BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate them into research in Geometry. We will begin with a very rough introduction to Riemannian Geometry so that first and second year graduate students will be able to understand the talk, at least heuristically. We will then cover the necessary materials to define the Bergman kernel on polarized manifolds and use it to prove Tian s theorem on the convergence of polarized metrics. Come for the pizza, stay for the geometry! Contents 1. Introduction 1 2. Kähler Geometry 3 2.1. Complex Manifolds 3 2.2. Covariant Derivative and Curvature 5 2.3. Example: Complex Projective Space with Fubini-Study Metric 6 2.4. Line Bundles 6 3. Bergman Kernel 7 3.1. Bergman kernel on domains 7 3.2. Bergman kernel on manifolds 8 4. Applications 9 4.1. Tian s Theorem 9 References 9 1. Introduction Differential geometry is a very interesting field, which uses tools from almost every other field of mathematics, including PDEs, Functional Analysis, Real and Complex Analysis, Algebra, Topology, etc. The primary objects of study are manifolds and their metrics. Let us begin with a definition of a manifold. Definition 1.1 (Definition 1.1.1). Let M be a connected paracompact and Hausdorff space. M is a manifold of dimension n if every point has a neighborhood U that is homeomorphic to an open subset Ω of R n. Such a homeomorphism x : U Ω is called a (coordinate) charts. An atlas is a family {U α, x α } of charts for which the U α constitute an open covering of M. In the overlap, we require that the transition functions x β x 1 α : x α (U α U β ) x β (U α U β ) to be smooth, (or C k ). Then M is a smooth (C k ) manifold. Date: January 6, 2015. 1

2 SHOO SETO In a more intuitive language, we mean a space that locally looks like an open subset of the Euclidean space R n. Definition 1.2. At each point p M, we can define the tangent space T p M. Intuitively, it gives the possible directions that one can move on the manifold. The tangent space can be viewed as the space of point derivations, hence given a coordinate system (x 1,..., x n ), we have a basis at the point p { } x 1,, p x n We require that under the change of coordinates, say (y 1,..., y n ), the vectors transform according to the rule x i = yα x i y α Remark 1.3. Here we are using the Einstein summation convention where y α x i y α := n α=1 p y α x i y α so that when we have a repeated index in the upper and lower position, we are taking the summation. Definition 1.4. We define the dual to the tangent space, cotangent space as Tp M. coordinates (x 1,..., x n ), we have a basis: Given where {dx 1,..., dx n } ( ) dx i x j = δ ij We can equip this space with a product called the wedge product, defined by so that dx i dx j = dx i dx j dx j dx i dx i dx j = dx j dx i for i, j = 1,..., n. Another name for elements of T M is differential form. Under a change of coordinates, they transform according to the rule dx i = xi y α dyα Definition 1.5. We define the Riemannian metric on the tangent space T p M as a symmetric positive definite bilinear form that depends smoothly on p M. We can write this as g(p) : T p M T p M R ( ) g x i, x j := g ij g = g ij dx i dx j

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS 3 2. Kähler Geometry 2.1. Complex Manifolds. We now consider complex manifolds, where locally they look like C n. More precisely, Definition 2.1 (Complex Manifold). Let M be a smooth manifold of (real) dimension 2n. We say that M is a complex manifold of dimension n if M can be covered by charts (U, z) where U is an open subset of M and z : U C n is a homeomorphism onto an open subset z(u) C n, with the following property: if (V, w) is another chart with U V, then the transition maps and are holomorphic. w z 1 : z(u V ) w(u V ) z w 1 : w(u V ) z(u V ) Definition 2.2. A smooth function f on M is holomorphic if for each coordinate chart (U, z), we have f z 1 z i = 0, for i = 1,..., n, on U. By abuse of notation, we write f := f z 1. To check that this is well-defined, consider another chart (V, w) with U V : f w j = f z k z k w j + f z k z k w j = 0, where we use the fact that the transition maps are holomorphic. On complex manifolds, we work with the complexified tangent bundle (T M) C = T M C. At a point p, it is given by the span over C of z 1,..., z n, z 1,..., z n We decompose the tangent space into its holomorphic and anti-holomorphic components (T p M) C = Tp 1,0 M Tp 0,1 M This decomposition is independent of the choice of complex coordinate charts: w i = zk w i z k + zk w i since z k is anti-holomorphic. Similar for the conjugate term. We can now define a linear map on the tangent space z k = zk w i z k, Definition 2.3. An almost complex structure on a smooth manifold is an endomorphism J : T M T M of the tangent bundle such that J 2 = I, where I is the identity map. The almost complex structure equips the tangent space at each point with a linear map which behaves like multiplication by 1. Example 2.4. Given a complex manifold, with holomorphic coordinates (z 1,..., z n ), we can define J at each tangent space T p M by ( ) J z i = 1 z i, ( ) J z i = 1 z i

