03.12.44 · modern-geometry / homotopy

Mixed Hodge structures on rational homotopy theory (Morgan's theorem)

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Anchor (Master): Morgan 1978 *The algebraic topology of smooth algebraic varieties* (Publ. Math. IHÉS 48, 137–204); Deligne 1971 *Théorie de Hodge II* (Publ. Math. IHÉS 40, 5–57) and 1974 *Théorie de Hodge III* (Publ. Math. IHÉS 44, 5–77); Hain 1987 *The de Rham homotopy theory of complex algebraic varieties I–II* (K-Theory 1, 271–324 and 481–497); Deligne-Griffiths-Morgan-Sullivan 1975 *Real homotopy theory of Kähler manifolds* (Invent. Math. 29); Peters-Steenbrink *Mixed Hodge Structures* (Springer 2008)

Intuition Beginner

The cohomology ring of a space records its holes. The Sullivan minimal model records the holes and the relations among them, and from it you can read off the rational homotopy groups — homotopy up to torsion. For an arbitrary space that is the end of the story. But when the space is an algebraic variety — a shape cut out by polynomial equations over the complex numbers — something extra appears. The rational homotopy is not just an algebra. It secretly carries a hidden grading, a weight attached to each piece, and a finer splitting into Hodge types. This hidden symmetry severely constrains what shapes algebraic varieties can have.

Think of a stained-glass window. Two windows can have the same outline of holes between the lead strips, yet one is sorted by colour into orderly bands and the other is a random jumble. The outline is the bare homotopy. The colour-banding is the weight grading: an extra layer of structure that only some shapes carry. Algebraic varieties always come pre-sorted by colour, and you cannot sort an arbitrary shape that way.

The weight grading was introduced by Pierre Deligne in 1971 for the cohomology of varieties, and lifted to the rational homotopy by John Morgan in 1978. It is the reason certain shapes — for instance the smooth compact ones from complex geometry — turn out to be remarkably rigid, with no hidden higher-order linking among their cohomology classes.

Visual Beginner

A mixed Hodge structure layers two pieces of data onto each vector space attached to a variety: a weight grading that sorts the space into bands indexed by an integer $k$ , and inside each band a Hodge splitting that further sorts by a pair of types. The picture to hold is a staircase: each step is a weight band, and the colour-shading across a step is the Hodge splitting on that step.

The picture captures the structural shape: the weight filtration builds the staircase, the Hodge filtration colours each step, and the defining requirement is that each step — taken on its own — looks like the clean, single-weight cohomology of a compact variety. A reader who internalises this staircase will recognise the template every time a mixed Hodge structure appears: sort by weight first, then split each weight band by Hodge type.

Worked example Beginner

Take the thrice-punctured projective line $U = P^{1} ∖ {0, 1, \infty}$ — the Riemann sphere with three points removed. Its first cohomology $H^{1} (U; Q)$ is two-dimensional, with a basis given by the two forms $\frac{d z}{z}$ (which circles the puncture at $0$ ) and $\frac{d z}{z - 1}$ (which circles the puncture at $1$ ). The puncture at $\infty$ contributes no new class because the three residues around the three punctures sum to zero.

Compare this with a compact curve, say an elliptic curve $E$ . Its $H^{1} (E; Q)$ is also two-dimensional, but it sits in weight $1$ : it splits into a holomorphic piece and an anti-holomorphic piece, types $(1, 0)$ and $(0, 1)$ , adding to weight $1$ . This is the clean, single-weight Hodge structure of a compact shape.

The punctured line behaves differently. Each puncture forces a logarithmic pole, and a pole pushes the weight up by one. So $H^{1} (U)$ lives in weight $2$ , of pure Hodge type $(1, 1)$ — both basis classes are type $(1, 1)$ , adding to weight $2$ . The numbers: $dim H^{1} = 2$ in both cases, but weight $1$ for the compact $E$ versus weight $2$ for the open $U$ . The single removed point at $0$ raises one generator's weight from $1$ to $2$ .

What this tells us: removing points from a curve does not just add cohomology — it adds cohomology of higher weight. The weight is a bookkeeping device counting how far a shape is from being compact. For the punctured line every $H^{1}$ class sits one weight-step above where a compact curve would put it, and this single fact propagates all the way up into the rational homotopy groups.

Check your understanding Beginner

Exercise (easy, multiple choice).

What two pieces of data make up a mixed Hodge structure on a vector space?

A. A single grading by degree only. B. An increasing weight filtration and a decreasing Hodge filtration on the complexification. C. A choice of basis and an inner product. D. A cup product and a cap product.

Hint

One filtration sorts the space into integer weight bands; the other refines each band by Hodge type. Recall the staircase picture.

Answer

B. A mixed Hodge structure is a rational vector space equipped with an increasing weight filtration $W_{∙}$ and a decreasing Hodge filtration $F^{∙}$ on the complexified space, subject to the compatibility that each weight band, on its own, carries a clean single-weight Hodge structure. Feedback if you picked A: a single grading is not enough — the colouring of each band by Hodge type is the second, essential ingredient. Feedback if you picked C or D: those are metric and product structures, not the filtration data of a mixed Hodge structure.

Exercise (easy, multiple choice).

Morgan's 1978 theorem says that for a smooth complex algebraic variety the weight-and-Hodge data extends from cohomology onto which further object?

A. The Riemannian metric. B. The Sullivan minimal model, hence the rational homotopy groups. C. The set of singular points. D. The fundamental class only.

Hint

Deligne put the structure on cohomology; Morgan lifted it one level higher, onto the algebraic gadget that records homotopy.

