Fano varieties with torsion in the third cohomology group

We construct first examples of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues of the Artin--Mumford threefold. As an application, we answer a question of Voisin on the coniveau and strong coniveau filtrations of rationally connected varieties.


Introduction
If X is a nonsingular complex projective variety, the torsion subgroup of the integral cohomology group H 3 (X, Z) is an important stable birational invariant.It was introduced by Artin and Mumford in [2], where they used the invariant to show that a certain unirational threefold is not rational.
For rationality questions, perhaps the most interesting class of varieties is that of Fano varieties, that is, smooth varieties with ample anticanonical divisor.In dimension at most 2, these are all rational, with H 3 (X, Z) = 0.In dimension 3, there are 105 deformation classes of Fano varieties [20,21,23], and direct inspection shows that in each class the group H 3 (X, Z) is torsion free.Beauville asked on MathOverflow whether the same statement holds for Fano varieties in all dimensions [1]. 1 In this paper, we answer the question in the negative.
Theorem 1.1.For each even d ≥ 4, there is a d-dimensional Fano variety X of Picard rank 1 with H 3 (X, Z) = Z/2.
As a consequence, by [7,15], the variety X is rationally connected but not stably rational.We do not know if it is unirational.
The X in the theorem is a complete intersection in a double cover of the space of rank ≤ 4 quadrics in P d/2+2 .The families of maximal linear subspaces of these quadrics give Brauer-Severi varieties over X, and via the isomorphism Br(X) ∼ = Tors H 3 (X, Z), the associated Brauer class maps to the nonzero element in H 3 (X, Z).Our examples can be regarded as higher-dimensional analogues of the Artin-Mumford threefold from [2], whose construction is closely related to that of our X (see Section 4.3).
Starting in dimension 6, the Fano varieties we consider have a further exotic property.
Theorem 1.2.When d ≥ 6, the d-dimensional Fano variety X from Theorem 1.1 has the property that the coniveau and strong coniveau filtrations differ.More precisely, (1.1) Date: November 17, 2023. 1 An incorrect counterexample is proposed in the answer to [1]; see Section 4.4.
The two coniveau filtrations N c H l (X, Z) ⊆ N c H l (X, Z) of H l (X, Z) were introduced in the paper [4].The subgroups of the filtrations contain the cohomology classes in H l (X, Z) obtained via pushforward from smooth projective varieties (resp.possibly singular projective varieties) of codimension at least c.In the case c = 1, l = 3, they are described as follows.The group N 1 H 3 (X, Z) consists of classes in H 3 (X, Z) supported on some divisor of X.Its subgroup N 1 H 3 (X, Z) consists of pushforwards f * β of classes β ∈ H 1 (S, Z) via proper maps f : S → X where S is nonsingular of dimension dim X − 1.
An inequality of the two coniveau filtrations is particularly interesting for c = 1 because for each l ≥ 0, the quotient is a stable birational invariant for smooth projective varieties [4,Proposition 2.4].
While the examples of [4] show that this quotient can be non-zero in general, it is known to be zero for certain classes of varieties.Voisin [30] proved that for a rationally connected threefold, any class in H 3 (X, Z) modulo torsion lies in N 1 H 3 (X, Z).Tian [27,Theorem 1.23] strenghtened this to show that H 3 (X, Z) = N 1 H 3 (X, Z) for any rationally connected threefold.Theorem 1.2 shows that the quotient (1.2) can be nonzero for rationally connected X of higher dimension, answering a question of Voisin (see [30,Question 3.1] and [4,Section 7.2]).
The paper is organised as follows.Section 2 begins with background on the geometry of symmetric determinantal loci and their double covers.In Section 2.2, we explain how these symmetric determinental loci and their double covers are GIT quotients of affine space by an action of an orthogonal similitude group.In Section 2.3 (more specifically Definition 2.13), we define the main examples in Theorem 1.1 as linear sections of the double covers of symmetric rank loci.
In Section 3, we use the presentation of the double symmetric rank loci as GIT quotients to show that their smooth part has non-trivial torsion classes α ∈ H 3 (X, Z).Taking a linear section and applying a generalised Lefschetz hyperplane theorem then proves Theorem 1.1, restated more precisely as Theorem 4.1.In Section 4, we study some special examples appearing in our construction and compute their geometric invariants, in particular, the "minimal" example of a 4-dimensional Fano variety.
