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Mixed Hodge Structures on Smooth Algebraic Varieties

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Mixed Hodge Structures on Smooth Algebraic Varieties 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.
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