Answer

B. Deligne (1971) placed a mixed Hodge structure on the cohomology of a smooth variety. Morgan (1978) lifted it to the Sullivan minimal model and therefore to the rational homotopy groups. The weight grading and Hodge splitting become invariants of the rational homotopy type, not merely of the cohomology ring. Feedback if you picked A or C: a metric and the singular locus are not what the mixed-Hodge package controls here.

Formal definition Intermediate+

Throughout, $V$ denotes a finite-dimensional vector space over $Q$ and $V_{C} = V \otimes_{Q} C$ its complexification, carrying the complex-conjugation involution $v \otimes λ \mapsto v \otimes \overset{ˉ}{λ}$ .

Definition (pure Hodge structure). A pure Hodge structure of weight $k$ on $V$ is a direct-sum decomposition $$ V_{\mathbb{C}} = \bigoplus_{p+q = k} V^{p,q}, \qquad \overline{V^{p,q}} = V^{q,p}, $$ or equivalently a decreasing Hodge filtration $F^{∙}$ on $V_{C}$ with $V_{C} = F^{p} \oplus \overline{F^{k - p + 1}}$ for every $p$ ; one recovers $V^{p, q} = F^{p} \cap \overline{F^{q}}$ . This is the structure carried by the degree- $k$ cohomology of a compact Kähler manifold (see 04.09.01), where $V^{p, q} = H^{p, q}$ is the space of harmonic forms of bidegree $(p, q)$ .

Definition (mixed Hodge structure; Deligne 1971). A mixed Hodge structure (MHS) on $V$ consists of:

an increasing weight filtration $W_{∙}$ of $V$ by $Q$ -subspaces, $\dots \subseteq W_{k - 1} \subseteq W_{k} \subseteq W_{k + 1} \subseteq \dots$ , with $W_{m} = 0$ for $m ≪ 0$ and $W_{m} = V$ for $m ≫ 0$ ;
a decreasing Hodge filtration $F^{∙}$ of $V_{C}$ by complex subspaces;

subject to the single axiom that the filtration induced by $F$ on each weight-graded piece $gr_{k}^{W} V := W_{k} / W_{k - 1}$ defines a pure Hodge structure of weight $k$ on $gr_{k}^{W} V$ . Explicitly, writing $F^{p} gr_{k}^{W} V_{C}$ for the image of $F^{p} \cap (W_{k})_{C}$ , the requirement is $$ \mathrm{gr}^W_k V_{\mathbb{C}} = F^p , \mathrm{gr}^W_k V_{\mathbb{C}} \oplus \overline{F^{k-p+1} , \mathrm{gr}^W_k V_{\mathbb{C}}} \quad \text{for all } p. $$ A morphism of mixed Hodge structures $f : V \to V^{'}$ is a $Q$ -linear map with $f (W_{k} V) \subseteq W_{k} V^{'}$ and $f (F^{p} V_{C}) \subseteq F^{p} V_{C}^{'}$ .

The canonical bigrading (Deligne). A mixed Hodge structure determines a canonical splitting $V_{C} = ⨁_{p, q} I^{p, q}$ with $W_{k}_{C} = ⨁_{p + q \leq k} I^{p, q}$ and $F^{p} = ⨁_{p^{'} \geq p} I^{p^{'}, q}$ , where $$ I^{p,q} = (F^p \cap W_{p+q}) \cap \left( \overline{F^q \cap W_{p+q}} + \sum_{j \ge 1} \overline{F^{q-j} \cap W_{p+q-j-1}} \right). $$ The splitting satisfies $\overline{I^{p, q}} \equiv I^{q, p} mod ⨁_{r < q, s < p} I^{r, s}$ ; it is the precise sense in which a mixed Hodge structure is "almost" a bigrading, failing to be a clean conjugate-symmetric bigrading only by lower terms. The graded piece $gr_{k}^{W} V_{C} = ⨁_{p + q = k} I^{p, q}$ recovers the pure Hodge structure on each step.

The motivating geometric example. For $X$ a smooth projective complex variety, $H^{k} (X; Q)$ is pure of weight $k$ — its mixed Hodge structure has a single weight band, the classical Hodge decomposition of 04.09.01. For $U = X ∖ D$ with $D = ⋃ D_{i}$ a normal-crossings divisor (each $D_{i}$ smooth, meeting transversally), Deligne's mixed Hodge structure on $H^{k} (U; Q)$ is computed from the logarithmic de Rham complex $Ω_{X}^{∙} (lo g D)$ — holomorphic forms on $X$ allowed at worst simple poles $\frac{d z _{i}}{z _{i}}$ along the components $D_{i}$ . The weight filtration $W$ counts the number of log poles ( $W_{k} Ω_{X}^{∙} (lo g D)$ = forms with at most $k$ log factors) and the Hodge filtration $F$ is the stupid filtration by holomorphic degree. The result: $H^{k} (U)$ carries weights in the range $k \leq w \leq 2 k$ , with $W_{k} H^{k} (U) = im (H^{k} (X) \to H^{k} (U))$ the part pulled back from the compactification. Functoriality in $U$ is automatic from functoriality of the log complex under maps of pairs $(X, D)$ .

Counterexamples to common slips

A mixed Hodge structure is not a bidegree decomposition on the nose. The clean conjugate-symmetric splitting $V^{p, q}$ exists only for pure structures. In the mixed case the canonical $I^{p, q}$ satisfies conjugate symmetry only modulo lower-weight pieces; assuming $\overline{I^{p, q}} = I^{q, p}$ exactly is a common error.
Weight is not cohomological degree. For a smooth open variety $H^{k} (U)$ generally spreads across several weights $k, k + 1, \dots, 2 k$ . Identifying "weight" with the degree $k$ holds only in the pure (compact) case.
The weight filtration lives over $Q$ ; the Hodge filtration lives over $C$ . Placing $W_{∙}$ on $V_{C}$ rather than on the rational space $V$ discards the rational structure that makes weights an arithmetic invariant. The defining data is $(V, W_{∙})$ rational and $F^{∙}$ on $V_{C}$ .
A morphism need not preserve the canonical bigrading $I^{p, q}$ strictly, but it is automatically strict for $W$ and $F$ . Strictness — that $f (V) \cap W_{k} V^{'} = f (W_{k} V)$ and likewise for $F$ — is a theorem (the morphisms form an abelian category), not part of the definition, and is the engine behind weight-degeneration arguments.

