Mixed Hodge Structures for Minimal Models

The first half of this section deals with generalities on extensions of mixed Hodge structures. The second half of the section is devoted to proving there is a family of mixed Hodge structures on the minimal model of a mixed Hodge diagram.

Let A and G be A-vector spaces {kC R) with given mixed Hodge structures (A, W, F) and (C, W, F). We wish to classify all short exact sequences (extensions):

(*) o - ^ A - ^ B - ^ C - ^ o

188

where B has a mixed Hodge structure and 9 and ^ are morphisms of mixed Hodge structures. The extension o - ^ A ^ B ' - ^ C - ^ o is equivalent to (*) if and only if there is a commutative diagram of mixed Hodge structures:

.B,

We will classify extensions up to equivalence.

If X and Y are vector spaces with nitrations, W(X), F(X), W(Y), and F(Y), denote by Hom^X, Y) and Hom^(X, Y) the subspace of homomorphisms compatible with W, and those compatible with both W and F.

Proposition (8.1). — There is a natural one-to-one correspondence between equivalence classes of extensions of G by A and:

Hom^Cc, Ac)/{Hom^(Cc, A^+CHom^C, A))c}.

Proof. — Let o -> A -^ B -> G -> o be an extension of mixed Hodge structures.

Then <p and ^ are strictly compatible with W. We choose (unnaturally) a splitting:

o —> A -^-> A ® G -"^ C _> o

such that Ii sends the direct sum filtration W(A©C) isomorphically onto W(B). In particular C ^ A Q C - ^ B is strictly compatible with the nitrations W.

Over the complex numbers we choose any splitting:

o —> Ac ~^> Ac©Cc ~^> Gc —> o

such that:

(+) l 2 : ( A C ® C C , W , F ) ^ ( B C , W , F )

189

i go J O H N W . M O R G A N

is an isomorphism of bifiltered vector spaces. To show that such an Ig exists consider the decomposition:

A <p -n

————-^ ^ — - V

—> ©A^ -^> ©I

P,q P,q

^c ' ^XJ Gr ——> o

©C^ o P.?

Since 9 and ^ preserve the direct sum structures, we can choose a splitting for ^ : Bc-^Cp which sends C^ into B^. Let 13 :A®C->B be the isomorphism induced by this splitting. It induces Ig : A^QC^ ^ B^ for all p and y. The composition:

G ^ A C ® C ^ B C

is strictly compatible with both W and F. It is not, however, necessarily a map of mixed Hodge structures since it is not defined over k. This proves that a map as required by (f) exists.

The difference of the two splittings (I^c and 12 is a homomorphism d : Cc->\.

Since both I^ and 12 are strictly compatible with W and since W(A) is the restriction of W(B) to A, it follows that d is compatible with W. The difference, d, is in general only a complex linear map since 12 is only defined over C. We are free to vary I^ exactly by any element a in Hom^G, A), and to vary 12 exactly by any element [B in Hom^(Cc, Ap). Changing I^ and 12 in this manner changes d by a+(B. Thus:

(8.3) E^eHom^Cc, Ac)/{(Hom^(Gc, A^+^om^C, A))c}

is a well-defined invariant of the extension. Clearly, it remains unchanged if we replace the extension by an equivalent one.

Conversely, given two extensions o - > A - ^ B - ^ G - > o and o - ^ A - ^ B ' - ^ C — ^ o whose difference invariants (8.2) are the same, we can choose splittings for ^ and <(/

A ® G ^ B , A®G4.B', Ac®Cc-^Bc and Ac®Cc-^Bc as before, such that the differ-ence homomorphisms d : Cc->Ac and d' : Cc->Ac are equal. Let I : B->B' be given by B —> A® C —> B'. This composition is a ^-linear isomorphism of filtered vector spaces (B, W)->(B', W). Since the difference element for (Ii)c-l2 equals that for (I^c-I^

I : Bc->Bc is also the composition B(, -^ A^OCc-^ Be. Thus Ig is an isomorphism which is compatible with F. Consequently I : B->B' is an isomorphism and a morphism of mixed Hodge structures, i.e. it is an isomorphism of mixed Hodge structures. Clearly 4''I==^ and 19=9'.

