1 2. Chern connections and Chern curvatures1 Let V be a complex vector space with dimC V = n. A hermitian metric h on V is h : V £ V ¡¡! C such that h(av; bu) = abh(v; u) h(a1v1 + a2v2; u) = a1h(v1; u) + a2h(v2; u) h(v; u) = h(u; v) h(u; u) > 0; u 6= 0 where v; v1; v2; u 2 V and a; b; a1; a2 2 C. If we ¯x a basis feig of V , and set hij = h(ei; ej) then ¤ ¤ ¤ ¤ h = hijei ­ ej 2 V ­ V ¤ ¤ ¤ ¤ where ei 2 V is the dual of ei and ei 2 V is the conjugate dual of ei, i.e. X ¤ ei ( ajej) = ai It is obvious that (hij) is a hermitian positive matrix. De¯nition 0.1. A complex vector bundle E is said to be hermitian if there is a positive de¯nite hermitian tensor h on E. r Let ' : EjU ¡¡! U £ C be a trivilization and e = (e1; ¢ ¢ ¢ ; er) be the corresponding frame. The r hermitian metric h is represented by a positive hermitian matrix (hij) 2 ¡(­; EndC ) such that hei(x); ej(x)i = hij(x); x 2 U Then hermitian metric on the chart (U; ') could be written as X ¤ ¤ h = hijei ­ ej For example, there are two charts (U; ') and (V; Ã). We set g = à ± '¡1 :(U \ V ) £ Cr ¡¡! (U \ V ) £ Cr and g is represented by matrix (gij). On U \ V , we have X X X ¡1 ¡1 ¡1 ¡1 ¡1 ei(x) = ' (x; "i) = à ± à ± ' (x; "i) = à (x; gij"j) = gijà (x; "j) = gije~j(x) j j For the metric X ~ hij = hei(x); ej(x)i = hgike~k(x); gjle~l(x)i = gikhklgjl k;l that is h = g ¢ h~ ¢ g¤ 12008.04.30 If there are some errors, please contact to: [email protected] 2 Example 0.2 (Fubini-Study metric on holomorphic tangent bundle T 1;0Pn). On the n @ trivialization of T P , for example U0 = fZ0 6= 0g, and basis of the ¯ber ei = , we could set @zi 2 Pn 2 @ log(1 + i=1 jzi j) hij(z) = hei(z); ej(z)i = @zi@zj Or X h(z) = hijdzi ­ dzj When E is a hermitian vector bundle, there is a natural sequilinear map Ap(M; E) £ Aq(M; E) ¡¡! Ap+q(M; C); (s; t) ¡¡!fs; tg P i P j In local coordinate, if s = σ ­ ei and t = ¿ ­ ej, X X i j i j fs; tg = σ ^ ¿ hei; eji = hijσ ^ ¿ i;j i;j De¯nition 0.3. A connection r on E is compatible with the hermitian structure of E, or a metric connection if for any s 2 Ap(M; E) and t 2 Aq(M; E) we have dfs; tg = frs; tg + (¡1)pfs; rtg Proposition 0.4. If r is a metric connection, then £¤ = ¡£ with respect to any trivialization. Proof. It is obvious that 0 = d2fs; tg = fr2s; tg + fs; r2tg =) £¤ = ¡£ In the local holomorphic coordinates fzig of M, we have @ p @ @ i fs; tg = fr s; tg + (¡1) fs; r tg @z @zi @zi If (E; r; h) is a Hermitian complex vector bundle with a metric connection, there is a decom- position r = r0 + r00, where