In the final Section 5 we prove Theorem 1.2, restated precisely as Theorem 5.3.The key point is that the mod 2 reduction of the generator α of H 3 (X, Z) satisfies α 2 = 0 (mod 2), which implies that α is not of strong coniveau 1 by a topological obstruction described in [4].
We would like to thank N. Addington, O. Benoist, J. Kollár, S. Schreieder, F. Suzuki and C. Voisin for useful discussions.The work on this paper was begun at the Oberwolfach workshop Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects in the summer of 2022.J.V.R. is funded by the Research Council of Norway grant no.302277.1.1.Notation.We work over the complex numbers C. We use the notation for projective bundles where P(E ) consists of lines in E .
By a Fano variety we mean a nonsingular projective variety with ample anticanonical bundle.

Symmetric determinantal loci and related varieties
Here we survey basic facts on symmetric determinal loci.Some of these are well known; we in particular follow the works of Hosono-Takagi [13, Section 2] and Tyurin [28].
Let V = C n .We identify P(Sym 2 V ∨ ) with the space of quadrics in P(V ) and let Z r,n ⊂ P(Sym 2 V ∨ ) denote the subset of quadrics of rank r.Z r,n is a quasi-projective variety; its closure Z r,n parameterizes the quadrics of rank ≤ r and is defined by the vanishing of the (r + 1) × (r + 1)-minors of a generic n × n symmetric matrix.These give a nested chain of subvarieties of P(Sym 2 V ∨ ), where Z 1,n is the 2nd Veronese embedding of P n−1 , and Z n−1,n is the degree n hypersurface defined by the determinant.
Proposition 2.1.The variety Z r,n is irreducible of dimension This proposition can be checked using the incidence variety Z r,n parameterizing (n − r − 1)-planes contained in the singular loci of quadrics For [L] ∈ Gr(n − r, V ), the fiber of the first projection π 1 , can be identified with the space of quadrics in P(V /L) ≃ P r−1 , so π 1 is a P r(r+1)/2−1 -bundle over Gr(n − r, V ).It follows that Z r,n is nonsingular, and its dimension is given by (2.1).Moreover, it is straightforward to check that the second projection gives a desingularization π 2 : Z r,n → Z r,n .For the claim about the singular locus, see [13,Section 2].
• Z 1,5 is the 2nd Veronese embedding of P 4 in P 14 ; it is a fourfold of degree 16.
2.1.Double covers.We will only be interested in the case when the rank r is even.In this case, we can define a double cover which is ramified exactly over the locus Z r−1,n , of codimension n − r + 1 in Z r,n .The construction is based on the classical fact that for a quadric Q of rank r in n variables, the variety of (n − r/2 − 1)-planes in Q ⊂ P n−1 is isomorphic to the orthogonal Grassmannian OG(r/2, r), which has two connected components.
The formal construction of W r,n from this observation starts with the incidence variety Taking the Stein factorisation of the projection U r,n → Z r,n we get a new variety W r,n and morphisms where η has connected fibres and σ is finite.The fibre of η at a general point of W r,n is isomorphic to a connected component of OG(r/2, r).The morphism σ is a double cover, ramified exactly along Z r−1,n (see [13,Proposition 2.3]).