Lastly, we show all classes [d] in (8.2) arise as the invariants of extensions of mixed Hodge structures. We will make use of the next lemma.

190

Lemma (8.3). — Suppose given o - > A - ^ B ^ G - > o an exact sequence over k andfil-trations W(A), W(B), W(C), F(AJ, F(Bc), and F(Cc) .. that y and ^ are compatible with all filtrations. Suppose also that (A, W, F) and (C, W, F) define mixed Hodge structures. Then for (B, W, F) to be a mixed Hodge structure it is necessary and sufficient that 9 and ^ strictly preserve W and that on the associated graded objects GrwW and Gr^) be strictly compatible with the jiltrations induced by F.

Proof. — Necessity follows immediately from theorem (1.12). We consider sufficiency. If 9 and ^ are strictly compatible with W, then:

o -. Gr^(A) ^ Gr-(B) ^ Gr;(C) -> o

is a short exact sequence. Thus to prove sufficiency we need only show that if

o-> x->Y - ^ Z - > o i s a short exact sequence strictly compatible with filtrations F(Xc), F(Yc), and F(Zc), and if (X, F) and (Z, F) are Hodge structures of weight n, then

(Y, F) is a Hodge structure of weight n.

(Y, F) is a Hodge structure of weight n if and only if for any p + q == n + i (FTO®P(Yc))=Yc (see [4], (1.2.5)). We show first that for p+q==n+i:

FTO^FTO=O.

Since g preserves F and F, we have that:

^(Yc) nFTO) C F^(Zc) nP(Zc) =o.

Thus_ FP(Yc)nP(Yc)CIm(/). Since / is injective and strictly compatible with F and F, we have:

FTO nP(Yc) C/(F^(Xc)) n/(FTO) =/(F^(Xc) nP(Xc)) =/(o) =o.

Now we show_that if p+q=n+i^ then F^(Yc)+P(Yc)=Yc. Let j^eYc Then ^(^eF^(Zc)+F?(Zc)=^(F^(Yc))+^(F^(Yc)). This allows us to assume that

^00=o. Then ^Im(/)=/(F^(X))+/(FTO). Consequently:

j/eFTO+FTO.

This completes the proof of the lemma.

To complete the proof of (8.1), let d : C^A^ be any complex linear homo-morphism compatible with W. Form:

/ H A d \ I—— —— I = = a ,

\ 0 Ida/ d

Ac®Cc ——————> (A®G)c

Endow A®G with the direct sum filtration W. Push the direct sum filtration F*

on AC®GC forward via ^ to one F^((A©C)c).

Claim. — (A®C, W, F^) is a mixed Hodge structure defined over k. Its difference-element is [d].

191

192 J O H N W . M O R G A N

Proof. — A®C is a ^-vector space with a filtration W defined over k. Since d is compatible with W, o^ is strictly compatible with W. We have a commutative diagram:

o —> Ac -^ (A®G)c -^> Gc —^ o

o —> Ac -^> Ac®Cc ——> Cc -^ o

If we use the direct sum filtrations W and F in the lower sequence then z\ and TCg are strictly compatible with W and the filtration induced by F on Gr^ Since o^ sends W to W and F to F^, the same is true for W and F^ in the upper sequence. Applying lemma (8.3) proves that (Ac 5 W, F^) is a mixed Hodge structure. Clearly, its difference element is d.

Definition (8.4). — If A is a A-vector space with a filtration W(A), and X is a complex vector space with a bi-grading X^OX^, then an isomorphism I : X—^Ac defines a mixed Hodge structure on A i f : '

a) I( © X^)==(W,(A))c, and

P+q^f

b) if we define FP(Ac)=I( © X'*8), then (A, W, F) is a mixed Hodge structure.

r^p

Corollary (8.5). — Let o->^—^^->A.^->o be a short exact sequence of k-vector spaces with each A^ having a weight filtration W(A^), i==i, 2, 3. Let o—>^^->X.^->^K^->o be a short exact sequence of complex vector spaces with each X^ having a bi-grading X^= © X^. Let:

P,<?

o —> (Ai)c —> (A2)c —> (A3)c —> o

X, X, X,

be a commutative diagram. If I^ and 13 induce mixed Hodge structures and U © X^)=(W,(A,))c,

P+q^_£

then Ig also induces a mixed Hodge structure.

