HODGE NUMBERS OF COMPLETE INTERSECTIONS

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1 HODGE NUMBERS OF COMPLETE INTERSECTIONS LIVIU I. NICOLAESCU 1. Holomorphic Euler characteristics Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p dim C X we have a sheaf Ω p (E) whose sections are holomorphic (p, 0)- forms with coefficients in E. We set H p,q (X, E) : H q (X, Ω p (E) ), h p,q (X, E) : dim C H p,q (X, E), and we define the holomorphic Euler characteristics χ p (X, E) : q 0( 1) q h p,q (X, E). It is convenient to introduce the generating function of these numbers χ y (X, E) : p 0 y p χ p (X, E). Observe that Ω p (E) Ω 0 (Λ p T X 1,0 ) h p,q (X, E) h 0,q (X, Λ p T X 1,0 E) and χ p (X, E) χ 0 (X, Λ p T X 1,0 E). If E is the trivial holomorphic line bundle C then we write χ y (X) instead of χ y (X, C). Observe that χ y (X) y 1 ( 1) p+q h p,q (X) χ(x), p,q Hence for n 1 we have while for n we have χ n p (X) ( 1) n χ p (X). χ(x) χ 0 (X), χ(x) χ 0 (X) χ 1 (X). Example 1.1. X P N then { h p,q (P N 1 if 0 p q N ) 0 if p q Hence χ p (P N ) ( 1) p, ; χ y (P N ) N p0. y p yn+1 1 y 1. Date: March,

2 LIVIU I. NICOLAESCU. The Riemann-Roch-Hirzebruch formula The main tool for computing the holomorphic Euler characteristics χ p (X, E) is based on the following fundamental result. Theorem.1 (Riemann-Roch-Hirzebruch). where χ 0 (X, E) td(x) ch(e), [X], td(x) k 0 td k (X), td k (X) H k (X, Q), denotes the Todd genus of the complex tangent bundle of X, ch(e) k 0 ch k (E), ch k (E) H k (X, Q), denotes the Chern character of E and, denotes the Kronecker pairing between cohomology and homology. We have the following immediate corollary. Corollary.. χ p (X, E) χ 0 (X, Λ p T X 1,0 E) td(x) ch(λ p T X 1,0 ) ch(e), [X]. For a vector bundle V X we define ch y (V ) p 0 y p ch(λ p V ). For simplicity we set T X T X 1,0. The result in the previous corollary can be rewritten as χ y (X, E) td(x) ch y (T X) ch(e), [X]. To use this equality we need to recall a few basic properties of the Todd genus and the Chern character of a complex vector bundle. Proposition.3. (a) Suppose L X is a complex line bundle and set x c 1 (L) H (X, Z). Then (b) If td(l) x 1 e x x/ ex/ sinh(x/), ch(l) ex, ch y (L) 1 + ye x. 0 E 0 E E 1 0. is a short exact sequence of complex vector bundles then td(e) td(e 0 ) td(e 1 ), ch(e) ch(e 0 ) + ch(e 1 ), Moreover ch y (E 0 E 1 ) ch y (E 0 ) ch y (E 1 ). ch(e 0 E 1 ) ch(e 0 ) ch(e 1 ).

3 HODGE NUMBERS OF COMPLETE INTERSECTIONS 3 Example.4. Suppose X P N. We want to compute td(x) and ch y (T X 1,0 ). Denote by H P N the hyperplane line bundle. Its sections can be identified with linear maps C N+1 C. We denote its first Chern class by h. As is known we have the equality h N, [P N ] h N 1 P N and an isomorphisms of rings H (P N, Z) Z[h]/(h N+1 ). The dual H of H is the tautological line bundle which is a subbundle of the trivial bundle C N+1 P N. We denote by Q the quotient C N+1 /H. The tangent bundle of P N can be identified with Hom(H, Q) H Q. By tensoring the short exact sequence with H we obtain the short exact sequence 0 H C N+1 Q 0 0 C H N+1 T X 0 and ( td(h N+1 ) td(c) td(t X) td(t X) td(h) N+1 h ) N+1. (.1) 1 e h Similarly, from the sequence 0 T X 1,0 (H ) N+1 C 0 (1 + y) ch y (T X) ch y (H ) N+1 (1 + ye h ) N+1 ch y (T X) (1 + ye h ) N+1. (.) 1 + y 3. Hodge numbers of hypersurfaces in P N. Consider the line bundle mh H m on X P N. Its sections can be identified with degree m homogeneous polynomials in the variables (z 0,, z N ). For a generic section s the zero set is a smooth hypersurface of degree m. We denote it by Z. We want to compute h p,q (Z). We follow closely the approach in [1]. For this we need to use the following fundamental result. Theorem 3.1 (Lefschetz). For k < N 1 1 dim R Z the induced morphism H k (P N, C) H k (Z, C) is an isomorphism. Moreover the morphism H N 1 (P N, Z) H N 1 (Z, C) is one-to-one. We deduce that for k < (N 1) b k (Z) b k (P N ) { 1 if k Z 0 if k Z + 1