Key theorem with proof Intermediate+

Theorem (Morgan 1978, Publ. Math. IHÉS 48). Let $U$ be a smooth complex algebraic variety. Then the Sullivan minimal model $M_{U}$ of the complex piecewise-polynomial CDGA $A_{P L} (U) \otimes C$ (see 03.12.06) carries a canonical mixed Hodge structure: each graded piece $M_{U}^{n}$ is equipped with a weight filtration $W_{∙}$ and a Hodge filtration $F^{∙}$ , multiplicative in the sense that the product $M_{U}^{a} \otimes M_{U}^{b} \to M_{U}^{a + b}$ and the differential are morphisms of mixed Hodge structures, and the induced structure on $H^(\mathcal{M}_U) = H^(U; \mathbb{C}) $a g r ees w i t h D e l i g n e^{'} s mi x e d H o d g es t r u c t u r e . C o n se q u e n tl y, f or$ U $ni l p o t e n tt h e d u a l r a t i o na l h o m o t o p y g r o u p s$ \big(\pi_n(U) \otimes \mathbb{Q}\big)^\vee $inh er i t a f u n c t or ia l mi x e d H o d g es t r u c t u r eco m p a t ib l e w i t h t h e W hi t e h e a d - p r o d u c t s t r u c t u r e . T h e w h o l e p a c k a g e i s f u n c t or ia l : am or p hi s m$ U \to U'$ of smooth varieties induces a morphism of mixed-Hodge minimal models.

Proof.

Step 1: a bigraded model from the logarithmic complex. Choose a smooth compactification $X \supseteq U$ with $D = X ∖ U$ a normal-crossings divisor (Hironaka resolution guarantees one exists). The logarithmic de Rham complex $(Ω_{X}^{∙} (lo g D), d)$ computes $H^{*} (U; C)$ (Deligne, Hodge II §3) and is a CDGA under wedge product. It carries the weight filtration $W$ (by number of log poles) and the Hodge filtration $F$ (stupid filtration by holomorphic degree), and these are multiplicative: $W_{a} \land W_{b} \subseteq W_{a + b}$ and $F^{a} \land F^{b} \subseteq F^{a + b}$ . The complex $Ω_{X}^{∙} (lo g D)$ is the mixed-Hodge CDGA model of $U$ .

Step 2: transport to a real/rational model. The complex $Ω_{X}^{∙} (lo g D)$ is connected to $A_{P L} (U) \otimes C$ by a zigzag of CDGA quasi-isomorphisms (the holomorphic log complex maps into the $C^{\infty}$ log complex, which maps into the full de Rham complex of $U$ , a model for $A_{P L} (U) \otimes C$ ). Morgan's bigraded-model construction produces, functorially, a CDGA with two filtrations whose underlying CDGA is a model of $U$ and whose filtrations induce Deligne's mixed Hodge structure on cohomology. The key technical device is that one may choose, at each inductive stage of the minimal-model construction of 03.12.06, the new generators and the differential to be morphisms of bifiltered complexes, using the strictness of the differential with respect to $W$ and $F$ (Deligne's two-filtration lemma, Hodge II §1.3).

Step 3: descend the bifiltration to the minimal model. Run the inductive construction of the minimal model $M_{U} \sim Ω_{X}^{∙} (lo g D)$ of 03.12.06. At stage $n$ one adjoins generators dual to $H^{n}$ and to the kernel/cokernel of the previous differential. Because the obstruction classes governing the choice of differential are computed in cohomology groups that themselves carry pure Hodge structures (by Step 1 and the strictness lemma), the generators can be assigned weights and Hodge types so that $d$ and the product are bifiltered morphisms. The strictness of $d$ with respect to $(W, F)$ is exactly what makes this choice consistent stage by stage — without it the bigrading would fail to close up. This equips each $M_{U}^{n}$ with $(W_{∙}, F^{∙})$ defining a mixed Hodge structure, and the construction is independent of the compactification because any two are dominated by a third (resolution of indeterminacy), giving canonicity and functoriality.

Step 4: pass to homotopy groups. For $U$ nilpotent, the main theorem of rational homotopy theory (03.12.06) identifies the space of indecomposables $Q M_{U}^{n} = M_{U}^{n, +} / (M_{U}^{+} \cdot M_{U}^{+})$ with $Hom (π_{n} (U), C)$ . The decomposable ideal $M_{U}^{+} \cdot M_{U}^{+}$ is a sub-mixed-Hodge structure (the product is bifiltered), so the quotient $Q M_{U}^{n}$ inherits a mixed Hodge structure. Dualising, $π_{n} (U) \otimes Q$ carries a mixed Hodge structure, and the Whitehead bracket — dual to the quadratic part of $d$ — is a morphism of mixed Hodge structures because $d$ is. $□$

Synthesis. Three facts combine here, and together they make the homotopy theory of algebraic varieties strictly finer than the homotopy theory of bare spaces. The foundational reason the construction closes up is Deligne's strictness lemma: a differential that is a morphism for both filtrations is automatically strict, so the bigrading survives every inductive step of the minimal-model build. This is exactly the mechanism by which the weight grading on cohomology propagates onto the generators of the model and then, by dualising the indecomposables, onto the homotopy groups. Putting these together, Deligne's mixed Hodge structure on $H^{*} (U)$ generalises from cohomology to the entire rational homotopy type; the central insight is that the minimal model — built only from cohomology and higher operations — is rigid enough that the Hodge-theoretic weight has nowhere to go but onto $π_{*}$ . The bridge is the indecomposables functor $Q$ , which is dual to $π_{*}$ and carries the mixed Hodge structure across.