For the remainder of the paper, we will let be the pullback of the polarization from P(Sym 2.2.(Double) symmetric determinantal loci as GIT quotients.In this section, we explain how the varieties Z r,n and W r,n can be presented as GIT quotients of affine spaces, which is a key ingredient in the cohomology computations needed in Theorems 1.1 and 1.2.Let r be even, let S = C r , and let ω S ∈ Sym2 S ∨ be a nondegenerate quadratic form.The orthogonal similitude group GO(S) ⊂ GL(S) consists of the linear automorphisms of S which preserve ω S up to scaling. 2 In other words, an invertible linear map φ : S → S lies in GO(S) if there exists a χ(φ) ∈ C * such that for all v ∈ S, The map χ : GO(S) → C * defined by this relation is a group homomorphism, and we have an exact sequence The group GO(S) naturally acts on the orthogonal Grassmannian OG(r/2, S).The variety OG(r/2, S) has two connected components, and the action of GO(S) on this two-element set gives an exact sequence where GO(S) • is connected.We further have SO(S) = O(S) ∩ GO(S) • , and an exact sequence Consider now the affine space Hom(V, S) ≃ A rn .The group GO(S) acts on Hom(V, S) via GO(S) × Hom(V, S) → Hom(V, S) We have a morphism of affine spaces τ : Hom(V, S) → Sym 2 V ∨ , defined by, for any f ∈ Hom(V, S) and v, w ∈ V , Let CZ r,n ⊂ Sym 2 V ∨ be the subset of Sym 2 V ∨ corresponding to quadratic forms of rank r, so that Z r,n = CZ r,n /C * .The set τ −1 (CZ r,n ) ⊂ Hom(V, S) consists of the f : V → S such that f * ω S has rank r.
Proof.The previous lemma shows that GO(S) • acts freely on τ −1 (CZ r,n ).So let f ∈ τ −1 (CZ r−1,n ), and let φ ∈ GO(S) be an element which fixes f .We will show that φ is the identity.
Since f * ω S has rank r − 1, we may find a basis v 1 , . . ., v n of V such that The elements f (v 1 ), . . ., f (v r−1 ) ∈ S are orthonormal, so we can choose a vector e ∈ S such that f (v 1 ), . . ., f (v r−1 ), e is an orthonormal basis for S. Since φ fixes f , we have and, for each i, This implies that φ(e) = ±e, and then the fact that φ ∈ GO(S) • forces φ(e) = e.This means that φ is the identity element.
. The set of maps f of rank r − 2 has codimension 2(n − r + 2), while the set of f with rank r − 1 has codimension n − r + 1.The further requirement that P(f (V )) is tangent to the quadric gives codimension n − r + 2.
Proof.Let R be the coordinate ring of Hom(V, S).The GIT quotient Hom(V, S) ss GO(S) is given by Proj R O(S) , where the ring R O(S) is graded by the action of GO(S), an action which factors through χ : GO(S) → C * .Any linear function ) , and the first fundamental theorem of invariant theory for orthogonal groups says that these τ * (x) generate R O(S) [24, p. 390].This shows that Hom(V, S) us = τ −1 (0), and moreover that τ gives a closed embedding Hom(V, S) ss GO(S) → P(Sym 2 V ∨ ).It is easy to see that its image is Z r,n .
Thinking of χ as a character of GO(S) • , we get a GO(S) • -linearisation of O Hom(V,S) .The associated GIT semistable locus in Hom(V, S) is the same as for the GO(S)linearisation, since GO(S) • has finite index in GO(S).
Lemma 2.9.The GIT quotient Hom(V, S) ss GO(S) • is isomorphic to W r,n .
The open subset τ −1 (CZ r,n ) GO(S) • ⊂ Hom(V, S) ss GO(S) • is isomorphic to σ −1 (Z r,n ) ⊂ W r,n by the following construction.Fix an r/2-dimensional isotropic linear subspace L ⊂ S. Recall the variety U r,n from (2.2) and define a morphism and the linear subspace This morphism is GO(S) • -invariant, and one checks that it gives a bijection between the GO(S) • -orbits in τ −1 (CZ r,n ) and the points of The birational map ψ fits in the following commutative diagram: Let L be the function field of Hom(V, S) ss GO(S) • , identified with the function field of W r,n .Since these two varieties are normal and finite over Z r,n , they are both equal to the relative normalisation of Z r,n in Spec L, and so ψ extends to an isomorphism of varieties.
