Symplectic Geometry, Lecture 18

SYMPLECTIC GEOMETRY, LECTURE 18 Let (M, ω, J) be a compact Kähler manifold, [ω] ∈ H 1,1 (M ) ∩ H 2 (M, Z). Then we can find a line bundle L → M with first Chern class c1 (L) = [ω]. Choose a Hermitian metric on L along with a Hermitian connection � with R� = −2πiω. More explicitly, starting with any hermitian connection �, R� is a closed imaginary 2-form: in a trivialization, � = d + A, so R� = dA + [A, A] = dA. Thus, (1) [R� ] = −2πic1 (L) = −2πi[ω] =⇒ ∃a ∈ Ω1 (M ) s.t. R� = −2πiω + ida Letting �� = � − ia, we find that R� = R − ida = −2πiω. Next, recall that �0,1 defines a holomorphic structure on L iff (R� )0,2 = 0. Since ω is a (1, 1)-form and � R = −2πiω, we get a holomorphic line bundle structure for L. We will furthermore see that L⊗k has ”enough holomorphic sections”, i.e. the number of such sections → ∞. Given this, consider a basis of holomorphic � 0. Then we can define sections s0 , . . . , sN ∈ H 0 (L) (or H 0 (L⊗k )). Assume that, ∀p ∈ M, ∃s ∈ H 0 (L) s.t. s(p) = a map f : M → CPn , p → [s0 (x) : · · · : sN (x)] (2) More intrinsically, we obtain a map (3) M → P(H 0 (L)∗ ), p �→ Hp = {s ∈ H 0 (L)|s(p) = 0} ⊂ H 0 (L) Here, Hp is the kernel of the linear form given by evaluation at p, well-defined up to scaling. Definition 1. L is very ample if f : M → P(H 0 (L)∗ ) is a well-defined embedding, and ample if L⊗k is very ample for some k. We can reformulate this using the Kodaira embedding theorem: Theorem 1 (Kodaira). A holomorphic line bundle is ample ⇔ it has a holomorphic connection whose curvature is a Kähler form. The traditional proof of the Kodaira embedding theorem requires the Kodaira vanishing theorem. Instead, we will prove this using Donaldson’s argument. For simplicity, replace ω by 2ωπ , so [ 2ωπ ] = c1 (L). We will explicitly construct holomorphic sections of L⊗k for all k >> 0. � • First, fix p ∈ M , and choose local Darboux coordinates s.t. ω = 2i dzj ∧ dzj and J = J0 + O(|z|) (we can’t assume that J is the natural complex structure, because that would imply the Kähler metric was flat). • Next, choose a unitary trivialization of L⊗k , so that � corresponds to k� (4) d + iA0 = d + zj dzj − zj dzj 4 To see that we can choose A in this way, note that, in any trivialization, � = d+iA, so −ikω = R = idA. We have k� (5) idA0 = dzj ∧ dzj = −ikω0 = idA 4 Thus, A−A0 is closed and locally exact. Moreover, changing the trivialization by f = eiφ ∈ C ∞ (U, U (1)) changes the connection 1-form to A� = A + dφ. Thus a suitable change of trivialization ensures that the connection form becomes iA0 . 1 2 Remark. Baby model: assume J = J0 in our coordinates (so that the Kähler metric is flat), and consider 2 s(z) = exp(− k4 |z| ): this function arises from considering the curvature � (7) � 2 R1,1 = ∂ � ∂ + ∂ ∂ � = ∂∂ log |σ| (6) for σ a holomorphic section. We claim that s is holomorphic w.r.t. �. To see this, note that k� k� −k � �s = ds + iA0 s = (− zj dzj + zj dzj )s + ( zj dzj − zj dzj )s = ( zj dzj )s 4 4 2 � so ∂ s = 0 as desired. • In our case, � � 2 J = J0 + O(|z|) =⇒ ��s0,1 � = O(|z| · |�s|) = O(k |z| · |s|) (8) while (9) � � sup ��s0,1 � 1 |�s| = O(k |z| |s|) =⇒ = O( √ ) sup |�s| k We say that s is ”approximately holomorphic”. Definition 2. A family of sections sk ∈ C ∞ (L⊗k ) is uniformly bounded if it satisfies the uniform bounds r (10) sup |�r sk |g ≤ Cr k 2 x∈M and approximately holomorphic if (11) � � r−1 sup ��r−1 ∂sk �g ≤ Cr k 2 x∈M for all r. Furthermore, sk is uniformly concentrated at p if ∃ a polynomial P and a constant λ > 0 s.t. � � √ � 1 � t � (12) ∀x ∈ M, � t/2 � s(x)�� ≤ P ( kd(p, x)) exp(−λkdist(p, x)2 ) k for t ∈ {0, . . . , r}. Proposition 1. If (M, ω) is a compact symplectic manifold with a compatible almost complex structure, then ∃ a family of sections (σk,p )k>>0,p∈M which are uniformly bounded, approximately holomorphic, uniformly con­ centrated, and |σk,p | ≥ c > 0 over B(p, √1k ). In the Kähler case, we also have the following approximation theorem. Proposition 2. Given a family of sections {σk,p } as above, ∃{σ̃k,p } holomorphic s.t. (13) sup(k r/2 |�r σk,p − �r σ̃k,p |) ≤ Ce−λk/3 That is, any estimate you make via σ can also be applied to σ̃, so you can assume that your approximately holomorphic sections are holomorphic and obtain the desired embedding. To use these sections to prove Kodaira embedding, note that ∀p ∈ M, ∃s ∈ H 0 (L⊗k ) s.t. s(p) �= 0 since |σ̃k,p (p)| ≈ 1 (that is, L⊗k is base point free). Moreover, given p �= q ∈ M, ∃s, s� ∈ H 0 (L⊗k ) s.t. |s(p)| > |s(q)| and |s� (p)| < |s� (q)|: e.g., if p, q are distant by 1 more than k − 2 we can take s = σ̃k,p and s� = σ̃k,q (that is, our sections separate points). Finally, at every point p, ∀v ∈ Tp M, ∃σ1 , σ2 ∈ H 0 (L⊗k ) s.t. dv ( σσ12 ) �= 0 (that is, our sections separate tangent vectors). This is done by choosing a local holomorphic coordinate so that v = Re ∂z∂1 and perturbing z1 σk,p to a holomorphic section; setting σ2 = σ̃k,p gives the desired nonzero derivative.