DIFFERENTIAL FORMS AND COHOMOLOGY

Transcription

1 DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem - a generalization of the fundamental theorem of calculus. Develop a various cohomology theories. Remark 1. he standard definition of differential forms is essentially a real variable topic on smooth manifolds. We will follow this usual approach and also define pull-backs, cohomologies, etc. in this setting. hen via a standard trick define it on complex manifolds Alternating forms. Definition 1. An m-linear function f which maps the m-fold cartesian product V m of a vector space V into some other vector space W is called alternating if f(v 1,..., v m ) = 0 whenever v 1,..., v m V and v i = v j for i j. We let (V m, W ) be the vector space of m- linear alternating functions mapping V m into W. We then define m V by the property that if f m (V, W ) then there exists a unique h: m V W with f(v 1,..., v m ) = h(v 1 v m ). hen associating f with h we obtain the linear isomorphism m (V, W ) = Hom( m V, W ). We will usually think of W as R. Proposition 1. A linear map f : V V induces a map m (f, W ): m (V, W ) m (V, W ) defined by h f (m) V m f (m) (V ) m h W where f (m) (v 1,..., v m ) = (f(v 1 ),..., f(v m )) for (v 1,..., v m ) V m. Example 1. Let V = V in the proposition with dim(v ) = n <. hen ( m f)φ = det(f)φ whenever φ n (V, W ) Differential forms. Remark 2. Recall that there is a canonical isomorphism: m p (M) = m ( Hom( p(m), R) ) = Hom ( m p (M), R ) = [ m p(m) ] Date: March 3, 2010.

2 Definition 2. A m-form at p U M is a (continuous) function ω : U m p (M). he tangent bundle is (M) = p M p(m) and the cotangent bundle is (M) = p M p (M). hen m (M) = m p M p (M). Let Hom(M, m (M)) be the space of m-forms, and requiring that the maps be differentiable, i.e. C, gives us the space of differential m-forms ) m Ω m (M) = C (M (M). he space of differential forms on M is Ω(M) = m N Ωm (M), this is a graded something or other. Definition 3 (Helpful Jargon). he set (M) = p M p(m) is a vector bundle over M, that is, to each point p M there is an assigned vector space p(m). For every vector bundle E over M there is (by definition) a canonical projection π : E M. hen a section of E is a continuous map s : U E defined in open sets U M, such that πs = id U. One can say a section is holomorphic, smooth, C r, etc. if the function s is. In this jargon, we define a differential k-form to be a smooth section of k p (M). Definition 4. Let ω Ω k 1 (M) and ν Ω k 2 (M). he exterior product of ω ν Ω k 1+k 2 (M) is defined by (ω ν)(p) := ω(p) ν(p) Pull backs. Definition 5. Let f : M N be a holomorphic map, we define a pull back f : Ω k (N) Ω k (M) by (f η)(p)(v 1 v k ) = η f(p) ( pf v 1 pf v k ), for tangent vectors v 1,..., v k p(m) (or v 1 v k k p(m)). hen f extends in the obvious way to f : Ω k (N) Ω k (M). Proposition 2. Let f : M N and η 1, η 2 Ω(N). hen f (η 1 η 2 ) = f η 1 f η 2. Proposition 3. Let f : M N and g : N P, then (f g) = g f Exterior derivatives. he following heorem can be found [4, heorem 3.1, pp22]. heorem 1. Let M be a smooth manifold of dimension n. here exists a unique R-linear endomorphism d of Ω(M), called the exterior derivative or exterior differential satisfying the following axioms: i. d maps p-forms to (p + 1)-forms. ii. d is the usual differential on functions as elements of Ω 0 (M). iii. (Anti-derivation) If η is a p-form and ω is a q-form, then d(η ω) = dη ω + ( 1) p η ω. iv. d d = 0. v. (Naturality) If ψ : M N is a smooth map and ω is a smooth form, then d(ψ ω) = ψ (dω). 2