Proposition 2.10.Étale locally near a point p ∈ σ −1 (Z r−2,n ), the pair (W r,n , p) is isomorphic to (C × A M , (0, 0)) where C is the affine cone over the Segre embedding of P n−r+1 × P n−r+1 , 0 ∈ C is the singular point, and Proof.We use the isomorphism Hom(V, S) ss GO(S) • ≃ W r,n .Let f ∈ Hom(V, S) ss be a point whose orbit maps to σ −1 (Z r−2,n ) under this isomorphism.Then f ∈ τ −1 (CZ r−2,n ), and we can choose a basis v 1 , . . ., v n for V such that This means that the elements f (v 1 ), . . ., f (v r−2 ) are orthonormal in S, and we extend this sequence to a basis of S by adding vectors e 1 , e 2 such that The isotropic subspaces of e 1 , e 2 are e 1 and e 2 .Reordering the e i , we may assume that f (v r−1 ), . . ., f (v n ) are all contained in e 1 .After linearly transforming the v i , we may assume that f (v r−1 ) = γe 1 for some γ ∈ C and f There are now two cases to consider: Let us write A i (j) for a T -representation of dimension i with weight j.We then have an isomorphism of T -representations The Luna étale slice theorem implies that étale locally near the orbit of f , the variety Hom(V, S) ss GO(S) • is isomorphic to gives for some M .The quotient N f T is isomorphic to C × A M with C the cone over P n−r+1 × P n−r+1 , so this completes the proof.
where the H i are general divisors in |H|.In other words, X is a ramified double cover of a linear section of Z r,n , We are particularly interested in the case when X is also a Fano variety.This can happen only when r < 6: Lemma 2.12.Let X denote a general linear section of W r,n .If 6 ≤ r ≤ n, then either X is singular, or K X is base-point free.
Proof.Write X as in (2.5) for divisors H i ∈ |H|.As the H i are general, and W r,n is Gorenstein with canonical singularities, it follows that the same holds for X.By Proposition 2.3 and adjunction, the canonical divisor is given by Therefore, if c > rn/2, X is of general type, and for c = rn/2, it is Calabi-Yau.If c < rn/2, we note that which is non-negative for our choices of r and n.This means that X meets the singular locus of W r,n , and hence it must be singular.
By Lemma 2.12, we obtain Fano varieties as linear sections of W r,n only when r = 2 or r = 4.The case r = 2 gives W 2,n = P n−1 × P n−1 , and many linear sections of P n−1 × P n−1 are indeed Fano, but these varieties do not have interesting cohomology groups from the point of view of this paper.
We therefore focus on the case r = 4.In this case the existence of the double cover σ : W 4,n → Z 4,n is explained as follows.A smooth quadric surface in P 3 contains two families of lines; thus a quadric of rank 4 in n variables contains two families of (n − 2)-planes, each parameterised by a P 1 .Thus W 4,n parameterises quadrics plus a choice of one of the two families.
The dimensions of the first few rank loci Z i are given by By Corollary 2.11, the double cover W 4,n is singular along σ −1 (Z 2,n ), which has codimension 2n − 5 in W 4,n .By (2.4), the canonical divisor of W 4,n equals Definition 2.13.Given n ≥ 4 and c ≥ 0, let X n,c be a general complete intersection (2.6) The varieties X in Theorems 1.1 and 1.2 are X n,2n−1 with n ≥ 5 and n ≥ 6, respectively.

Cohomology computations
Let X sm n,c be the smooth part of X n,c .In this section we compute the low degree cohomology of X sm n,c .In Proposition 3.1 we compute the low degree cohomology of BGO(4) • , and in Proposition 3.5 we show that this agrees with low degree cohomology of X sm n,c .We summarise the consequences for the cohomology of X sm n,c in Corollary 3.6.In order to prove Theorem 1.1, we want a non-zero 2-torsion cohomology class of degree 3, and for Theorem 1.2, the class should furthermore have a non-zero square modulo 2 (this will be explained in Proposition 5.2).

Cohomology of BSO(4).
The cohomology rings with integer coefficients of the classifying spaces BSO(n) were computed by Brown [6] and Feshbach [10].For n = 4, the ring is given by , where e is the Euler class (of degree 4), p is the Pontrjagin class (degree 4), and ν is a 2-torsion class of degree 3. Thus the low-degree cohomology groups of BSO(4) are given by The cohomology ring of BSO(4) with Z/2-coefficients is given by where w 2 , w 3 , w 4 ∈ H * (BSO(4), Z/2) denote the Stiefel-Whitney classes [19].

3.3.