4 4 LIVIU I. NICOLAESCU Using the equalities b k (Z) k h p,k p (Z) h p,q (Z) h p,q (P N ), p + q < (N 1). Hence if we set ν (N 1) dim C Z p0 χ 0 (Z) 1 ( 1) ν h 0,ν (Z), χ 1 (Z) 1 + ( 1) ν 1 h 1,ν 1, etc. Thus to compute the Hodge numbers it suffices to compute χ y (Z). The first Chern class of mh is mh ch(mh) e mh. If we denote by N Z Z the normal bundle of the embedding Z X then we have the adjunction formula (mh) Z NZ and a short exact sequence 0 T Z T X Z (mh) Z 0. Using Proposition.3(b), and the identities (.1) and (.) that ( h ) N+1 td(z) td((mh) Z ) td(x) Z td(z) Z 1 e mh 1 e h Z mh and ch y (T Z) ch y (T X) Z ch y ( mh) 1 Z (1 + ye h ) N+1 (1 + y)(1 + ye mh ) Z. Hence χ y (Z) td(z) ch y (T Z), [Z] ( h ) N+1 Z 1 e mh (1 + ye h ) N+1 1 e h Z mh (1 + y)(1 + ye mh ) Z, [Z]. Since the cohomological class Poincaré dual to [Z] in X is mh ( h ) N+1 1 e mh (1 + ye h ) N+1 χ y (Z) 1 e h mh (1 + y)(1 + ye mh ) mh, [X] h N+1( 1 + ye h ) N+1 1 e mh 1 e h (1 + y)(1 + ye mh ), [X] We deduce that χ y (Z) can be identified with the coefficient of z 1 in the Laurent expansion of the function ( 1 + ye z ) N+1 1 e mz z 1 e z (1 + y)(1 + ye mz ). This coefficient can be determined using the residue formula χ y (Z) 1 ( 1 + ye z ) N+1 1 e mz πi 1 e z (1 + y)(1 + ye mz ) dz. z ε If we make the change in variables ζ 1 e z e z 1 ζ, e mz (1 ζ) m, e z dz dζ dz 1 1 ζ dζ

5 HODGE NUMBERS OF COMPLETE INTERSECTIONS 5 that χ y (Z) is given by χ y (Z) 1 ( 1 + y(1 ζ) ) N+1 1 (1 ζ) m πi ζ (1 + y)(1 ζ)(1 + y(1 ζ) m ) dζ, C ε where C ε is a small closed path with winding number around 0 equal to 1. This residue is equal to the coefficient of ζ N in the ζ-taylor expansion of the function ( ) N y(1 ζ) 1 (1 ζ) m f N (y, ζ) (1 ζ)(1 + y) 1 + y(1 ζ) m For our purposes it is perhaps more productive to understand the y-expansion on f N (y, ζ) f N (y, ζ) p 0 f N,p (ζ)y p, and then compute the coefficient of ζ N in f N,p (ζ). Set u (1 ζ). We have 1 (1 ζ) m (1 + y)(1 + y(1 ζ) m ) 1 u m ( (1 + y)(1 + u m y) (1 um ) ( 1) i y i)( ( 1) j u mj y j) (1 u m ) ( 1) k( k u mj) y k ( 1) k (1 u m(k+1) )y k. k 0 j0 k 0 N+1 ( ) N + 1 (1 + uy) N+1 u j y j. j From the equalities f N,p (ζ)y p f y (ζ) 1 ( ( 1) k (1 u m(k+1) )y k)( N+1 ( N + 1 )u j y j) u j Note that p 0 f N,p (ζ) 1 u k 0 j0 i 0 j0 p ( ) N + 1 ( 1) k (1 u m(k+1) )u p k. p k k0 f N,0 (ζ) (1 um ) u 1 (1 ζ)m (1 ζ) j ζ (1 ζ)m 1. f N,1 (ζ) (N + 1)(1 u m ) (1 um ) (N + 1) ( 1 (1 ζ) m ) (1 ζ) 1 + (1 ζ) m 1. u The holomorphic Euler characteristic χ p (Z) is the coefficient of ζ N in f N,p (ζ). For any power series a(x) n 0 a nx n we set T n (a(x)) : a n. We want to discuss a few special cases. N. In this case Z is a plane curve and we have χ 0 (Z) 1 h 0,1 (Z) 1 χ(z). Then T (f,0 ) χ 0 (Z) T (1 ζ) 1 T (1 ζ) m 1 (m 1)(m ) 1 We deduce h 0,1 (Z) h 1,0 (Z) (m 1)(m ) 1 b 1(Z)

