MIXED HODGE STRUCTURES ON SMOOTH ALGEBRAIC VARIETIES LIVIU I. NICOLAESCU Abstract. We discuss some of the details of Deligne's proof on the existence of a functorial mixed Hodge structure on a smooth quasiprojective variety. Contents Notations 1 1. Formulation of the problem 2 2. The logarithmic complexes 2 3. The weight ¯ltration and the associated spectral sequence 6 4. Spectral sequences 11 ² 5. The degeneration of the spectral sequence Er(A (X; log D);W ) 14 6. Functoriality 15 7. Examples 15 Appendix A. Gysin maps and the di®erential d1 21 References 27 Notations ² H²(S) := H²(S; C). ² ² If W² is an increasing ¯ltration on the vector space V then we denote by W¡ the decreasing ¯ltration de¯ned by ` W¡ = W¡`: If F ² is a decreasing ¯ltration then we can associate in a similar fashion the increasing ¡ ¯ltration F² . ² For an increasing ¯ltration W² and k 2 Z we de¯ne the shifted ¯ltration W [n]² = Wn+²: We de¯ne the shifts of decreasing ¯ltrations in a similar way. Note that ² ² W [n]¡ = W¡[¡n] : ² n ² ² For every graded object C = ©n2ZC and every integer k we denote by C [k] the graded object de¯ned by Cn[k] := Cn+k: ² For a bigraded object C²;² and every integers (k; m) the shifted complex C²;²[k; m] is de¯ned in a similar fashion. Date: April, 2005. 1 2 LIVIU I. NICOLAESCU 1. Formulation of the problem We assume X¤ is a smooth, complex, n-dimensional algebraic variety. According to Hi- ronaka it admits a smooth compacti¯cation X such that the complement D = X n X¤ is a normal crossings divisor. This means that every point p 2 D there exist an integer 1 · k · n and an open neighborhood O bi-holomorphic to an open ball B in Cn centered at 0 such that » D \ O = f(z1; ¢ ¢ ¢ ; zn) 2 B; z1 ¢ ¢ ¢ zk = 0g: We assume X is projective and we explain how to produce a mixed Hodge structure on X¤ using the Hodge structure on X. The mixed Hodge structure thus produced will be independent of the compacti¯cation X. The strategy we will employ is easy to describe. Denote by j : X¤ ,! X the natural inclusion of X¤ as an open subset of X. We observe that ² ¤ » ² H (X ) = H (X; j¤C): The construction of a mixed Hodge structure on H²(X¤; Z) is carried on in several steps. ² Step 1. We construct a complex of sheaves S on X quasi-isomorphic to j¤C. This complex will be equipped with a natural decreasing ¯ltration F p and a natural increasing ¯ltration ²;² W`. We then produce a hypercohomology spectral sequence Er associated to the decreasing ` ² ¤ ² » ² ¤ ¯ltration W¡ := W¡` and converging to H (X ; S ) = H (X ; C). The increasing ¯ltration ² ¤ W` induces an increasing ¯ltration on H (X ; C) which, up to a shift, will be the weight ¯ltration. p;q Step 2. We will show that the ¯ltration F induces pure Hodge structures on E1 and the di®erential d1 is a morphism of pure Hodge structures of a given bidegree (0; 0). In particular, we deduce that E2 is equipped with a mixed Hodge structure. Step 3. We will show that for every r ¸ 2 the di®erential dr on Er vanishes so that W m ¤ » ¡`;m+` Gr` H (X ) = E2 is equipped with a natural pure Hodge structure induced by F of weight m + `. We deduce ² m ¤ that the ¯ltrations (F ;W [¡m]²) de¯ne a mixed Hodge structure on H (X ). We have m ¤ ¤ m m ¤ m ¤ m ¤ ; = Wm¡1H (X ) ½ j H (X) = WmH (X ) ½ ¢ ¢ ¢ ½ W2mH (X ) = H (X ): Step 4. We will show that a holomorphic map between smooth quasiprojective varieties induces a morphism of mixed Hodge structures. The implementation of Step 1 requires the introduction of smooth and holomorphic log complexes. Step 2 requires the use of the Poincar¶eresidue. Step 3 is based on a clever algebraic argument of P. Deligne known as "le lemme de deux ¯ltrations". The last step makes heavy use of Hironaka's resolution of singularities theorem. 