Cohomology of hyperplane sections of W sm r,n .Let S be a quadratic r-dimensional vector space and L ⊆ P(Sym 2 V ∨ ) is a codimension c linear subspace.We analyse the natural homomorphism Z) and show that it is an isomorphism in low degrees.To define the homomorphism, begin with the pullback maps , with τ and CZ r−2,n as defined in Section 2.2.By Lemma 2.5 and Corollary 2.11, the variety W sm r,n is isomorphic to (Hom(V, S) − τ −1 (CZ r−2,n )/ GO(S) • , where the group action is free, so we get an isomorphism . Finally, we have the pullback homomorphism Z), and composing these maps gives (3.2).Lemma 3.2.Let G be an algebraic group on an affine space A N .Let Z ⊂ A N be a closed, G-invariant subset of codimension c, and let U = A N − Z. Then the natural homomorphisms ) are isomorphisms for l < 2c − 1, and injective for l = 2c − 1.
Proof.The Leray-Serre spectral sequence for equivariant cohomology [18, p. 501] has E 2 -page H i G (pt, H j (U )) and converges to H i+j G (U ).Since H j (U ) = 0 for 0 < j ≤ 2c−2, there are no non-trivial differentials whose domain is of degree (i, j) with i+ j ≤ 2c− 2. The claim of the theorem follows from this.
Proof.Combine Lemma 2.7 and Lemma 3.2.Lemma 3.4.Let L ⊆ P(Sym 2 V ∨ ) be a generic codimension c linear subspace.The homomorphisms Proof.The generalised Lefschetz theorem of Goresky-MacPherson [11,Thm p.150] states that we have isomorphisms on the level of homotopy groups for low degrees.Combining this with the Hurewicz theorem gives the statement for cohomology groups.Proposition 3.5.Let L ⊆ P(Sym V ∨ ) be a generic codimension c subspace.The homomorphisms If moreover c ≤ 4n − 13, then the square of the non-zero class in H 3 (X sm n,c , Z) does not vanish modulo 2.

The varieties X n,c
We now analyse a few particularly interesting choices of n and c.Proof.The singular locus in W 4,n has dimension 2n − 2 by Proposition 2.1 and Corollary 2.11, so X is nonsingular by Bertini's theorem.The proof of Lemma 2.12 gives K X = −H.Finally, H 3 (X, Z) is computed in Corollary 3.6.
Proposition 4.2.The fourfold X is a Fano variety with invariants (1) Pic(X) = ZH, with H 4 = 10. ( Proof.(1), ( 2) and ( 4 4.2.1.Homological projective duality.In the paper [25], the second named author studies derived categories of linear sections of the stack Sym 2 P n−1 from the perspective of homological projective duality [17].When n is odd, the paper defines a noncommutative resolution Y n of W n−1,n , and shows that linear sections of this noncommutative resolution are related to dual linear sections of Sym 2 P n−1 in precisely the way predicted by HP duality, which strongly suggest that Y n is HP dual to Sym 2 P n−1 . Specialising to the case n = 5 and linear sections of the appropriate dimensions gives the following result.Let V = C 5 , let L 1 , . . ., L 9 be general hyperplanes in P(Sym 2 V ), and let L be their intersection.Let be the orthogonal complement.
In this language, X = L × P(Sym 2 V ) W 4,5 .Since X avoids the singular locus of W 4,5 , the noncommutative resolution Y 5 of W 4,5 is equivalent to W 4,5 , and the main theorem of [25] applies.On the other side of the HP duality we find (4.1) which is the intersection of 6 general (1, 1)-divisors in Sym 2 P(V ∨ ) = Sym 2 P 4 .The following is a slight amplification of the main result of [25].
Proposition 4.3.The category D(X) admits a semiorthogonal decomposition where the E i are exceptional objects.
The amplification consists in the fact that [25] only proves that D(S) includes as a semiorthogonal piece in D(X).The fact that the orthogonal complement is generated by 4 exceptional objects is not difficult to show using the techniques of the paper.
