Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory Complex manifolds and K¨ahlerGeometry Lecture 3 of 16: Exterior forms on complex manifolds Dominic Joyce, Oxford University September 2019 2019 Nairobi Workshop in Algebraic Geometry These slides available at http://people.maths.ox.ac.uk/∼joyce/ 1 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory Plan of talk: 3 Exterior forms on complex manifolds 3.1 Exterior forms on real and complex manifolds 3.2 The @ and @¯ operators 3.3 Exterior forms on almost complex manifolds 3.4 (p; q)-forms in terms of representation theory 2 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory 3.1. Exterior forms on real and complex manifolds Let X be a real manifold of dimension n. It has cotangent bundle T ∗X and vector bundles of k-forms Λk T ∗X for k = 0; 1;:::; n, with vector space of sections C 1(Λk T ∗X ). A k-form α in 1 k ∗ C (Λ T X ) may be written αa1···ak in index notation. The exterior derivative is d : C 1(Λk T ∗X ) ! C 1(Λk+1T ∗X ), given by k+1 j+1 X (−1) @αa1···aj−1aj+1···ak+1 (dα)a1···ak+1 = · k + 1 @xa j=1 j in index notation, where d2 = 0. 3 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory The de Rham cohomology of X is 1 k ∗ 1 k+1 ∗ k Ker d : C (Λ T X ) ! C (Λ T X ) HdR(X ; R) = : Imd : C 1(Λk−1T ∗X ) ! C 1(Λk T ∗X ) k It is isomorphic to the usual cohomology H (X ; R) of the underlying topological space X . It will be helpful to consider complex forms on X , that is, sections of the complex vector bundles of complexified k-forms k ∗ Λ T X ⊗R C. 1 k ∗ A complex k-form α in C (Λ T X ⊗R C) is of the form β + iγ for β; γ real k-forms. The complex conjugate isα ¯ = β − iγ. The 1 k ∗ exterior derivative extends to d : C (Λ T X ⊗R C) ! 1 k+1 ∗ C (Λ T X ⊗R C) by d(β + iγ) = dβ + idγ. Complex de Rham k cohomology HdR(X ; C) is defined using complex forms. 4 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory Exterior forms on complex manifolds Now let (X ; J) be a complex manifold in the sense of x2, so that X b is a real 2n-manifold, and J = Ja is a complex structure. We will use J to decompose k-forms into components. Define an action of J on complex 1-forms α on X by b 2 (Jα)a = Ja αb, in index notation. Then J = −1, so J has ∗ eigenvalues ±i, and T X ⊗R C splits as a direct sum of complex subbundles ∗ ∗ 1;0 ∗ 0;1 T X ⊗R C = T X ⊕ T X ; where T ∗X 1;0, T ∗X 0;1 are the eigenspaces of J with eigenvalues i; −i. Both have complex rank n. Sections of T ∗X 1;0; T ∗X 0;1 are called (1,0)-forms and (0,1)-forms. If α = β + iγ for β; γ real b 1-forms, then α is a (1,0)-form if γa = −Ja βb, and a (0,1)-form if b γa = Ja βb. If α = β + iγ is a (1,0)-form thenα ¯ = β − iγ is a (0,1)-form, and vice versa. 