Bridge. This result builds toward the modern picture in which homotopy types of varieties are objects of Hodge-theoretic, even motivic, nature, and it appears again in the study of the Malcev completion of $π_{1}$ (Hain 1987), where the same mixed-Hodge-CDGA machine equips the pro-unipotent fundamental group with weights. The construction generalises the DGMS formality theorem of 04.09.05: formality of a simply-connected compact Kähler manifold is the pure-weight special case ( $D = \emptyset$ , so $H^{k}$ is pure of weight $k$ and the weight grading forces the model differential to vanish on cohomology generators). The whole apparatus is dual to the cohomological mixed Hodge theory of 04.09.01, lifted one categorical level from $H^{*}$ to $π_{*}$ , and the foundational reason it works — strictness of bifiltered differentials — is the same two-filtration lemma that makes Deligne's cohomological theory functorial.

Exercises Intermediate+

Exercise 3 (medium, short-answer).

Show that for a smooth projective variety $X$ (so $D = \emptyset$ ), Morgan's mixed Hodge structure on the minimal model is pure: each cohomology generator in degree $k$ has weight $k$ . Deduce that the model differential vanishes on these generators, recovering formality.

Hint

With $D = \emptyset$ the log complex is the ordinary de Rham complex and $H^{k} (X)$ is pure of weight $k$ . The differential is weight-preserving, and it would have to raise degree by one while preserving weight.

Answer

When $D = \emptyset$ , $Ω_{X}^{∙} (lo g D) = Ω_{X}^{∙}$ and $H^{k} (X)$ is pure of weight $k$ (the classical Hodge decomposition, 04.09.01). The minimal-model generators dual to $H^{k}$ therefore carry weight $k$ . The differential $d$ raises cohomological degree by $1$ but is a morphism of mixed Hodge structures, hence weight-preserving; a generator of degree $k$ and weight $k$ would map to degree $k + 1$ , where the available targets (products of lower generators) have weight equal to their degree, i.e. weight $k + 1 \neq = k$ . So $d$ must vanish on the cohomology generators. The minimal model thus equals that of $(H^{*} (X), 0)$ , which is the definition of formality (compare the DGMS route of 04.09.05). Rubric: full credit for purity of generators, the weight-vs-degree mismatch forcing $d = 0$ , and the formality conclusion.

Exercise 4 (medium, short-answer).

Use the weight-strictness of the differential to show that a nonzero triple Massey product $⟨ a, b, c ⟩$ on a smooth variety $U$ is constrained: it can only be nonzero between classes whose weights are compatible with raising total weight by the weight of the model differential. Sketch the constraint.

Hint

Massey products are computed from bounding cochains $a^{'}$ with $d a^{'} = ab$ . Since $d$ preserves weight, $a^{'}$ must carry the same weight as the product $ab$ .

Answer

The triple Massey product cocycle is $m = a^{'} c \pm a b^{'}$ with $d a^{'} = ab$ , $d b^{'} = b c$ (see 03.12.51). Because $d$ is weight-preserving and the product is weight-additive, the bounding cochain $a^{'}$ has weight $w (a) + w (b)$ , and $m$ has weight $w (a) + w (b) + w (c)$ . So $⟨ a, b, c ⟩$ , if nonzero, lives in the weight- $(w (a) + w (b) + w (c))$ part of $H^{∣ a ∣ + ∣ b ∣ + ∣ c ∣ - 1} (U)$ . On a pure (compact projective) variety, weight equals degree, so $w (a) + w (b) + w (c) = ∣ a ∣ + ∣ b ∣ + ∣ c ∣$ , whereas the target degree is $∣ a ∣ + ∣ b ∣ + ∣ c ∣ - 1$ with weight $∣ a ∣ + ∣ b ∣ + ∣ c ∣ - 1$ — a mismatch of one, forcing the Massey product into the indeterminacy ideal: this re-proves vanishing of Massey products on compact Kähler manifolds. On an open $U$ the higher weights leave room, so Massey products can survive. Rubric: full credit for the weight-bookkeeping and the pure-case vanishing.

Exercise 5 (medium, short-answer).

Explain why the decomposable ideal $M_{U}^{+} \cdot M_{U}^{+}$ is a sub-mixed-Hodge structure of $M_{U}$ , and why this is exactly what is needed to put a mixed Hodge structure on the homotopy groups.

Hint

The product map is a morphism of mixed Hodge structures, and a quotient by a sub-MHS is again an MHS.

Answer

The product $M_{U}^{+} \otimes M_{U}^{+} \to M_{U}^{+}$ is a morphism of mixed Hodge structures (the wedge product on the log complex is bifiltered, $W_{a} \land W_{b} \subseteq W_{a + b}$ and $F^{a} \land F^{b} \subseteq F^{a + b}$ ). Its image, the decomposable ideal, is therefore a sub-MHS. The category of mixed Hodge structures is abelian with strict morphisms, so the quotient $Q M_{U} = M_{U}^{+} / (M_{U}^{+} \cdot M_{U}^{+})$ inherits a mixed Hodge structure. Since $Q M_{U}^{n} ≅ Hom (π_{n} (U), C)$ by the main theorem of 03.12.06, dualising transfers the MHS onto $π_{n} (U) \otimes Q$ . Rubric: full credit for the bifiltered product, the abelian-category quotient, and the dualisation onto $π_{*}$ .

Exercise 6 (hard, short-answer).

