Quadric bundles and hyperbolic equivalence

We introduce the notion of hyperbolic equivalence for quadric bundles and quadratic forms on vector bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties. Furthermore, when the base is a projective space we show that two quadratic forms are hyperbolic equivalent if and only if their cokernel sheaves are isomorphic up to twist, their fibers over a fixed point of the base are Witt equivalent, and, in some cases, certain quadratic forms on intermediate cohomology groups of the underlying vector bundles are Witt equivalent. For this we show that any quadratic form over $\mathbb{P}^n$ is hyperbolic equivalent to a quadratic form whose underlying vector bundle has many cohomology vanishings; this class of bundles, called VLC bundles in the paper, is interesting by itself.


Introduction
Let Q → X be a quadric bundle, that is a proper morphism which can be presented as a composition Q ֒→ P X (E) → X, where P X (E) → X is the projectivization of a vector bundle E and Q ֒→ P X (E) is a divisorial embedding of relative degree 2 over X.A quadric bundle is determined by a quadratic form q : Sym 2 E → L ∨ with values in a line bundle L ∨ , or, equivalently, by a self-dual morphism (1.1) q : E ⊗ L → E ∨ .
Conversely, the quadratic form q is determined by Q up to rescaling and a twist transformation where M is a line bundle on X.Furthermore, with a quadric bundle one associates the coherent sheaf on X, which we call its cokernel sheaf and which is determined by Q up to a line bundle twist.We will usually assume that X is integral and the general fiber of Q → X is non-degenerate, or equivalently, that q is an isomorphism at the general point of X, so that Ker(q) = 0 and C(q) is a torsion sheaf on X.
Then the sheaf C(q) is endowed with a "shifted" self-dual isomorphism where C(q) ∨ is the derived dual of C(q) and [1] is the shift in the derived category (see §4.1 for a discussion of sheaves enjoying this property).
The main question addressed in this paper is: what properties of quadric bundles are determined by their cokernel sheaves (we restate this question below in a more precise form as Question 1.2)?A prioiri it is hard to expect that the cokernel sheaf determines a lot; for instance because it is supported only on the discriminant divisor of Q/X.However, the main result of this paper, is that in the case where X is a projective space and some mild numerical conditions discussed below are satisfied, the cokernel sheaf determines the quadric bundle up to a natural equivalence relation, which we call hyperbolic equivalence, and which itself preserves the most important geometric properties of quadric bundles.
Hyperbolic equivalence is generated by operations of hyperbolic reduction and hyperbolic extension.The simplest instance of a hyperbolic reduction (over the trivial base) is the operation that takes a quadric Q ⊂ P r and a smooth point p ∈ Q and associates to it the fundamental locus of the linear projection Bl p (Q) → P r−1 , which is a quadric Q − ⊂ P r−2 ⊂ P r−1 of dimension by 2 less than Q.From the above geometric perspective it is clear that the hyperbolic reduction procedure is invertible: the inverse operation, which we call a hyperbolic extension, takes a quadric Q ⊂ P r and a hyperplane embedding P r ֒→ P r+1 and associates to it the quadric Q + ⊂ P r+2 obtained by blowing up Q ⊂ P r+1 and then contracting the strict transform of P r ⊂ P r+1 .
The operations of hyperbolic reduction and extension can be defined in relative setting, i.e., for quadric bundles Q ⊂ P X (E) → X over any base X, and, moreover, can be lifted to operations on quadratic forms.For the reduction a smooth point is replaced by a section X → Q that does not pass through singular points of fibers, or more generally, by a regular isotropic subbundle F ⊂ E, and for the extension a hyperplane embedding is replaced by an embedding E ֒→ E ′ of vector bundles of arbitrary corank.We define these operations for quadratic forms and quadric bundles in §2.1 and §2.2 and say that quadratic forms (E, q) and (E ′ , q ′ ) or quadric bundles Q and Q ′ over X are hyperbolic equivalent if they can be connected by a chain of hyperbolic reductions and extensions.
While the construction of hyperbolic reduction is quite straightforward in the general case, this is far from true for hyperbolic extension.In fact, when we start with an extension 0 → E → E ′ → G → 0 of vector bundles, where the bundle G has rank greater than 1, this operation does not have a simple geometric description (as in the rank 1 case); moreover, the set HE(E, q, ε) of all hyperbolic extensions of (E, q) with respect to an extension class ε ∈ Ext 1 (G, E) is empty unless a certain obstuction class q(ε, ε) ∈ Ext 2 ( 2 G, L ∨ ) vanishes, and when the obstruction is zero, HE(E, q, ε) is a principal homogeneous space under the natural action of the group Ext 1 ( 2 G, L ∨ ).This can be seen even in the simplest case where the extension is split, i.e., E ′ = E ⊕ G -in this case the obstruction vanishes and the corresponding hyperbolic extensions have the form E + = E ⊕ G + , where G + is an arbitrary extension of G by L ∨ ⊗ G ∨ with the class in the subspace Ext 1 ( 2 G, L ∨ ) ⊂ Ext 1 (G, L ∨ ⊗ G ∨ ).For a discussion of a slightly more complicated situation see Remark 2.10.In general the situation is similar but even more complicated.The construction of hyperbolic extension explained in §2.2 (see Theorem 2.9) is the first main result of this paper.
As we mentioned above hyperbolic equivalence does not change the basic invariants of a quadratic form.In §2.3 we prove the following (for the definition of the Clifford algebra Cliff 0 (E, q) we refer to [11]).
Proposition 1.1.Let (E, q) and (E ′ , q ′ ) be hyperbolic equivalent generically non-degenerate quadratic forms over X and let Q → X and Q ′ → X be the corresponding hyperbolic equivalent quadric bundles, where X is a scheme over a field k of characteristic not equal to 2. Then (0) One has dim(Q/X) ≡ dim(Q ′ /X) mod 2.
(1) The cokernel sheaves C(Q) = C(q) and C(Q ′ ) = C(q ′ ) are isomorphic up to twist by a line bundle on X and their isomorphism is compatible with the shifted quadratic forms (1.3).(2) The discriminant divisors Disc Q/X ⊂ X and Disc Q ′ /X ⊂ X of Q and Q ′ coincide.
(3) The even parts of Clifford algebras Cliff 0 (E, q) and Cliff 0 (E ′ , q ′ ) on X are Morita equivalent.
in the Grothendieck ring of varieties K 0 (Var/k).
(5) If the base scheme X is integral the classes of general fibers q K(X) and q ′ K(X) in the Witt group of quadratic forms over the field of rational functions K(X) on X are equal.If, moreover, In the rest of the paper we explore if the converse of Proposition 1.1(1) is true.More precisely, we discuss the following Question 1.2.Does the cokernel sheaf endowed with its shifted quadratic form (1.3) determine the hyperbolic equivalence class of quadratic forms?At this point it makes sense to explain the relation of hyperbolic equivalence to Witt groups.Recall that the Witt group W(K) of a field K is defined as the quotient of the monoid of isomorphism classes of non-degenerate quadratic forms (V, q), where V is a K-vector space and q ∈ Sym 2 V ∨ is a non-degenerate quadratic form, by the class of the hyperbolic plane K ⊕2 , ( 0 1 1 0 ) .Similarly, the Witt group W(X) of a scheme X is defined [10] as the quotient of the monoid of isomorphism classes of unimodular, i.e., everywhere non-degenerate quadratic forms (E, q), where E is a vector bundle on X and q ∈ Hom(O X , Sym 2 E ∨ ) is everywhere non-degenerate, by the classes of metabolic forms F ⊕ F ∨ , 0 1 1 q ′ .As it is explained in the survey [4], modifying the standard duality operation on the category of vector bundles on X one can define the Witt group W(X, L) that classifies classes of line bundle valued non-degenerate quadratic forms q : Sym 2 E → L ∨ .Moreover, a trick described in [5] allows one to define the Witt group W nu (X, L) of non-unimodular quadratic forms (i.e., forms that are allowed to be degenerate) as the usual Witt group of the category of morphisms of vector bundle.Thus, quadratic forms (1.1) define elements of W nu (X, L).
It is well known that hyperbolic reduction (as defined above) does not change the class of a quadratic form (E, q) in the Witt group W nu (X, L) (see, e.g., [4, §1.1.5],where it is called sublagrangian reduction).On the other hand, Witt equivalence may change the cokernel sheaf of a quadratic form, e.g., for any morphism ϕ : E 1 → E 2 of vector bundles the class of the quadratic form in the Witt group W nu (X, O X ) is zero, but the corresponding cokernel sheaf C ∼ = Coker(ϕ) ⊕ Coker(ϕ ∨ ) is non-trivial unless ϕ is an isomorphism.Therefore, Question 1.2 does not reduce to a question about Witt groups.
To answer Question 1.2 (in the case X = P n ) we define the following two basic hyperbolic equivalence invariants of quadratic forms that take values in the non-unimodular Witt group W nu (k) of the base field k.Here and everywhere below we assume that the characteristic of k is not equal to 2.
To define the first invariant, assume X is a k-scheme with a k-point x ∈ X(k).We fix a trivialization of L x and define to be the class of the quadratic form q x obtained as the composition Sym where the second arrow is given by the trivialization of L x (we could also define w x (E, q) to be the class of the quotient of (E x , q x ) by the kernel; then it would take values in W(k)).The class w x (E, q) depends on the choice of trivialization, but this is not a problem for our purposes.If the scheme X has no k-points, we could take x to be a k ′ -point for any field extension k ′ /k and define w x (E, q) ∈ W nu (k ′ ) in the same way.
