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Volume 29, Issue 1 (Jan 2017)

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Rational homotopy of complex projective varieties with normal isolated singularities

David Chataur
  • Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Université de Picardie Jules Verne 33, rue Saint-Leu, 80039 Amiens Cedex 1, France
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/ Joana Cirici
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  • Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
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Published Online: 2016-06-18 | DOI: https://doi.org/10.1515/forum-2015-0101

Abstract

Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper, we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological space. This result applies to a large class of examples, such as normal surface singularities, varieties with ordinary multiple points, hypersurfaces with isolated singularities and, more generally, complete intersections with isolated singularities. We obtain analogous results for contractions of subvarieties.

Keywords: Rational homotopy; mixed Hodge theory; weight filtration; fundamental groups; formality,singular varieties; isolated singularities; contractions

MSC 2010: 55P62; 32S35

1 Introduction

The rational homotopy type of a topological space X is the commutative differential graded algebra (cdga for short) 𝒜pl(X) in the homotopy category Ho(CDGA) defined by inverting quasi-isomorphisms, where 𝒜pl:𝖳𝗈𝗉CDGA is Sullivan’s functor of rational piece-wise linear forms. A topological space X is said to be formal if its rational algebra of piece-wise linear forms is a formal cdga: there is a string of quasi-isomorphisms from 𝒜pl(X) to its cohomology H*(𝒜pl(X))H*(X;) considered as a cdga with trivial differential. In particular, if X is formal then its rational homotopy type is completely determined by its cohomology ring, and higher order Massey products vanish. Using Hodge theory, Deligne, Griffiths, Morgan and Sullivan [9] proved that smooth projective varieties or, more generally, compact Kähler manifolds, are formal.

Simpson [25] showed that every finitely presented group G is the fundamental group π1(X) of an irreducible projective variety X. Later, Kapovich and Kollár [19] showed that X can be chosen to be a complex projective surface with simple normal crossing singularities only. This implies the existence of non-formal complex projective varieties. For instance, one may take G to be the fundamental group of the complement of the Borromean rings, which has non-trivial triple Massey products. In this paper, we use mixed Hodge theory to show that a large class of projective varieties with normal isolated singularities are formal topological spaces. We also study the mixed Hodge structures on the rational homotopy type of a contraction of a subvariety and prove analogous results in this setting.

We next outline the contents of this paper. For a complex algebraic variety X, the complex homotopy type 𝒜pl(X) can be computed from the differential bigraded algebra defined by the first term E1*,*(X) of the multiplicative weight spectral sequence, a multiplicative analogue of Deligne’s weight spectral sequence [8]. This result was proven by Morgan [21] for smooth quasi-projective varieties and by Cirici–Guillén [6] in the possibly singular case. In Section 2, we study the multiplicative weight spectral sequence of a projective variety with isolated singularities and provide a simple description of its algebra structure in terms of a resolution of singularities of the variety. This gives an upper bound on the homotopy theoretic complexity of the variety.

The idea that purity of the weight filtration implies formality is part of the folklore on mixed Hodge theory and goes back to [9]. In Section 3, we give a simple proof of a refinement of this idea: we show that the purity of the weight filtration of a complex projective variety up to a certain degree, implies formality of the variety up to the same degree. We prove the main results of this paper in Section 4. We first show, using the multiplicative weight spectral sequence, that every normal complex projective surface is formal. We then generalize this result to arbitrary dimensions and prove formality for normal projective varieties with isolated singularities whose link is (n-2)-connected, where n is the dimension of the variety. In particular, complete intersections with isolated singularities are formal. Using similar techniques, we show that if X is a projective variety with normal isolated singularities admitting a resolution of singularities with smooth exceptional divisor, then X is a formal topological space. Lastly, in Section 5 we prove analogous results for contractions of subvarieties. In particular, we show that if YX is a closed immersion of smooth projective varieties, then X/Y is a formal topological space.

The results of this paper arose from a more general study (see [4]) of the vanishing of Massey products on the intersection cohomology of projective varieties with only isolated singularities. However, the simplicity of the statements and proofs in the classical setting of rational homotopy encouraged us to write them up separately. Our more general study follows the intersection-homotopy treatment of [5], where Chataur, Saralegi and Tanré prove intersection-formality for certain spaces. In particular, they show that any nodal hypersurface in 4 is formal.

2 Multiplicative weight spectral sequence

Deligne proved that the k-th cohomology space Hk(X;) of every complex algebraic variety X carries a functorial mixed Hodge structure: this is given by an increasing filtration

0=W-1Hk(X;)W0Hk(X;)W2kHk(X;)=Hk(X;)

of the rational cohomology of X, called the weight filtration, together with a decreasing filtration F of the complex cohomology Hk(X;), called the Hodge filtration, such that F and its complex conjugate induce a Hodge decomposition of weight p on each graded piece GrpWHk(X;). If X is smooth then Wk-1Hk(X;)=0, while if X is projective then WkHk(X;)=Hk(X;). Let us briefly explain how W is defined in the latter case. Let XX be a simplicial resolution of X: this is a smooth simplicial variety X together with an augmentation morphism XX satisfying cohomological descent (see [8], see also [24, Section II.5]). Deligne showed that the associated spectral sequence E~1p,q(X):=Hq(Xp;)Hp+q(X;) degenerates at the second stage, and that the induced filtration on the rational cohomology of X is well-defined (does not depend on the chosen resolution) and is functorial for morphisms of varieties. The weight filtration W on H*(X;) is then defined by décalage of the induced filtration. We have GrqWHp+q(X;)E~2p,q(X).

Let k0. The weight filtration on Hk(X;) is said to be pure of weight k if

0=Wk-1Hk(X;)WkHk(X;)=Hk(X;).

For instance, if X is smooth and projective then W on Hk(X;) is pure of weight k, for all k0. A well-known consequence of the Decomposition Theorem of intersection homology is that a projective variety X with only isolated singularities satisfies semi-purity: the weight filtration on Hk(X;) is pure of weight k, for all k>n, where n=dim(X) (see [26], see also [22] for a more direct proof).