4 SHOO SETO J is well defined since the transition maps are holomorphic. We are now ready to define Kähler metrics. We are interested in Riemannian metrics on M which are compatible with the complex structure. Definition 2.5. A Riemannian metric g on M is Hermitian if for any X, Y T p M, g(jx, JY ) = g(x, Y ), or in other words, we require J to be an orthogonal transformation on each tangent space. Definition 2.6. Given a Hermitian metric g, we define ω(x, Y ) = g(jx, Y ). ω is called the fundamental 2 form or sometimes Kähler form Definition 2.7. A Hermitian metric g is Kähler if the Kähler form ω is closed, i.e. (2.1) dω = 0 We give a local description of the above: Definition 2.8. A Hermitian metric g on M is a Hermitian inner product on the n-dimensional complex vector space Tp 1,0 M for each p, which varies smoothly in p. Locally g is given by an n n positive definite Hermitian matrix. We denote its (i, j)th entry as g ij := g( z i, z j ). Furthermore, have the following symmetries: g ij = g ji Given T 1,0 vector fields X = X i z i and Y = Y j z j, we define their pointwise inner product and we write (X, Y ) g := g(x, Y ) = g ij X i Y j X 2 g = (X, X) g for the norm of X with respect to g. In local coordinates, the Kähler form is given by ω = 1g ij dz i dz j. Remark 2.9. Other authors may have a factor of 1 2 or 1 2π. The Kähler condition (2.1) written locally implies the following symmetries: g ij z k = g kj z i and g ij z k = g ik z j Furthermore, using these Kähler symmetries, it is possible to find holomorphic normal coordinates. Lemma 2.10. Let (M, g) be a Kähler manifold. For any fixed point x M, there exists a holomorphic coordinate chart (U, z) centered at p such that, at p, for all i, j, k = 1, 2,..., n. g ij (p) = δ ij, and g ij (p) = 0, zk

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS 5 2.2. Covariant Derivative and Curvature. To differentiate tensors, we define the notion of covariant differentiation. Let be the Levi-Civita connection, which means that g = 0. By using the Kähler condition, it can be shown that J = 0. We use the following notation convention: i := z i, j := z j. By using the compatibility of the connection with the complex structure ( J = 0), we have J ( i k ) = i (J k ) = 1 i k, so that i k T 1,0 M. By a similar argument, it can be shown that j k T 0,1 M. Furthermore, we can show that i k T 0,1 M and k i T 1,0 M, however, using the torsion free condition: i k = k i. Since they are of different types, they must both vanish. Christoffel symbols: Definition 2.11. i j = Γ k ij k, i j = Γ k ij k. Using these facts, we can define the In terms of the metric, using the metric compatibility condition ( g = 0) 1, we have or i g jk = g( i j, k ) = Γ l ijg lk Γ p ij = gpk i g jk With the above Christoffel symbols, we can take the covariant derivative of a (1, 0)-vector field X = X i i ( ) X k jx = z j + X i Γ k ji Next, we define the Riemannian curvature tensor as Definition 2.12. k Written out in coordinates, we have R(X, Y, Z, W ) := ( X Y Z Y X Z [X,Y ] Z, W ) g R αβδγ := R( α, β, δ, γ ) = ( α β δ β α δ, γ ) g where α, β, δ, γ = 1,..., 2n. However, since holomorphic vectors are parallel to anti-holomorphic vectors, the non-vanishing components of the curvature tensor are In terms of the metric, we have R ijkl = ( j i k, l ) g R ijkl = 2 g kl z i z j + g gpq kq g pl z i z j We also define the Ricci and the scalar curvature as: Definition 2.13. Ric ij = g kl R ijkl ρ = g ij Ric ij 1 Metric compatibility means ig(x, Y ) = ( i g)(x, Y ) + g( i X, Y ) + g(x, i Y ) = g( i X, Y ) + g(x, i Y )