For the second invariant, assume X is smooth, connected and proper k-scheme, n = dim(X) is even, and L ⊗ ω X ∼ = M 2 for a line bundle M on X, where ω X is the canonical line bundle of X.Then we define the bilinear form on the cohomology group H n/2 (X, E ⊗ M) which we denote H n/2 (q) or H n/2 (Q).This form, of course, depends on the choice of the line bundle M (if Pic(X) has 2-torsion, there may be several choices), but we suppress this in the notation.The bilinear form H n/2 (q) is symmetric if n/2 is even (and skewsymmetric otherwise) and possibly degenerate.Anyway, if n is divisible by 4, we denote its class in the non-unimodular Witt group by (again, we could define hw(E, q) to be the class of the quotient of H n/2 (q) by its kernel; then it would take values in W(k)).As before, the class hw(E, q) depends on the choice of isomorphism L ⊗ ω X ∼ = M 2 , but this is still not a problem.Note that when k is alegbraically closed, W(k) ∼ = Z/2 and so, if the corresponding forms are nondegenerate, the invariants w x (E, q) and hw(E, q) take values in Z/2, and do not depend on extra choices.In this case w x (E, q) is just the parity of the rank of E and hw(E, q) is the parity of the rank of H n/2 (q).
The second main result of this paper is the affirmative answer to Question 1.2 in the case X = P n .Recall that Pic(P n ) = Z, hence any line bundle L has the form L = O(−m) for some m ∈ Z.We need to define the following two "standard" types of unimodular quadratic forms with values in O(m): where q is the sum of tensor products of the natural pairings (the second is given by wedge product, hence it is symmetric if n/2 is even and skew-symmetric if n/2 is odd) and of non-degenerate bilinear forms q W i : W −i ⊗ W i → k which for i = 0 are symmetric in the case (1.7) and (1.8) with n/2 even and skew-symmetric in the case (1.8) with n/2 odd.
Recall that the cokernel sheaf C(q) of a quadratic form (E, q) is endowed with the shifted self-duality isomorphism q, see (1.3).In conditions ( 1) and ( 2) of the theorem we use the same trivialization of O(−m) x and the same isomorphism O(−m) ⊗ ω P n ∼ = M 2 for (E 1 , q 1 ) and (E 2 , q 2 ).Theorem 1.3.Let k be a field of characteristic not equal to 2 and let X = P n be a projective space over k.Let E 1 (−m) −−→ E ∨ 2 be generically non-degenerate self-dual morphisms over P n .Assume there is an isomorphism of sheaves C(q 1 ) ∼ = C(q 2 ) compatible with the quadratic forms q1 and q2 .Then (E 1 , q 1 ) is hyperbolic equivalent to the direct sum of (E 2 , q 2 ) and one of the standard quadratic forms (1.7) or (1.8), where W i = 0 for i = 0 and q W 0 is anisotropic.
If, moreover, the following conditions hold true: (1) if m is even then w If k is algebraically closed and x is chosen away from the support of C(q i ), condition (1) in the theorem just amounts to E 1 and E 2 having ranks of the same parity.Similarly, condition (2) amounts to the forms hw(E i , q i ) having ranks of the same parity.
Note also that adding a standard summand of type (1.7) with W i = 0 for i = 0 and dim(W 0 ) = 1 corresponds geometrically to replacing a quadric bundle Q ⊂ P P n (E) → P n by the quadric bundle Q → P n , where Q → P P n (E) is the double covering branched along Q (note that this operation changes the parity of the rank of E).The geometric meaning of adding a trivial summand of type (1.8) is not so obvious.
Remark 1.4.The condition of compatibility of an isomorphism C(q 1 ) ∼ = C(q 2 ) with the shifted quadratic forms q1 and q2 may seem subtle, but in many applications it is easy to verify.For instance, if the sheaves C(q i ) are simple, i.e., End(C(q i )) ∼ = k, then a non-degenerate shifted quadratic form on C(q i ) is unique up to scalar, so if k is quadratically closed then any isomorphism of C(q i ) after appropriate rescaling is compatible with the shifted quadratic forms.In §4 we apply this technique to the case of resolutions of symmetric sheaves (see Definition 4.1).Any cokernel sheaf C(q) is symmetric, and conversely, if X = P n then under a mild technical assumption any symmetric sheaf is isomorphic to C(q) for some self-dual morphism q : E(−m) → E ∨ (see [7] or Theorem 4.8 and Remark 4.9 in §4).
Our main technical result here is the Modification Theorem (Theorem 4.17) in which we show that any self-dual morphism over P n is hyperbolic equivalent to the sum of a self-dual VHC morphism and a standard unimodular self-dual morphism of type (1.7) or (1.8).This implies Theorem 1.3, see §4.4 for the proof.
Combining Theorem 1.3 with Proposition 1.1 we obtain the following corollary, which for simplicity we state over an algebraically closed ground field.
Corollary 1.5.Let k be an algebraically closed field of characteristic not equal to 2. Let Q → P n and Q ′ → P n be generically smooth quadric bundles such that there is an isomorphism of the cokernel sheaves C(Q) ∼ = C(Q ′ ) compatible with their shifted quadratic forms.If n is divisible by 4 and m is odd assume also that rk(H n/2 (Q)) ≡ rk(H n/2 (Q ′ )) mod 2, where the quadratic forms H n/2 (Q) and H n/2 (Q ′ ) are defined by (1.5).Then (1) If dim(Q/P n ) and dim(Q ′ /P n ) are even then the corresponding discriminant double covers S → P n and S ′ → P n are isomorphic over P n , and the Brauer classes β S ∈ Br(S ≤1 ) and β ′ S ∈ Br(S ′ ≤1 ) on the corank ≤ 1 loci inside S and S ′ are equal.
(2) If dim(Q/P n ) and dim(Q ′ /P n ) are odd then the corresponding discriminant root stacks S → P n and S ′ → P n are isomorphic over P n , and the Brauer classes β S ∈ Br(S ≤1 ) and β ′ S ∈ Br(S ′ ≤1 ) on the corank ≤ 1 loci inside S and S ′ are equal.
To finish the Introduction it should be said that this paper was inspired by the recent paper [6], where similar questions were discussed.In particular, assertions (1) and ( 4) of Corollary 1.5 in case n = 2 have been proved there.We refer to [6] for various geometric applications of these results.
On the other hand, we want to stress that the approach of the present paper is completely different: the results of [6] are based on an explicit computation of the Brauer class of a quadric bundle using the technique developed in [9].It is unclear whether these methods can be effectively generalized to higher dimensions.
It also makes sense to mention that the technique of hyperbolic extensions and VHC resolutions developed in this paper can be used for other questions related to quadric bundles over arbitrary schemes and vector bundles on projective spaces.
Convention: Throughout the paper we work over an arbitrary field k of characteristic not equal to 2. that allowed me to improve significantly the results of Proposition 1.1(4) and Corollary 1.5 (3) and the anonymous referee for many useful comments about the first version of the paper.

Quadric bundles and hyperbolic equivalence
Recall from the Introduction the definition of a quadric bundle, of its associated quadratic form and self-dual morphism (1.1) (which we assume to be generically non-degenerate), of the cokernel sheaf (1.2) and of its shifted self-duality (1.3).Conversely, we denote by the quadric bundle associated with a quadratic form (E, q) or a morphism (1.1).
2.1.Hyperbolic reduction.We start with the notion of hyperbolic reduction, which is well known, see [3,13].For the reader's convenience we remind the definition in a slightly different form.
Let (1.1) be a self-dual morphism of vector bundles on a scheme X.We will say that a vector subbundle φ : is surjective and vanishes on the subbundle F ⊂ E, i.e., F is contained in the subbundle (2.1) If F is regular isotropic, the restriction of q to F ⊥ contains F in the kernel, hence induces a quadratic form on F ⊥ /F.We summarize these observations in the following Lemma 2.1.Let (1.1) be a self-dual morphism of vector bundles on a scheme X.Let φ : F ֒→ E be a regular isotropic subbundle.Denote The restriction of q to F ⊥ induces a self-dual morphism q − : E − ⊗ L → E ∨ − such that there is an isomorphism C(q − ) ∼ = C(q) of the cokernel bundles compatible with their shifted self-dualities q and q− .Proof.The result follows from the argument of [13,Lemma 2.4].Indeed, it is explained in loc.cit.that the cokernel sheaf C(q − ) is isomorphic to the cohomology of the bicomplex (cf.[13, (2)]) (2.2) Its left and right columns are acyclic, while the middle one coincides with (1.1), hence C(q − ) ∼ = C(q).Furthermore, using the self-duality of q, we see that the dual of (2.2) twisted by L is isomorphic to (2.2), and moreover, this isomorphism is compatible with the isomorphism of the dual of (1.1) twisted by L with (1.1).This means that the isomorphism of the cokernel sheaves C(q − ) ∼ = C(q) is compatible with their shifted self-dualities.

The operation
defined in Lemma 2.1 is called hyperbolic reduction of a quadratic form (resp. of a quadric bundle) with respect to the subbundle F. As explained in [13,Proposition 2.5], this operation can be interpreted geometrically in terms of the linear projection of Q ⊂ P X (E) from the linear subbundle P X (F) ⊂ Q ⊂ P X (E).The next simple lemma motivates the terminology.
Lemma 2.2.Assume X is integral and K(X) is the field of rational functions on X.If Q/X is a generically non-degenerate quadric bundle and Q − /X is its hyperbolic reduction, then the quadratic forms q K(X) and (q − ) K(X) corresponding to their general fibers are equal in the Witt group W(K(X)) of K(X).
Proof.Hyperbolic reduction commutes with base change, so the question reduces to the case where the base is the spectrum of K(X), i.e., to the case of hyperbolic reduction of a quadric Q K(X) ⊂ P(E K(X) ) with respect to a linear subspace F K(X) ⊂ E K(X) .In this case q − is the induced quadratic form on F ⊥ K(X) /F K(X) (the orthogonal is taken with respect to the quadratic form q). It is easy to see that the quadratic form q is isomorphic to the orthogonal sum q − ⊥ q 0 of q − with the hyperbolic form q 0 = 0 1 dim(F ) 1 dim(F ) 0 , hence q = q − in the Witt group W(K(X)).
The following obvious lemma shows that hyperbolic reduction is transitive.
Lemma 2.3.Let (E − , q − ) be the hyperbolic reduction of (E, q) with respect to a regular isotropic subbundle F ֒→ E and let (E −− , q −− ) be the hyperbolic reduction of (E − , q − ) with respect to a regular isotropic subbundle F − ֒→ E − .Then (E −− , q −− ) is a hyperbolic reduction of (E, q).Proof.Let F ⊂ F ⊥ be the preimage of F − ⊂ E − under the map F ⊥ ։ F ⊥ /F = E − , so that there is an exact sequence 0 → F → F → F − → 0 and an embedding F ֒→ E. Then F is regular isotropic and the hyperbolic reduction of (E, q) with respect to F is isomorphic to (E −− , q −− ).