Lemma 4.4.The surface S in (4.1) is smooth of degree 35 with respect to the embedding S ⊂ P(Sym 2 V ∨ ).It has Hodge numbers Proof.The map P 4 × P 4 → Sym 2 (P 4 ) induces an étale double cover π : T → S where T is a general complete intersection of 6 symmetric (1, 1)-divisors in P 4 × P 4 .In particular, T is simply connected by the Lefschetz theorem.Furthermore, we find that S = 35 and hence χ top (S) = 85 by Noether's formula.From this we find that h 1,1 (S) = 65.Corollary 4.5.With S and X as above, we have Proof.The semiorthogonal decomposition in Proposition 4.3 gives the relation of Hochschild homology groups Expressing Hochschild homology via Hodge numbers through and using the fact that h 0,i (X) = 0 since X is Fano gives the result.
Example 4.6.The fact that Tors H 3 (X, Z) = 0 can be seen as a consequence of the fact that the conic bundle η : U 4,5 → W 4,5 does not admit a rational section.
To see this, recall that U 4,5 is a projective bundle over the Grassmannian G = Gr(3, V ).Explicitly, U 4,5 = P(E) where E is the rank 9 vector bundle appearing as the kernel of the natural map S 2 (V ∨ ⊗ O G ) → S 2 (U ∨ ), and where U is the universal subbundle of rank 3. Now, if D ⊂ P(E) is the divisor determined by a rational section of η, D is linearly equivalent to a divisor of the form aL + bG, where L = O P(E) (1) and G is the pullback of O Gr(3,V ) (1).We must also have D • L 13 = 10 (as the 1-cycle L 13 is represented by 10 fibers of P(E) → W 4,5 ).On the other hand, using the Chern classes of S 2 (U ∨ ), we compute that D • L 13 = −20b, contradicting the condition that b is an integer.
This shows that the Brauer group of W sm 4,5 is non-trivial.In our case, we may identify the Brauer group with Tors H 3 (W sm 4,5 , Z) because H 2 (W sm 4,5 , Z) = Z is generated by algebraic classes [3,Proposition 4].Finally, Lemma 3.4 shows that H 3 (W sm 4,5 , Z) → H 3 (X, Z) is an isomorphism, so the latter group has non-trivial torsion part as well.
For an alternative approach to the absence of rational sections, see Claim A.2 in the Appendix.

4.3.
The case c = 2n − 2. Let X = X n,2n−2 .Then X has dimension 2n − 5, isolated singularities in σ −1 (Z 2,n ) ∩ X, and K X = −2H.Let X → X be the blow-up at the singular points.Then the exceptional divisor E is a disjoint union of components E 1 , . . ., E s , all of which are isomorphic to P n−3 × P n−3 , by Proposition 2.10.
By Corollary 3.6, we have H 3 (X sm , Z) = Z/2.Since X sm ≃ X −E, we get a pullback map H 3 ( X, Z) → H 3 (X sm , Z).This map is an isomorphism by the exact sequence using also that H 1 (E, Z) = 0 and H 2 (E, Z) is torsion free.Proposition 4.7.For each n ≥ 4, X is a smooth projective variety of dimension 2n − 5 with Tors H 3 ( X, Z) = 0.The variety X is unirational, but not stably rational.
Proof.Only the unirationality remains to be proved.The incidence variety U 4,n of (2.3) is a P 2n−2 -bundle over the Grassmannian Gr(n − 2, V ).This means that if X is a complete intersection of 2n − 2 divisors in W 4,n , the preimage U X = η −1 (X) is birational to Gr(n − 2, V ).Therefore U X is rational, and hence X is unirational.
Example 4.8.When n = 4, X is a double cover of P 3 branched along a singular quartic surface.This is the example famously studied by Artin and Mumford in [2], and for which they prove Proposition 4.7.Here X has 10 ordinary double points and the blow-up X contains 10 exceptional divisors isomorphic to P 1 × P 1 .4.4.The case n = 4, c < 6.The Artin-Mumford examples of X 4,6 can also naturally be generalised to X 4,c with c < 6.We will explain that, at least when c = 4 or 5, these do not have torsion in H 3 in their smooth models (correcting a claim made in a MathOverflow answer [1]).