6 SHOO SETO 2.3. Example: Complex Projective Space with Fubini-Study Metric. The complex projective space CP n is defined to be the space of complex lines in C n+1. We can describe points of CP n by homogeneous coordinates, [Z 0 : : Z n ], where we identify [Z 0 : : Z n ] = [λz 0 : : λz n ] for all λ C\{0}. To define the complex structure, we will use n + 1 charts. Let { } U i = [Z 0 : : Z n ] Z i 0 and ϕ i : U i C n ( ) Z 0 [Z 0 : : Z n ],..., Ẑi,..., Z n Z i Z i Z i where the Z i Z i term is omitted. To see that the transition functions are holomorphic, let w 1,..., w n on C. Then ( ) 1 ϕ 1 ϕ 1 (w 1,..., w n ) = w 1, w2 w 1,..., wn w 1. CP n has a natural Kähler metric ω F S called the Fubini-Study metric. It is defined as 1 2π log( Z 0 2 + Z 1 2 + + Z n 2 ). In local coordinates, say on U 0, it is given by g ij = i j log(1 + w 1 2 + + w n 2 ) 2.4. Line Bundles. A holomorphic vector bundle E over a complex manifold M is a holomorphic family of complex vector spaces parameterized by M. When E is a 1 dimensional complex vector space, we say that it is a line bundle. Vector bundles come with a projection map π : E M and an open cover {U α } such that we have biholomorphisms (trivializations) ϕ α : π 1 (U α ) U α C k for some integer k > 0, called the rank of E. transition maps The trivializations are related by holomorphic ϕ β ϕ 1 α : (U α U β ) C k (U α U β ) C k (p, v) (p, ϕ βα (p)v) where at each point p U α U β gives a linear isomorphism ϕ βα (p) : C k C k. Definition 2.14. A holomorphic section of a line bundle L is a holomorphic map s : M L such that π s is the identity map. The local trivialization ϕ α gives rise to a nonzero local section e α, which we call a local frame. All other local holomorphic sections over U α can be written as f = f α e α where f α is a holomorphic function on U α. We write the space of global holomorphic sections as H 0 (M, L).

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS 7 3. Bergman Kernel 3.1. Bergman kernel on domains. We first define the Bergman kernel on a domain Ω C n. The Bergman kernel lives and interacts with functions in the following space: Definition 3.1. The Bergman space is { A 2 (Ω) := f holomorphic on Ω Ω } f(z) 2 dv (z) = f A 2 (Ω) <, in other words, holomorphic square integrable (L 2 ) functions. We equip the space with the L 2 norm f, g := f(z)g(z)dv. We prove a key inequality Ω Lemma 3.2. Let K Ω C n be compact. There is a constant C K > 0, depending on K and on n, such that sup f(z) C K f A 2 (Ω), f A 2 (Ω). z K Proof. Since K is compact, there is an r(k) = r > 0 such that, for any z K, B(z, r) Ω. For each z K, apply the mean value property of holomorphic functions, 1 f(z) = f(w)dv (w) Vol(B(z, r)) B(z,r) = 1 f(w)χ Vol(B(z, r)) B(z,r) dv (w) Ω Vol(B(z, r)) 1 2 f A 2 (Ω), where we use the Cauchy-Schwarz inequality in the last line. By using Lemma 3.2 and a normal family argument, one can show that the space A 2 (Ω) with the L 2 inner product is a Hilbert space. 2 Next consider the evaluation functional ev z (f) = f(z), f A 2 (Ω). Lemma 3.3. For each fixed z Ω, the evaluation functional ev z is a continuous linear functional on A 2 (Ω). Proof. By Lemma 3.2, we have ev z (f) = f(z) C f A 2 (Ω), for some C > 0. Hence it is continuous with respect to the A 2 (Ω) norm. Since ev z is continuous, by Riesz representation theorem, there exists an element K z A 2 (Ω) such that ev z (f) = f, K z Definition 3.4. The Bergman kernel is the holomorphic function K(z, w) := K Ω (z, w) = K z (w), z, w Ω, with the reproducing property: f(z) = K(z, w)f(w)dv (w). f A 2 (Ω) 2 Complete inner product space Ω

