Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016

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1 Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces and projective submanifolds. We conclude by establishing a relationship between the negativity of a line bundle and the non-existence of holomorphic sections. Contents 1 Kähler manifolds 1 2 Holomorphic line bundles 4 3 Positive line bundles 7 4 First Chern class 9 5 Ricci curvature 11 6 Riemann surfaces 12 7 Subvarieties of projective space 14 8 Connections on vector bundles 17 9 Curvature First Chern class and sections 22 1 Kähler manifolds We follow somewhat closely the notation of [GH]. Throughout, denotes a complex manifold of complex dimension n. This means in particular, there is an open cover U α of together with continuous maps ϕ α : U α C n mapping U α homeomorphically onto an open subset of C n in such a way that the transition maps ϕ α ϕ 1 β : ϕ β(u α U β ) ϕ α (U α U β ) are biholomorphisms between open subsets of C n. 1

2 Example 1.1. For example, the space C n is a complex manifold of dimension n with the usual global coordinates. Example 1.2. Let CP n denote the set of complex lines through the origin in C n+1. For a point (Z 0,..., Z n ) in C n+1 \ {0}, we let [Z 0,..., Z n ] denote the line it determines. If U j denotes the subset of CP n given by U j = {[Z 0,..., Z n ] : Z j 0} then we have complex coordinates on U j given by ϕ j : U j C n ( ) Z 0 [Z 0,..., Z n ],..., Ẑj,..., Z n. Z j Z j Z j Example 1.3. Suppose that f 1,..., f k C[x 0,..., x n ] are homogeneous polynomials and V is the welldefined subset of CP n given by V = {[Z] = [Z 0,..., Z n ] : f(z) = 0}. If V enjoys the structure of a smooth submanifold of CP n, then V is a complex submanifold by the implicit function theorem [GH]. We call such a complex submanifold a projective submanifold. Example 1.4. If is a complex manifold, then there is an endomorphism J : T T of the tangent bundle of defined in the following way. If (z 1,..., z n ) are local complex coordinates on, write z j = x j + 1y j to obtain local real coordinates on. Then let J be the unique linear map satisfying J ( ) x j = y j J ( ) y j = x j. It follows that J does not depend on the choice of local coordinates, and hence defines a global section of End(T ). oreover, it is apparent that J 2 = id. Thus, if we extend J to an endomorphism of the complexified tangent bundle T C = T C, then we obtain a decomposition T C = T 1,0 T 0,1 where T 1,0 is the +i eigenspace for J and T 0,1 is the i eigenspace for J. We call T 1,0 the holomorphic tangent bundle, and it enjoys the structure of a complex manifold with local coordinates given by {z j, / z j } where z j = 1 ( 2 x j 1 ) y j. 2

3 The complexified cotangent bundle Ω 1 = T C = T C decomposes similarly into Ω 1 = Ω 1,0 Ω 0,1 where Ω 1,0 = (T 0,1 ) and Ω 0,1 = (T 1,0 ). We call Ω 1,0 the holomorphic cotangent bundle, and it enjoys the structure of a complex manifold with local coordinates {z j, dz j } where dz j = dx j + 1dy j. Let g be a Riemannian metric on. Suppose that g is hermitian in the sense that g(jx, JY ) = g(x, Y ) for each pair of vector fields X, Y on. Extend g to a metric on the complexified tangent bundle T C. One can readily check that g is completely determined by the functions ( ) g kj = g z k, z j and in fact a local expression for g is Define a (1, 1)-form ω by the rule g = g kj dz j d z k. ω(x, Y ) = g(jx, Y ). Then ω is compatible with J in the sense that ω(jx, JY ) = ω(x, Y ) and a local expression for ω is given by ω = 1g kj dz j d z k. Definition 1.5. We say a hermitian metric g on is Kähler if the corresponding (1, 1)-form ω is closed, that is, dω = 0. Note 1.6. A closed (1, 1)-form ω can determine a Kähler metric in the following way. Suppose that ω is compatible with J in the sense that ω(jx, JY ) = ω(x, Y ). If the (0, 2)-tensor defined by g(x, Y ) = ω(x, JY ) is symmetric and positive definite, then the metric g is Kähler. Example 1.7. The complex manifold C n with the metric determined by ω = 1 j dzj d z j is Kähler. Example 1.8. We describe a (1, 1)-form on CP n called the Fubini-Study metric ω F S. If U is an open subset of CP n and Z : U C n+1 \0 is a local holomorphic section of the projection map π : C n+1 \0 CP n, 3