Morgan [21] introduced mixed Hodge diagrams of differential graded algebras and proved the existence of functorial mixed Hodge structures on the rational homotopy groups of smooth complex algebraic varieties. His results were extended to the singular setting by Hain [16] and Navarro-Aznar [23] independently. In particular, Navarro-Aznar introduced the Thom–Whitney simple functor and developed the construction of algebras of piece-wise linear forms associated to simplicial varieties. As a result, for any complex algebraic variety X, one has a multiplicative weight spectral sequence E1*,*(X) which is a well-defined algebraic invariant of X in the homotopy category of differential bigraded algebras, and is homotopy equivalent, when forgetting the algebra structure, to Deligne’s weight spectral sequence E~1*,*(X). Furthermore, the multiplicative weight spectral sequence carries information on the rational homotopy type of the variety.

2.1 Mixed Hodge diagrams and multiplicative weight spectral sequence

We next recall the definition of the multiplicative weight spectral sequence associated with a complex algebraic variety.

A filtered cdga (A,W) over a field 𝐤 is a cdga A over 𝐤 together with an (increasing) filtration {WpA} indexed by such that Wp-1AWpA, d(WpA)WpA, and WpAWqAWp+qA.

Every filtered cdga (A,W) has an associated spectral sequence, each of whose stages (Er*,*(A,W),dr) is a differential bigraded algebra with differential of bidegree (r,1-r).

A mixed Hodge diagram (of cdgas over ) consists of a filtered cdga (𝒜,W) over , a bifiltered cdga (𝒜,W,F) over , together with a string of filtered quasi-isomorphisms from (𝒜,W) to (𝒜,W). In addition, the following axioms are satisfied:

  • (MH1)

    The weight filtration W is regular and exhaustive. The Hodge filtration F is biregular. The cohomology H*(𝒜) has finite type.

  • (MH2)

    For all p, the differential of GrpW𝒜 is strictly compatible with F.

  • (MH3)

    For all k0 and all p, the filtration F induced on Hk(GrpW𝒜) defines a pure Hodge structure of weight p+k on Hk(GrpW𝒜).

Such a diagram is denoted as 𝒜=((𝒜,W)φ(𝒜,W,F)).

By forgetting the multiplicative structures we recover the original notion of mixed Hodge complex introduced by Deligne (see [8, Section 8.1]). In particular, the k-th cohomology group of every mixed Hodge diagram is a mixed Hodge structure.

The following is a multiplicative version of Deligne’s Theorem [8] on the existence of functorial mixed Hodge structures in cohomology.

([23, §9], see also [16])

For every complex algebraic variety X there exists a mixed Hodge diagram A(X) such that A(X)QApl(X) and for all k0, the cohomology Hk(A(X)) is isomorphic to Deligne’s mixed Hodge structure on Hk(X;Q). This construction is well-defined and functorial for morphisms of varieties in the homotopy category of mixed Hodge diagrams.

The rational component (𝒜,W) of every mixed Hodge diagram 𝒜 is a filtered cdga over the rationals. Hence it has an associated spectral sequence E1*,*(𝒜,W)H*(X;).

([6, Theorem 3.23])

Let A be mixed Hodge diagram such that H0(AQ)Q. There is a string of quasi-isomorphisms of complex cdgas from (AQ,W)C to E1(AQ,W)C compatible with the filtration W.

The proof of the above result uses minimal models in the sense of rational homotopy. This is why we ask that mixed Hodge diagrams are cohomologically connected. Under some finite type conditions the above result is also valid over (see [6, Theorem 2.26]). We remark that in [6], a weaker notion of mixed Hodge diagram is used. This does not affect the above result.

Let X be a complex algebraic variety. The multiplicative weight spectral sequence of X is the spectral sequence

E1*,*(X):=E1*,*(𝒜(X),W)H*(X;)

associated with the filtered cdga (𝒜(X),W) given by the rational component of 𝒜(X).

This is a well-defined algebraic invariant of X in the homotopy category of differential bigraded algebras. When forgetting the multiplicative structures, there is a homotopy equivalence of bigraded complexes between E1*,*(X) and Deligne’s weight spectral sequence E~1*,*(X).

The main advantage of the multiplicative weight spectral sequence with respect to Deligne’s weight spectral sequence is that by Theorem 2.4, the former carries information about the rational homotopy type of X.

2.2 Thom–Whitney simple

Theorem 2.3 relies on the Thom–Whitney simple functor, which associates, to every strict cosimplicial cdga A over a field 𝐤 of characteristic zero, a new cdga sTW(A) over 𝐤. This cdga is homotopically equivalent, as a complex, to the total complex of the original cosimplicial cdga. Hence the Thom–Whitney simple can be viewed as a multiplicative version of the total complex. We next recall its construction. For every α0, denote by Ωα the cdga given by

Ωα:=Λ(t0,,tα,dt0,,dtα)ti-1,dti,

where Λα:=Λ(t0,,tα,dt0,,dtα) denotes the free cdga over 𝐤 generated by ti in degree 0 and dti in degree 1. The differential on Λα is defined by d(ti)=dti and d(dti)=0. For all 0iα, define face maps δαi:ΩαΩα-1 by letting

δαi(tk)={tk,k<i,0,k=i,tk-1,k>i.

These definitions make Ω={Ωα,δαi} into a strict simplicial cdga (the adjective strict accounts for the fact that we do not require degeneracy maps).

Recall that the total complex s(K) of a strict cosimplicial cochain complex K is given by

s(K):=αKαC*(Δ|α|)=αKα[-|α|],

where C*(Δ|α|) denotes the cochain complex of Δ|α|. Analogously, we have:

([23, §3])

Let A be a strict cosimplicial cdga over 𝐤. The Thom–Whitney simple of A is the cdga over 𝐤 defined by the end

sTW(A):=αAαΩα.

Let f,g:AB be morphisms of cdgas. Then sTW(f,g) is given by the pull-back

where δx:BΛ(t,dt)B is the evaluation map given by tx and dt0.