6 LIVIU I. NICOLAESCU Z is a Riemann surface of genus (m 1)(m ). Note that if m N we have h N 1,0 h 1,0 1, Z is an elliptic curve. Z is a 1-dimensional Calabi-Yau manifold. N 3 Z is an algebraic surface. We have We have χ 0 (Z) 1 + h 0, (Z), χ 1 (Z) h 1,1 (Z), χ (Z) χ 0 (Z) h,0 (Z) + 1. ( ) m 1 χ 0 (Z) T 3 (f 3,0 ) T 3 (1 ζ) 1 T 3 (1 ζ) m (m 1)(m )(m 3) 1 + χ 1 (Z) T 3 (f 3,1 ) 4T 3 (1 ζ) m T 3 (1 ζ) 1 + T 3 (1 ζ) m 1 ( ) ( ) m m m(m 1)(m ) Hence h,0 (Z) h 0, (Z) (m 1)(m )(m 3) (m 1)(m )(m 3), 4m3 1m + 14m. h 1,1 (Z) 4m3 1m + 14m, ( ) ( ) ( ) m 1 m m 1 b (Z) h 0, + h 1,1 (Z) m 3 4m + m Note that if m (N + 1) 4 we have h N 1,0 h,0 1, h 1,1 0. In this case we say that X is a K3 surface. It is a -dimensional Calabi-Yau manifold. 4. Hodge numbers of complete intersection curves in P 3 Consider two generic hypersurfaces Z m, Z n P 3 of degree m and respectively n. Their intersection is a curve C C m,n. We want to compute its genus. Denote by N C the normal bundle of the embedding C m,n P 3. N C C is a rank vector bundle over C and we have the adjunction isomorphism N C (mh nh) C. Hence we obtain a short exact sequence 0 T C (T P 3 ) C (mh nh) C c 1 (T C) ch(t C) ch(t P 3 ) C ch(mh) C ch(nh) C. From the exact sequence 0 C H 4 T P 3 0 ch(t P 3 ) 4e h c 1 (T C) (4e h 1 e mh enh) C

7 HODGE NUMBERS OF COMPLETE INTERSECTIONS 7 e(c) c 1 (T C) (4 m n)h C. We deduce that χ(c) e(c), [C] (4 m n)h C, [C]. Since [C] is the homology class Poincaré dual to mnh n 0 χ(c m,n ) (4 m n)mnh 3, [P 3 ] (m + n 4)mn g(c m,n ) h 1,0 (C m,n ) 1 + mn(m + n 4). Remark 4.1. If X Xm n 1,,m k P n+k is a generic intersection of k hypersurfaces of degrees m 1,, m k (so dim C X n) then χ y (Xm n 1,,m k )z n+k 1 k (1 + zy) m i (1 z) m i (1 + zy)(1 z) (1 + zy) m. i + y(1 z) m i For a proof, we refer to [1, Appendix I]. References [1] F. Hirzebruch: Topological Methods in Algebraic Geometry, Springer Verlag, New York, 19. Department of Mathematics, University of Notre Dame, Notre Dame, IN address: nicolaescu.1@nd.edu j1