In the next subsection we will describe a construction inverse to hyperbolic reduction, and in the rest of this subsection we introduce the input data for that construction.
Assume F ⊂ E is a regular isotropic subbundle with respect to a quadratic form q and let (E − , q − ) be the hyperbolic reduction of (E, q) with respect to F. Consider the length 3 filtration Its associated graded is gr In particular, we have two exact sequences The next lemma describes a relation between their extension classes.
Lemma 2.4.Let ε ∈ Ext 1 (F ∨ ⊗L ∨ , E − ) be the extension class of (2.5).Then the extension class of (2.4) is equal to q − (ε), the Yoneda product of ε with the map q − : E Proof.Tensoring diagram (2.2) by L ∨ and taking quotients by F we obtain a morphism of exact sequences This is a pushout diagram and the extension class of the top row is ε, hence the extension class of the bottom row is q − (ε).It remains to note that the bottom row is the twisted dual of (2.4).Since the sequences (2.4) and (2.5) come from a length 3 filtration of E, the Yoneda product of their extension classes vanishes.
We axiomatize the property of the class ε observed in Lemma 2.4 as follows (recall that for s ∈ Z we denote by [s] the shift by s in the derived category).Definition 2.5.Let (1.1) be a self-dual morphism, let G be a vector bundle on X, and let ε ∈ Ext 1 (G, E) be an extension class.We define the classes q(ε) We say that ε is q-isotropic if q(ε, ε) = 0.
Using this terminology we can reformulate Lemma 2.4 by saying that the class of (2.5) is q − -isotropic.
Remark 2.6.It is easy to see that q(ε, ε) and it remains to note that Sym

Hyperbolic extension.
The following definition is central for this section.
Definition 2.7.Given a self-dual morphism (1.1) and a q-isotropic extension class ε ∈ Ext 1 (G, E) we say that (E + , q + ) is a hyperbolic extension of (E, q) with respect to ε if there is a regular isotropic embedding L ∨ ⊗ G ∨ ֒→ E + such that the hyperbolic reduction of (E + , q + ) with respect to L ∨ ⊗ G ∨ is isomorphic to (E, q) and the induced extension 0 We denote by HE(E, q, ε) the set of isomorphism classes of all hyperbolic extensions of (E, q) with respect to a q-isotropic extension class ε.The main goal of this section is to show that HE(E, q, ε) is non-empty; we will moreover see that this set may be quite big.
We start, however, with a simpler case, where the set HE(E, q, ε) consists of a single element.
Proposition 2.8.Let (1.1) be a self-dual morphism of vector bundles.If G is a line bundle then for any extension class ε ∈ Ext 1 (G, E) there exists a unique (up to isomorphism) hyperbolic extension of (E, q) with respect to ε.
Proof.We start by proving the existence of a hyperbolic extension.The construction described below is an algebraic version of the geometric construction sketched in the Introduction.Let be an extension of class ε and consider its symmetric square 0 tensor product with G ∨ , and its pushout along the map Sym defining a vector bundle E + and a morphism φ.We will show that E + comes with a natural quadratic form q + such that the embedding L ∨ ⊗ G ∨ ֒→ E + in the bottom row of (2.7) is regular isotropic and the corresponding hyperbolic reduction is isomorphic to (E, q).For this we consider a component of the symmetric square of φ: We will show that its cokernel is canonically isomorphic to L ∨ , and we will take the cokernel morphism Sym 2 E + → L ∨ as the definition of the quadratic form q + .Indeed, considering (2.6) as a length 2 filtration on E ′ and taking its fourth symmetric power we obtain a length 5 filtration on Sym Similarly, the combination of the bottom row of (2.7) with (2.6) provides E + with a length 3 filtration which induces a length 5 filtration on Sym 2 E + with factors It is easy to check that the morphism (2.8) is compatible with the filtrations, induces isomorphisms of the last two factors, epimorphisms on the first two factors, and the morphism on the middle factors.Therefore, the cokernel of (2.8) is canonically isomorphic to Coker(q, id) ∼ = L ∨ .This induces a canonical morphism q + : Sym 2 E + → L ∨ which vanishes on the first two factors of (2.10) and restricts to the morphism (− id, q) on the middle factor.Since the morphism q + vanishes on the first factor Similarly, since the morphism q + vanishes on the second factor of (2.10) and nowhere vanishes on the summand regular isotropic, the underlying vector bundle of the hyperbolic reduction of (E + , q + ) is isomorphic to E, and the induced extension of G by E coincides with (2.6).Finally, since the restriction of q + to the summand Sym 2 E of the middle factor of (2.10) equals q, the induced quadratic form on E is equal to q.Thus, (E + , q + ) is a hyperbolic extension of (E, q) with respect to ε.Now we prove that the constructed hyperbolic extension is unique.For this it is enough to show that for any hyperbolic extension (E + , q + ) of (E, q) with respect to ε there is a diagram (2.7) such that q + is the cokernel of Sym 2 (φ).
First, consider the morphism where e i are sections of E + and f is a section of G ∨ that we consider as a subbundle in E + ⊗ L. The symmetric square of the exact sequence 0 where the first map takes e ⊗ f 1 ⊗ f 2 to ef 1 ⊗ f 2 .The composition of this map with φ + acts as The second summand is zero because L ∨ ⊗ G ∨ ⊂ E + is isotropic and the first summand cancels with the last because the rank of G is 1, hence f 1 and f 2 are proportional.Therefore, the map φ + factors through a map φ : Moreover, it is easy to see that this map fits into the diagram (2.7).Finally, it is straightforward (but tedious) to check that the composition vanishes, and since q + is a hyperbolic extension of q, it vanishes on the first two factors of (2.10) and induces the morphism L ∨ ⊕ Sym 2 E → L ∨ of the third factor which is equal to q on Sym 2 E, hence equal to (− id, q) on this third factor, and thus coincides with the canonical cokernel of Sym 2 (φ).
Note that the general case (where the rank of G is greater than 1) does not immediately reduce to a rank 1 case, because a general vector bundle does not admit a filtration by line bundles.Besides, even if such a filtration exists, it is hard to trace what happens with the obstructions and to see how the nontrivial space of extensions shows up.So, in the proof of the theorem below we use the projective bundle trick.
Theorem 2.9.For any self-dual morphism (1.1) and a q-isotropic extension class ε ∈ Ext 1 (G, E) the set HE(E, q, ε) of hyperbolic extensions of (E, q) with respect to ε is non-empty and is a principal homogeneous variety under an action of the group Ext 1 ( 2 G, L ∨ ).
The action of the group Ext 1 ( 2 G, L ∨ ) on the set HE(E, q, ε) will be constructed in course of the proof.
Proof.Consider the projectivization π : P X (G) → X and the tautological line subbundle O(−1) ֒→ π * G.Note that the quotient bundle π * G/O(−1) can be identified with T π (−1), where T π is the relative tangent bundle for the morphism π.We denote by γ ∈ Ext 1 (T π (−1), O(−1)) the extension class of the tautological sequence By Proposition 2.8 there is a unique hyperbolic extension of (π * E, π * q) with respect to ε which is given by an extension of vector bundles We denote the extension class of the above sequence by ε′ Note that by Lemma 2.4 the restriction of ε′ to π * E ⊂ Ẽ′ is π * q(ε); in particular, Ẽ+ has a length 3 filtration with gr and the extension classes linking its factors are (π * q)(ε) and ε, respectively.
It would be natural at this point to consider a hyperbolic extension of ( Ẽ+ , q+ ) by T π (−1) (note that the rank of T π (−1) is less than G) and then show that the result descends to a self-dual morphism on X.However, it turns out to be more convenient to use a simpler construction by "adding" the (twisted) dual bundle π * L ∨ ⊗ Ω π (1) to the kernel space of q and then applying another version of descent.
Consider the product of extension classes (recall that γ is the extension class of (2.11)): (where Ω π = T ∨ π is the relative sheaf of Kähler differentials).We claim that γ • ε′ = 0. Indeed, using (2.12) and taking into account isomorphisms for all p ∈ Z, and note that under this isomorphism the product γ ), and hence vanishes as ε is assumed to be q-isotropic.
Consider the tensor product of the dual sequence of (2.11) with π * L ∨ : its extension class is also γ.The vanishing of the product γ • ε′ implies that the class ε′ lifts to a class in Ext which shows that such a lift of γ is unique up to the natural free action of the group In other words, the set of such lifts is a principal homogeneous space under an action of Ext 1 ( 2 G, L ∨ ).
The lifted classes define a vector bundle Ê+ that fits into two exact sequences We consider the quadratic form on Ê+ defined by the following composition where the latter embedding is induced by the surjection Ê+ ։ Ẽ+ from (2.13).Note that by construction Ê+ has a length 4 filtration with gr and the extension classes linking its adjacent factors are γ, (π * q)(ε), and ε, respectively.Furthermore, the subbundle π * L ∨ ⊗ Ω π (1) ⊂ Ê+ is contained in the kernel of the quadratic form q+ .Now we explain how to descend the quadratic form ( Ê+ , q+ ) over P X (G) to a quadratic form (E + , q + ) over X.
Consider the subbundle Ker( Ê+ → O(−1)) ⊂ Ê+ generated by the first three factors of the filtration.Since the first two factors a linked by the class γ of the twisted dual of (2.11), this bundle is an extension of π * E by π * (L ∨ ⊗ G ∨ ).Since the functor π * is fully faithful on the derived category of coherent sheaves, its extension class is a pullback, hence there exists a vector bundle E ′′ on X and exact sequences where the right vertical arrow is the tautological embedding.The embedding of bundles Ê+ ֒→ π * E + in the middle column is identical on the subbundle π * E ′′ hence the induced morphism is the blowup with center P X (E ′′ ) ⊂ P X (E + ), i.e., we have and therefore the derived pullback functor ρ * is fully faithful.