We shall need the following filtered version of the Thom–Whitney simple. Given r0, consider on the cdga Ωα the multiplicative increasing filtration σ[r] defined by letting ti be of weight 0 and dti be of weight -r, for all 0iα. Note that σ[0] is the trivial filtration, while σ[1] is the bête filtration.

Let (A,W) be a strict cosimplicial filtered cdga. The r-Thom–Whitney simple of (A,W) is the filtered cdga (sTW(A),W(r)) defined by

W(r)psTW(A):=αq(Wp-qAασ[r]qΩα).

To study the weight spectral sequence we shall mostly be interested in the behavior of the filtration W(1) of the Thom–Whitney simple, which in the setting of complexes corresponds to the diagonal filtration δ(W,L) of s(-) introduced in [8, Section 7.1.6] (see also [7, Section 2]). The following is a matter of verification.

Let (A,W) be a finite strict cosimplicial filtered cdga. The spectral sequence associated with (sTW(A),W(1)) satisfies

E1p,q(sTW(A),W(1)):=Hp+q(Gr-pW(1)sTW(A))αm(E1p-m,q(Aα,W)Ωαm).

Deligne’s simple of a cosimplicial mixed Hodge complex 𝒦 is the diagram of complexes given by

sD(𝒦):=((s(𝒦),δ(W,L))s(φ)(s(𝒦),δ(W,L),F)),

which by [8, Theorem 8.1.15], is again a mixed Hodge complex. The following Lemma is a multiplicative analogue of this result and gives a Thom–Whitney simple in the category of mixed Hodge diagrams.

([23, §7.11])

Let A be a strict cosimplicial mixed Hodge diagram. The diagram of cdgas given by

sTW(𝒜):=((sTW(𝒜),W(1))sTW(φ)(sTW(𝒜),W(1),F(0)))

is a mixed Hodge diagram, which is homotopy equivalent, as a complex, to Deligne’s simple sD(A).

2.3 Main examples

We next give a description of the multiplicative weight spectral sequence in some particular situations of interest for this paper.

Smooth projective varieties

Let X be a smooth projective variety. A mixed Hodge diagram 𝒜(X) for X is given by the data 𝒜(X):=𝒜pl(X) and 𝒜(X):=𝒜dR(X;) the complex de Rham algebra, with W the trivial filtration and F the classical Hodge filtration. The multiplicative weight spectral sequence satisfies E10,*(X)=H*(X;) and E1p,*(X)=0 for all p>0.

Varieties with normal crossings

Let DX~ be a simple normal crossings divisor in a smooth projective variety X~ of dimension n. We may write D=D1DN as the union of irreducible smooth varieties meeting transversally. Let D(0)=X~ and for all p>0, denote by D(p)=|I|=pDI the disjoint union of all p-fold intersections DI:=Di1Dip where I={i1,,ip} denotes an ordered subset of {1,,N}. Since D has normal crossings, it follows that D(p) is a smooth projective variety of dimension n-p. For 1kp, denote by jI,k:DIDI{ik} the inclusion and let jp,k:=|I|=pjI,k:D(p)D(p-1). This defines a simplicial resolution D={D(p),jp,k}D, called the canonical hyperresolution of D. Deligne’s weight spectral sequence is the first quadrant spectral sequence given by (see for example [14, §4])

E~1p,q(D)=Hq(D(p+1);)Hq+p(D;)

with the differential d~1p,*=jp+2* defined via the combinatorial restriction morphisms

jp*:=k=1p(-1)k-1(jp,k)*:H*(D(p-1);)H*(D(p);).

By computing the cohomology of E~1*,*(D) we obtain

GrqWHp+q(D;)E~2p,q(D)=Ker(d1p,q)/Im(d1p-1,q){Ker(j2*)q,p=0,Ker(jp+2*)q/Im(jp+1*)q,p>0.

Let D={D(p),jp,k}D be the canonical hyperresolution of a simple normal crossings divisor D. The multiplicative weight spectral sequence of D is given by

E1p,q(D)αHq(D(α+1);)ΩαpHp+q(D;).

Proof.

Since D(p) is smooth and projective for each p, there is a strict cosimplicial mixed Hodge diagram 𝒜(D):={𝒜(D(p)),jp,k*} with trivial weight filtrations. A mixed Hodge diagram for D is then given by the Thom–Whitney simple 𝒜(D):=sTW(𝒜(D)). Indeed, cohomological descent for rational homotopy gives a quasi-isomorphism 𝒜(D)𝒜pl(X). Furthermore, the filtrations of 𝒜(D) given in Lemma 2.10 make 𝒜(D) into a mixed Hodge diagram. By Lemma 2.10, when forgetting the multiplicative structures, 𝒜(D) is quasi-isomorphic to the canonical mixed Hodge complex for D (see also [14, §4]). Hence it induces Deligne’s mixed Hodge structure on the cohomology of D. The result follows from Lemma 2.9, by noting that E10,q(D(α))=Hq(D(α);) and E1p,q(D(α))=0 for p>0. ∎

Let D be a simple normal crossings divisor of complex dimension 1. Then D(1) is a disjoint union of smooth projective curves, D(2) is a collection of points and D(p)= for all p>2. Deligne’s weight spectral sequence is given by

where j2*:=j2,1*-j2,2*. The multiplicative weight spectral sequence is given by

where the term E10,0(D) is given by the pull-back

and the differential d1:E10,0(D)E11,0(D) is given by (a,b(t))b(t)dt. The products E10,0×E10,kE10,k and E10,0×E11,0E11,0 are given by

(a,b(t))c=acand(a,b(t))c(t)dt=b(t)c(t)dt

respectively. The unit is (1D(1),1D(2))E10,0(D). The maps H0(D(1);)E10,0 and H0(D(2);)E11,0 given by

a(a,j21*(a)(1-t)+j22*(a)t)andb-bdt

define an inclusion E~1(D)E1(D) of bigraded complexes which is a quasi-isomorphism.