4 we let ω F S be the (1, 1)-form with local expression 1 log Z 2. This does not depend on the choice of holomorphic section Z because if Z is another such section, then Z = fz for some local holomorphic C-valued function f, and we find that 1 log Z 2 = 1 log Z log f 2, but the last expression 1 log f 2 is zero since f is holomorphic. It follows that ω F S exists globally. Now the group U(n + 1) acts transitively on CP n and preserves the form ω F S. Thus to check whether the corresponding symmetric bilinear form is positive definite, it suffices to check at one point, such as [1, 0,..., 0]. In the coordinate chart U 0, the form ω F S has local expression ω F S = 1 log(1 + z 2 ) where z = (z 1,..., z n ) are the coordinates on U 0. At the point z = 0, this is 1 j dzj d z j. Hence the corresponding hermitian matrix at this point is the identity matrix, which is positive definite. Example 1.9. If V is a complex submanifold of a Kähler manifold (, ω), then the induced metric on V is Kähler as well, since the differential d commutes with the pullback i corresponding to the inclusion i : V. 2 Holomorphic line bundles Definition 2.1. Let be a complex manifold. A holomorphic line bundle over consists of a family {L p } p of one-dimensional complex vector spaces together with the structure of a complex manifold on the disjoint union L = p L p in such a way that the natural projection map π : L is holomorphic and such that for each point p, there is an open neighborhood U of p in and a biholomorphism ϕ : π 1 (U) U C which is linear on each fiber L p for p U and which commutes with the projection maps π = proj 1 ϕ, where proj 1 : U C U is the projection onto the first factor. Definition 2.2. A smooth section s : L is called holomorphic if s is holomorphic as a map of complex manifolds. We let H 0 (, L) denote the space of holomorphic sections. Note 2.3. If L is a holomorphic line bundle, then there is an open cover U α of together with biholomorphisms ϕ α : π 1 (U α ) U α C which are linear on the fibers and which commute with the projection maps. Thus, on the intersections U α U β, we obtain holomorphic maps called transition functions 4

5 g αβ : U α U β GL(1, C) = C satisfying ϕ β ϕ 1 α (p, v) = (p, g βα (p)v) for p U α U β and v C. The collection {g αβ } satisfies the cocycle relations g αβ g βγ = g αγ g αα = 1, and hence determines an element of the Čech cohomology group H1 (, O ). Conversely, it can be shown that any 1-cocyle {g αβ } determines a holomorphic line bundle, and two such cocycles determine isomorphic line bundles if and only if they differ by an exact 1-cycle. Thus, we obtain an identification of the group of isomorphism classes of holomorphic line bundles with H 1 (, O ). Definition 2.4. If L is a holomorphic line bundle over, then there is a holomorphic line bundle L 1 over which is dual to L in the sense that each fiber L 1 p functions for L, then transition functions for L 1 are given by g 1 αβ. is the dual vector space to L p. If g αβ are transition Note 2.5. Suppose L has transition functions g αβ with respect to an open cover U α of with trivializations ϕ α. If s is a holomorphic section, then the function ϕ α s : U α U α C can be expressed as ϕ α s(p) = (p, s α (p)) for some holomorphic function s α : U α C. oreover, from the definitions of the transitions functions g αβ, we find that s β = g βα s α on the overlap U α U β. Conversely, given a collection of holomorphic functions s α : U α C satisfying s β = g βα s α on the overlaps, we obtain a holomorphic section of L. Example 2.6. We describe a line bundle O( 1) on CP n in the following manner: let O( 1) be the subbundle of the trivial bundle CP n C n+1 whose fiber over the point determined by the line l is the line l itself. If U j denotes the chart on CP n given by U j = {[Z 0,..., Z n ] : Z j 0}, then it can be shown that transition functions for the line bundle O( 1) relative to these charts are g jk ([Z 0,..., Z n ]) = Z j Z k. We let O(1) denote the dual to O( 1). oreover, by taking tensor powers, we obtain holomorphic line bundles O(l) for each l in Z. One can show that the transition functions for O(l) satisfy g (l) jk ([Z 0,..., Z n ]) = It can be shown that every holomorphic line bundle on CP n is of the form O(l) for some l, but a proof of ( Zk Z j ) l. 5