The singular locus of X 4,c has codimension 3 and is a smooth Enriques surface or a smooth genus 6 curve when c = 4 and c = 5, respectively.There is a resolution π : X → X 4,c obtained by blowing up the singular locus, where the exceptional divisor is a P 1 × P 1 -bundle over the singular locus.Proposition 4.9.With X as above, we have that the group H 3 ( X, Z) is torsion free for c = 5 and 0 for c = 4.
Proof.To show that H 3 ( X, Z) is torsion free, we first remark that H 3 (X, Z) has no torsion by Corollary 4.11 below.Next, we consider the Leray spectral sequence associated to the blow-up π : X → X, with E 2 -page H p (X, R q π * Z) converging to H p+q ( X, Z).Let S ⊂ X be the singular locus.We have R 0 π * Z X = Z X , R 1 π * Z = 0, R 2 π * Z X = F and R 3 π * Z = 0, where F is a rank two local system.More explicitly, we have F = R 2 π * Z E , and since E is P 1 × P 1 -bundle over S, this means F ∼ = R 0 f * Z S ′ , where f : S ′ → S is the étale double cover of S corresponding to the two families of lines in each fibre of E → S.
By Corollary 4.11, we have H 3 (X, Z) = 0, and so the only non-vanishing term of the E 2 -page of the spectral sequence is Running the spectral sequence then gives Since H 1 (S ′ , Z) is torsion free, the same is true for H 3 ( X, Z).When c = 4, the variety S is an Enriques surface, so that S ′ is either a K3 surface or two copies of S; in either case H 1 (S ′ , Z) = 0 which gives H 3 ( X, Z) = 0.
In the argument above, we used the following version of the Weak Lefschetz hyperplane theorem for singular varieties.Proposition 4.10.Let V be a projective variety of dimension n + 1 and let D be an ample divisor which is disjoint from the singular locus sing(V ).Then the natural maps are isomorphisms for i < n and surjective for i = n.
Proof.Letting U = V − D, the relative cohomology sequence takes the form . Now, using that U is affine of dimension n + 1, the cohomology groups H i (U, Z) vanish for all i > n+1, by Artin's vanishing theorem.
Corollary 4.11.Let σ : X → P n be a ramified double cover.Then for each i < n Proof.Note that X can be defined by an equation of the form z 2 = f (x 0 , . . ., x n ) in the weighted projective space V = P(1, . . ., 1, d 2 ).Thus X is an ample divisor, disjoint from the one singular point of V .Thus the conditions of Proposition 4.10 hold, and we find that H j (X, Z) = H 2n−j (V, Z) when j < n.The cohomology of V is computed in [14, Theorem 1], which gives claim (i), and claim (ii) follows by the Universal Coefficient theorem.

Proof of Theorem 1.2
In this section we state and prove a precise version of Theorem 1.2.We first recall some general background on the coniveau filtrations on cohomology of algebraic varieties, referring to [4] for details.We restrict ourselves to the case of cohomology with integral coefficients H i (X, Z) on a smooth projective variety X over C.
A cohomology class α ∈ H l (X, Z) is said to be of coniveau ≥ c if it restricts to 0 on X − Z where Z is a closed subset of codimension at least c in X.These classes give the coniveau filtration N c H l (X, Z) ⊂ H l (X, Z).Equivalently, viewing H l (X, Z) as H 2n−l (X, Z) via Poincaré duality, a class α ∈ H 2n−l (X, Z) is of coniveau ≥ c if and only if α = j * β for some β ∈ H 2n−l (Y, Z), where j : Y → X is the inclusion of a closed algebraic subset of X of codimension at least c.So for example, N c H 2c (X, Z) consists of exactly the algebraic classes in H 2c (X, Z).
A class α ∈ H l (X, Z) is said to be of strong coniveau ≥ c if α = f * β where f : Z → X is a proper morphism, Z is a smooth complex variety of dimension at most n − c, and We have N c H l (X, Z) ⊂ N c H l (X, Z) for every c.Moreover, the quotient is a birational invariant among smooth projective varieties [4].This invariant is particularly interesting for rationally connected varieties X.In this case, all cohomology classes are of coniveau ≥ 1: Proposition 5.1.Let X be a rationally connected smooth projective complex variety.Then for any l > 0, N 1 H l (X, Z) = H l (X, Z).