Smooth quasi-projective varieties

Let X be a smooth projective variety and let XX~ be a smooth compactification of X such that the complement D:=X~-X is a union of divisors with normal crossings. A mixed Hodge diagram for X is defined via the algebra of forms on X~ which have logarithmic poles along D (see [21, §3] for details, see also [23, §8]). In this case, the multiplicative weight spectral sequence for X coincides with Deligne’s weight spectral sequence, given by

E1-p,q(X)=Hq-2p(D(p);)Hq-p(X;).

The differential d1-p,*:E1-p,*(X)E1-p+1,*(X) is given by the combinatorial Gysin map

γp:=k=1p(-1)k(jp,k)!:H*-2(D(p);)H*(D(p-1);),

where jp,k:D(p)D(p-1) are the inclusion maps. The algebra structure of E1*,*(X) is induced by the combinatorial restriction morphisms jp*=k=1p(-1)k-1(jp,k)* together with the cup product of H*(D(p);) for p>0.

Isolated singularities

Let X be a complex projective variety of dimension n with only isolated singularities and denote by Σ the singular locus of X. By Hironaka’s Theorem on resolution of singularities there exists a cartesian diagram

where X~ is smooth, f:X~X is a proper birational morphism which is an isomorphism outside Σ and D=f-1(Σ) is a simple normal crossings divisor. Since both X~ and Σ are smooth and projective, there are mixed Hodge diagrams 𝒜(X~) and 𝒜(Σ) for X~ and Σ respectively with trivial weight filtration. Let 𝒜(D)=sTW(𝒜(D)) be a mixed Hodge diagram for D as constructed in the proof of Proposition 2.11. The maps j1:D(1)X~ and g1:D(1)Σ defined by composing the map D(1)D with j and g respectively induce morphisms of mixed Hodge diagrams j1*:𝒜(X~)𝒜(D) and g1*:𝒜(Σ)𝒜(D).

Let X be a complex projective variety with only isolated singularities. With the above notation we have:

  • (1)

    A mixed Hodge diagram for X is given by 𝒜(X)=sTW(𝒜(Σ)×𝒜(X~)𝒜(D)).

  • (2)

    For q0 , the term E10,q(X) is given by the pull-back

    while for p>0 and q0 we have

    E1p,q(X)E1p,q(D)Λ(t)E1p-1,q(D)Λ(t)dt.

    The differential of E1*,*(X) is defined component-wise, via the differential of E1*,*(D) and the differentiation with respect to t.

  • (3)

    Denote by jp*:H*(D(p-1);)H*(D(p);) the combinatorial restriction maps, with D(0)=X~ . Let

    τ:H0(X~;)×H0(Σ;)H0(D(1);)

    be given by τ(x,σ)=j1*(x)-g1*(σ) . Then

Proof.

By Lemma 2.10 the Thom–Whitney simple of a strict cosimplicial mixed Hodge diagram, with the filtrations W(1) and F(0), is a mixed Hodge diagram. Hence 𝒜(X) is a mixed Hodge diagram, which by cohomological descent satisfies 𝒜(X)𝒜pl(X). By forgetting the multiplicative structures, we obtain a mixed Hodge complex for X (see for example [11, Section 2.9]). This proves (1). Assertion (2) now follows from (1) and Lemma 2.9. Assertion (3) is a matter of verification. ∎

3 Purity implies formality

In this section we give a simple proof of the fact that the purity of the weight filtration of a complex projective variety up to a certain degree, implies formality of the variety up to the same degree. A direct application is the formality of the Malcev completion of the fundamental group of projective varieties with normal isolated singularities.

Let r0 be an integer. A morphism of cdgas f:AB is called r-quasi-isomorphism if the induced morphism in cohomology Hi(f):Hi(A)Hi(B) is an isomorphism for all ir and a monomorphism for i=r+1.

A cdga (A,d) over 𝐤 is called r-formal if there is a string of r-quasi-isomorphisms from (A,d) to its cohomology (H*(A;𝐤),0) considered as a cdga with trivial differential. We will say that a topological space X is r-formal if the rational cdga 𝒜pl(X) is r-formal.

The case r=1 is of special interest, since 1-formality implies that the rational Malcev completion of π1(X) can be computed directly from the cohomology group H1(X;), together with the cup product

H1(X;)H1(X;)H2(X;).

In this case we say that π1(X) is formal. For r= we recover the usual notion of formality, which in the case of simply connected (or more generally, nilpotent) spaces, implies that the higher rational homotopy groups πi(X), with i>1 can be computed directly from the cohomology ring H*(X;). Note that if X is formal, then π1(X) is also formal.

Let X be a complex projective variety and let r0 be an integer. If the weight filtration on Hk(X;Q) is pure of weight k, for all 0kr, then X is r-formal.

Proof.

We prove formality over and apply independence of formality on the base field for cdgas with cohomology of finite type (see [27], see also [17]). Since the disjoint union of r-formal spaces is r-formal, we may assume that X is connected, so it has a mixed Hodge diagram 𝒜(X) with H0(𝒜(X)). By Theorem 2.4 it suffices to define a string of r-quasi-isomorphisms of differential bigraded algebras from (E1*,*(X),d1) to (E2*,*(X),0). Let M be the bigraded vector space given by M0,q=Ker(d10,q) for all q0 and Mp,q=0 for all p>0 and all q0, where d1p,q:E1p,q(X)E1p+1,q(X) denotes the differential of E1*,*(X). Then M*,* is a bigraded sub-complex of (E1*,*(X),d1) with trivial differential. Denote by φ:(M,0)(E1*,*(X),d1) the inclusion. Since Ker(d10,*)×Ker(d10,*)Ker(d10,*), the multiplicative structure induced by φ on M is closed in M. Hence φ is an inclusion of differential bigraded algebras. On the other hand, we have

E20,q(X)M0,q=Ker(d10,q).