Proof. — This is an immediate consequence of lemma (8.3).

At this point let us summarize our results to date. Let 9 : (E, W)c -> (<^, W, F) be a mixed Hodge diagram, and let e^->E be a minimal model for E, and ^T->(?0 ^

a complex minimal model for <?.

192

1) ^ has a bigrading ^T= © ^T^ so that, if we define WJ^T) to be © ./T^

P,<?^0 ^K' p+q^k

and F^) to be ©^S then (e/T, W, F) -t (^, Dec W, F) is a quasi-isomorphism of bifiltered algebras. Given another such bigraded minimal model for €, it is isomorphic to ^ by an isomorphism well-defined up to bigraded homotopy (6.6).

2) J( has a minimal filtration W(^) so that p : (^, W) -> (E, Dec W) is a quasi-isomorphism. Any other such filtered minimal model { J K ' , W) is isomorphic to (e^f, W) by an isomorphism unique up to homotopy compatible with the filtration (7.6).

Now we wish to meld these two results together. The fact that 9 : Ep-x? induces an isomorphism on cohomology implies that there is an isomorphism I :^f->^ well-defined up to homotopy. Since (E, Dec W)c -^ (<?, Dec W) is a quasi-isomorphism, the map I can be taken to be an isomorphism of filtered minimal models (7.5). Such a filtered isomorphism is well-defined up to homotopy compatible with the filtrations.

Theorem (8.6). — Any such isomorphism I as above defines a mixed Hodge structure on ej^.

The induced mixed Hodge structure on H(e^) agrees via p* with the mixed Hodge structure that the mixed Hodge diagram defines on H(E).

Proof. — We prove that any such I induces a mixed Hodge structure by induction on the canonical series {eJ^} for J( and {^} for ^T. Since the series are canonical, I restricted to ^ induces an isomorphism I ^^"^(^Jc ^or every a. We assume that for some fixed a, I : ^ -> (^a)c induces a mixed Hodge structure. We can write .Xc+i^^a0^^ ^d ^+l=t^a0dA(V')fe. We can suppose that the bigrading of e^+i induces one on V and that the bigrading of ^a+i induces the bigrading on ^ (6.9). Likewise, we can assume that the filtration o n ^ + i is the multiplicative extension of the one on ^ and the one induced on V (7.2).

We know already that the map induced by I on cohomology, I : H^) -> H(c^)c (which is the map 9* when we make the natural identifications H(^)c==H(E; C) and H(^r)=H(<?; C)) induces a mixed Hodge structure on cohomology. The reason is that the filtration W(^) becomes the filtration Dec W on H(E; QJ and the bigrading of^T becomes the bigrading associated with the mixed Hodge structure on H(^$ C). We have a commutative ladder of exact sequences:

^c H^+i( _^c

H^ H^)c H^^c V.

i*

H^) H^^T) • V H^W H^+^^T) By the above discussion the second and fifth vertical arrows induce mixed Hodge structures. By the inductive hypothesis I : ^a^-(^a)c induces a mixed Hodge 193

i94 J O H N W . M O R G A N

structure on ^. The map d : ^ ->^a is compatible with both filtrations and hence is a morphism of mixed Hodge structures; by ( i . 12), 4) it follows that there is induced on H(e<^) a mixed Hodge structure. This means that P : H(J^)-> H(e<,)o induces a mixed Hodge structure. Thus the first and fourth vertical arrows also induce mixed Hodge structures. Using ( i . 12), 4) and (8.5) we see that I : V'->Vc induces a mixed Hodge structure. Now to finish the proof that ^a +1 -> GXc +i)c induces a mixed Hodge structure we filter both algebras as follows:

S^a+i)^2^^--'^!^5 oee<, and z^eV}

Si(^a+l)⁼{S(oA^A...AyJJ^^, oe^, and ^eV'}.

Then I induces isomorphisms I : S,(^+i) -> S,(^+i)c- one proves by induction on i that I induces a mixed Hodge structure, using (8.5).

Note. — i) For forms of degree n in Ji the first possible non zero weight in the mixed Hodge structure is W^.

2) The mixed Hodge structure on Ji will depend in general on the choice of the