8 SHOO SETO Consider an orthonormal basis {ϕ α } for the space A 2 (Ω). Since K z (w) A 2 (Ω), we have K z (w) = a i ϕ i (w). Using the reproducing property and recovering the coefficients, we have Hence the Bergman kernel is given by Proposition 3.5. a i = ϕ i, K z = K z, ϕ i = ϕ i (z). K z (w) = ϕ i (z)ϕ i (w). Example 3.6 (Bergman kernel on the unit disk). This motivates our definition of the Bergman kernel on manifolds. 3.2. Bergman kernel on manifolds. Let (M, ω) be a compact Kähler manifold with a positive line bundle (L, h) such that Ric(h) = ω, that is at a point p M, with local coordinates (z 1,..., z n ) and frame e L of L in a neighborhood U α, the Kähler metric of the manifold and the hermitian metric of the line bundle satisfy the following relation locally: 1 ω = 2π g ij dzi dz j = 1 log h α where h α is the local representation of the hermitian metric h α = h(e L, e L ). Let L k be the k-th tensor power of L, extending the Hermitian metric h to h k as well. Consider the space of L k -valued holomorphic sections H 0 (M, L k ). We equip the space with the following L 2 inner product induced by the Hermitian metric h: S, T L 2,h := h(s, T )dv g By the Hodge theorem, this space is finite-dimensional. Consider an orthonormal basis {S i }. Definition 3.7. We define the Bergman kernel on manifolds as where dim(h 0 (M, L k )) = d k B k (x, y) = d k i=1 M S i (x) S i (y) The Bergman kernel admits the following asymptotic expansion on the diagonal Theorem 3.8 (Tian-Yau-Zelditch Expansion). d k ( S i (x) 2 h k n a k 0 (x) + a 1(x) + a ) 2(x) k k 2 + i=1 More precisely, for any m, d k i=1 S i (x) 2 h k k n j<n a j (x) k j C m C N,m k n N The coefficients themselves are very difficult to compute. Prof. Z. Lu computed the first 4 terms in 1999, and developed an algorithm to compute any number of terms. The idea is to first construct peak sections, first considered by Tian, which are locally just monomials concentrated near the

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS 9 origin, and to Taylor expand the metric terms, then compute them by brute force. The first 3 terms are a 0 = 1 a 1 = ρ 2 a 2 = ρ 3 + 1 24 ( R 2 4 Ric 2 + 3ρ 2 ) 4. Applications 4.1. Tian s Theorem. Given a compact Kähler manifold with a positive hermitian holomorphic line bundle, (this set up is sometimes referred to as polarized manifold, and the line bundle sometimes called ample line bundle), we have by the Kodaira embedding theorem, a map ϕ : M CP N such that ϕ : x [S 0 (x) : : S N (x)] The Bergman kernel can be used in the following way: Lemma 4.1. ϕ ω F S = ω + 1 log B k where ω F S is the Fubini-Study metric. Proof. On the subset of M where S 0 0, we have ϕ ω F S = ( 1 log 1 + 2 S 1 S 0 + + S dk S 0 ) = ( 1 log 1 + S 1 2 h S 0 2 + + S d k 2 h h S 0 2 h = 1 log(b k ) 1 log( S 0 2 h ) = 1 log(b k ) + ω We can use the asymptotic expansion to prove Tian s Theorem: Theorem 4.2. For large k, an orthonormal basis of H 0 (M, L k ) gives a map ϕ k : M CP d k 1 and 1 k ϕ k ω F S ω C = O( 1 k 2 ) According to the above theorem, any Kähler metric in c 1 (L) can be approximated by algebraic metrics obtained as pullbacks of Fubini-Study metrics under projective embeddings. Another application of the asymptotic expansion gives us the special case of the Hirzebruch- Riemann-Roch Theorem: Theorem 4.3. As k, we have dim H 0 (M, L k ) = k n M 2 ) ω n n! + kn 1 ρ(ω) ωn 2 M n! + O(kn 2 ) References [1] J. Jost. Riemannian Geometry and Geometric Analysis. Universitext. Springer, 2008. [2] Zhiqin Lu and Gang Tian. Lecture notes in kähler geometry. preprint, 2014. [3] Jian Song and Ben Weinkove. An introduction to the Kähler-Ricci flow. In An introduction to the Kähler-Ricci flow, volume 2086 of Lecture Notes in Math., pages 89 188. Springer, Cham, 2013. [4] Gábor Székelyhidi. An introduction to extremal Kähler metrics, volume 152 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2014. [5] Ben Weinkove. The kähler-ricci flow on compact kähler manifolds. preprint, 2013. E-mail address: shoos@uci.edu