Let π + : P X (E + ) → X and π+ : P P X (G) ( Ê+ ) → X be the projections, so that π+ = π + • ρ, and we have a commutative diagram Let furthermore H + and Ĥ+ be the relative hyperplane classes of P X (E + ) and P P X (G) ( Ê+ ), respectively, so that ρ * O(H + ) ∼ = O( Ĥ+ ).Note that the quadratic form q+ can be represented by a section of the line bundle π * . Thus, q+ is (in a unique way) the pullback of a section q + of the line bundle π * + L ∨ ⊗ O(2H + ) on P X (E + ), i.e., q+ = ρ * (q + ).Furthermore, q + induces a morphism First, note that a combination of (2.14) and the second row of (2.16) shows that E + has a filtration , and G, respectively.In particular, there is an exact sequence and the diagram (2.16) implies that the sequence (2.12) is its pullback.Using the natural isomorphism Ext 1 (O(−1), π * E) ∼ = Ext 1 (G, E) and the definition of (2.12) we conclude that the extension class of the above sequence is ε.So, we only need to show that the subbundle L ∨ ⊗ G ∨ ֒→ E + is regular isotropic and that the induced quadratic form on E coincides with q.
The first follows immediately from the fact that π * L ∨ ⊗ Ω π (1) ⊂ Ê+ is contained in the kernel of the quadratic form q+ (as it was mentioned above) and that the subbundle π * L ∨ ⊗ O(1) ⊂ Ẽ+ is isotropic for the quadratic form q+ (because ( Ẽ+ , q+ ) is a hyperbolic extension).Moreover, by the same reason the induced quadratic form on π * E coincides with π * q.
To finish the proof of the theorem we must check that any hyperbolic extension of (E, q) comes from the above construction.So, assume that (E + , q + ) is a hyperbolic extension of (E, q) with respect to ε. Define the bundle Ê+ from the diagram (2.16), consider the blowup morphism ρ as above, and the pullback q+ = ρ * (q + ) of the quadratic form q + .It defines a quadratic form on Ê+ over P X (G).It is easy to see that π * L ∨ ⊗ Ω π (1) is contained in the kernel of q+ and that the quotient ( Ẽ+ , q+ ) (where Ẽ+ is defined by the first sequence in (2.13)) is a hyperbolic extension of π * E with respect to (2.12).Therefore, by the uniqueness result in Proposition 2.8 this quadratic form coincides with the one constructed in the proof and the rest of the construction shows that (E + , q + ) coincides with one of the hyperbolic extensions of the theorem.
The non-triviality of the construction of hyperbolic extension is demonstrated by the following.
is in general non-trivial; one can identify it with the Massey product µ(ε, q, ε).
The operation of hyperbolic extension is transitive in the following sense.Lemma 2.11.Let (E + , q + ) be a hyperbolic extension of (E, q) with respect to a q-isotropic extension class ε ∈ Ext 1 (G, E) and let (E ++ , q ++ ) be a hyperbolic extension of (E + , q + ) with respect to a q + -isotropic extension class ε + ∈ Ext 1 (G + , E + ).Then (E ++ , q ++ ) is a hyperbolic extension of (E, q).Proof.By definition the hyperbolic reduction of (E ++ , q ++ ) with respect to L ∨ ⊗ G ∨ + ֒→ E ++ is (E + , q + ) and the hyperbolic reduction of (E + , q + ) with respect to L ∨ ⊗G ∨ ֒→ E + is (E, q).Therefore, by Lemma 2.3 we see that (E, q) is a hyperbolic reduction of (E ++ , q ++ ), hence by definition we conclude that (E ++ , q ++ ) is a hyperbolic extension of (E, q).2.3.Hyperbolic equivalence.We combine the notions of hyperbolic reduction and extension defined in the previous sections into the notion of hyperbolic equivalence.Definition 2.12.We say that two quadratic forms are hyperbolically equivalent if they can be connected by a chain of hyperbolic reductions and hyperbolic extensions.
Since the operations of hyperbolic reduction and hyperbolic extension are mutually inverse by definition, this is an equivalence relation.In this subsection we discuss hyperbolic invariants, i.e., invariants of quadratic forms and quadric bundles with respect to hyperbolic equivalence.
Recall the invariants (1.4) and (1.6) with values in the (non-unimodular) Witt group W nu (k) defined in the Introduction.The hyperbolic invariance of (1.4) is obvious.Lemma 2.13.For any k-point x ∈ X and a fixed trivialization of the fiber L x of the line bundle L the class w x (E.q) = [(E x , q x )] ∈ W nu (k) is hyperbolic invariant.In particular, the parity of rk(E) is hyperbolic invariant.
Proof.This follows immediately from the fact that if (E − , q − ) is the hyperbolic reduction of (E, q) with respect to a regular isotropic subbundle F then (E −,x , q −,x ) is the sublagrangian reduction of (E x , q x ) with respect to the subspace Applying the rank parity homomorphism W nu (k) → Z/2 we deduce the invariance of the parity of rk(E) from that of w x (E, q); alternatively, this invariance can be seen directly from the construction.
The hyperbolic invariance of (1.6) requires a bit more work.Lemma 2.14.If X is smooth and proper, L ⊗ ω X is a square in Pic(X), and n = dim(X) is divisible by 4, the class hw(E, q) ∈ W nu (k) is hyperbolic invariant.In particular, the parity of the rank of the form H n/2 (q) defined by (1.5) is hyperbolic invariant.
Proof.Let M be a square root of L ⊗ ω X .By Serre duality we have Therefore, the pairing (1.5) can be rewritten as the composition of the morphism (2.17) and the Serre duality pairing.Now assume that F ֒→ E is a regular isotropic subbundle and (E − , q − ) is the hyperbolic reduction.It is enough to check that hw(E, q) = hw(E − , q − ).Note that E − ⊗ M and E ∨ − ⊗ L ∨ ⊗ M by definition are the cohomology bundles (in the middle terms) of the complexes Therefore, the morphism of cohomology H n/2 (X, ) is computed by the morphism of the spectral sequences whose first pages look like (dotted arrows show the directions of the only higher differentials d 2 ) and Moreover, the morphism of spectral sequences is equal to the identity on the first and last columns and is induced by q on the middle column.On the other hand, by Serre duality hence the morphism of spectral sequences is self-dual.It follows that (H n/2 (X, E − ⊗ M), H n/2 (q − )) is obtained from (H n/2 (X, E ⊗ M), H n/2 (q)) by a composition of the hyperbolic reduction with respect to the regular isotropic subspace followed by a hyperbolic extension with respect to the space Therefore, we have the required equality hw(E − , q − ) = hw(E, q) in the Witt group W nu (k).
Applying the rank parity homomorphism W nu (k) → Z/2 we deduce the invariance of the parity of the rank of H n/2 (q) from that of hw(E, q).Other hyperbolic invariants of quadric bundles have been listed in Proposition 1.1.We are ready now to prove this proposition.
Proof of Proposition 1.1.Since assertion (0) is clear from the definition (or follows from Lemma 2.13), it is enough to prove assertions (1)-( 5) of the proposition.Moreover, in most cases it is enough to prove the assertions for a single hyperbolic reduction.So, assume that (1.1) is a self-dual morphism and (E − , q − ) is its hyperbolic reduction with respect to a regular isotropic subbundle F ֒→ E.
By Lemma 2.1 we have C(q) ∼ = C(q − ), an isomorphism compatible with the shifted quadratic forms; this proves assertion (1).Furthermore, the equality of the discriminant divisors follows as well and proves (2).Similarly, (5) follows from Lemma 2.2 and Witt's Cancellation Theorem.Now we prove (3).We refer to [11] for generalities about sheaves of Clifford algebras and modules.Here we just recall that for a vector bundle E with a quadratic form q : L → Sym 2 E ∨ we denote and set Cliff i+2 (E, q) = L ∨ ⊗ Cliff i (E, q).The Clifford multiplication (see [12, §3]) (induced by q and the wedge product on • E) provides Cliff 0 (E, q) with the structure of O X -algebra (called the sheaf of even parts of Clifford algebras) and each Cliff i (E, q) with the structure of Cliff 0 (E, q)bimodule.In the case where the line bundle L is trivial, the sum Cliff(E, q) = Cliff 0 (E, q) ⊕ Cliff 1 (E, q) also acquires a structure of O X -algebra (called the total Clifford algebra), which is naturally Z/2-graded.
Now consider the subbundle F ⊥ ⊂ E defined by (2.1).It comes with the quadratic form q F ⊥ , the restriction of the form q, so that the subbundle F ⊂ F ⊥ is contained in the kernel of q F ⊥ and the induced quadratic form on the quotient F ⊥ /F = E − coincides with q − .Thus, the maps F ⊥ ֒→ E and F ⊥ ։ E − are morphisms of quadratic spaces.Therefore, they are compatible with the Clifford multiplications and induce O X -algebra morphisms of sheaves of even parts of Clifford algebras The kernel of the second morphism is the two-sided ideal where the arrow is the natural morphism induced by the embedding F ֒→ F ⊥ ֒→ Cliff 1 (F ⊥ , q F ⊥ ) and the Clifford multiplication.Now we denote k = rk(F) and consider the right ideal in Cliff 0 (E, q) defined as Since F ⊥ is the orthogonal of F with respect to q, the subalgebra Cliff 0 (F ⊥ , q F ⊥ ) ⊂ Cliff 0 (E, q) anticommutes with k F ⊂ Cliff k (E, q), hence P is invariant under the left action of Cliff 0 (F ⊥ , q F ⊥ ) on Cliff 0 (E, q).Furthermore, since F is isotropic, the Clifford multiplication vanishes on F ⊗ k F, hence the ideal R annihilates P. Therefore, P has the structure of a left module over the algebra This structure obviously commutes with the right Cliff 0 (E, q)-module structure, hence P is naturally a (Cliff 0 (E − , q − ), Cliff 0 (E, q))-bimodule.We show below that P defines the required Morita equivalence.