The fundamental group of $U = P^{1} ∖ {0, 1, \infty}$ is free on two generators $x, y$ (loops around $0$ and $1$ ). Describe the mixed Hodge structure on the first two layers of its lower central series, and explain in what sense the weights make $π_{1}$ "motivic".

Hint

The abelianisation $π_{1}^{ab} \otimes Q ≅ H_{1} (U; Q)$ is dual to $H^{1} (U)$ , pure of weight $- 2$ (homology weights are negatives of cohomology weights). The next graded piece of the lower central series is built from the commutator $[x, y]$ .

Answer

Work with the Malcev (pro-unipotent) completion, whose associated graded is the free Lie algebra on $H_{1} (U; Q)$ . The first layer $gr_{1} = H_{1} (U; Q)$ is dual to $H^{1} (U)$ , hence pure of weight $- 2$ (two generators, types $(- 1, - 1)$ ). The second layer $gr_{2} = Λ^{2} H_{1}$ is spanned by the commutator $[x, y]$ , of weight $- 4$ , type $(- 2, - 2)$ — it is one-dimensional and accounts for the residue relation around $\infty$ . The weights are even and pure of Tate type $(- n, - n)$ at every level: this is the hallmark of a mixed Tate motive. The class of $[x, y]$ at weight $- 4$ , paired against the de Rham side, produces the period $ζ (2) = π^{2} /6$ ; deeper commutators produce $ζ (n)$ , linking $π_{1}$ of the thrice-punctured line to the Drinfeld-Ihara theory and the motivic fundamental group (Hain, Deligne-Goncharov). Calling $π_{1}$ "motivic" means precisely that its weight-graded pieces are mixed Tate, with periods the multiple zeta values. Rubric: full credit for the weight $- 2$ / weight $- 4$ Tate identification and the connection to $ζ (n)$ .

Exercise 7 (hard, short-answer).

Deligne's two-filtration (strictness) lemma is the technical engine of Morgan's construction. State the strictness property and explain why, without it, the inductive assignment of weights to minimal-model generators would fail.

Hint

Strictness says a bifiltered morphism's image meets each filtration step in exactly the image of that step: $f (V) \cap W_{k} V^{'} = f (W_{k} V)$ , and likewise for $F$ . The minimal-model build computes obstruction classes in cohomology groups that must inherit clean Hodge structures.

Answer

Strictness: for a morphism $f$ of mixed Hodge structures, $f (W_{k} V) = f (V) \cap W_{k} V^{'}$ and $f (F^{p} V_{C}) = f (V_{C}) \cap F^{p} V_{C}^{'}$ ; consequently kernels, images, and cohomology of bifiltered complexes inherit mixed Hodge structures with the induced filtrations, and the spectral sequences of $W$ and $F$ degenerate appropriately (Deligne, Hodge II §1.3). In the inductive minimal-model construction, at each stage the new generators are dual to cohomology groups of the partially-built model relative to the log complex; the differential is chosen to kill an obstruction class. If the relevant cohomology groups did not carry well-defined Hodge structures with induced filtrations — which is exactly what strictness guarantees — there would be no consistent weight to assign the new generators, and the bigrading would not propagate. Strictness is what makes "induced filtration on cohomology" a legitimate mixed Hodge structure rather than an ill-defined filtration. Rubric: full credit for the strictness statement and the role of induced-MHS-on-cohomology in closing the induction.

Advanced results Master

The mixed Hodge structure on the minimal model is the first instance of a systematic Hodge theory of homotopy types, and several refinements sharpen Morgan's theorem.

Weight bounds on homotopy. For a smooth variety $U$ , the weights on $π_{n} (U) \otimes Q$ satisfy $- 2 n \leq w \leq - n - 1$ in the dual convention (equivalently, the weights on the indecomposables $Q M_{U}^{n}$ lie in $[n + 1, 2 n]$ ). For $U$ smooth projective the weights are pure: $Q M_{U}^{n}$ has weight exactly $n$ — recovering, via Exercise 3, the DGMS formality theorem of 04.09.05 as the boundary case where the weight interval collapses to a point. The width of the weight interval is a precise measure of how far $U$ is from being formal: a one-point interval is formality, and any spread signals surviving higher operations.

Hain's bar-construction refinement. Hain (1987) re-derived Morgan's theorem through the reduced bar construction $B (M_{U})$ on a mixed-Hodge CDGA model, obtaining a mixed Hodge structure not only on the homotopy groups but on the full homotopy coalgebra and, for the fundamental group, on the Malcev Lie algebra $p = Lie (π_{1}^{un})$ . The bar construction computes $H^{0} (B (M_{U}))$ as the dual of the universal enveloping algebra of $p$ , and the weight filtration on the bar complex descends to the lower central series filtration of $p$ . This is the framework in which the relation $[x, y]$ of Exercise 6 acquires its weight $- 4$ .

The motivic fundamental group. For $U = P^{1} ∖ {0, 1, \infty}$ the mixed Hodge structure on $π_{1}^{un}$ is of mixed Tate type: all weights even, all Hodge types $(p, p)$ . Deligne and Goncharov upgraded this to a motivic fundamental group $π_{1}^{mot}$ , an object in a Tannakian category of mixed Tate motives over $Z$ , whose periods are the multiple zeta values $ζ (n_{1}, \dots, n_{r})$ . The Hodge realisation of $π_{1}^{mot}$ is exactly the mixed Hodge structure that Morgan's theorem produces. This places the rational homotopy of one of the simplest open varieties at the centre of modern arithmetic geometry.

Non-formality is the generic situation. In contrast to the compact Kähler case, smooth quasi-projective varieties are generically not formal. The complement of a generic hyperplane arrangement, for instance, is formal (Brieskorn, Orlik-Solomon), but adding tangencies or higher-multiplicity intersections destroys formality, and the mixed Hodge structure on the model is then the canonical replacement for the cohomology ring as a complete rational-homotopy invariant. The weight filtration detects exactly the failure: nonformality occurs precisely when the weight grading on the minimal model is not concentrated in the lowest possible weights.