6 this is beyond the scope of these notes. The following characterization of the space H 0 (CP n, O(l)) of global sections is not too difficult to obtain. For l < 0, we have H 0 (CP n, O(l)) = 0. For l 0, the space H 0 (CP n, O(l)) can be identified with the space of homogeneous polynomials of degree l in the n + 1 variables Z 0,..., Z n. Indeed, let us consider the case when n = 1 and l = 1. In this case, a global section of O(1) consists of two holomorphic functions s 0 : U 0 C and s 1 : U 1 C satisfying s 1 = ( Z0 Z 1 ) s 0. A holomorphic function s 1 : U 1 C has a taylor series expansion s 1 = a 0 + a 1 Z 0 Z 1 + a 2 ( Z0 Z 1 ) 2 +. We can also write The relation between s 0 and s 1 implies that s 0 = b 0 + b 1 Z 1 Z 0 + b 2 ( Z1 Z 0 ) 2 +. b 0 Z 0 Z 1 + b 1 + b 2 Z 1 Z 0 + b 3 ( Z1 Z 0 ) 2 + = a 0 + a 1 Z 0 Z 1 + a 2 ( Z0 Z 1 ) 2 +. We thus find that b 0 = a 1 and b 1 = a 0, and all other a j, b k are zero. If we set f = a 1 Z 0 + a 0 Z 1, then we find that s 1 = f/z 0 and s 0 = f/z 1. Conversely, any such f determines a holomorphic section of O(1). Example 2.7. If has dimension n, the top exterior power K := Λ n Ω 1,0 of the holomorphic cotangent bundle is a holomorphic line bundle on called the canonical bundle of. Locally a trivialization of K is given by dz 1 dz n where (z 1,..., z n ) are complex coordinates on. Example 2.8. We claim that the canonical bundle of CP n is isomorphic to O( n 1). In the chart U 0, we use the coordinates z j = Z j Z 0, j = 1,..., n. A local frame for K CP n in these coordinates is given by dz 1 dz n, which gives a trivialization of K CP n on the chart U 0. For i 0, we use the coordinates z j = Z j Z i, j i. 6

7 A local frame for K CP n in these coordinates is given by ( 1) i dz 0 dz i dz n, which gives a trivialization of K CP n on U i. The coordinates z j on the chart U i satisfy z j = z j z 0, j i z i = 1 z 0. This implies that dz j = dz j z 0 z j dz 0 (z 0 )2, j i dz i = dz 0 (z 0 )2. It follows that dz 1 dz n has local expression in the chart U i as dz 1 dz n = ( 1) i 1 (z 0 )n+1 dz 0 dz i dz n ( ) n 1 = ( 1) i Z0 dz 0 Z dz i dz n i Thus the transition functions for K CP n relative to the cover U j satisfy g i0 ([Z 0,..., Z n ]) = ( Z0 Z i ) n 1. One can show that more generally, the transition functions satisfy g jk ([Z 0,..., Z n ]) = That is, K CP n is isomorphic to O( n 1). ( Zk Z j ) n 1. 3 Positive line bundles Definition 3.1. A hermitian metric h on a holomorphic line bundle L consists of a family {h p } p of hermitian inner products, each h p an inner product on L p, which is smooth in the sense that for each pair of local smooth sections s 1, s 2 of L, the local C-valued function p h p (s 1 (p), s 2 (p)) is smooth. We sometimes 7

8 write the metric as s 1, s 2 h = h(s 1, s 2 ). We often write s 2 h = s, s h. Definition 3.2. Let L be a holomorphic line bundle with hermitian metric h. The curvature of (L, h) is the (1, 1)-form on with local expression 1 log s 2 h for a local non-vanishing holomorphic section s of L. Remark 3.3. The curvature is well-defined for the following reason. If s is another non-vanishing holomorphic section, then there is a local holomorphic function f such that s = fs, and hence s 2 h = f 2 s 2 h. We then find that 1 log s 2 h = 1 log s 2 h 1 log f 1 log f. Since f is holomorphic, so is log(f) and the second term on the right-hand side is therefore zero. Since f is anti-holomorphic, so is log( f), and hence the third term is 1 log f = 1 log f = 0. Thus 1 log h(s ) = 1 log h(s), meaning that the curvature is independent of the choice of the non-vanishing holomorphic section s. Example 3.4. Let O( 1) denote the tautological line bundle over CP n. Recall that we can view O( 1) as the sub-bundle of the trivial bundle CP n C n+1 whose fiber over the point determined by the line l is the line l itself. The standard Euclidean metric on the trivial bundle induces a metric on O( 1). The curvature of this metric is defined to be 1 log Z 2 where Z is a local non-vanishing holomorphic section of O( 1). It follows immediately that the curvature is given by ω F S where ω F S is the Fubini-Study Kähler form on CP n. Definition 3.5. We say that a (1, 1)-form α on is positive if the 2-tensor defined by (X, Y ) α(x, JY ) is positive definite. A negative (1, 1)-form is defined in an analogous way. Example 3.6. Say that a (1, 1)-form α has local expression α = 1α kj dz j d z k. 8