5 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory For complex k-forms we have (Λk T ∗X ) ⊗ = Λk (T ∗X ⊗ ), R C C R C that is, we can take exterior powers of the real vector bundle and then complexify, or complexify and then take complex exterior powers. Therefore (Λk T ∗X ) ⊗ = Λk (T ∗X 1;0 ⊕ T ∗X 0;1). R C C k ∼ L p q Now Λ (U ⊕ V ) = p+q=k Λ U ⊗ Λ V . Thus we have a natural k ∗ L p;q splitting (Λ T X ) ⊗R C = p+q=k Λ X , with Λp;qX =∼ Λp (T ∗X 1;0) ⊗ Λq (T ∗X 0;1): C C Sections of Λp;qX are called (p; q)-forms. We also have 1 k ∗ M 1 p;q C (Λ T X ) ⊗R C = C (Λ X ): p+q=k 6 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory If α is a (p; q)-form thenα ¯ is a (q; p)-form. Thus a (p; q)-form α can only be real (α =α ¯) if p = q. We can also express Λp;qX as an eigenspace. Let J act on complex k-forms α = αa1···αk by k X J(α) = Jb α : a1···ak aj a1···aj−1baj+1···ak j=1 As αa1···ak is antisymmetric in a1;:::; ak , so is J(α)a1···ak , so J(α) is a k-form. It turns out that if α is a (p; q)-form then J(α) = i(p − q)α, so Λp;qX is the eigensubbundle of J in p+q ∗ (Λ T X ) ⊗R C with eigenvalue i(p − q). 7 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory (p; q)-forms in complex coordinates A good way to write (p; q)-forms is in complex coordinates on X . This also fits in with the definition of complex manifolds in x1. Let (z1;:::; zn) be holomorphic coordinates on some open U ⊆ X . Then dz1;:::; dzn and their complex conjugates dz¯1;:::; dz¯n are complex 1-forms on U. It turns out that dz1;:::; dzn are (1,0)-forms, and dz¯1;:::; dz¯n are (0,1)-forms. More generally, if 1 6 a1 < ··· < ap 6 n and 1 6 b1 < ··· < bq 6 n then dza1 ^ · · · ^ dzap ^ dz¯b1 ^ · · · ^ dz¯bq is p;q a (p; q)-form on U, and these form a basis of sections of Λ X jU , so every (p; q)-form α on U may be written uniquely as X α = αabdza1 ^ · · · ^ dzap ^ dz¯b1 ^ · · · ^ dz¯bq 16a1<···<ap6n 16b1<···<bq6n for functions αab : U ! C, with a = (a1;:::; ap), b = (b1;:::; bq). p;q n n The rank of Λ X , as a complex vector bundle, is p · q . 8 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory 3.2. The @ and @¯ operators Let (X ; J) be a complex manifold and f : X ! C be a smooth function. Then df is a complex 1-form on X . Since 1 ∗ 1 ∗ 1;0 1 ∗ 0;1 C (T X ⊗R C) = C (T X ) ⊕ C (T X ), we may write df ¯ 1 uniquely as df = @f + @f , where @f = 2 (df − iJ(df )) is a ¯ 1 (1; 0)-form, and @f = 2 (df + iJ(df )) is a (0; 1)-form. Note that @¯f = 0 iff J(df ) = idf , that is, iff f is holomorphic. Now let α be a (p; q)-form. As in x3.1, let (z1;:::; zn) be holomorphic coordinates on U ⊆ X , with P αjU = a;b αabdza1 ^ · · · ^ dzap ^ dz¯b1 ^ · · · ^ dz¯bq : As dzaj ; dz¯bj are closed, we see that P dαjU = a;b dαab ^ dza1 ^ · · · ^ dzap ^ dz¯b1 ^ · · · ^ dz¯bq P = a;b @αab ^ dza1 ^ · · · ^ dzap ^ dz¯b1 ^ · · · ^ dz¯bq (3.1) P ¯ + a;b @αab ^ dza1 ^ · · · ^ dzap ^ dz¯b1 ^ · · · ^ dz¯bq : 9 / 44 Dominic Joyce, Oxford University Lecture 3: Exterior forms on complex manifolds Exterior forms on real and complex manifolds Exterior forms on complex manifolds The @ and @¯ operators K¨ahlermetrics Exterior forms on almost complex manifolds (p; q)-forms in terms of representation theory The second sum in (3.1) is a (p + 1; q)-form, and the third a (p; q + 1)-form. Hence, if α is a (p; q)-form then dα is the sum of a (p + 1; q)-form and a (p; q + 1)-form.