Synthesis. The central insight is that the minimal model is rigid enough to carry the entire Hodge-theoretic weight structure, and putting these refinements together shows the weight interval on $π_{*}$ is the foundational reason the homotopy theory of varieties is finer than that of bare spaces. This is exactly the boundary phenomenon of 04.09.05: the DGMS formality theorem is the collapse of the weight interval to a point, and the generalises-to-open-varieties direction is Morgan's spreading of that interval. The bridge from cohomology to homotopy is dual to the bar construction; the bar construction generalises the indecomposables functor, and Hain's mixed Hodge structure on the Malcev Lie algebra is dual to the mixed Hodge structure on cohomology. The motivic fundamental group is the deepest layer, where this picture connects to periods and multiple zeta values — the central insight being that even the homotopy of $P^{1} ∖ {0, 1, \infty}$ encodes arithmetic.

Full proof set Master

Proposition 1 (purity forces formality of smooth projective varieties). Let $X$ be a smooth projective complex variety. Then Morgan's mixed Hodge structure on the minimal model $M_{X}$ is pure — the indecomposables $Q M_{X}^{n}$ have weight exactly $n$ — and $X$ is formal.

Proof. For $X$ projective, $D = \emptyset$ , so the log complex is $Ω_{X}^{∙}$ and $H^{k} (X; C)$ is pure of weight $k$ (04.09.01). The minimal-model generators in degree $n$ dual to $H^{n}$ carry weight $n$ . The differential $d$ preserves weight (it is a morphism of mixed Hodge structures) and raises cohomological degree by $1$ . A generator $x$ of degree $n$ and weight $n$ has $d x \in M_{X}^{n + 1}$ ; the decomposable part of $M_{X}^{n + 1}$ is spanned by products of generators of total degree $n + 1$ and hence, by purity, total weight $n + 1$ . Since $w (d x) = w (x) = n \neq = n + 1$ , the component of $d x$ in the decomposables vanishes, and by minimality $d x \in M_{X}^{+} \cdot M_{X}^{+}$ already, so $d x = 0$ . Thus every cohomology generator is closed and $M_{X}$ has the form of the minimal model of $(H^{*} (X), 0)$ , which is formality. $□$

Proposition 2 (the weight filtration on $H^{1}$ of a curve complement). Let $X$ be a smooth projective curve of genus $g$ and $U = X ∖ {p_{1}, \dots, p_{m}}$ with $m \geq 1$ . Then $H^{1} (U; Q)$ has dimension $2 g + m - 1$ , with $W_{1} H^{1} (U)$ of dimension $2 g$ (pure weight $1$ , the part from $H^{1} (X)$ ) and $gr_{2}^{W} H^{1} (U)$ of dimension $m - 1$ (pure weight $2$ , type $(1, 1)$ , spanned by residues at the punctures modulo the global residue relation).

Proof. The Gysin/residue sequence of the pair $(X, D)$ with $D = {p_{1}, \dots, p_{m}}$ reads $$ 0 \to H^1(X) \to H^1(U) \xrightarrow{\ \mathrm{Res}\ } \bigoplus_{i=1}^m H^0(p_i)(-1) \xrightarrow{\ \Sigma\ } H^2(X) \to 0, $$ where $(- 1)$ denotes a Tate twist raising weight by $2$ . The leftmost map is the pullback, injective with image the weight- $1$ part $W_{1} H^{1} (U) ≅ H^{1} (X)$ , of dimension $2 g$ and pure weight $1$ . The residue map lands in $⨁_{i} Q (- 1)$ , pure of weight $2$ and type $(1, 1)$ ; its image is the kernel of the sum-of-residues map $Σ$ , which is the global residue relation $\sum_{i} Res_{p_{i}} = 0$ (a single linear condition since $H^{2} (X) = Q (- 1)$ ). Hence $dim im (Res) = m - 1$ , giving $gr_{2}^{W} H^{1} (U)$ of dimension $m - 1$ , and $dim H^{1} (U) = 2 g + (m - 1)$ . The filtration $0 \subseteq W_{1} \subseteq W_{2} = H^{1} (U)$ with the stated graded pieces is Deligne's mixed Hodge structure. $□$

For $X = P^{1}$ ( $g = 0$ ) and $m = 3$ : $dim H^{1} (U) = 0 + 3 - 1 = 2$ , all of weight $2$ , recovering the worked example.

Proposition 3 (functoriality and well-definedness). The mixed Hodge structure on $M_{U}$ is independent of the chosen normal-crossings compactification, and a morphism $f : U \to U^{'}$ of smooth varieties induces a morphism of mixed-Hodge minimal models.

Proof. Given two compactifications $(X_{1}, D_{1})$ and $(X_{2}, D_{2})$ of $U$ , resolution of singularities yields a third $(X_{3}, D_{3})$ dominating both, with proper maps $X_{3} \to X_{i}$ that are isomorphisms over $U$ . Pullback of log forms $Ω_{X_{i}}^{∙} (lo g D_{i}) \to Ω_{X_{3}}^{∙} (lo g D_{3})$ is a bifiltered quasi-isomorphism (Deligne, Hodge II §3.2: the log complex is independent of the compactification in the filtered derived category). Hence the induced mixed Hodge structures on $H^{*} (U)$ , and by the strictness-driven minimal-model construction on $M_{U}$ , agree. For a morphism $f : U \to U^{'}$ , extend to a morphism of compactifications $\overset{ˉ}{f} : (X, D) \to (X^{'}, D^{'})$ after blowing up (possible by Hironaka), inducing $\overset{ˉ}{f}^{*} : Ω_{X^{'}}^{∙} (lo g D^{'}) \to Ω_{X}^{∙} (lo g D)$ , a bifiltered CDGA morphism. Functoriality of the minimal-model construction then yields a bifiltered morphism $M_{U^{'}} \to M_{U}$ , i.e. a morphism of mixed Hodge structures on each graded piece, dualising to $π_{*} (U) \to π_{*} (U^{'})$ . $□$