9 Then α is positive if and only if α kj defines a positive definite matrix. For example, the Fubini-Study form ω F S on CP n is a positive (1, 1)-form. Definition 3.7. A holomorphic line bundle L is called positive if it admits a hermitian metric whose corresponding curvature form is positive. The notion of negative line bundle is defined in an analogous way. Example 3.8. Example 3.4 shows that the line bundle O( 1) over CP n admits a hermitian metric whose corresponding curvature form is negative. Hence O( 1) is a negative line bundle. If O(1) denotes the dual to O( 1), then the curvature form corresponding to the metric dual to the above metric on O( 1) is given by ω F S, and hence is positive. Thus we find that O(1) is a positive line bundle. oreover, the curvature of O(l) with the induced metric is given by l ω F S. It follows that the line bundle O(l) is positive if and only if l is a positive integer. In particular, the canonical bundle K CP n O( n 1) is negative. 4 First Chern class Definition 4.1. Let L be a holomorphic line bundle. The first Chern class c 1 (L) of L is the cohomology class determined by the (1, 1)-form with local expression 1 2π log s 2 h for some hermitian metric h on L and some local holomorphic non-vanishing section s of L. Remark 4.2. The first Chern class does not depend on the choice of hermitian metric for the following reason. If h is another hermitian metric on L, then s 2 h = λ s 2 h for some positive local function λ. We compute that 1 2π log s 2 h = 1 2π log s 2 h 1 2π log λ. But the second term is exact as log λ = ( + ) log λ = d( log λ). Remark 4.3. One can show that c 1 (L) is an integral cohomology class, that is, c 1 (L) H 2 (, Z). But a proof of this fact is beyond the scope of these notes. ore specifically, there is the following interpretation of the first Chern class. The exact sequence of sheaves over 0 Z O exp O 0 gives rise to a long exact sequence in cohomology, and the first connecting homomorphism c 1 : H 1 (, O ) H 2 (, Z) 9

10 is the map taking a line bundle to its first Chern class. Here we identify H 1 (, O ) with the group of isomorphism classes of holomorphic line bundles over. From this viewpoint, it is clear that c 1 satisfies c 1 (L L ) c 1 (L) + c 1 (L ) and c 1 (L ) = c 1 (L) where L denotes the dual to L. The following lemma is a key tool in complex geometry. Lemma 4.4 ( -Lemma). Let be a compact, Kähler manifold. If ω and η are two real (1, 1)-forms representing the same cohomology class, then there is a smooth function f : R such that η = ω + 1 f. It is clear that if such a function f exists, then η and ω represent the same cohomology class. However, the crux of the matter is that there is a converse type of statement showing the existence of such an f. A standard proof of the -lemma involves defining adjoints, and using Hodge theory. Corollary 4.5. If η is a real (1, 1)-form representing the cohomology class c 1 (L), then there is a metric h on L such that 2πη is the curvature of h. Proof. Say that c 1 (L) is represented by ω = 1 2π log s 2 h for a metric h on L. By the -lemma, there is a smooth function f : R such that η = ω + 1 f. Let h be the metric determined by h = e 2πf h. Then note that the curvature of h is 1 log s 2 h = 1 log s 2 h + 2π 1 f = 2πω + 2π 1 f = 2πη. This completes the proof. Definition 4.6. Say that the first Chern class c 1 (L) is positive if c 1 (L) is represented by a positive (1, 1)- form. 10

11 Lemma 4.7. A line bundle L is positive if and only if c 1 (L) is positive. Proof. If L is positive, then there is a hermitian metric h on L whose curvature form is positive, and hence c 1 (L) is represented by a positive (1, 1)-form. Conversely, if c 1 (L) is represented by a positive (1, 1)-form, then the previous lemma shows that there is a metric h on L whose curvature is this positive (1, 1)-form, and hence L is positive. Definition 4.8. For a complex manifold, we define the first Chern class of by the rule c 1 () = c 1 (K 1 ) where K 1 denotes the anti-canonical bundle, that is, the line bundle dual to the canonical bundle K. Example 4.9. The canonical bundle of C n is trivial, and hence the first Chern class c 1 (C n ) is zero. This follows from the isomorphism K 1 C n K C n and then the observation c 1(K 1 C n ) = c 1(K C n) = c 1 (K 1 C n ). Example We have seen that K CP n O( n 1), from which we find K 1 CPn O(n + 1). Example 3.8 shows that there is a metric on O(n + 1) whose curvature form is (n + 1)ω F S. It follows that the first Chern class c 1 (CP n ) = c 1 (K 1 CP n) = c 1(O(n + 1)) is positive. 5 Ricci curvature Let (, g) be a Kähler manifold of dimension n. Viewing g as a metric on the holomorphic tangent bundle T 1,0, we obtain a metric det(g) on the anti-canonical bundle K 1 = Λn T 1,0. Definition 5.1. The Ricci curvature of (, g) is the curvature of det(g) on K 1 : Ric(g) = 1 log det(g). If we are given a Kähler form ω on, we sometimes write Ric(ω) to denote Ric(g), where g is the corresponding Kähler metric. Definition 5.2. A Kähler manifold (, ω) is called Kähler-Einstein if there is a constant k such that Ric(ω) = kω. Theorem 5.3. The Kähler manifold (CP n, ω F S ) is Kähler-Einstein with Ric(ω F S ) = (n + 1)ω F S. Proof. Remember that on the coordinate chart U 0, the Fubini-Study form has local expression ω F S (z 1,..., z n ) = 1 log(1 + z 2 ) = ( i 1 dz ( i d z i i 1 + z 2 z ) ( idz i j z )) j z j (1 + z 2 ) 2. 11