3 Proof. As with most canonical definitions/theorems, we just define d is the obvious way and check that the properties are satisfied. Let x 1,... x n be the standard coordinates on R n and take I = {i 1,... i p } with i 1 < < i p we denote dx I = dx i1 dx ip. hen every p-form can be written as as we define dω by ω = I dω = I ω I dx I, dω I dx I. Remark 3. By the usual differential on functions as elements of f Ω 0 (M) := C (M), we mean df = f x i dx i. More formally let (U, φ) be a chart. hen in U f φ 1 : f(u) R n R is analytic (holomorphic, smooth, C r, whatever) and so we interpret f x i := (f φ 1 ) x i 2. Cohomology and the Complexification of Differential forms 2.1. de Rham cohomology. Definition 6. A differential form α is closed if dα = 0, and exact if α = dβ for some form β. Remark 4. Since d d = 0, every exact form is closed. Definition 7. We define the sets Z k (M), B k (M) Ω k (M) as Z k (M) = {closed k-forms} = ker{d: Ω k (M) Ω k+1 (M)} B k (M) = {exact k-forms} = image{d: Ω k 1 (M) Ω k (M)} Remark 5. Observe that the previous remark implies that B k (M) Z k (M). Remark 6. Georges de Rham Definition 8. he k th de Rham cohomology group (vector space even) is given as the quotient group H k DR(M) = Zk (M) B k (M), that is, we look at closed k-forms and identify any whose difference is an exact k-form. Let α Z k (M) Ω k (M) then the equivialence class that contains α is denoted [α] and called the cohomology class of α 3

4 Proposition 4. If f : M N is smooth and [β] H k (N) then the pull back f defines a linear map (group homomorphism, v.s. homom.) by f ([β]) = [f β]. f : H k (N) H k (M) Proof. Suppose [β] H k (N). If β is closed, then f β is closed. herefore [f β] is a cohomology class in H k (M). Now it needs to be seen that [f β] depends only on the cohomology class of β. o see this, let β β dη B k (N), and so f β f β = df η B k (M). Remark 7 (Poincaré s Lemma). On a contractable domain every closed form is exact, that is, dα = 0 if and only if α = dβ for some β. herefore the de Rham cohomology is locally trival. However this is also one of the more interesting properties of this cohomology as it allows one to obtain topological information using purely differential methods. Since at this moment we are primarily interested in complex geometry, I ll have to save this interesting result for coffee time Return to Griffiths and Harris with a change of notation. In [2], they use A p (M, R) := Ω p (M) to denote the set of differential p-forms on M, Z p (M, R) = Z p (M) the closed p-forms, and d(a p 1 (M, R)) := B p (M) the exact p-forms. hen the quotient group H p DR (M, R) = Zp (M, R)/d(A p 1 (M, R)) form the de Rham cohomology groups of M. Let A p (M) := A p (M, C) to denote the set of complex differential p-forms on M, Z p (M) := Z p (M, C) the closed p-forms, and da p 1 (M) := d(a p 1 (M, C)) the exact p-forms. hen the quotient group H p DR (M) := Zp (M)/dA p 1 (M) form the de Rham cohomology groups of M. hen H p DR (M) = Hp DR (M, R) C. Recall that C,p (M) = p(m) p (M). herefore by taking duals C,p(M) = p (M) p (M) m C,p(M) = ( i p (M) i+j=m A m (M) = i+j=m A i,j (M) j p (M) Definition 9. A form φ A p,q (M) is said to be of type or bidegree (p, q). By the way of notation, we denote by π (p,q) the projection maps so that for φ A (M), A (M) A p,q (M), φ = π (p,q) φ; we usually write φ (p,q) for π (p,q) φ. If φ A p,q (M), then dφ A p+1,q (M) A p,q+1 (M). 4 )

5 We can then define two maps: by and so : A p,q (M) A p+1,q (M) : A p,q (M) A p,q+1 (M) = π (p+1,q) d, = π (p,q+1) d, d = +. In local coordinates z = (z 1,..., z n ), a form φ A m (M) is of type (p, q) if we can write φ(z) = φ IJ (z)dz I dz J, I =p, J =q where for each multiindex I = {i 1,..., i p }, he operators and are then given by φ(z) = I,J,j φ(z) = I,J,j dz I = dz i1 dz ip. z j φ IJ (z)dz j dz I dz J, z j φ IJ (z)dz j dz I dz J. In particular, we say that a form φ of type (p, 0) is holomorphic if φ = 0; which is the case if and only if φ(z) = φ I (z)dz I with φ I (z) holomorphic. Proposition 5. Let f : M N be a holomorphic map of complex manifolds, and f = f on A p,q (N). I =p f (A p,q (N)) A p,q (M) Definition 10. Let Z p,q (M) denote the space of -closed forms of the type (p, q). Proposition 6. 2 = 0, 2 = 0 and =. Proof. Let ω A p,q (M). hen 0 = d 2 ω = ( + ) 2 ω = 2 ω + ( + )ω + 2 ω However 2 ω A p+2,q (M), ( + )ω A p+1,q+1 (M) and 2 ω A p,q+2 (M), and are therefore of three distinct types. If they are to sum to zero, then each must be zero. 5