This gives an inclusion of bigraded algebras ψ:ME2*,*(X). Assume that the weight filtration on Hk(X;) is pure of weight k, for all kr. We next show that both φ and ψ are r-quasi-isomorphisms. Indeed, for every p>0 and every q0 such that p+qr we have E2p,q(X)=0, while for every q0, we have E20,q(X)M0,qHq(X;). Therefore the induced maps Hk(φ) and Hk(ψ) are isomorphisms for all kr and the maps Hk+1(φ) and Hk+1(ψ) are monomorphisms. ∎

We highlight the two extreme cases r=1 and r= in the following corollary.

Let X be a complex projective variety.

  • (1)

    If the weight filtration on H1(X;) is pure of weight 1 , then π1(X) is formal.

  • (2)

    If the weight filtration on Hk(X;) is pure of weight k , for all k0 then X is formal.

(Q-homology manifolds)

Let X be a complex projective variety of dimension n. Assume that X is a -homology manifold (for all xX, H{x}k(X;)=0 for k2n and H{x}2n(X;)). Then the weight filtration on Hk(X;) is pure of weight k, for all k0 (see [8, Theorem 8.2.4]). Hence X is formal. Examples of such varieties are given by weighted projective spaces or more generally V-manifolds (see [10, Appendix B]), surfaces with A1-singularities, the Cayley cubic or the Kummer surface.

In fact, purity of the weight filtration is strongly related to Poincaré duality: if X is a complex projective variety whose rational cohomology satisfies Poincaré duality, then the weight filtration on Hk(X;) is pure of weight k, for all k0. Indeed, the Poincaré duality maps Hk(X;)(H2n-k(X;)*)(-n) are compatible with mixed Hodge structures. The weights on the left-hand (resp. right-hand) side are k (resp. k), hence equal to k. Therefore such varieties are formal (cf. [16, Theorem 5]). Another well-known result relating purity and Poincaré duality is the purity of the weight filtration on the (middle perversity) intersection cohomology IH*(X;) of a projective variety X. Furthermore, Weber [29] showed that for a complex projective variety X, the image of the map Hk(X;)IHk(X;) is isomorphic to the pure term GrkWHk(X;).

The purity of the weight filtration in cohomology does not imply Poincaré duality, as shown by the following example.

(Projective cone over a smooth curve)

Let CN be a smooth curve of genus g and consider the projective cone X=PcC over C. The Betti numbers of X are b0=1, b1=0, b2=1, b3=2g and b4=1. The weight filtration on Hk(X;) is pure of weight k, for all k0, and hence X is formal, but H*(X;) does not satisfy Poincaré duality.

In [1], it is shown that the weight filtration on H1(X;) of a normal complex projective variety X is pure of weight 1. In the case of isolated singularities, the proof is a standard argument in mixed Hodge theory.

Let X be a normal complex projective variety with isolated singularities. Then the weight filtration on H1(X;Q) is pure weight 1.

Proof.

Let Σ denote the singular locus of X and f:X~X a resolution such that D:=f-1(Σ) is a simple normal crossings divisor. Since X is normal, by Zariski’s main Theorem we have H0(D;)H0(Σ;). Since dimΣ=0, we have Hk(Σ)=0 for all k>0. This gives a Mayer–Vietoris long exact sequence

0H1(X;)f*H1(X~;)j*H1(D;)H2(X;)f*

which is strictly compatible with the weight filtration (see for example [24, Corollary-Definition 5.37]). Since X~ is smooth and projective, its weight filtration is pure. Hence the weight filtration on H1(X;) is pure weight 1. ∎

The fundamental group π1(X) of every normal complex projective variety X with isolated singularities is formal.

4 Formality of projective varieties with isolated singularities

By purely topological reasons we know that every simply connected, 4-dimensional CW-complex is formal. We also know there exist non-formal 4-dimensional CW-complexes. As mentioned in the introduction, thanks to deep results of Simpson and Kapovich–Kollár we know that there exist non-formal complex projective surfaces. In this section we prove that every complex projective surface with normal singularities is formal. We generalize this result in two directions. First, we prove formality for projective varieties of dimension n with only isolated singularities whose link is (n-2)-connected. Second, we prove formality for those projective varieties with normal isolated singularities admitting a resolution of singularities with smooth exceptional divisor.

Every normal complex projective surface is a formal topological space.

Proof.

Let X be a normal complex projective surface, which we may assume to be connected. We use the formulas for E1*,*(X) given in Proposition 2.13. Since normal singularities have codimension 2, we have dim(Σ)=0. Since dim(D(p))=1-p, we have E1p,q(X)=0 for all q>4-2p. By Lemma 3.7 together with semi-purity, the weight filtration on Hk(X;) is pure of weight k for all k2. We have

For all p,q0 take a section E2p,q(X)Ker(d1p,q)E1p,q(X) of the projection Ker(d1p,q)E2p,q(X). This defines a morphism ρ:(E2*,*(X),0)(E1*,*(X),d1) of bigraded complexes which is a quasi-isomorphism. We next show that ρ is multiplicative. By bidegree reasons, the only non-trivial products in E2*,*(X) are the products

Ker(d10,q)×Ker(d10,q)Ker(d10,q+q)

induced by the cup product of H*(X~;). Since ρ is the identity on Ker(d10,q), it preserves these products. It also preserves the unit 1Ker(d10,0). It only remains to see that the diagram

commutes. By Proposition 2.13, the term E10,1(X) is given by the pull-back

and E11,k(X)Hk(D(1);)Λ(t)dt for k{1,2}. Moreover, the differential d10,1:E10,1(X)E11,1(X) is given by (x,a(t))a(t)dt and the product E10,1(X)×E11,1(X)E11,2(X) is given by (x,a(t))b(t)dt=a(t)b(t)dt.

Let (x,a(t))E10,1(X). Since a(0)=0, it follows that (x,a(t))Ker(d10,1) if and only if a(t)=0. Therefore we have Ker(d10,1)E11,1(X)=0, and the above diagram commutes. This proves that the map

ρ:(E2*,*(X),0)(E1*,*(X),d1)

is multiplicative. Since X is connected, it has a mixed Hodge diagram 𝒜(X) with H0(𝒜(X)). Hence by Theorem 2.4 we have a string of quasi-isomorphisms of complex cdgas

(𝒜pl(X),d)(E1*,*(X),d1)(E2*,*(X),0)(H*(X;),0).