12 It follows that on U 0, the corresponding metric g has local expression g j k = δ j k(1 + z 2 ) z j z k (1 + z 2 ) 2. The matrix z j z k is symmetric of rank one with nonzero eigenvalue given by its trace z 2. Thus, under a suitable linear change of coordinates, we find that g has local expression It follows that det(g) has local expression The Ricci form of ω F S is defined by 1 ( g = (1 + z 2 (1 + z 2 ) 2 )I n diag( z 2, 0,..., 0) ) 1 = (1 + z 2 ) 2 diag(1, 1 + z 2,..., 1 + z 2 ). det(g) = (1 + z 2 ) n 1 (1 + z 2 ) 2n = 1 (1 + z 2 ) n+1. Ric(ω F S ) = 1 log det(g). Using our expression of det(g) above, we find that Ric(ω F S ) has local expression in U 0 as Ric(ω F S ) = (n + 1) 1 log(1 + z 2 ) = (n + 1)ω F S. A similar computation shows the equality Ric(ω F S ) = (n + 1)ω F S in any coordinate chart U j. 6 Riemann surfaces So far, we have only computed the first Chern class of C n and CP n. The goal of this section is to understand the first Chern class of a Riemann surface in terms of its genus. Let be a compact, connected Riemann surface. The fundamental class [] is a generator of the homology group H 2 (, Z) Z. So the linear functional given by pairing with [] describes an isomorphism H 2 (, Z) Z. Definition 6.1. Define the degree of a line bundle L to be the integer c 1 (L), []. In other words, the degree of L is equal to c 1 (L) under the isomorphism H 2 (, Z) Z. To proceed further, we need the following classical result. Theorem 6.2 (Gauss-Bonnet). Let be a compact Riemann surface with Gaussian curvature k and volume form Φ. Then k Φ = 2πχ() = 2π(2 2g) 12

13 where χ() denotes the Euler characteristic of and g denotes the genus of. This result is somewhat surprising, since the left-hand side seems to depend upon the choice of Riemannian metric, while the right-hand side is expressed purely in terms of topological invariants of. Theorem 6.3. Let be a compact Riemann surface with Gaussian curvature K and Ricci curvature Ric. Then Ric = K Φ. Proof. Say that the metric is h 2 dz d z. We then compute Ric = 1 log(h 2 ) = 2 1 log(h) = 2 ( ) 2 1 z z log(h) dz d z = log(h)dz d z ( ) ( ) log(h) 1 = 1h 2 h 2 dz d z 2 = k Φ, where is the usual Gaussian curvature and k = log(h) h 2 Φ = 1 2! 1h 2 dz d z is the volume form corresponding to the metric. Corollary 6.4. For a compact, Riemann surface of genus g, we have deg(k ) = χ() = 2g 2. Hence If g > 1, then c 1 () < 0. If g = 1, then c 1 () = 0. If g = 0, then c 1 () > 0. Proof. The first Chern class of satisfies c 1 () = c 1 (T 1,0 ) = c 1 (K 1 ) = [ 1 2π Ric]. 13

14 By Gauss-Bonnet, the degree of K 1 is The result follows. deg(k 1 ) = 1 Ric = χ() = 2 2g. 2π 7 Subvarieties of projective space To conclude our list of examples, we study the first Chern class c 1 (V ) of a smooth projective submanifold V CP n. It turns out that we can understand the first Chern class in terms of the degrees of the polynomials defining V. Definition 7.1. For a complex submanifold V, define the normal bundle N V to be the quotient 0 T 1,0 V T 1,0 V N V 0 of the holomorphic tangent bundle T 1,0 V restricted to V. The conormal bundle NV is the dual to the normal bundle. Note that the conormal bundle consists of all cotangent vectors to that vanish on the holomorphic tangent bundle T 1,0 V. Lemma 7.2. For a complex submanifold V of codimension 1, the canonical bundle K V satisfies K V = (K V ) N V Proof. We have an exact sequence of the form 0 T V T V N V 0. Taking duals, we get an exact sequence 0 N 1 V T 1 V T V 1 0. Using the properties of the determinant, we conclude that K V K V N 1 V, as desired. Lemma 7.3. Let L be a holomorphic line bundle with holomorphic global section s such that V = 14