6 Definition 11. he result of the last proposition is that forms a cohomology theory, called the Dolbeault cohomology, that is, the (p, q) th Dolbeault cohomology group is defined as H p,q Zp,q (M) = (M) (A p,q 1 (M)). Corollary 1. If f : M N is a holomorphic map of complex manifolds, then f induces a map f : H p,q (N) Hp,q(M). heorem 2 (-Poincaré Lemma). Let be a polycylinder in C n, that is, = Ω 1 Ω n with each Ω j bounded, open and non-empty in C, then for all q 1. H p,q ( ) = 0 3. Calculus on Manifolds Definition 12. Formally a Riemannian metric on M is a section of the of the vector bundle S 2 (M), that is, the (positive) symmetric bilinear forms on (M). hat is a function p g p (, ) =:, p where g p : p (M) p (M) R. A Hermitian metric metric on M will be a Riemannian metric on M which is compatible with the complex structure of M. So for each p M we have a (positive) definite Hermitian inner product, p : C,p (M) C,p (M) C on C,p (M) = p(m) p (M). he compatibility condition is α, β = iα, iβ, where α, β C,p (M). More generally for any almost complex structure J on M, the compatibility condition is α, β = Jα, Jβ. A compatible Riemann metric on M will be symmetric and satisfy the following three conditions: α, β = α, β ; α, β = 0 α, α > 0 for α, β in p(m); unless α = 0 (if positive). Vice versa, a symmetric complex bilinear form, on C,p (M) satisfying these three conditions is the complex bilinear extension of a compatible (positive) Riemannian metric. hat is to say if h is a Hermitian metric and g is a Riemannian metric we have the condition h(x, y) = g(x, y) + g(jx, Jy), for any almost complex structure J on M. In [2] a Hermitian metric on M is usually denoted ds 2. he real part of a Hermitian metric is a Riemannian metric, i.e. Re ds 2 : R,p (M) R,p (M) R 6

7 and often called the induced Riemannian metric on M. When we speak of distance, area, or volume on a complex manifold with Hermitian metric, we always refer to the induced Riemannian metric. We also may observe that Im ds 2 : R,p (M) R,p (M) R is alternating, and so it represents a real differential form of degree 2; ω = 1 2 Im ds2 is called the associated (1, 1)-form of the metric. When ω is closed, ω is a Kähler form (to be defined later). A real differential (1, 1)-form ω is called a positive (1, 1)-form if for every p M and ν p(m) we have 1 ω(p), ν ν > 0. In local coordinates z on M a form is positive if 1 ω(z) = h i,j (z)dz i dz j, 2 i,j with H(z) = (h i,j (z)) i,j a positive definite Hermitian matrix for every z. A coframe for the Hermitian metric is an n-tuple of (1, 0)-forms (φ 1,..., φ n ) such that ds 2 = i φ i φ i. Proposition 7. Let f : M N be a holomorphic map such that f := p f : p(m) p(n) is injective, then a Hermitian metric on N induces a Hermitian metric on M. Furthermore the pull-back of the associated (1, 1)-form of the metric on N is associated (1, 1)-form of the induced metric on M. Note that [2] denotes p f by f and that the pull-back goes the opposite direction from f, i.e. f : A 1,1 (N) A 1,1 (M) Example 2. he Hermitian metric on C n given by ds 2 = n dz i dz i is called the Euclidean or standard metric on C n = R 2n. i=1 Example 3. Let Z 0,..., Z n be coordinates in C n+1 and denote by π : C n+1 \ {0} P n be the standard projection map. Let U P n be an open set and Z : U C n+1 \ {0} a lifting of U, i.e. a holomorphic map with π Z = id; consider the differential form ω = 1 2π log Z 2. 7