Connections Master

The construction sits directly on top of 03.12.06 (Sullivan minimal models and rational homotopy theory): Morgan's theorem equips that unit's minimal model and its identification $Q M^{n} ≅ Hom (π_{n}, Q)$ with a mixed Hodge structure. Every step of the bigraded construction runs the inductive minimal-model build of 03.12.06, adding weight and Hodge data compatible with the differential and product.
The pure-weight special case is the DGMS formality theorem of 04.09.05 (the $d d^{c}$ -lemma and Kähler formality). When the divisor $D$ is empty the weight grading collapses to a single value per degree, the model differential is forced to vanish on cohomology generators (Proposition 1), and formality follows. Morgan's theorem is the open-variety generalisation in which the weight interval is allowed to spread.
The Hodge-theoretic input is the pure Hodge decomposition of 04.09.01 (Hodge decomposition): each weight-graded piece $gr_{k}^{W}$ of the mixed structure is a pure Hodge structure of weight $k$ , exactly the structure that 04.09.01 places on the cohomology of a compact Kähler manifold. The motivating example of $H^{k} (U)$ for open $U$ is built from the pure pieces of the compactification via the logarithmic complex.
Laterally, the weight filtration constrains the Massey products and formality of 03.12.51 (Massey products and the formality condition): a surviving Massey product on a smooth variety must respect the weight bookkeeping (Exercise 4), and on a compact Kähler manifold the weight mismatch forces all rational Massey products into the indeterminacy ideal, re-proving their vanishing by Hodge-theoretic rather than $d d^{c}$ means.
The construction also connects to 03.12.45 (arithmetic square and integral fracture): the mixed Hodge structure is a rational-homotopy refinement, living at the rational (Hodge) localisation, while the arithmetic square assembles rational and $p$ -adic homotopy types; the motivic fundamental group of $P^{1} ∖ {0, 1, \infty}$ is the object where the Hodge and $ℓ$ -adic realisations are compared.

Historical & philosophical context Master

Pierre Deligne introduced mixed Hodge structures in Théorie de Hodge II ^{[Deligne 1971]} (Publ. Math. IHÉS 40, 1971), constructing the canonical functorial mixed Hodge structure on the cohomology of a smooth open complex algebraic variety via the logarithmic de Rham complex, and completed the theory for arbitrary singular varieties in Théorie de Hodge III ^{[Deligne 1974]} (Publ. Math. IHÉS 44, 1974) through simplicial resolutions. The motivation came from the Weil conjectures: weights on cohomology are the Hodge-theoretic shadow of the weights of Frobenius eigenvalues in the $ℓ$ -adic theory, and Deligne's yoga of weights organised both.

John Morgan, in The algebraic topology of smooth algebraic varieties ^{[Morgan 1978]} (Publ. Math. IHÉS 48, 1978), lifted Deligne's cohomological mixed Hodge structure onto the Sullivan minimal model and hence onto the rational homotopy groups, generalising the Deligne-Griffiths-Morgan-Sullivan formality theorem ^{[DGMS 1975]} (Invent. Math. 29, 1975) from compact Kähler manifolds to arbitrary smooth varieties. Richard Hain, in The de Rham homotopy theory of complex algebraic varieties ^{[Hain 1987]} (K-Theory 1, 1987), re-derived and extended the result via the bar construction, placing a mixed Hodge structure on the Malcev completion of the fundamental group; this fed directly into the theory of the motivic fundamental group of Deligne and Goncharov and the appearance of multiple zeta values as periods of $π_{1} (P^{1} ∖ {0, 1, \infty})$ .

Bibliography Master

@article{Deligne1971HodgeII,
  author  = {Deligne, Pierre},
  title   = {Th\'eorie de Hodge, II},
  journal = {Publications Math\'ematiques de l'I.H.\'E.S.},
  volume  = {40},
  pages   = {5--57},
  year    = {1971}
}

@article{Deligne1974HodgeIII,
  author  = {Deligne, Pierre},
  title   = {Th\'eorie de Hodge, III},
  journal = {Publications Math\'ematiques de l'I.H.\'E.S.},
  volume  = {44},
  pages   = {5--77},
  year    = {1974}
}

@article{Morgan1978,
  author  = {Morgan, John W.},
  title   = {The algebraic topology of smooth algebraic varieties},
  journal = {Publications Math\'ematiques de l'I.H.\'E.S.},
  volume  = {48},
  pages   = {137--204},
  year    = {1978}
}

@article{DGMS1975,
  author  = {Deligne, Pierre and Griffiths, Phillip and Morgan, John and Sullivan, Dennis},
  title   = {Real homotopy theory of K\"ahler manifolds},
  journal = {Inventiones Mathematicae},
  volume  = {29},
  number  = {3},
  pages   = {245--274},
  year    = {1975}
}

@article{Hain1987,
  author  = {Hain, Richard M.},
  title   = {The de Rham homotopy theory of complex algebraic varieties I, II},
  journal = {K-Theory},
  volume  = {1},
  pages   = {271--324, 481--497},
  year    = {1987}
}

@book{PetersSteenbrink2008,
  author    = {Peters, Chris A. M. and Steenbrink, Joseph H. M.},
  title     = {Mixed Hodge Structures},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {52},
  publisher = {Springer},
  year      = {2008}
}