15 s 1 (0) is a complex submanifold. Then N V = L V. Proof. We show that the line bundle N 1 V L V is trivial by constructing a global non-vanishing section. Say that the transition functions of L are g αβ with respect to some open cover U α with trivializations ϕ α, and let s α : U α C be the corresponding local holomorphic functions determining the section s. Because s is constant along V, the functions ds α vanish along the tangent bundle T V. oreover, we note that on (U α U β ) V, we have ds α = d(g αβ s β ) = (dg αβ )s β + g αβ (ds β ) = g αβ (ds β ). It follows that the collection ds α assembles to form a global section of N 1 V L V. oreover, because V is smooth everywhere, none of the ds α ever vanish, and this global section is nonvanishing, as desired. Corollary 7.4. Suppose that V CP n is a projective hypersurface of degree d > 0, that is, V is the zero locus of a global section of O(d). Then K V O(d n 1) V. Hence, If d > n + 1, then the first Chern class of V is positive c 1 (V ) < 0. If d < n + 1, then the first Chern class of V is negative c 1 (V ) > 0. If d = n + 1, then the first Chern class of V is zero c 1 (V ) = 0. Proof. Because V is the divisor of zeros of a global section of O(d), the previous lemma shows that N V = O(d) V. The adjunction formula gives K V (O( n 1) O(d)) V O(d n 1) V. The rest of the claims now follow from the definition of the sign of the first Chern class. Lemma 7.5. ore generally, suppose that V is a complex submanifold of codimension r. Then we have K V K V r N V 15

16 Proof. Again we have an exact sequence 0 N 1 V T 1 V T V 1 0. Again using properties of the determinant, because N 1 V is of rank r, we have K V K V r N 1 V. Rearranging completes the proof. Lemma 7.6. If d 1,..., d r are integers, then r (O(d 1 ) O(d r )) = O(d d r ). Proof. We use induction on r. The claim is obvious when r = 1. For r > 1, we have an exact sequence of the form 0 O(d r ) O(d 1 ) O(d r ) O(d 1 ) O(d r 1 ) 0. Using properties of the determinant and the inductive hypothesis, we conclude that r (O(d 1 ) O(d r )) = r 1 (O(d 1 ) O(d r 1 )) 1 O(d r ) = O(d d r 1 ) O(d r ) = O(d d r ). This completes the inductive step, and the proof. Corollary 7.7. Suppose that V CP n is a projective submanifold of codimension r determined by the transverse intersection of r hypersurfaces of degrees d 1,..., d r. If d = d d r, then K V O(d n 1) V. Hence, If d > n + 1, then the first Chern class of V is positive c 1 (V ) < 0. If d < n + 1, then the first Chern class of V is negative c 1 (V ) > 0. If d = n + 1, then the first Chern class of V is zero c 1 (V ) = 0. Proof. We first claim that the normal bundle to V is (O(d 1 ) O(d r )) V. If V j denotes the hypersurface of degree d j, then the normal bundle to V j is O(d j ) Vj. Since T V T V j for each j, we get natural surjective 16

17 maps N V N Vj for each j and hence a natural map N V N V1 N Vr O(d 1 ) O(d r ), which we claim is an isomorphism. Indeed at each point, the fibers of both are r dimensional, so it suffices to show that the map is injective. Because the intersection is smooth, we have r T p V = T p V j. j=1 If a vector in the fiber of N V over p mapped to zero in the fiber of N V1 N Vr, this would mean that the vector belongs to each T p V j, which would imply that the vector belongs to T p V, and hence is zero in the fiber of N V. Now the adjunction formula implies that K V K n P V Λ r (O(d 1 ) O(d r ) V ) O( n 1) V O(d d r ) V O(d n 1) V. The claims follow. 8 Connections on vector bundles Definition 8.1. A complex vector bundle of rank r over consists of a family {E p } p of r- dimensional complex vector spaces together with the structure of a smooth manifold on the disjoint union E = p E p in such a way that the projection map π : E is smooth and for each point p, there is a neighborhood U of p in and a diffeomorphism ϕ : π 1 (U) U C r which is linear on the fibers and which commutes with the projection maps to U. Definition 8.2. A connection on a complex vector bundle E is a complex linear map D : Ω 0 (E) Ω 1 (E) satisfying Leibniz rule D(fs) for f C (, C) and s Ω 0 (E). = df s + f(ds) Note 8.3. If (z 1,..., z n ) are complex coordinates for, then for each k, a connection D gives rise to a C-linear map D k : Ω 0 (E) Ω0 (E) defined by ( ) D k s = (Ds) z k. 17