8 If Z : U C n+1 \ {0} is another lifting, then Z = f Z with f a nonzero holomorphic function, so that 1 1 2π log Z 2 = 2π log( Z 2 + log f + log f) 1 = ω + ( log f log f) 2π = ω herefore ω is independent of the lifting chosen; since liftings always exist locally, ω is a globally defined differential form in P n. Clearly ω is of type (1, 1). o see that ω is positive, first note that the unitary group U(n + 1) acts transitively ( ) on P n and leaves the form ω invariant, so that ω is positive everywhere if it is positive at one point. Now let {w i = Z i /Z 0 } be coordinates on the open set U 0 = (Z 0 0) in P n and use the lifting Z = (1, w 1,..., w n ) on U 0 ; we have 1 ω = 2π log(1 + w i w i ) ( ) 1 = 2π wi dw i 1 + w i w i ( 1 dwi dw i = 2π 1 + ( wi dw i ) ( ) w i dw i ) w i w i (1 +. w i w i ) 2 At the point [1, 0,..., 0], 1 ω = dwi dw i > 0. 2π hus ω defines a Hermitian metric on P n, called the Fubini-Study metric. Remark 8. ( ) he unitary group U(n + 1) is the set of all n + 1 n + 1 matrices U such that UU = I n+1. A group G acts transitively on a set X if for any two x, y X there is a g G such that gx = y. heorem 3 (Wirtinger s heorem). vol(s) = 1 d! Proof. he interplay between the real and imaginary parts of a Hermitian metric now gives us the Wirtinger theorem, which expresses another fundamental difference between Riemannian and Hermitian differential geometry. Let M be a complex manifold, z = (z 1,..., z n ) local coordinates on M, and S ω d. ds 2 = φ i φ i a Hermitian metric on M with associated (1, 1)-form ω. Write φ i = α i + 1β i ; then the associated Riemannian metric on M is Re ds 2 = i α i α i + β i β i 8

9 and the volume associated to Re ds 2 is given by On the other hand we have so that the n th exterior power dµ = α 1 β 1 α n β n. ω = α i β i ω n = n! α 1 β 1 α n β n = n! dµ Now let S M be a complex submanifold of dimension d. he (1, 1)-form associated to the metric induced on S by ds 2 is just ω S and apply the above to the induced metric on S. Remark 9. In particular this shows us that ω = 0 E when dim(e) <deg(ω). Furthermore with a slight abuse of notation this can be stated as A p,q (E) = 0 when either p or q is greater than the dimension of E. Here E is any set where the integration is defined, i.e. manifolds, submanifolds, analytic varieties, subvarieties. Remark 10. he fact that the volume of a complex submanifold S of the complex manifold M is expressed as the integral over S of a globally defined differential form on M is quite different from the real case. For a C arc t (x(t), y(t)) in R n the element of arc length is given by x (t) 2 + y (t) 2 dt which is not, in general, the pullback of any differential form in R 2. ( he coefficient of dt is not C. ) We now get the manifold version of the fundamental theorem of calculus b a f(x) dx = F (b) F (a) heorem 4 (Stokes heorem). Let K be a compact subset of M with piecewise C 1 boundary, and u any differential form of class C 1 with degree m 1 on M. We then have u = du. K heorem 5 (Stokes heorem for Analytic Varieties). For M a complex manifold and V M an analytic subvariety of dimension k, and φ a differential form of degree 2k 1 with compact support in M, dφ = 0. K V 9

10 Proof. I m just going to point out the main idea. First observe that if φ has degree 2k 1 then its either p or q is k where (p, q) is the bidegree of φ. We now apply the usual version of Stokes heorem (a little work is needed to make it apply in this case) dφ = φ. V However dim(v ) = k 1, and so by the remark following Wirtinger s heorem, we have that this right integral must be 0. Remark 11. echnically we haven t defined integration over an analytic variety yet. Recall that the smooth part of the variety V = V \ V s is a submanifold. herefore one defines the integration over V by integration over V. V References [1] D ly, J.P., Complex Analytic and Algebraic Geometry, fr/~d ly/books.html [2] Griffiths, P., Harris, J., Principles of Algebraic Geometry, 1994 [3] Lee, J. L., Manifolds and Differential Geometry, 2009 [4] Moroianu, A., Lectures on Kähler Geometry,