To conclude that X is formal it suffices to apply descent of formality of cdgas from to . ∎

The following is an example of a normal projective surface with isolated singularities and non-trivial weight filtration on H2 (cf. [28, §7], see also [3]).

Let C be a curve of degree d3 with n>0 nodes in 2. The genus of C is given by g=(d-1)(d-2)/2-n. Choose a smooth projective curve C of degree d=d+1 intersecting C transversally at smooth points of C, so that |CC|=dd and consider the blow-up X~=BlCC2 of 2 at the dd points of CC. Then the proper transform C~ of C has negative self-intersection |C~C~|=d(d-d)=-d and we may consider the blow-down X of C~ to a point. Explicitly, assume that the curve C is given by f(x,y,z)=0, and that C is given by g(x,y,z)=0. Then X is the projective variety defined by the equation wf(x,y,z)+g(x,y,z)=0, which has a normal isolated singularity at (0,0,0,1). Here (x,y,z,w) are the homogeneous coordinates in 3. The normalization of C~ is a smooth projective curve of genus g and X~ is homeomorphic to the connected sum of dd+1 projective planes. Deligne’s weight spectral sequence can be written as

Hence H2(X;) has a non-trivial weight filtration:

Gr0WH2(X;)n,Gr1WH2(X;)2gandGr2WH2(X;)dd.

Since X is simply connected (see for example [10, Corollary V.2.4]), we may compute the rational homotopy groups of X with their weight filtration from a bigraded minimal model ρ:ME2*,*(X) of the bigraded algebra E2*,*(X). The weight filtration on πi:=πi(X) satisfies GrqWπp+qHom(Q(M)p,q,), where the term Q(M)p,q denotes the indecomposables of M of bidegree (p,q).

The cohomology ring of X~ is given by H*(X~;)[a,b1,,bdd] with a2=T, bi2=-T and bibj=0 for all ij. Here T denotes the top class of X~, a is the hyperplane class and bi correspond to the exceptional divisors. Let γi:=a-dbi. Then E20,2(X)Ker(d10,2)[γ1,,γdd] with γi2=T(1-d2) and γiγj=d2T for ij. Hence we may write

E2*,*(X)[α1,,αn,β1,,β2g,γ1,,γdd],

where the generators have bidegree |αi|=(2,0), |βi|=(1,1) and |γi|=(0,2). By bidegree reasons, the only non-trivial products are given by γi2=T(1-d2) and γiγj=d2T, for all ij. We compute the first steps of a minimal model for E2*,*(X). Let M2 be the free bigraded algebra

M2=Λ(α¯1,,α¯n,β¯1,,β¯2g,γ¯1,,γ¯dd)

with trivial differential generated by elements of bidegree |α¯i|=(2,0), |β¯i|=(1,1) and |γ¯i|=(0,2). Then the map ρ2:M2E2*,*(X) given by x¯x is a 2-quasi-isomorphism of bigraded algebras. Hence we have

Gr0Wπ2n,Gr1Wπ22gandGr2Wπ2dd.

Let M3=M2dΛ(V3,0,V2,1,V1,2,V0,3,V-1,4) where Vi,j are the graded vector spaces of pure bidegree (i,j), and d:Vi,jM2i+1,j are the differentials given by

V3,0=xij,dxij=α¯iα¯j,1ijn,V2,1=yij,dyij=α¯iβ¯j,1in,1j2g,V1,2=zij,wkl,dzij=α¯iγ¯j,dwkl=β¯kβ¯l,1in,1jdd,1kl2g,V0,3=τij,dτij=β¯iγ¯j,1in,1j2g,V-1,4=ξij,dξij=γ¯iγ¯j,1idd,(i,j)(1,1).

Then the extension ρ3:M3E2*,*(X) of ρ2 given by Vi,j0 is a 3-quasi-isomorphism. The formula GrpWπ3Hom(V3-p,p,) gives

Gr0Wπ3n(n+1)2,Gr1Wπ32gn,Gr2Wπ3ddn+g(2g+1),Gr3Wπ3dd2g,Gr4Wπ3dd(dd+1)2-1.

For example, we may take C to be the nodal cubic curve given by f(x,y,z)=y2z-x2z-x3 and C a smooth plane quartic. Then dd=12, g=0 and n=1. This gives

Gr0Wπ2,Gr1Wπ2=0andGr2Wπ212;Gr0Wπ3,Gr1Wπ3=0,Gr2Wπ312,Gr3Wπ3=0andGr4Wπ377.

The following is a generalization of Theorem 4.1 to projective varieties of arbitrary dimension.

Let X be a complex projective variety of dimension n with normal isolated singularities. Denote by Σ the singular locus of X, and for each σΣ let Lσ denote the link of σ in X. If H~k(Lσ;Q)=0 for all kn-2 for every σΣ, then X is a formal topological space.

Proof.

The link Lσ of σΣ in X is a smooth connected real manifold of dimension 2n-1. Let L=σΣLσ. Then H0(L;)H0(Σ;). Assume that H~k(Lσ;)=0 for all kn-2. By Poincaré duality the only non-trivial rational cohomology groups of L are in degrees 0,n-1,n and 2n-1. Let Xreg=X-Σ. From the Mayer–Vietoris exact sequence

Hk-1(L;)Hk(X;)Hk(Xreg;)Hk(Σ;)Hk(L;)

it follows that the map Hk(X;)Hk(Xreg;) is an isomorphism whenever k<n-1 or n+1<k<2n-1, and injective for k=n-1. Since Hk(X;) has weights in {0,1,,k} and Hk(Xreg;) has weights in {k,k+1,,2k}, and the morphism Hk(X;)Hk(Xreg;) is strictly compatible with the weight filtrations, it follows that for kn,n+1, the weight filtration on Hk(X;) is pure of weight k. Furthermore, by semi-purity we have that Hn+1(X;) is pure of weight n+1. Therefore the only non-trivial weights of H*(X;) are in degree k=n. The weight spectral sequence for X has the form

where the bullets denote the non-trivial elements. Consider the quasi-isomorphism of complexes