18 oreover, the map D k satisfies the Leibniz rule in the sense that D k (fs) = f z k s + f(d ks) for each f C (, C). We obtain similar mappings D k : Ω 0 (E) Ω0 (E) which satisfy analogous Leibniz rule relations. Conversely, if we are given such D k and D k satisfying Leibniz rule, then we obtain a connection D on E defined by Ds = dz k (D k s) + d z k (D ks). Note 8.4. The decomposition Ω 1 = Ω 1,0 Ω 0,1 implies that any connection D on E decomposes as D = D + D where D : Ω 0 (E) Ω 1,0 (E) D : Ω 0 (E) Ω 0,1 (E). Definition 8.5. A hermitian metric h on a complex vector bundle E consists of a family {h p } p of hermitian inner products, each h p an inner product on E p, which is smooth in the sense that for each pair of local smooth sections s 1, s 2 of E, the local C-valued function p h p (s 1 (p), s 2 (p)) is smooth. We sometimes write the metric as s 1, s 2 h = h(s 1, s 2 ). Note 8.6. If h is a hermitian metric on E, then we obtain a bundle morphism h : E E 1 defined in the following way. For a section s of E, we let h(s) : E C denote the bundle map defined by h(s)(t) = s, t h. In this way, we can view h as a section of the bundle Hom(E, E 1 ). Definition 8.7. A connection D on a hermitian vector bundle (E, h) is called unitary if k h(s 1, s 2 ) = h(d k s 1, s 2 ) + h(s 1, D ks 2 ) kh(s 1, s 2 ) = h(d ks 1, s 2 ) + h(s 1, D ks 2 ) for each pair of sections s 1, s 2 of E. Definition 8.8. If D is a connection on E, then we obtain a connection on Hom(E, E 1 ) defined in the 18

19 following way. For sections s, t of E and a section h of Hom(E, E 1 ), we require k (h(s)(t)) = (D k h)(s)(t) + h(d k s)(t) + h(s)(d k t) k(h(s)(t)) = (D kh)(s)(t) + h(d ks)(t) + h(s)(d kt). Lemma 8.9. A connection D on a hermitian vector bundle (E, h) is unitary if and only if Dh = 0, where we view h as a section of End(E). Proof. The statement Dh = 0 is equivalent to D k h = D kh = 0 for each k. According to the above definition, this is equivalent to k (h(s)(t)) = h(d k s)(t) + h(s)(d k t) with a similar equality for k. But according to the definition of h, this is equivalent to s, t h = D k s, t h + s, D k t h = D k s, t h + s, D k t h, with a similar expression for k. This is clearly equivalent to the requirement that D is unitary. Definition A complex vector bundle E is called holomorphic if E admits the structure of a complex manifold in such a way that the projection map π : E is holomorphic and for each point p, there is a neighborhood U of p in and a biholomorphism ϕ : π 1 (U) U C r which is linear on the fibers and which commutes with the projection maps. Definition A section s : E of a holomorphic vector bundle E is called holomorphic if s is holomorphic as a map of complex manifolds. We let H 0 (, E) denote the space of holomorphic sections of E. Definition If E is holomorphic, then there is a natural map : Ω p,q (E) Ωp,q+1 (E) described in the following way. For a section s Ω p,q (E), write s = s α e α where s α are local smooth (p, q) forms and e α is a local holomorphic frame for E. Then set s = ( s α ) e α. This definition does not depend on the representation of s, because if f α is another holomorphic frame in which s = t α f α, then there are local holomorphic functions g α β on such that f β = g α β e α, and hence ( t β ) f β = ( t β ) g α β e α = (t β g α β ) e α = ( s α ) e α. 19

20 Definition A connection D = D + D on a holomorphic vector bundle E is said to be compatible with the holomorphic structure if D =. Theorem If E is a holomorphic vector bundle with hermitian metric h, then there is a unique connection D on E which satisfies the following two properties. (i) D is unitary (ii) D is compatible with the holomorphic structure. Proof. Assume that D is a connection on E satisfying (i) and (ii). Let {e α } be a local holomorphic frame for E and write De α = θαe β β for some local matrix of 1-forms θα. β Also define local smooth functions h αβ by h αβ = h(e α, h β ). To say that D satisfies (ii) implies that θ β α are of type (1, 0). The fact that D is unitary implies that dh αβ = θ γ αh γβ + θ γ β h αγ. Decomposing into types, we find that h αβ = θ γ αh γβ h αβ = θ γ β h αγ. Writing this in terms of matrices, we find that h = θh h = hθ. This implies that θ = ( h)h 1, hence the connection D is uniquely determined. On the other hand, such a connection D can be shown to exist by defining θ locally by θ = ( h)h 1. Definition The unique connection which is compatible with the metric and the holomorphic structure is called the Chern connection on E, and we often denote this connection by. Note A hermitian metric g on T determines a Chern connection on the holomorphic tangent bundle T 1,0. One can show that the metric is Kähler if and only if the Chern connection agrees with the Levi-Civita connection on T 1,0. 20