The question now is local over X, so we may assume that L = O X and there is an orthogonal direct sum decomposition where E 0 = F ⊕ F ∨ and the quadratic form q 0 is given by the natural pairing F ⊗ F ∨ → O X .Furthermore, as L = O X , we can consider the total Z/2-graded Clifford algebras.On the one hand, the orthogonal direct sum decomposition (2.19) implies the natural isomorphism Cliff(E, q) ∼ = Cliff(E − , q − ) ⊗ Cliff(E 0 , q 0 ) (where the right-hand side is the tensor product in the category of Z/2-graded algebras), compatible with the gradings.On the other hand, since F ⊂ E 0 is Lagrangian, the algebra is Morita trivial, and its Z/2-grading is induced by the natural Z/2-grading of • F. It follows that the (Cliff(E − , q − ), Cliff(E, q))-bimodule defines a Morita equivalence of Cliff(E − , q − ) and Cliff(E, q), compatible with the grading.Therefore, the even part of defines a Morita equivalence between the even Clifford algebras Cliff 0 (E − , q − ) and Cliff 0 (E, q).Finally, a simple computation shows that the globally defined bimodule P is locally isomorphic to the bimodule P0 , hence it defines a global Morita equivalence.
In conclusion we prove (4).To show that [Q] = [Q ′ ] we will first show that for any point x ∈ X there is a Zariski neighborhood , and after that we will use this local equality to deduce the global one.Since we are going to work locally, we may assume that the line bundle L is trivial and the base is affine.Then two things happen with hyperbolic extension -first, any extension class ε ∈ Ext 1 (G, E) vanishes (in particular, any such class is q-isotropic), and second, the group Ext 1 ( 2 G, E) vanishes as well, so that the result of hyperbolic extension becomes unambiguous.Moreover, it is clear that this result becomes isomorphic to E + = E ⊕ (G ⊕ G ∨ ), the orthogonal direct sum of E and G ⊕ G ∨ , with the quadratic form on G ⊕ G ∨ induced by duality.Similarly, hyperbolic reduction reduces to splitting off an orthogonal summand F ⊕ F ∨ .Thus, locally, hyperbolic equivalence turns into Witt equivalence (in the non-unimodular Witt ring of the base scheme).Therefore, a hyperbolic equivalence between Q and Q ′ locally can be realized by a single quadric bundle Q such that both Q and Q ′ are obtained from Q by hyperbolic reduction.In other words, we may assume that the quadrics Q and Q ′ correspond to quadratic forms obtained from a single quadratic form ( Ê, q) by isotropic reduction with respect to regular isotropic subbundles F ⊂ Ê and F ′ ⊂ Ê of the same rank.Below we prove isomorphism of Q and Q ′ in a neighborhood of x by induction on the rank of F and F ′ .
First assume that the rank of F and F ′ is 1 and q(F, F ′ ) = 0 at x (hence also in a neighborhood of x).Since F and F ′ are isotropic, the restriction of q to F ⊕ F ′ is non-degenerate, hence there is an orthogonal direct sum decomposition Ê = Ē ⊕ (F ⊕ F ′ ).
Then obviously F ⊥ = Ē ⊕ F, hence the hyperbolic reduction of ( Ê, q) with respect to F is isomorphic to ( Ē, q|Ē).Similarly, the hyperbolic reduction of ( Ê, q) with respect to F ′ is isomorphic to ( Ē, q|Ē) as well.In particular, the two hyperbolic reductions are isomorphic.On the other hand, assume that the rank of F and F ′ is 1 and q(F, F ′ ) vanishes at x. Then we find (locally) yet another regular isotropic subbundle F ′′ ⊂ Ê such that q(F, F ′′ ) = 0 and q(F ′ , F ′′ ) = 0 at x. Let v, v ′ ∈ Êx be the points corresponding to F, F ′ .Let v ′′ ∈ Êx be a point such that qx (v, v ′′ ) = 0 and qx (v ′ , v ′′ ) = 0.The existence of a regular subbundle F implies rationality of Q over X, hence (maybe over a smaller neighborhood of x) there esists a regular isotropic subbundle F ′′ corresponding to the point v ′′ .Now, when we have such F ′′ , we apply the previous argument and conclude that the hyperbolic reduction of ( Ê, q) with respect to F ′′ is isomorphic to the hyperbolic reductions with respect to F and F ′ , hence the latter two reductions are mutually isomorphic.Now assume the rank of F and F ′ is bigger than 1.Shrinking the neighborhood of x if necessary, we may split F = F 1 ⊕F 2 and , where the rank of F 1 and F ′ 1 is 1.The above argument shows that the isotropic reductions of ( Ê, q) with respect to F 1 and F ′ 1 are isomorphic.Hence Q and Q ′ correspond to hyperbolic reductions of the same quadratic form with respect to regular isotropic subbundles F 2 and F ′ 2 , which have smaller rank than F and F ′ , and therefore by induction Q and Q ′ are isomorphic.
Finally, we deduce the global result from the local results obtained above.Indeed, the argument above and quasi-compactness of X imply that X has an open covering {U i } such that over each U i we have an isomorphism in the Grothendieck ring of varieties.For any finite set I of indices set U I = ∩ i∈I U i .Then inclusion-exclusion gives and since by base change we have isomorphisms Remark 2.15.The same technique proves the following more general formula for any hyperbolic equivalent quadric bundles Q/X and Q ′ /X, where n = dim(Q/X) and we assume that it is greater or equal than dim(Q ′ /X), which we write in the form dim(Q ′ /X) = n − 2d.Indeed, first (2.20) can be proved over a small neighborhood of any point of X; for this the same argument reduces everything to the case where Q ′ is a hyperbolic reduction of Q, in which case the formula is proved in [13,Corollary 2.7].After that the inclusion-exclusion trick proves (2.20) in general.

VHC resolutions on projective spaces
From now on we consider the case X = P n .This section serves as a preparation for the next one.Here we introduce a class of locally free resolutions (which we call VHC resolutions) which plays the main role in §4 and show that on P n any sheaf of projective dimension 1 has a (essentially unique) VHC resolution, see Corollary 3.18 for existence and Theorem 3.15 for uniqueness.

Complexes of split bundles.
For each coherent sheaf F on P n = P(V ) (and more generally, for any object of the bounded derived category D(P n ) of coherent sheaves) and each integer p we write (3.1) This is a graded module over the homogeneous coordinate ring For a sheaf F we will often consider the S-module of intermediate cohomology as a bigraded S-module; with index p corresponding to the homological and index t corresponding to the internal grading.We will use notation [p] and (t) for the corresponding shifts of grading.
Recall the following well-known result.
Lemma 3.1.Let F be a coherent sheaf, so that the S-module H 0 * (F) is finitely generated.The minimal epimorphism S(t i ) → H 0 * (F) of graded S-modules gives rise to an epimorphism O(t i ) → F such that the induced morphism S(t i ) = H 0 * ( O(t i )) → H 0 * (F) coincides with the original epimorphism.
by split bundles.It follows that the complex L ℓ → • • • → L 1 → L 0 (where the morphism L 1 → L 0 is defined as the composition L 1 ։ E ′ ֒→ L 0 ) is a resolution of E of length ℓ by split bundles.The converse statement follows immediately from the hypercohomology spectral sequence applied to the resolution since the intermediate cohomology of split bundles vanishes.
The following obvious observation about complexes is quite useful.
for some i ∈ Z and t ∈ Z, and the differential Then there is an isomorphism of complexes Proof.By assumption the differential d i can be written in the form . After the modification of the direct sum decomposition of L i by the automorphism which implies (3.3),where L ′ j = L j for j ∈ {i, i − 1}.Now let be a complex of split bundles on P n .Since split bundles have no intermediate cohomology, the first page of the hypercohomology spectral sequence of L • has only two nontrivial rows: & & ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ . . .
We also set Lemma 3.4.If a complex L • of split bundles quasiisomorphic to an object F of the derived category D(P n ) has length ℓ = n then there is a canonical exact sequence Proof.This follows immediately from the hypercohomology spectral sequence.
The next two lemmas are crucial for the rest of the paper.
Lemma 3.5.If an acyclic complex L • of split bundles has length ℓ = n + 1 then the following conditions are equivalent Proof.Since L • is acyclic, its hypercohomology spectral sequence converges to zero, hence the canonical morphism d n+1 : 1), (2), and (3) are equivalent.Now we prove (3) =⇒ (4).So, assume (3) holds.Then for each t the hypercohomology spectral sequence of L • (−t) degenerates on the second page; in particular the bottom row of the first page is exact.Let t be the maximal integer such that O(t) appears as one of summands of one of the split bundles L i .Then the bottom row of the first page of the hypercohomology spectral sequence of L • (−t) is nonzero and takes the form where m i is the multiplicity of O(t) in L i .Since this complex is exact, it is a direct sum of shifts of trivial complexes k id − − → k.Since Hom(O(t), O(t ′ )) = 0 for all t ′ < t, it follows that L • contains the subcomplex H 0 (P n , L • (−t)) ⊗ O(t); this subcomplex is isomorphic to a direct sum of shifts of trivial complexes O(t) id − − → O(t), and each of its terms is a direct summand of the corresponding term of L • .Applying Lemma 3.3 to one of these trivial subcomplexes we obtain the direct sum decomposition (3.3).The condition (3) holds for L ′ • (because it is a direct summand of L • ), hence by induction L ′ • is the sum of shifts of trivial complexes, and hence the same is true for L • ; which means that (4) holds.
Lemma 3.6.Assume objects F and F ′ in D(P n ) are quasiisomorphic to complexes L • and L ′ • of split bundles of length ℓ.If ℓ < n then any morphism ϕ : F → F ′ is induced by a morphism of complexes If ℓ = n the same is true for a morphism ϕ : F → F ′ if and only if the composition vanishes, where the first and last morphisms are defined in Lemma 3.4.Moreover, in both cases a morphism of complexes ϕ • inducing a morphism ϕ as above is unique up to a homotopy h Proof.Obviously, the first page of the spectral sequence is non-zero only when −ℓ ≤ p ≤ ℓ and q ∈ {0, n}.Consequently, we have an exact sequence and (under the assumption ℓ ≤ n) the last term is non-zero only if ℓ = n.Furthermore, we have hence a morphism ϕ : F → F ′ can be represented by a morphism of complexes ϕ • : L • → L ′ • if and only if it comes from E 0,0 ∞ .In particular, this holds true for ℓ < n since in this case where the middle arrow is (3.5)) is given by ∂ϕ.Thus, if (3.5) vanishes then ∂ϕ = 0, and it follows that ϕ is in the image of E 0,0 ∞ , hence is induced by a morphism of complexes.Conversely, if ϕ is given by a morphism of complexes ϕ • , the commutative diagram where the rows are the exact sequences of Lemma 3.4, shows that (3.5) is zero.The uniqueness up to homotopy of ϕ • in both cases follows from the above formula for E 0,0 ∞ .