r0 : Ap;q(M; E) ¡! Ap+1;q(M; E); r00 : Ap;q(M; E) ¡! Ap;q+1(M; E) under the decomposition M Al(M; E) = Ap;q(M; E); p+q=l and r0(f ^ s) = @f ^ s + (¡1)deg f f ^ r0s; r00(f ^ s) = @f ^ s + (¡1)deg f f ^ r00s: 3 i In the local holomorphic coordinates fz gi=1;¢¢¢ ;n on M, and local holomorphic frame fe®g®=1;¢¢¢ ;r, we set re = (¡¯ dzi + ¡¯ dzl) ­ e ® ®i ®l ¯ then r0e = ¡¯ dzi ­ e ; r00e = ¡¯ dzl ­ e ® ®i ¯ ® ®l ¯ The formal adjoint operators are denoted by ±0 : Ap+1;q(M; E) ¡! Ap;q(M; E); ±00 : Ap;q+1(M; E) ¡! Ap;q(M; E) with respect to the inner product induced by h. The Laplacian operators are de¯ned by ¢0 = r0±0 + ±0r0; ¢00 = r00±00 + ±00r00; ¢ = r± + ±r For the metric connection r on the Hermitian complex vector bundle (E; h), we have the decom- position of the curvature £ = £2;0 + £1;1 + £0;2 where £2;0 = r02 2 ¡(M; ¤2;0T ¤M ­ E¤ ­ E) £1;1 = r0r00 + r00r0 2 ¡(M; ¤1;1T ¤M ­ E¤ ­ E) £2;0 = r002 2 ¡(M; ¤0;2T ¤M ­ E¤ ­ E) i In the local holomorphic coordinates (z ) of M and coordinates (e®) of E, we can write £2;0 = R¯ dzi ^ dzj ­ e®¤ ­ e ; £0;2 = R¯ dzi ^ dzj ­ e®¤ ­ e ij® ¯ ij® ¯ £1;1 = R¯ dzi ^ dzj ­ e®¤ ­ e ij® ¯ Here and henceforth we sometimes adopt the Einstein convention for summation. On the holomorphic vector bundle E ¡!¼ M, we can de¯ne the @-operator: @ : Ap(M; E) ¡! Ap+1(M; E) p If we take the holomorphic frame e = (e1; ¢ ¢ ¢ ; en) for E over U, then write s 2 A (M; E) as X i s = σ ­ ei and set i @s = @σ ­ ei Proposition 0.5. The @-operator is well de¯ned. 0 0 0 Proof. If e = (e1; ¢ ¢ ¢ ; en) is any other holomorphic frame of E over U, with 0 ei = gijej i i0 0 i0 P j If s = σ ­ ei = σ ­ ei, then σ = j gjiσ then we have i0 0 j 0 j 0 j @s = @σ ­ ei = @(gjiσ ) ­ ei = gij@σ ­ ei = @σ ­ ej since gij is holomorphic. 4 Lemma 0.6. If E is a hermitian holomorphic vector bundle, then there exists a unique connection such that: i. r is compatible with the complex structure, i.e. r00 = @ ii. r is compatible with the hermitian structure, i.e. dfs; tg = frs; tg + (¡1)deg sfs; rtg for any s; t 2 A²(M; E). Proof. Let e = (e1; ¢ ¢ ¢ ; er) be a holomorphic frame for E under some trivialization, and let h®¯ = he®; e¯i. If the metric connection r exists, then its connection matrix ! must be type (1, 0), for the (0; 1) part of ! for the @ is zero, i.e. ¡¯ = 0. Then we have: ®l γ γ dh®¯ = dhe®; e¯i = !®hγ¯ + h®γ!¯ so we have γ γ γ¯ @h®¯ = !®hγ¯; i:e: !® = @h®¯h γ γ γ¯ @h®¯ = h®γ!¯; i:e: !