ρ:(E2*,*(X),0)(E1*,*(X),d1)

defined by taking sections of the projections Ker(d1p,q)E2p,q(X). We next show that ρ is multiplicative. Note that by bidegree reasons, the only non-trivial products of E2*,*(X) are between elements of the first column. Since ρ is the identity on E20,*(X)Ker(d10,q), it preserves these products. It also preserves the unit 1E20,0(X). Since E1p,q(X)=0 for all q>2(n-p), we have E1p,n-p(X)E1p,n-p(X)=0 for all p,p>0. Therefore it only remains to show that for p,q>0, the following diagram commutes:

By Proposition 2.13 and since Hq(Σ)=0 for q>0, the term E10,q(X) is given by

while for p>0 we have E1p,n-p(X)E1p,n-p(D)Λ(t)E1p-1,n-p(D)Λ(t)dt. The proof now follows as in the proof of Theorem 4.1. ∎

(Complete intersections)

Let X be a complete intersection of dimension n>1. Assume that the singular locus Σ={σ1,,σN} is a finite number of points. The link of σi in X is (n-2)-connected (this result is due to Milnor [20] in the case of hypersurfaces and to Hamm [18] for general complete intersections). Therefore by Theorem 4.3, X is formal. Note that in particular, every complex hypersurface with isolated singularities is formal.

Let X be a projective variety with only isolated singularities. Assume that there exists a resolution of singularities f:X~X such that the exceptional divisor D=f-1(Σ) is smooth. Then X is a formal topological space.

Proof.

We may assume that X is connected. By Proposition 2.13 the multiplicative weight spectral sequence is given by

where

The differential d1:E10,*(X)E11,*(X) is given by (x,a(t))a(t)dt. The non-trivial products of E1*,*(X) are the maps E10,q(X)×E10,q(X)E10,q+q(X) given by (x,a(t))(y,b(t))=(xy,a(t)b(t)) and the maps E10,q(X)×E11,q(X)E11,q+q(X) given by (x,a(t))b(t)dt=a(t)b(t)dt. The unit is (1X~,1D)E10,0(X). By computing the cohomology of E1*,*(X) we find

with the non-trivial products being Ker(j*)q×Ker(j*)qKer(j*)q+q. Now define a quasi-isomorphism ρ:(E2*,*(X),0)(E1*,*(X),d1) of bigraded complexes as follows: Let ρ:E20,q(X)E10,q(X) be defined by the inclusion, for all q0. Define ρ:E21,q(X)E11,q(X) by taking a section Coker(j*)qHq(D) of the projection Hq(D)Coker(j*)q, for all q>0.

To see that ρ is a morphism of bigraded algebras it suffices to observe that it preserves the unit and that ρ(Ker(j*))E11,*(X)=0. Since X is connected, it has a mixed Hodge diagram 𝒜(X) with H0(𝒜(X)). Hence by Theorem 2.4 we have a string of quasi-isomorphisms of complex cdgas

(𝒜pl(X),d)(E1*,*(X),d1)(E2*,*(X),0)(H*(X;),0).

To conclude that X is formal it suffices to apply descent of formality of cdgas from to . ∎

Projective varieties with only isolated ordinary multiple points are formal. Projective cones over smooth projective varieties are formal.

(Segre cubic)

The Segre cubic S is a simply connected projective threefold with ten singular points, and is described by the set of points (x0:x1:x2:x3:x4:x5) of 5:

S:{x0+x1+x2+x3+x4+x5=0,x03+x13+x23+x33+x43+x53=0}.

A resolution f:¯0,6S of S is given by the moduli space ¯0,6 of stable rational curves with six marked points, and f-1(Σ)=i=1101×1, where Σ={σ1,,σ10} is the singular locus of S. We have

Hence S has a non-trivial weight filtration, with 0Gr2WH3(S;)5. By Theorem 4.5 and since S is simply connected, we may compute the rational homotopy groups π*(S) with their weight filtration from a minimal model of E2*,*(S). Since S is a hypersurface of 4, the map S4 induces an isomorphism Hk(4)Hk(S) for k3,4 (see [10, Theorems V.2.6 and V.2.11]). We deduce that

E2*,*(S)[a,b1,,b5,c0,,c5,e]

with the only non-trivial products a2=c0 and a3=e. The bidegrees are given by |a|=(0,2), |bi|=(1,2), |ci|=(0,4) and e=(0,6). In low degrees we obtain

Gr2Wπ2,Gr2Wπ35,Gr4Wπ45,Gr3Wπ510,Gr5Wπ55,Gr4Wπ625,Gr5Wπ625,Gr4Wπ740,Gr5Wπ750,Gr7Wπ726.

5 Contractions of subvarieties

Let YX be a closed immersion of complex projective varieties. Assume that Y contains the singular locus of X and denote by X/Y the space obtained by contracting each connected component of Y to a point. In general, X/Y is not an algebraic variety. For instance, the contraction of a rational curve in a smooth projective surface is a complex algebraic variety if and only if the self-intersection number of the curve is negative. For contractions of divisors in higher-dimensional varieties there are general conditions on the conormal line bundle, which ensure the existence of contractions in the categories of complex analytic spaces and complex algebraic varieties respectively (see for example [13, 2]). In the general situation in which X/Y is a pseudo-manifold with normal isolated singularities, we can endow the rational homotopy type of X/Y with a mixed Hodge structure, coming from the mixed Hodge structures on X and Y as follows.

Let YX be a closed immersion of complex projective varieties. Assume that Y contains the singular locus of X and that X is connected.

  • (1)

    The rational homotopy type of X/Y carries mixed Hodge structures.