21 Note If D 1 is a connection on E 1 and D 2 is a connection on E 2, then the map D described by D = D 1 id E2 + id E1 D 2 is a connection on E 1 E 2. Definition If E is a holomorphic vector bundle and is a Kähler manifold, then we obtain a connection on the bundle T 1,0 E by using the Chern connection on both factors. We denote this connection by when no confusion will arise. 9 Curvature Definition 9.1. We can extend any connection D on E to a C-linear map D : Ω p (E) Ωp+1 (E) by forcing Leibniz rule D(ψ s) = dψ s + ( 1) p ψ (Ds) for ψ Ω p and s Ω 0 (E). The curvature of the connection D is then the composition F D = D 2 : Ω 0 (E) Ω2 (E). Lemma 9.2. The curvature F D is C ()-linear and hence corresponds to a section F D Ω 2 (End(E)). Proof. This is a simple computation using Leibniz rule. If f C () and s is a smooth section of E, then F D (fs) = D(df s + f (Ds)) = ddf s df (Ds) + df (Ds) + f (F D s) = f(f D s) where we used the fact that d 2 = 0. Lemma 9.3. If L is a holomorphic line bundle with hermitian metric h, then the curvature F of the Chern connection has local expression F lk = l k log s 2 h where s is a holomorphic non-vanishing section of L. Proof. Let s be a holomorphic non-vanishing section. The curvature of the Chern connection is of type (1, 1) and satisfies F s = + s = s where we used the fact that s is holomorphic and hence s = s = 0. Say that k s = A k s for some 21

22 smooth functions A k. Then note that s has local expression s = d z l l(dz k ( k s)) = (d z l dz k ) l(a k s) = (d z l dz k )( la k )s. where again we used that s is holomorphic. It follows that F lk has local expression F lk = la k s. However, because is unitary, we know that k s 2 h = ks, s + s, ks = A k s 2 h. It follows that The claim now follows. A k = k s 2 h s 2 h = k log s 2 h. 10 First Chern class and sections We conclude by discussing the implications of a line bundle being negative. Essentially, the result is that such a line bundle has no holomorphic sections at all. Theorem If L is a holomorphic line bundle over a compact Kähler manifold such that c 1 (L) < 0, then there are no nonzero global holomorphic sections of L. We require a lemma about integration by parts on a Kähler manifold. Lemma Let h be a metric on L. Then for each holomorphic section s of L, we have g k l l k s, s h dv = g k l k s, l s h dv. Proof. Let s be a holomorphic section of L. Let g : Ω 1,0 T 0,1 denote the isomorphism determined by the Kähler metric g and let h : L L 1 denote the isomorphism determined by the metric h. Define a vector field X by the rule X = g(h( s)( s)) Note that X has local expression where X = X l z l X l = g k lh( k s)( s). 22

23 Since is compatible with both g and h, note that m X = g(h( m s)( s)) + g(h( s)( m s)). We find that m X has local expression The divergence is then m X = ( g k l m k s, s h + g k l k s, m s h ) z l ( lx) l = g k l l k s, s h + g k l k s, l s h The result now follows from Stoke s theorem: (divx)dv = d(ι X dv ) = 0. We now prove Theorem Proof. Because c 1 (L) < 0, there is a hermitian metric on L whose curvature form F k l = k l log s 2 h is negative definite. The Chern connection satisfies k l = l k + F k l. If s is a holomorphic section, then 0 = g k lh( k ls)s = g k lh( l k + F k ls)s = g k lh( l k )s + g k lf k l s 2 h g k lh( l k )s c s 2 h. for some constant c > 0, because F k l is negative definite and g k l is positive definite. Integrating this over and using the previous lemma gives 0 g k lh( k s) l s dv c s 2 hdv = s 2 g hdv c s 2 hdv. This implies that s 2 h = 0 and we conclude that s = 0 from the positivity of h. 23

24 Lemma Let s be a holomorphic section of a line bundle such that s = 0. Then the norm s 2 h is constant. Proof. We compute that k s, s h = k s, s + s, ks = 0. The result follows. Proposition Let L be a holomorphic line bundle over a compact, Kähler manifold with c 1 (L) = 0. If L is not the trivial bundle, then H 0 (, L) = 0. Proof. The 2-form which is identically zero is a class representing 2πc 1 (L). By Corollary 4.5, there is a metric h on L whose corresponding curvature form is 0. It now follows from the proof of the previous proposition that for a holomorphic section s we have 0 = g k l k ls, s = g k l l k s, s. Integrating over and using integration by parts gives that 0 = s 2 g h dv. Hence the norm of s is constant. If the norm of s is nonzero, then s defines a global non-vanishing section and hence L is trivial. Otherwise, the norm of s is zero, and s is the zero section. References [GH] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York (1978). 24