Methods of Functional Analysis and Topology Vol. 24 (2018), no. 1, pp. 41{52 ADJUNCTION FORMULA, POINCARE´ RESIDUE AND HOLOMORPHIC DIFFERENTIALS ON RIEMANN SURFACES A. LESFARI Abstract. There is still a big gap between knowing that a Riemann surface of genus g has g holomorphic differential forms and being able to find them explicitly. The aim of this paper is to show how to construct holomorphic differential forms 1 on compact Riemann surfaces. As known, the dimension of the space H (D; C) of holomorphic differentials of a compact Riemann surface D is equal to its genus, 1 dim H (D; C) = g(D) = g. When the Riemann surface is concretely described, we show that one can usually present a basis of holomorphic differentials explicitly. We apply the method to the case of relatively complicated Riemann surfaces. 1. Adjunction formula and Poincare´ residue Let M be a compact complex manifold of dimension n and let D ⊂ M be a smooth hypersurface. Let ND be the normal bundle on D which can be defined as the quotient 0 TM jD 0 0 ND = 0 , where TM and TD are the tangent bundles on M and D respectively (The TD 0 sign is added to recall the adjective holomorphic). The dual of ND is called the conormal ∗ 0∗ bundle and we denote it by ND. This is a sub-bundle of the cotangent bundle TM jD; i.e., 0 0 the set of cotangent vectors on M vanishing on TD ⊂ TM jD. We suppose that D is locally defined by functions fα 2 O(Uα). Therefore, the line bundle [D] on M is defined by the fα transition functions gαβ = . Since fα ≡ 0 on D\Uα, dfα is a section of the conormal fβ ∗ bundle ND on D\Uα. Moreover, by assumption, D is smooth so dfα does not vanish on D\Uα. Moreover, we have on Uα \Uβ \D, that dfα = d(gαβfβ) = gαβdfβ, and since D is 1 ∗ smooth, dfα is non-zero on D. So this shows that the sections dfα 2 H (Uα \D; OD(ND)) ∗ determine a global section, which does not vanish on D, of the line bundle ND ⊗ [D]jD. Therefore, the latter is trivial (a bundle that has its holomorphic sections is trivial) and we have the adjunction formula ∗ (1) ND = [−D]jD: n Now let KM = Λ TM be the canonical bundle of M where n is the dimension of M. This is a line bundle contrary to tangent and cotangent bundles. Recall that the sections n of KM are the holomorphic n-forms, i.e., O(KM ) = ΩM . To determine the canonical bundle KD of D as a function of KM , one proceeds as follows: consider the exact sequence ∗ 0∗ 0∗ 0 −! ND −! TM jD −! TD −! 0: We deduce that 0∗ ∗ 0∗ n 0∗ n−1 0∗ ∗ ∗ TM jD = ND ⊕ TD ; Λ TM ' Λ TD ⊗ ND;KM jD = KD ⊗ ND; D 2010 Mathematics Subject Classification. 34M45, 34M05, 11G10, 14H40, 14K02, 14H70. Key words and phrases. Adjunction formula, Poincar´e residue, Riemann surfaces, holomorphic differentials. 41 42 A. LESFARI and taking account of formula (1), one obtains KD = KM jD ⊗ ND = KM jD ⊗ [D]jD = (KM ⊗ [D])jD: Therefore, we get the following. Proposition 1. Let D ⊂ M be a smooth hypersurface, N ∗D be the dual bundle of the normal bundle ND on D and KD (resp. KM ) the canonical bundle on D (respectively M). Then we have the following adjunction formulas: ∗ ND = [−D]jD;KD = (KM ⊗ [D])jD: n Example 1. Let Z0;:::;Zn be homogeneous coordinates in P (C) and Z1 Z2 Zn x1 = ; x2 = ; : : : ; xn = ; Z0 Z0 Z0 n the affine coordinates in P (C)nfZ0 = 0g. Let us define the divisor of the n-meromorphic form dx dx ! = 1 ^ · · · ^ n : x1 xn n To do this, consider on P (C)nfZn = 0g the following affine coordinates: Z0 Z1 Zn−1 u0 = ; u1 = ; : : : ; un−1 = : Zn Zn Zn We have 1 x1 xn−1 u0 = ; u1 = ; : : : ; un−1 = ; xn xn xn and 1 ! = n+1 du0 ^ du1 ^ · · · ^ dun−1: u0 n Hence the divisor of ! is (!) = −(n + 1)H1, where H1 is a hyperplane in P (C) and n KP (C) = [(!)] = [−(n + 1)H1]: n Let ! 2 ΩM (D) be a meromorphic n-form on D with at worst a simple pole along D. Let (z1; : : : ; zn) be complex local coordinates on an open subset of M and assume that D is defined locally by functions f(z1; : : : ; zn). Therefore, g(z1; : : : ; zn)dz1 ^ · · · ^ dzn ! = 2 KM ; f(z1; : : : ; zn) where g(z1; : : : ; zn) is a holomorphic function. Taking into account the second adjunction formula (Proposition 1), one can write ! in the form df ! = ^ ! 