3.2.
VHC resolutions and uniqueness.The notion of a VHC resolution is based on the following Definition 3.7.We will say that a vector bundle E on P n has • the vanishing lower cohomology property if • the vanishing upper cohomology property if We will abbreviate these properties to VLC and VUC, respectively.Proof.Follows from the definition, Serre duality, and Horrock's Theorem.
Below we give a characterization of VLC and VUC bundles in terms of resolutions by split bundles.
Lemma 3.10.A vector bundle E on P n is VLC if and only if there is an exact sequence where L i are split bundles.A vector bundle E on P n is VUC if and only if there is an exact sequence where L i are split bundles.
Proof.First assume that E is a VLC vector bundle.
; by Lemma 3.2 this is equivalent to the existence of a resolution of length ℓ = k = ⌊(n − 1)/2⌋ by split bundles.
The case of a VUC bundle follows from this and Lemma 3.9 by duality.Definition 3.11.We will say that a locally free resolution 0 → E L → E U → F → 0 of a sheaf F has the VHC (vanishing of half cohomology) property (or simply is a VHC resolution), if E L is a VLC vector bundle and E U is a VUC vector bundle, see Definition 3.7.
The cohomology of bundles constituting a VHC resolution of a sheaf F are related to the cohomology of F as follows.
If n = 2k + 1 then Clearly, any VHC resolution is isomorphic to the direct sum of a linearly minimal VHC resolution and several trivial complexes U → F → 0 be linearly minimal VHC resolutions of the same sheaf F. If n = 2k + 1 assume also we have an equality with respect to the embeddings given by (3.7).Then the resolutions are isomorphic, i.e., there is a commutative diagram where ϕ L and ϕ U are isomorphisms.Moreover, an isomorphism (ϕ L , ϕ U ) of resolutions inducing the identity morphism of F is unique up to a homotpy h : and E U induced by any homotopy h are nilpotent.
Proof.Let k = ⌊(n − 1)/2⌋, so that n = 2k + 1 or n = 2k + 2. By Lemma 3.13 the object F[k] is quasiisomorphic to complexes of split bundles Using linear minimality we can assume that each of these complexes has no trivial complex O(t) id − − → O(t) as a direct summand.If n = 2k + 2, the lengths of the resolutions are less than n, hence the first part of Lemma 3.6 ensures that the identity morphism F → F is induced by a morphism of complexes.If n = 2k + 1 we use the second part of Lemma 3.6 (the composition (3.5) vanishes due to the assumption ) and Lemma 3.13) and obtain the same conclusion.Thus, we obtain a quasiisomorphism of complexes of split bundles (3.9) We prove below that it is necessarily an isomorphism, i.e., that each ϕ i is an isomorphism.For this we use the induction on the sum of ranks of the bundles L i .The base of the induction follows from Lemma 3.5.Indeed, if L • = 0 then L ′ • is acyclic, hence is the sum of trivial complexes.But by assumption it has no trivial summands, hence L ′ • = 0. Now assume that L • = 0.The totalization of (3.9) is the following acyclic complex of split bundles of length 2k + 2. If n = 2k + 2 we can formally add the zero term on the right and obtain an acyclic complex of length ℓ = n + 1 of split bundles for which the condition (1) of Lemma 3.5 holds true.If n = 2k + 1 the condition (1) of Lemma 3.5 follows from the assumption . In both cases Lemma 3.5 implies that (3.10) is isomorphic to a direct sum of shifts of trivial complexes.
To make this direct sum decomposition more precise, we consider as in the proof of Lemma 3.5 the maximal integer t such that O(t) appears as one of summands of one of the split bundles L i or L ′ i .Twisting (3.9) by O(−t) and applying the functor H 0 (P n , −) we obtain a nonzero bicomplex (as before, m i and m ′ i are the multiplicities of O(t) in L i and L ′ i , respectively) with acyclic totalization.If any of the horizontal arrows in this bicomplex is nontrivial, Lemma 3.3 implies that the trivial • , which contradicts the linear minimality assumption.Therefore, the horizontal arrows are zero, and hence the vertical arrows are all isomorphisms.
This means that m i = m ′ i for all i and we can write and that φ• : L• → L′ • is a quasiisomorphism of complexes of split bundles which have no trivial summands.Moreover, we have rk( Li ) < rk(L i ).By induction, we deduce that φi is an isomorphism for each i, hence so is ϕ i .
Since ϕ • is an isomorphism of complexes, it induces an isomorphism of resolutions of E L and E ′ L and of E U and E ′ U , compatible with the maps E L → E U and E ′ L → E ′ U , hence an isomorphism (ϕ L , ϕ U ) of the original VHC resolutions.This proves the first part of the theorem.
Further, recall that by Lemma 3.6 the morphism ϕ • in (3.9) inducing the identity of F is unique up to a homotopy h . Note that first part (h i ) 0≤i≤k−1 of such a homotopy replaces the morphism (ϕ i ) 0≤i≤k of the right resolutions of E U and E ′ U by a homotopy equivalent morphism, hence it does not change ϕ U , and a fortiori does not change ϕ L .Similarly, the last part (h i ) k+1≤i≤2k of a homotopy does not change (ϕ L , ϕ U ). Finally, it is clear that the middle component h k : L k → L ′ k+1 of a homotopy modifies (ϕ L , ϕ U ) by the homotopy of the VHC resolutions.This proves the second part of the theorem.So, it only remains to check the nilpotence of the induced endomorphisms of E L and E U .For this let us write and for each c ∈ Z define finite filtrations of these bundles by Then the morphism L k+1 → L k induced by f takes F ≥c L k+1 to F ≥c+1 L k (because f is assumed to be linearly minimal) and obviously any morphism h :

3.3.
Existence of VHC resolutions.The results of this subsection are not necessary for §4, but the technique used in their proofs is similar.
Definition 3.16.Let F be a coherent sheaf and 1 ≤ k ≤ n − 1.We will say that a graded S- ) for any t > t 0 .Similarly, for any 1 ≤ p 0 ≤ n − 1 and any t 0 ∈ Z we define the shadow of (p 0 , t 0 ) as the set and say that a bigraded S-submodule A ⊂ n−1 p=1 H p * (F) is shadowless if for any (p 0 , t 0 ) such that A p 0 t 0 = 0 we have A p t = H p (P n , F(t)) for any (p, t) ∈ Sh(p 0 , t 0 ).
To understand the meaning of this notion observe the following.Let T be the tangent bundle of P n .Recall the Koszul resolution of its exterior power where V is a vector space such that P n = P(V ).If F is a sheaf on P n , tensoring (3.12) by F(t) we obtain the hypercohomology spectral sequence The following picture shows the arrows d r , 1 ≤ r ≤ p, of the spectral sequence with source at the terms E t,p r , as well as the terms that in the limit compute the filtration on H p−s (P n , ∧ s T ⊗ F(t)) (these terms are circled), and the shadow of (p, t).
It is important that the arrows d r , 1 ≤ r ≤ p, applied to the terms E t,p r of the spectral sequence land in its shadow.This property will be used in Propositions 3.17 and 4.10 below.Proposition 3.17.For any coherent sheaf F on P n and any finite-dimensional shadowless S-submodule there exists a vector bundle E A and an epimorphism π A : Since A is shadowless we have H p (P n , F(t)) = A p t = 0 for all (p, t) ∈ Sh(p 0 , t 0 ) (the first equality holds because A is shadowless and the second follows from the above definition of (p 0 , t 0 )).In particular, the subspace A p 0 t 0 ⊂ H p 0 (P n , F(t 0 )) sits in the kernels of differentials d 1 , . . ., d p 0 −1 of the hypercohomology spectral sequence of F(t 0 ) tensored with the Koszul complex (3.12) for s = p 0 − 1.Moreover, H p 0 (P n , F(t 0 )) is the only nonzero subspace on the diagonal of the spectral sequence that in the limit computes the filtration on H 1 (P n , ∧ p 0 −1 T ⊗ F(t 0 )).Therefore, we obtain an inclusion ) is the natural embedding (the first identification uses (3.6)).Now the cohomology exact sequence implies that hence the quotient S-module A ′ := A/A p 0 t 0 is an S-submodule in H p * (F ′ ).Clearly dim(A ′ ) < dim(A) and it is straightforward to check that A ′ is shadowless.By the induction hypothesis there is a vector bundle E A ′ and an epimorphism π A ′ : E A ′ ։ F ′ inducing surjection on H 0 * and the natural embedding of A ′ into the intermediate cohomology of F ′ .We define E A as the kernel of the composition of epimorphisms By construction the map π A ′ lifts to a map π A that fits into a commutative diagram The surjectivity of π A ′ and H 0 * (π A ′ ) implies that of π A and H 0 * (π A ). Similarly, it follows that H p * (π A ) is an isomorphism onto A. Thus, the required result holds for A.
Corollary 3.18.If F is a sheaf on P n of projective dimension at most 1 then F has a VHC resolution.
Proof.Since the projective dimension of F is at most 1, there exists a locally free resolution where A k ⊂ H k * (F) is any finite-dimensional shadowless S-submodule (e.g., A k = 0).Note that we have k − 1 ≤ n − 2 as soon as n ≥ 1, hence A is finite-dimensional.Moreover, A is shadowless by construction.
Let π A : E A → F be the epimorphism constructed in Proposition 3.17 and let K A = Ker(π A ), so that is an exact sequence.First, H p * (E A ) = A p = 0 for ⌈n/2⌉ ≤ p ≤ n − 1 by definition, hence E A is VUC.Furthermore, K A is locally free because the projective dimension of F is at most 1.Finally, the cohomology exact sequence implies that H p * (K A ) = 0 for 1 ≤ p ≤ k = ⌊n/2⌋, hence K A is VLC.

