-closed (p, q)-forms. 5. 25) Remark. The decomposition formula shows that the so called Betti numbers k bk := dim HDR (X, C) satisfy the relation hp,q bk = p+q=k in terms of the Hodge numbers hp,q := dim H p,q (X, C). In addition to this, the complex conjugation u → u takes (p, q)-harmonic forms to (q, p)-harmonic forms, hence there is a canonical conjugate linear isomorphism H q,p (X, C) ≃ H p,q (X, C) and hq,p = hp,q .

Download PDF sample

Rated 4.11 of 5 – based on 12 votes