2 K ; f e M where !e is a holomorphic (n − 1)-form. Therefore, g(z1; : : : ; zn)dz1 ^ · · · ^ dzcj ^ · · · ^ dzn ! = (−1)j−1 ; e @f @zj @f where j is such that 6= 0. It is therefore possible to define an application (corre- @zj sponding to the second adjunction formula) by setting n n−1 PR :ΩM (D) −! ΩV ;! 7−! PR(!) = !ejf=0; and we call PR(!) the Poincar´eresidue of !. It is a useful formula as we shall see later that under certain conditions it allows explicitly to find holomorphic differentials on submanifolds. ADJUNCTION FORMULA, POINCARE´ RESIDUE AND HOLOMORPHIC DIFFERENTIALS ::: 43 From the exact sequence of sheaves n n PR n−1 0 −! ΩM −! ΩM (D) −! ΩD −! 0; we obtain the long exact cohomology sequence 0 n 0 n PR 0 n−1 0 −! H (M; ΩM ) −! H (M; ΩM (D)) −! H (D; ΩD ) 1 n 1 n −! H (M; ΩM ) −! H (M; ΩM (D)) −! · · · It should be noted that the kernel of the application of the Poincar´eresidue consists 1 n simply of an n-holomorphic form on M and this application is surjective if H (M; ΩM ) = 0. 2. How to construct holomorphic differential forms on some compact Riemann surfaces Let now M be an Abelian surface (a complex algebraic torus) and let D be a positive P divisor on M. Define L(D) = ff meromorphic onM :(f) ≥ −Dg, i.e., for D = kjDj a function f 2 L(D) has at worst a kj-fold pole along Dj: The divisor D is called ample N when a basis (f0; : : : ; fN ) of L(kD) embeds M smoothly into P (C) for some k via the map N M −! P (C); p 7−! [1 : f1(p): ::: : fN (p)]; then kD is called very ample. If we assume D to be very ample, then M is embedded into PN (C) using the functions of L(D) where N = dim L(D) − 1 and in particular D ⊆ PN (C), g = N + 2, (the geometric genus of D). n Letz _ = f(z), z 2 C , be a system of differential equations governing a flow X1 and let X2 be any other flow on M commuting with the first (X1 and X2 are linearly independent vector fields in a neighborhood of a point p in M). Let the vector field have Laurent solutions −1 (0) (1) (2) 2 (2) zj(t) = t zj + zj t + +zj t + ··· ; 1 ≤ j ≤ n whose coefficients are rational functions on an (n − 1)-dimensional affine variety. Con- sider the complex affine invariant manifold (thought of as lying in Cn) defined by the intersection of the constants of the motion of this system and let D be the pole solutions (2) restricted to this invariant manifold, i.e., the set of Laurent solution which remain confined to a fixed affine invariant manifold, related to specific values of the constants (generic constants). Consider a basis f0 = 1, f1,. ,fN of L(D); these functions will be polynomials in z1; : : : ; zn and if the Laurent series zj (see (2)) are substituted into these polynomials fj, they have the following property −1 (0) (1) (2) 2 (3) fj(z1(t); : : : ; zn(t)) = t fj + fj t + +fj t + ··· : (0) (0) Let W = ff1 ; : : : ; fN g be the span of the residues of the functions f0 = 1, f1,. ,fN in L(D), the residues being meromorphic functions on D. Let dt1, dt2 be the holomorphic 1-forms on M defined by dti(Xj) = δij. Giving the vector fields X1 and X2 above and associated times t1 and t2 defines on D the following differentials !i = dtijD, restrictions of dt1 and dt2 to D: (0) ! 1 f 1 ! = dt j = d j ;! = dt j = −f (0)X ! 1 1 D (0) (0) 2 2 D 2 1 fi ∆fi fi 44 A. LESFARI with 0 n 1 o n 1 o 1 X1 X2 fi fi ∆ = lim det @ n o n o A : t!0 fj fj X1 f X2 f i i Laurent solutions z(t;p;D) Taking into account the adjunction formula for the Poincar´eresidue applied to the 2-form ! = fjdt1 ^ dt2, with fj 2 L(D), we have dt1 ^ dt2 dt1 ! = −! R´es !jD = − ; 1 @ 1 fj @t f 2 j D dt2 = ; @ 1 @t f 1 j D dt2 = ; @ t1 2 (0) + ◦(t ) @t1 f 1 j D (0) = fj dt2 ; D (0) = fj !2; (0) @ where fj is the residue appearing in (3) and is the derivative according to the @tj (0) (0) vector field Xj. Similarly, using the second vector field X2, we find ! = fej !1, with fej the residue. Since M is an Abelian surface and D a positive divisor, we deduce from the Kodaira- 1 2 Nakano vanishing theorem [5, 11] that H (M; ΩM (D) = 0 and again from the above exact sequence of sheaves 2 2 PR 1 0 −! ΩM −! ΩM (D) −! ΩD −! 0; we get a long exact sequence 0 2 0 2 PR 0 1 1 2 0 −! H (M; ΩM ) −! H (M; ΩM (D)) −! H (D; ΩD) −! H (M; ΩM ) −! 0: Note that dt1 ^ dt2 is the only holomorphic 2-form on the Abelian surface M.