Hyperbolic equivalence on projective spaces
In this section we prove Theorem 4.17 on VHC modifications of quadratic forms and deduce from it Theorem 1.3 and Corollary 1.5 from the Introduction.In §4.1 we remind a characterization of cokernel sheaves of quadratic forms (symmetric sheaves), in §4.2 we define elementary modifications of quadratic forms with respect to some intermediate cohomology classes, and in §4.3 we state the Modification Theorem (Theorem 4.17) and prove it by applying an appropriate sequence of elementary modifications.Finally, in §4.4 we combine these results to prove Theorem 1.3 and Corollary 1.5.The goal of this subsection is to relate symmetric sheaves to cokernel sheaves of quadratic forms.Most of these results are well-known and not really necessary for the rest of the paper, but useful for the context.Lemma 4.2.If C is a (d, δ)-symmetric coherent sheaf on P n there is a self-dual isomorphism where m = d − δ.In particular, the sheaf C has projective dimension 1.
Proof.Let C = i * R. Using the definitions and Grothendieck duality we deduce This proves the required isomorphism.Moreover, it follows that this isomorphism is self-dual because so is the isomorphism R(−δ) ∼ = R ∨ .Finally, it follows that Ext 1 (C, O P n ) ∼ = C(m) and Ext i (C, O P n ) = 0 for i ≥ 2, which means that the projective dimension of C is 1.
The above lemma implies that symmetric sheaves can be understood as quadratic spaces in the derived category D(P n ) and define classes in the shifted Witt group W 1 (D(P n ), O(−m)) in the sense of [4, §1.4].
The following well-known lemma shows that cokernel sheaves of generically non-degenerate quadratic forms are symmetric.For reader's convenience we provide a proof.
in D(P n ).Since q is self-dual, we have q = q ∨ .In particular, it is generically an isomorphism, hence C ∨ [1] is a pure sheaf and, moreover, Let det(q) be the determinant of q which we understand as a global section of the line bundle Then for 0 ≤ i ≤ p there exists a sequence of classes y y s s s s s s s s s s . . .
(where τ i are the extension classes of the complexes (4.5)); in other words for each 0 ≤ i ≤ p.Moreover, for i ≥ 1 such ε i are unique, while ε 0 is unique up to a composition where the first arrow is the canonical embedding.Finally, if one of the following conditions is satisfied 2p ≤ n, or (4.8) 2p = n + 1 and 2t + m + n + 1 ≥ 0, (4.9) Proof.The existence of ε i satisfying (4.7) and their uniqueness follow by descending induction from the cohomology exact sequences of complexes (4.5) tensored with E(t) in view of the vanishing (4.6).
For the second assertion we also induct on i. Assume 1 ≤ i ≤ p − 1.We have Consider the tensor square of (4.5): Note that its extension class is . Furthermore, we note that Indeed, if (4.8) holds we use 1 ≤ 2(p − i) + 1 < n − i + 1 ≤ n together with (3.6) for the first vanishing and 1 ≤ 2(p − i) < n for the second.Similarly, if (4.9) holds and i ≥ 2 the same arguments prove the vanishings.Finally, if (4.9) holds and i = 1 the same arguments prove the second vanishing, while the first cohomology space is equal to H n (P n , O(2t + m + 1)), hence vanishes since 2t + m + 1 > −n − 1.
The cohomology vanishings that we just established imply that the morphism is injective, and hence the condition The following elementary modification procedure allows us to kill an isotropic cohomology class of a quadratic form by a hyperbolic extension (see §2.2).Recall that for a cohomology class ε p ∈ H p (P n , E(t)) we denote by q(ε p ) ∈ H p (P n , E ∨ (m + t)) the image of ε p under the map Using the class q(ε p ) we consider the map (4.11) where the first arrow is the projection to a direct summand.We denote by q(ε p ) ⊥ ⊂ n−1 i=1 H i * (E) the kernel of (4.11).
Proposition 4.11.Let ε p ∈ H p (P n , E(t)) be a cohomology class such that q(ε p , ε p ) = 0 and assume that the condition (4.6) holds and either (4.8) or (4.9) is satisfied.Let ε 1 ∈ Ext 1 (Ω p−1 (−t), E) be the extension class defined in Proposition 4.10.Then ε 1 is q-isotropic and for any hyperbolic extension (E + , q + ) of (E, q) with respect to ε 1 we have where in the second line ε + ∈ H n−p+1 (P n , E + (t + m + n + 1)) is a nonzero cohomology class that depends on the choice of (E + , q + ).
Consider the specral sequence of a filtered complex that computes the cohomolgoy of (twists of) E + ; the terms of its first page E •,• 1 which compute intermediate cohomology look like and the first differentials are given by ε 1 : E −1,i 1 → E 0,i 1 and q(ε 1 ) : E 0,i 1 → E 1,i 1 , respectively.In particular, there are only two possibly non-trivial differentials here: Since the spectral sequence is supported in three columns, the differential d 2 acts as , and using (3.6) we see that its source is nonzero only for i ∈ {1, n + 1} (note that ε 1 = 0), while its target is nonzero only in i ∈ {0, n + 1 − p, n}, hence d 2 = 0.The further differentials a fortiori vanish, so that E On the other hand, by (4.7) the image of the first map is kε p and the second map coincides with the map q(ε p ) defined in (4.11).Therefore, the totalization of E •,• 2 takes the form of the right-hand side of (4.12), where in the case q(ε p ) = 0 the class ε + comes from E 1,n−p 2 which survives exactly in this case.
As explained in Theorem 2.9 the construction of a hyperbolic extension might be ambiguous.In the situation described in Proposition 4.11 this happens precisely when the space Ext 1 ( 2 Ω p−1 , O(2t + m)) is non-zero.In the next lemma we determine when this happens.Lemma 4.12.Assume 2 ≤ 2p ≤ n + 1.Then the space Ext Taking the exterior square of (3.12) we see that 2 ( k T) is quasiisomorphic to the complex of split bundles of length 2k if k is odd and 2k − 1 if k is even.Since split bundles have no intermediate cohomology and since 2k ≤ n−1, the hypercohomology spectral sequence shows that H 1 (P n , 2 ( k T)⊗O(s)) = 0 unless k is odd and n = 2k + 1, and in the latter case we have where the morphism in the right side is induced by the tautological embedding O ֒→ V ⊗ O(1).Now it is easy to see that this space is zero unless s = −n − 1, in which case it is 1-dimensional.Now let C be the cokernel sheaf of a generically non-degenerate quadratic form (E, q).The next result shows that in the case where the construction of an elementary modification of Proposition 4.11 is ambiguous, i.e., Ext 1 ( 2 Ω p−1 , O(2t + m)) = 0, one can choose one such modification (E + , q + ), which has an additional nice property, namely, it has a prescribed image of H k * (E ∨ + ) in H k * (C).For our purposes it will be enough to consider the case where the bundle E is VLC, see Definition 3.7.
So, assume n = 2k + 1 and the bundle E in (4.1) is VLC.Note that E ∨ is VUC by Lemma 3.9.By Lemma 3.12 we have an exact sequence of graded S-modules E) be the extension class constructed from ε k+1 in Proposition 4.10 and consider the variety HE(E, q, ε 1 ) of all hyperbolic extensions of (E, q) with respect to ε 1 , i.e., the set of all elementary modifications of (E, q) with respect to ε k+1 .By (4.12) every (E + , q + ) ∈ HE(E, q, ε where LGr ε k+1 (H k * (C)) is the variety of all graded Lagrangian S-submodules A ⊂ H k * (C) satisfying (4.14).We will show that λ is an isomorphism onto the complement of the point E) denotes the extension class constructed from ε k+1 in Proposition 4.10 and let, as usual, q(ε 1 ) ∈ Ext 1 (E, Ω k+1 (t + m + n + 1)) be the class obtained from it by the application of q.Let (E + , q + ) be any hyperbolic extension of (E, q) with respect to ε 1 , so that (E, q) is the hyperbolic reduction of (E + , q + ) with respect to an embedding Ω k+1 (t + m + n + 1) ֒→ E + .Then we have the following commutative diagram (4.17) with the extension class of the bottom row being ε 1 and that of the left column being q(ε 1 ).Note that the cohomology exact sequence of the bottom row and the nontriviality of ε 1 imply that E ′ is VLC, hence E ′ ∨ is VUC.Similarly, the cohomology exact sequence of the left column implies that E ′′ is VLC.We will use these observations below.Consider the dual of the diagram (4.17) and the induced cohomology exact sequences: (the map ι is induced by the embedding Ω k+1 (2t + m + n + 1) → E ′′ (t) in the left column of (4.17)).Since E ′ ∨ is VUC, the upper arrow in the right column of (4.18) is surjective.From the commutativity of the diagram we conclude that the composition (of the right arrow in the middle row and the upper arrow in the middle column) is nontrivial, while middle arrow is an isomorphism by Serre duality.Therefore, the composition is injective, and since λ 2 (ε ′′ ) is determined by the image of ε ′′ under this composition, we conclude that λ 2 is injective.Finally, we note that HE(E, q, ε 1 ) comes with a transitive action of the group Therefore, to check the injectivity of λ 1 , it is enough to check the injectivity of the middle arrow in (4.19).
And for this, it is enough to check that the first arrow in (4.19) vanishes.To prove this vanishing consider the commutative square where the vertical arrows are induced by the extension class q(ε 1 ) of the left column of (4.17), and the horizontal arrows are induced by the morphisms in the dual of (3.12) with s = k.The space in the lower left corner is zero by (3.6) (recall that k ≥ 1), hence the compositions of arrows are zero.On the other hand, the argument of Proposition 4.10 shows that the top horizontal arrow is surjective.Therefore the right vertical arrow is zero, and as we explained above, this implies the injectivity of λ 1 , and hence of λ, and completes the proof of the proposition.