® = @h®¯h If we set ! = h¡1 ¢@h, we know ! satis¯es the condition above. Then the connection is determined by !. Remark 0.7. Here h¡1 = (h®¯) which is the inverse of ht, that is X ®γ γ h®¯h = ±¯ Now we compute the curvature in the local coordinates. In the local holomorphic coordinates i fz gi=1;¢¢¢ ;n on M, and local holomorphic frame fe®g®=1;¢¢¢ ;r, we have X X £ = £¯ e®¤ ­ e = R¯ dzi ^ dzj ­ e®¤ ­ e 2 ¡(M; ¤1;1T ¤M ­ E¤ ­ E) ® ¯ ij® ¯ ®;¯ By the relation £ = d! ¡ ! ^ !, we get ¯ ¯ γ ¯ £® = d!® ¡ !® ^ !γ On the other hand @h !¯ = h¯γ ®γ dzi ® @zi we get @2h @h @h @h @h d!¯ = ¡h¯γ( ®γ ¡ h±¹ ®¹ ±γ )dzi ^ dzj + hγ± ®± h¯¸ γ¸ dzi ^ dzj ® @zi@zj @zi @zj @zi @zj and @h @h !γ ^ !¯ = hγ± ®± h¯¸ γ¸ dzi ^ dzj ® γ @zi @zj then @2h @h @h R¯ = ¡h¯γ( ®γ ¡ h±¹ ®¹ ±γ ): ij® @zi@zj @zi @zj where we use the formula ®¯ ®γ ±γ @h = ¡h @h±γh 5 Here use the formal expression, we have ! = h¡1 ¢ @h then d! = @(h¡1@h) + @h¡1 ^ @h For ! ^ ! = @h¡1 ^ @h Then £ = @(h¡1 ¢ @h) If r = 1, that is E is a line bundle, then £ = @@ log h Now we set R = Rγ h ij®¯ ij® γ¯ then @h @h h R = ¡ ®¯ + h±γ ®γ ±¯ ij®¯ @zi@zj @zi @zj Theorem 0.8 (Normal frames on holomorphic vector bundles). For every point x0 of X and every local holomorphic coordinates (z1; ¢ ¢ ¢ ; zn) at x0, there exists a holomorphic frame (e1; ¢ ¢ ¢ ; er) in a neighborhood ­ of x0 such that Xn 3 he¸(z); e¹(z)i = ±¸¹ ¡ Rjk¸¹zjzk + O(jzj ) j;k=1 E where the (Rjk¸¹) are the coe±cients of the Chern curvature tensor £x0 under the above local frame. Proof. By linear transformations. Lemma 0.9. If L is a holomorphic line bundle over a complex manifold M with trivialization ¤ functions 'i : LjUi ¡¡! Ui £ C and transition functions gij : Ui \ Uj ¡¡! C . + ¡1 2 (1) If ( ; )L is a hermitian metric on L and ¹j : Uj ¡¡! R is given by ¹j(p) = k'j (p; 1)k . Then 2 ¹j = ¹ijgijj ; on Ui \ Uj + (2) Any collections ¹i : Ui ¡¡! R satis¯es 2 ¹j = ¹ijgijj ; on Ui \ Uj de¯nes a hermitian metric on L. (3) The unique Chern connection on L has the curvature p £ = ¡ ¡1@@ log ¹i on Ui 6 Proof. (1) If p 2 Ui \ Uj, ¡1 2 ¡1 ¡1 2 ¹j(p) = khj (p; 1)k = khi ± hi ± hj (p; 1)k ¡1 2 2 ¡1 2 = khi (p; gij(p))k = jgij(p)j khi (p; 1)k 2 = ¹i(p)jgij(p)j ¡1 (2) If p 2 Ui and v1; v2 2 Lp, then vi = 'j (p; ti) for i = 1; 2. We can de¯ne a hermitian metric ( ; )L on L by (v1; v2)L = ¹i(p)t1t2 It is a well-de¯ned hermitian metric. In fact, if p 2 Ui \ Uj, by the de¯nition of line bundles, 0 0 there exists t1; t2 such that ¡1 ¡1 0 vi = 'i (p; ti) = 'j (p; ti) for i = 1; 2 where 0 ti = gij(p)ti By de¯nition 0 0 ¡2 (v1; v2)L = ¹j(p)t1t2 = ¹j(p)jgij(p)j t1t2 = ¹i(p)t1t2 So the hermitian metric is well-de¯ned on L.