In [30, Question 3.1], Voisin asked whether N 1 H l (X, Z) = N 1 H l (X, Z) for X a rationally connected variety, i.e., whether all cohomology classes are of strong coniveau 1 (see also [4,Section 7.2]).In the same paper, she proved that any class in H 3 (X, Z) modulo torsion is of strong coniveau 1.This was extended by Tian [27,Theorem 1.23] who proved that H 3 (X, Z) = N 1 H 3 (X, Z) for any rationally connected threefold.Our Fano varieties give the first rationally connected examples where the two coniveau filtrations are different.
Proof.This is a special case of [4,Proposition 3.5].
Here is the precise version of Theorem 1.2: Theorem 5.3.For n ≥ 6, the variety X n,2n−1 from Definition 2.13 is a Fano variety of dimension 2n − 6 with K X = −H, such that Proof.Let X = X n,2n−1 .The computation of dim X, H 3 (X, Z) and K X is part of Theorem 4.1.Since X is Fano, it is rationally connected, so Proposition 5.1 gives N 1 H 3 (X, Z) = H 3 (X, Z).Corollary 3.6 shows that the class α = 0 ∈ H 3 (X, Z) is such that the mod 2 reduction of α 2 is non-zero.Proposition 5.2 then implies α ∈ N 1 H 3 (X, Z), so N 1 H 3 (X, Z) = 0.
Remark 5.4.We can obtain examples of other rationally connected varieties where N c H l = N c H l for any c ≥ 1 and l ≥ 2c + 1 by taking appropriate products with projective spaces (see e.g., [4,Theorem 4.3]).
Remark 5.5 (The Artin-Mumford example).In light of Theorem 1.2, it is natural to ask whether the 2-torsion class α ∈ H 3 (X, Z) the Artin-Mumford example has strong coniveau ≥ 1, i.e., whether the birational invariant (1.2) is zero.It turns out that this is indeed the case: Inspecting Artin-Mumford's 'brutal procedure' in [2, p. 82-83], shows that the class α is obtained from a cylinder map H 1 (C, Z) → H 3 (X, Z) from an elliptic curve C. In other words, α is the pushfoward from a class in H 1 from some ruled surface S over C. Note that this can also be seen as a special case of [27, Theorem 1.23].
5.1.Open questions.We conclude with two open questions regarding the two coniveau filtrations: Question 1. Are there rationally connected varieties X with N 1 H l (X, Z) = N 1 H l (X, Z) for some l > 0 and torsion free H l (X, Z)?
Question 2. Are there rationally connected varieties of dimension 4 or 5 where N 1 H l (X, Z) = N 1 H l (X, Z) for some l > 0?
Remark 5.6.Let X = X 5,9 be the fourfold from Section 4.2.Then we don't know if the generator α of H 3 (X, Z) has strong coniveau ≥ 1.We can show, however, that α 2 = 0 in H 6 (X, Z/2), so the topological obstruction of Proposition 5.2 vanishes.To see this, we use the fact that the third integral Steenrod square Sq 3 Z : H p (Z, Z) → H p+3 (X, Z) is naturally identified with the third differential d 3 in the Atiyah-Hirzebruch spectral sequence of topological K-theory, with E 2 -page H p (X, K q (pt)) = H p (X, Z) q even 0 otherwise converging to K p+q (X).Now H * (X, Z) has torsion only in degrees 3 and 6, with torsion part Z/2 in each of these degrees.It also has torsion Z/2 ⊕ Z/2 in its topological Ktheory, by Proposition 4.3 (because S is a general type surface with fundamental group Z/2).This implies d 3 = 0, since otherwise the Atiyah-Hirzebruch spectral sequence Most surfaces in P 4 , including general 6 × 5 determinantal surfaces, have only 1parameter families of 5-secants.

Theorem 4 . 1 .
The variety X is nonsingular of dimension 2n − 6 with K X = −H, and hence Fano.It has Picard number 1 and H 3 (X, Z) = Z/2.
by Proposition 2.10, and the claim follows since the singular locus is closed.
2.3.Linear sections of double symmetric determinental loci.The varieties appearing in Theorems 1.1 and 1.2 will be constructed by taking general linear sections of the double cover W r,n , i.e., complete intersections(2.5)