  • (2)

    Let f:X~X be a resolution of X such that D:=f-1(Y) is a simple normal crossings divisor. Denote by Σ the singular locus of X/Y . There is a string of quasi-isomorphisms from 𝒜pl(X/Y) to E1*,*(X/Y) , where E1*,*(X/Y) is the bigraded algebra given by

    E1*,*(X/Y)=sTW(H*(Σ;)×H*(X~;)E1*,*(D)).

Proof.

Take mixed Hodge diagrams 𝒜(Σ) and 𝒜(X~) with trivial weight filtrations, and let 𝒜(D) be a mixed Hodge diagram for D, as defined in the proof of Proposition 2.11. Let

𝒜(X/Y):=sTW(𝒜(Σ)×𝒜(X~)𝒜(D)).

By cohomological descent we have 𝒜(X/Y)𝒜pl(X/Y). Indeed, since the category of cdgas with the Thom–Whitney simple and the class of quasi-isomorphisms is a cohomological descent category (see [15, Proposition 1.7.4]), it suffices to show that H*(X~,D)H*(X/Y,Σ). This follows from excision, since the composition X~XX/Y is an isomorphism outside Σ. Hence by Lemma 2.10, 𝒜(X/Y) is a mixed Hodge diagram for X/Y. This proves (1). Assertion (2) follows from Lemma 2.9 and Theorem 2.4. ∎

The weight filtration on the cohomology of a projective variety with isolated singularities has some properties special to the algebraic case, which are not satisfied in the general setting of contractions. For instance, the weight filtration on the cohomology H*(X/Y;) is not semi-pure in general. Another feature of the weight filtration that is only applicable to algebraic varieties is the existence on a non-zero cohomology class wGr2WH2(X;), as a consequence of hard Lefschetz theory. Therefore the weight filtration on the cohomology of a contraction may serve as an obstruction theory for such contraction to be an algebraic variety.

Consider two projective lines in 2 intersecting at a point, and let X denote the topological space given by contraction of the two lines to a point in 2:

Then the weight spectral sequence is given by

Since Gr2WH3(X;)0, the weight filtration on H*(X) is not semi-pure. Furthermore, we have H2(X;)=0. This is a proof of the fact that X is not algebraic. Since X is a simply connected topological space of dimension 4, it is formal (see for example [12, Proposition 2.99]). We may compute the rational homotopy groups πi:=πi(X) with their weight filtration from a minimal model of E2*,*(X)Λ(a3,b4)/(a3b4,b42). In low degrees, we have

π2=0,Gr2Wπ3,Gr4Wπ4,π5=0,Gr6Wπ6andGr8Wπ7.

We now prove analogue statements of Theorems 3.3, 4.3 and 4.5 for contractions of subvarieties.

Let X be a connected complex projective variety of dimension n and let YX be a closed immersion such that Y contains the singular locus of X. If one of the following conditions is satisfied, then X/Y is a formal topological space.

  • (a)

    The weight filtration on Hk(X/Y) is pure of weight k , for each k0.

  • (b)

    The link Li:=L(σi,X/Y) of each singular point σiX/Y in X/Y is (n-2) -connected.

  • (c)

    The link Li:=L(Yi,X) of each connected component Yi of Y in X is (n-2) -connected.

  • (d)

    There is a resolution of singularities f:X~X such that D=f-1(X) is smooth.

Proof.

Assume that (a) is satisfied. By Proposition 5.1 (2), the proof of Theorem 3.3 is valid in this setting. Hence purity of the weight filtration implies formality. Note that (b) and (c) are equivalent, since, the links Li=L(σ,X/Y) and Li:=L(Yi,X) are homeomorphic (see [11, Application 4.2]). Assume that (b) is satisfied. Let L=iLi. From the Mayer–Vietoris exact sequence

Hk-1(X/Y-Σ)Hk-1(Σ)Hk-1(L)Hk(X/Y)Hk(X/Y-Σ)Hk(Σ)Hk(L)

we find that for kn,n+1, the weight filtration on Hk(X/Y;) is pure of weight k. Note that since X/Y is not algebraic in general, the weight filtration is not necessarily semi-pure. Hence Hn+1(X/Y;) may have non-trivial weights. The weight spectral sequence for X/Y has the form

where the bullets denote non-trivial elements. Consider the quasi-isomorphism of complexes

ρ:(E2*,*(X/Y),0)(E1*,*(X/Y),d1)

defined by taking sections of the projections Ker(d1p,q)E2p,q(X/Y). We next show that ρ is multiplicative. By bidegree reasons, the non-trivial products in E2*,*(X/Y) are

E20,q×E20,qE20,q+qfor q,q0

and

E20,1×E2p,n-pE2p,n+1-pfor 0<p<n.

Since ρ is the identity on E20,*(X)Ker(d10,q), it preserves the former products. It also preserves the unit 1E20,0(X). Arguing as in the proof of Theorem 4.3, and using the description of E1*,*(X/Y) given by Proposition 2.13, it is straightforward to see that the latter products are trivial and that the diagram

commutes. We may now proceed as in the proof of Theorem 4.3. Hence the equivalent conditions (b) and (c) imply that X/Y is formal. Lastly, assume that (d) is satisfied. The proof of Theorem 4.5 is valid for X/Y. Hence X/Y is formal. ∎

Let YX be a closed immersion of smooth projective varieties, with X connected. Then X/Y is a formal topological space.

Acknowledgements

We would like to thank J. W. Morgan for asking us about 1-formality, and A. Dimca for pointing out to us the results on the realization of fundamental groups. Thanks also to F. Guillén and V. Navarro-Aznar for useful comments. J. Cirici acknowledges the Simons Center for Geometry and Physics in Stony Brook at which part of this paper was written.

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About the article


Received: 2015-05-27

Revised: 2016-01-07

Published Online: 2016-06-18

Published in Print: 2017-01-01


Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: SPP-1786

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2013-42178-P

The second-named author would like to acknowledge financial support from the German Research Foundation through project SPP-1786 and partial support from the Spanish Ministry of Economy and Competitiveness through project MTM2013-42178-P.


Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0101.

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