@book{GriffithsMorgan1981,
  author    = {Griffiths, Phillip A. and Morgan, John W.},
  title     = {Rational Homotopy Theory and Differential Forms},
  series    = {Progress in Mathematics},
  volume    = {16},
  publisher = {Birkh\"auser},
  year      = {1981},
  note      = {2nd ed., Springer, 2013}
}

@article{DeligneGoncharov2005,
  author  = {Deligne, Pierre and Goncharov, Alexander B.},
  title   = {Groupes fondamentaux motiviques de Tate mixte},
  journal = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure},
  volume  = {38},
  number  = {1},
  pages   = {1--56},
  year    = {2005}
}

Prerequisites

03.12.06
04.09.05
04.09.01

Tier anchors

beginner: Griffiths-Morgan Ch. IX informal — the rational homotopy of an algebraic variety carries a hidden weight grading; the thrice-punctured line $\mathbb{P}^1 \setminus \{0,1,\infty\}$ as the first example
intermediate: Morgan 1978 *The algebraic topology of smooth algebraic varieties* (Publ. Math. IHÉS 48) §§1–8; Deligne 1971 *Théorie de Hodge II* (Publ. Math. IHÉS 40) §3 (weight + Hodge filtrations); Griffiths-Morgan *Rational Homotopy Theory and Differential Forms* Ch. IX; Peters-Steenbrink *Mixed Hodge Structures* §3
master: Morgan 1978 *The algebraic topology of smooth algebraic varieties* (Publ. Math. IHÉS 48, 137–204); Deligne 1971 *Théorie de Hodge II* (Publ. Math. IHÉS 40, 5–57) and 1974 *Théorie de Hodge III* (Publ. Math. IHÉS 44, 5–77); Hain 1987 *The de Rham homotopy theory of complex algebraic varieties I–II* (K-Theory 1, 271–324 and 481–497); Deligne-Griffiths-Morgan-Sullivan 1975 *Real homotopy theory of Kähler manifolds* (Invent. Math. 29); Peters-Steenbrink *Mixed Hodge Structures* (Springer 2008)

References

Morgan — The algebraic topology of smooth algebraic varieties · *Publications mathématiques de l'I.H.É.S.* 48 (1978), 137–204; the originator paper extending Deligne-Griffiths-Morgan-Sullivan from compact Kähler manifolds to arbitrary smooth complex algebraic varieties: the minimal model of $A_{PL}(U) \otimes \mathbb{C}$ for a smooth variety $U = X \setminus D$ (with $X$ smooth projective and $D$ a normal-crossings divisor) carries a canonical functorial mixed Hodge structure, built from the logarithmic de Rham complex $\Omega^\bullet_X(\log D)$, so that the rational homotopy groups $\pi_*(U) \otimes \mathbb{Q}$ inherit a mixed Hodge structure extending Deligne's mixed Hodge structure on $H^*(U)$; with the corollary that smooth quasi-projective varieties are generally not formal but the mixed Hodge structure is the canonical replacement
Deligne — Théorie de Hodge II · *Publications mathématiques de l'I.H.É.S.* 40 (1971), 5–57; the originator paper defining mixed Hodge structures (a $\mathbb{Q}$-vector space with an increasing weight filtration $W_\bullet$ and a decreasing Hodge filtration $F^\bullet$ on the complexification, $F$ inducing a pure Hodge structure of weight $k$ on each $\mathrm{gr}^W_k$), proving that the cohomology of a smooth open complex algebraic variety $U = X \setminus D$ carries a canonical functorial mixed Hodge structure via the logarithmic de Rham complex, with weights on $H^k(U)$ running from $k$ to $2k$, and establishing strictness of morphisms with respect to both filtrations
Deligne — Théorie de Hodge III · *Publications mathématiques de l'I.H.É.S.* 44 (1974), 5–77; the construction of the mixed Hodge structure on the cohomology of an arbitrary (possibly singular, possibly non-compact) complex algebraic variety via cohomological descent and simplicial smooth-projective resolutions, completing the functorial mixed-Hodge-theory package on which Morgan's homotopy-theoretic extension rests
Hain — The de Rham homotopy theory of complex algebraic varieties · *K-Theory* 1 (1987), Part I 271–324 and Part II 481–497; the construction of a mixed Hodge structure on the Malcev (pro-unipotent) completion of the fundamental group $\pi_1(U, x)$ of a smooth complex algebraic variety and on the higher rational homotopy groups via the bar construction on a mixed-Hodge CDGA model, refining and re-deriving Morgan 1978 and linking to periods and the motivic fundamental group
Deligne-Griffiths-Morgan-Sullivan — Real homotopy theory of Kähler manifolds · *Inventiones Mathematicae* 29 (1975), 245–274; the formality theorem for simply-connected compact Kähler manifolds (the rational de Rham CDGA is quasi-isomorphic to its cohomology ring with zero differential), the pure-weight special case ($D = \varnothing$) of Morgan's mixed-Hodge extension
Peters-Steenbrink — Mixed Hodge Structures · Ergebnisse der Mathematik 52, Springer 2008; §3 (the abelian category of mixed Hodge structures, Deligne's canonical bigrading $I^{p,q}$, strictness), §8 (mixed Hodge theory of the cohomology of varieties), the modern encyclopaedic reference
Griffiths-Morgan — Rational Homotopy Theory and Differential Forms · Progress in Mathematics 16, Birkhäuser, Boston 1981; second edition Birkhäuser 2013 (ISBN 978-1-4614-8467-7); Ch. IX develops the mixed-Hodge-theoretic refinement of rational homotopy theory for smooth algebraic varieties in the differential-form-flavoured Sullivan framework, with the thrice-punctured projective line as the worked example

Estimated time

beginner: 20m
intermediate: 50m
master: 85m