The elementary modification (E + , q + ) of (E, q) satisfying the properties of Proposition 4.13 for a given lift εk+1 of ε k+1 , will be referred to as the refined elementary modification associated with the class εk+1 .4.3.Modification theorem.Recall that a quadratic form (E, q) is called unimodular if the corresponding cokernel sheaf C vanishes, i.e., if q : E(−m) → E ∨ is an isomorphism.Recall the definitions (1.7) and (1.8) of standard unimodular quadratic forms.We will say that a standard unimodular quadratic form is anisotropic if W = W 0 and the form q W 0 is symmetric and anisotropic.
To prove the main result of this section we need the following simple observations.Recall the notion of linear minimality, see Definition 3.14 Lemma 4.14.Assume (E, q) is a generically non-degenerate quadratic form such that q : E(−m) → E ∨ is not linearly minimal.Then (E, q) is isomorphic to the orthogonal direct sum (E 0 , q 0 ) ⊕ (E 1 , q 1 ), where the second summand is a standard unimodular quadratic form (1.7) of rank 1 or 2.
Proof.Since q is not linearly minimal, it can be written as a direct sum of morphisms f : E ′ (−m) → E ′′ and id : O(t − m) → O(t − m) for some t ∈ Z; in particular E ∼ = E ′ ⊕ O(t), and the restriction of q to the summand O(t − m) of E(−m) is a split monomorphism.Consider the composition Let ϕ 0 : O(t − m) → E ′ ∨ and ϕ 1 : O(t − m) → O(−t) be its components.Since ϕ is a split monomorphism, there is a map ψ = (ψ 0 , ψ 1 ) : We consider the summand ψ 1 • ϕ 1 : O(t − m) → O(t − m).
First, assume ψ 1 • ϕ 1 = 0. Then it is an isomorphism, hence ϕ 1 is a split monomorphism, hence an isomorphism, hence t − m = −t and so m = 2t.Furthermore, it follows that the restriction of q to the subbundle E 1 = O(t) of E is unimodular.Taking E 0 = E ⊥ 1 to be the orthogonal of E 1 in E, we obtain the required direct sum decomposition.
Next, assume ψ 1 • ϕ 1 = 0. Then it follows that ψ 0 • ϕ 0 = 1, hence ϕ 0 is a split monomorphism.Therefore, we have E ′ ∼ = E 0 ⊕ O(m − t), so that E = E 0 ⊕ O(m − t) ⊕ O(t).Furthermore, it follows that the restriction of q to the subbundle E 1 = O(m − t) ⊕ O(t) of E is unimodular (the restriction to O(m − t) is zero and the pairing between O(m − t) and O(t) is a non-zero constant).Taking E 0 = E ⊥ 1 to be the orthogonal of E 1 in E, we obtain the required direct sum decomposition.Corollary 4.15.If (E, q) is a unimodular quadratic form and E is VLC then (E, q) is isomorphic to a standard unimodular quadratic form (1.7); in particular, E is split.
Proof.Since q is unimodular, we have E(−m) ∼ = E ∨ , so if E is VLC, and hence E ∨ is VUC, then E is both VLC and VUC, hence it is split by Lemma 3.9.Furthermore, q : E(−m) → E ∨ is an isomorphism of split bundles, hence it is not linearly minimal.Applying Lemma 4.14 we obtain a direct sum decomposition E = E 0 ⊕ E 1 , where E 1 is standard unimodular of type (1.7) and E 0 , being a direct summand of a unimodular VLC quadratic form, is itself unimodular and VLC.Iterating the argument, we conclude that E 0 is standard unimodular of type (1.7), hence so is E. Lemma 4.16.If (E, q) is a standard unimodular quadratic form of type (1.7) or (1.8), it is hyperbolic equivalent to one of the following • (W 0 , q W 0 ) ⊗ O(m/2), if m is even, or • (W 0 , q W 0 ) ⊗ Ω n/2 ((m + n + 1)/2), if m is odd and n is divisible by 4 (where in each case (W 0 , q W 0 ) is an anisotropic quadratic space), or to zero, otherwise.
Proof.By definition of standard unimodular quadratic forms for each i = 0 the summands W i ⊗O((m+i)/2)⊕W −i ⊗O((m−i)/2) or W i ⊗Ω n/2 ((m+m+1+i)/2)⊕W −i ⊗Ω n/2 ((m+n+1−i)/2) are hyperbolic equivalent to zero, hence any standard unimodular quadratic form is hyperbolic equivalent to the one with W = W 0 .It remains to note that by the standard Witt theory the bilinear form (W 0 , q W 0 ) is hyperbolic equivalent to an anisotropic form.Finally, in the case where m is odd and n ≡ 2 mod 4 the form q W 0 is skew-symmetric, so if it is anisotropic, it is just zero.Now we are ready to prove the main result of this section.Recall Definition 3.16.
Theorem 4.17.Any generically non-degenerate quadratic form q : E(−m) → E ∨ over P n is hyperbolic equivalent to an orthogonal direct sum (4.20) (E min , q min ) ⊕ (E uni , q uni ), where E min is a VLC bundle, (E min , q min ) has no unimodular direct summands, and (E uni , q uni ) is an anisotropic standard unimodular quadratic form which has type (1.7) if m is even, type (1.8) if m is odd and n ≡ 0 mod 4, and is zero otherwise.Moreover, if n = 2k + 1, C = C(q) is the cokernel sheaf of (E, q), and A k ⊂ H k * (C) is any shadowless subspace which is Lagrangian with respect to the bilinear form (4.3) then the quadratic form (E min , q min ) in (4.20) can be chosen in such a way that there is an equality H k * (E ∨ min ) = A k of S-submodules in H k * (C).Proof.We split the proof into a number of steps.
Step 1.First we show that q is hyperbolic equivalent to a quadratic form (E 1 , q 1 ) such that H i * (E 1 ) = 0 for each 1 ≤ i ≤ ⌊(n − 1)/2⌋ (if n is odd this is equivalent to the VLC property, and if n is even this is a bit weaker).For this we use induction on the parameter Note that ℓ 1 (E) < ∞ for any vector bundle E. Assume ℓ 1 (E) > 0. Let 1 ≤ p 0 ≤ ⌊(n − 1)/2⌋ be the minimal integer such that H p 0 * (E) = 0 and let t 0 be the maximal integer such that H p 0 (P n , E(t 0 )) = 0. Choose a non-zero element ε p 0 ∈ H p 0 (P n , E(t 0 )).
From now on we assume that ℓ 1 (E) = 0 and discuss separately the case of even and odd n.
Step 2. Assume that n = 2k.In this case ⌊(n − 1)/2⌋ = k − 1 < k = ⌊n/2⌋, hence by Step 1 the only non-trivial intermediate cohomology of E preventing it from being VLC is H k * (E) and it fits into the exact sequence 0 → H k−1 * (E ∨ ) → H k−1 of t.On the other hand, we have (the first equality follows from non-degeneracy of H k * (q ′ ) and the second from Serre duality), and as the right-hand side vanishes for −s − m − n − 1 > t, the left-hand side vanishes for s < −t − m − n − 1 = t.Now, replacing (E, q) by (E ′ , q ′ ), we may assume that H k * (q) is non-degenerate and H k (P n , E(t)) = 0 unless t = −(m + n + 1)/2.So, we set t := −(m + n + 1)/2 and let where ϕ L and ϕ U are isomorphisms.Moreover, such diagram is unique up to a homotopy represented by the dotted arrow.From now on we identify E 2 with E 1 by means of ϕ L , so we assume E 1 = E 2

4. 1 .
Reminder on symmetric sheaves.For a scheme Y and an object C ∈ D(Y ) we writeC ∨ := RHom(C, O Y )for the derived dual of C. Note that the cohomology sheaves H i (C ∨ ) of C ∨ are isomorphic to the local Ext-sheaves Ext i (C, O Y ).

Definition 4 . 1
(cf.[7, Definition 0.2]).We say that a coherent sheaf C on P n is (d, δ)-symmetric, if C ∼ = i * R, where i : D ֒→ P n is the embedding of a degree d hypersurface and R is a coherent sheaf on D endowed with a symmetric morphismR ⊗ R → O D (δ) such that the induced morphism R(−δ) → R ∨ (where the duality is applied on D) is an isomorphism.Note that d, δ, D, and R in Definition 4.1 are not determined by the sheaf C, see Remark 4.4.
To prove Theorem 1.3 we develop in §3 the theory of what we call VHC morphisms (here VHC stands for vanishing of half cohomology).These are morphisms of vector bundles E L → E U on P n such thatH p (P n , E L (t)) = 0 for 1 ≤ p ≤ ⌊n/2⌋ and all t ∈ Z, and H p (P n , E U (t)) = 0 for ⌈n/2⌉ ≤ p ≤ n − 1 and all t ∈ Z (we say then that E L is VLC as its lower intermediate cohomology vanishes, and E U is VUC as its upper intermediate cohomology vanishes).The main results of this section are Theorem 3.15, in which we prove the uniqueness (under appropriate assumptions) of VHC resolutions, and Corollary 3.18, proving the existence of VHC resolutions for any sheaf of projective dimension 1.
/ H 0 (L 0 ), one formed by H 0 (P n , L i ) and the other by H n (P n , L i ).The dashed arrows show the only non-trivial higher differentials d n+1 -these differentials are directed n steps down and n + 1 steps to the right.Therefore, if ℓ ≤ n there are no higher differentials, and if ℓ = n + 1 there is exactly one, which acts from H top (P / / . . ./ / . . ./ n , L • ) to H bot (P n , L • ), where we define (3.4) H top (P n , L • Example 3.8.Every split bundle is both VLC and VUC.Moreover,• every vector bundle on P 1 is both VLC and VUC since the conditions are void;• a vector bundle on P 2 is VLC if and only if it is VUC if and only if it is split;• if 1 ≤ p, q ≤ n − 1 and t ∈ Z we have p (t) is VLC if and only if p > ⌊n/2⌋ and it is VUC if and only if p < ⌈n/2⌉.Note that for even n the bundle Ω n/2 (t) is neither VLC nor VUC.Lemma 3.9.The properties VLC and VUC are invariant under twists, direct sums, and passing to direct summands.Moreover, a vector bundle E is VLC if and only if E ∨ is VUC.Finally, if a bundle E is VLC and VUC at the same time, it is split.