2. The logarithmic complexes For every subset S ½ X we de¯ne S¤ := S n D: m For every integer m ¸ 0 and every open set V ½ X we denote by AX (V; log D) the subspace m ¤ ¤ of AX (V ) consisting of smooth, complex valued m-forms on V with the propriety that for any coordinate neighborhood ( U; (zj)) ½ V such that D \ U = fz1 ¢ ¢ ¢ zk = 0g the forms ¤ ¤ z1 ¢ ¢ ¢ zk' and z1 ¢ ¢ ¢ zkd' on U extend to smooth forms on U . We de¯ne m m ¤ m ­X (V; log D) = ­X (V ) \ AX (V; log D): MIXED HODGE STRUCTURES ON SMOOTH ALGEBRAIC VARIETIES 3 The correspondences m m V ¡!7 AX (V; log D);U 7¡! ­X (V; log D) m m de¯ne sheaves AX (log D) and ­X (log D) on X. From the de¯nition we deduce ² ² ² ² dAX (V; log D) ½ AX (V; log D)[1];@­X (log D) ½ ­X (log D)[1]; ² and thus we obtain two complexes of sheaves on X:(AX (log D); d) called the smooth loga- ² rithmic complex, and (­X (log D);@) called the holomorphic logarithmic complex. ¤ ² Denote by j the natural inclusion j : X ,! X. By de¯nition, (­X (log D);@) is a subcom- ² ² plex of (AX (log D); d) which is a subcomplex of (j¤A ; d). Theorem 2.1. The inclusions ² ² (­X (log D);@) ,! (AX (log D); d) (2.1) and ² ² (­X (log D);@) ,! (j¤AX ; d) (2.2) are quasi-isomorphisms of complexes of sheaves. In particular, the inclusion ² ² (AX (log D); d) ,! (j¤AX ; d) (2.3) is also a quasi-isomorphism. Proof The proof of this result will occupy the remainder of this section. We use a com- bination of the approaches in [6, x5] and [8, Chap. 8]. We begin by giving an alternate ² description of AX (log D). Lemma 2.2. Suppose ( U; (zj)) is an open coordinate neighborhood on X such that D \ U = fz1 ¢ ¢ ¢ zk = 0g: m Then any ® 2 AX (U; log D) can be written as a combination Xk X dzi1 dzij ® = ®0 + ®i1¢¢¢ij ^ ^ ¢ ¢ ¢ ^ ; zi1 zij j=1 1·i1<¢¢¢·ij ·k where m m¡j ®0 2 AX (U); ®i1¢¢¢ij 2 AX (U): Proof For simplicity we consider only the case k = 1 and we write z = z1. Note ¯rst that we can write 1 ® = ¯; ¯ 2 Am(U): z X We write @ ¯ = ¯ + dz ^ ® ; ® 2 Am¡1(U); ¯ 2 Am(U); ¯ = 0: 0 1 1 X 0 X @z 0 m+1 On the other hand, since zd® 2 AX (U) we deduce ³ dz 1 ´ dz z ¡ ^ ¯ + d¯ = ¡ ^ ¯ + d¯ 2 Am+1(U): z2 z z X Hence dz ¡ ^ ¯ + d¯ ¡ dz ^ d® 2 Am+1(U) z 0 0 1 X 4 LIVIU I. NICOLAESCU Hence m ¯0 = z®0; ®0 2 AX (U): ut It is convenient to introduce some simplifying notations. If U is a coordinate neighborhood in which D is described by the monomial equation z1 ¢ ¢ ¢ zk then for every multi-index I = (1 · i1 < ¢ ¢ ¢ < ij · k) we set dz dzi jIj := j; (d log z)I := i1 ^ ¢ ¢ ¢ ^ j zi1 zij and X I ® = ®I ^ (d log z) jIj¸0 We will refer to such a representation as a local logarithmic representation. To prove the quasi-isomorphism (2.1) we will show that for every p ¸ 0 the sequence of sheaves over X p;0 p;0 @¹ p;1 @¹ 0 ! ­X (log D) ,! AX (log D) ! AX (log D) ! ¢ ¢ ¢ (2.4) is exact. To achieve this we will need a @¹-version of the Poincar¶elemma. We state below a n n more re¯ned version due to Nickerson, [7]. Denote by Dr the polydisk in C de¯ned by n Dr = f(z1; ¢ ¢ ¢ ; zn); jzjj < r; 8j = 1; ¢ ¢ ¢ ; ng: Lemma 2.3 (Dolbeault Lemma). For every integers n ¸ 1, 0 · p · n, 1 · q · n there exists a linear operator ²;² n ²;² n T0 : A (Dr ) ! A (Dr=2)[0; ¡1] p;q n such that 8® 2 Z (Dr ) we have ®j n = @T¹ ® + T @®:¹ Dr=2 0 0 ut n For every integers 0 · k · n we denote by Sk = Sk;n the normal crossings divisor in Dr de¯ned by the equation z1 ¢ ¢ ¢ zk = 0 if k > 0, S0 = ;, if k = 0. Set ³ ´ ²;² n ¹ ²;² n ²;² n Zk (Dr ) := ker @ : A (Dr ; log Sk) ! A (Dr ; log Sk)[0; 1] : To prove the exactness of the sequence (2.4) it su±ces to show the following. Lemma 2.4. For every k · n and every r > 0 there exists a linear map ²;² n ²;² n Tk : Zk (Dr ) ! A (Dr=2; log Sk)[0; ¡1]; such that ²;² n ®j n = @T¹ ®; 8® 2 Z (D ): Dr=2 k k r Proof We will argue by induction over k. For k = 0 this follows from the Dolbeault lemma. Let us prove the inductive step. Set p;q n z = zk+1. Suppose ® 2 Zk+1(Dr ). Then Lemma 2.2 implies that the forms dz ¯ = z@ ® 2 Ap¡1;q(Dn; log S ); γ = ® ¡ ^ ¯ 2 Ap;q(Dn; log S ) z r k z r k MIXED HODGE STRUCTURES ON SMOOTH ALGEBRAIC VARIETIES 5 contain no dz. By design, we have dz ® = ^ ¯ + γ: z From the equality @®¹ = 0 we deduce dz @γ¹ ¡ ^ @¯¹ = 0: z Since ¯ and γ contain no dz we deduce @γ¹ = 0 = @¯;¹ so that by induction, Tk¯ and Tkγ are well de¯ned. Now set dz T ® = ¡ ^ T ¯ + T γ: k+1 z k k Since ¯ and γ depend linearly on ® we deduce that Tk+1 is a linear operator.