MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE

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1 Dedicated to the memory of Andrei Nikolaevich Tyurin MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE ALEXANDER S. TIKHOMIROV 1. Introduction By a mathematical n-instanton vector bundle (shortly, a n-instanton) on 3-dimensional projective space P 3 we understand a rank-2 algebraic vector bundle E on P 3 with Chern classes (1) c 1 (E) =, c 2 (E) = n, n 1, satisfying the vanishing conditions (2) h (E) = h 1 (E( 2)) =. Denote by I n the set of isomorphism classes of n-instantons. This space is nonempty for any n 1 - see, e.g., [BT], [NT]. The condition h (E) = for a n-instanton E implies that E is stable in the sense of Gieseker-Maruyama. Hence I n is a subset of the moduli scheme M P 3(2;, 2, ) of semistable rank-2 torsion-free sheaves on P 3 with Chern classes c 1 =, c 2 = n, c 3 =. The condition h 1 (E( 2)) = for [E] I n (called the instanton condition) by the semicontinuity implies that I n is a Zariski open subset of M P 3(2;, 2, ), i.e. I n is a quasiprojective scheme. It is called the moduli scheme of mathematical n-instantons. In this paper we study the problem of the irreducibility of the scheme I n. This problem has an affirmative solution for small values of n, up to n = 5. Namely, the cases n = 1, 3, 3, 4 and 5 were settled in papers [B1], [H], [ES], [B3] and [CTT], respectively. The aim of this paper is to prove the following result. Theorem 1.1. For each n = 2m + 1, m, the moduli scheme I n of mathematical n- instantons is reduced and irreducible of dimension 8n 3. A guide to the paper is as follows. In section 3 we remind a well-known relation between mathematical n-instantons and nets of quadrics in arithmetic n-dimensional vector space k n. The nets of quadrics are considered as vectors of the space S n = S 2 (k n ) 2 V, where V = H (O P 3(1)), and those nets which correspond to n-instantons (we call them n-instanton nets) satisfy the so-called Barth s conditions - see definition (13). Thus the description of the moduli space I n of n-instantons reduces to that of the locally closed subset MI n S n of n-instanton nets of quadrics which is crucial for our study. In section 4 we prove one result of general position for the set of (2m + 1)-instanton nets of quadrics MI 2m+1, m 1. Essentially, this result means that the natural map MI 2m+1 S m+1 induced by a generic embedding k m+1 k 2m+1 is dominating - see Remark 8.1. Section 5 is a study of some linear algebra related to a direct sum decomposition ξ : k m+1 k m k 2m+1 giving the above embedding k m+1 k 2m+1. Using the result of section 4 we obtain here the relation (61) which is a key instrument for our further considerations. Also, the decomposition ξ enables us to relate (2m + 1)-instantons E to rank-(2m + 2) symplectic vector bundles E 2m+2 on P 3 satisfying the vanishing conditions h (E 2m+2 ) = h 2 (E 2m+2 ( 2)) =. In section 6 we introduce a new scheme X m as a locally closed subset of the vector space S m+1 Hom(k m, (k m+1 ) 2 V which is defined by linear algebraic data somewhat similar to Barth s conditions. We prove that X m as a reduced scheme is isomorphic to a certain dense

2 2 TIKHOMIROV open subset MI 2m+1 (ξ) of MI 2m+1 determined by the choice of the direct sum decomposition ξ above. This reduces the problem of the irreducibility of I 2m+1 to that of X m. The last ingredient in the proof of Theorem 1.1 is a scheme Z m introduced in section 7 as a closed subscheme of the vector space S m Hom(k m, (k m ) ) 2 V defined by explicit equations. We relate the scheme Z m to the so-called t Hooft instantons. Using the properties of t Hooft instantons (see subsection 5.2) we show that the scheme Z m is reduced and irreducible. In the last section 8 we finish the proof of Theorem 1.1. The proof is based on a study of certain scheme X m containing X m and fibred over the vector space Hom(k, k m+1 ) 2 V. We show that the zero fibre of this projection is scheme-theoretically isomorphic to a direct product of Z m and a certain vector space. This together with the irreducibility of Z m and some other results stated earlier leads to the irreducibility of X m. Acknowledgement. The author acknowledges the support and hospitality of the Max Planck Institute for Mathematics in Bonn where this paper was started during the authors stay there in Winter Notation and conventions Our notations are mostly standard. The base field k is assumed to be algebraically closed of characteristic. We identify vector bundles with locally free sheaves. If F is a sheaf of O X - modules on an algebraic variety or scheme X, then nf denotes a direct sum of n copies of the sheaf F, H i (F) denotes the i th cohomology group of F, h i (F) := dim H i (F), and F denotes the dual to F sheaf, i.e. the sheaf F := Hom OX (F, O X ). If Z is a subscheme of X, by I Z,X we denote the ideal sheaf corresponding to a subscheme Z. If X = P r and t is an integer, then by F(t) we denote the sheaf F O P r(t). [F] will denote the isomorphism class of a sheaf F. For any morphism of O X -sheaves f : F F and any k-vector space U (respectively, for any homomorphism f : U U of k-vector spaces) we will denote, for short, by the same letter f the induced morphism of sheaves id f : U F U F (respectively, the induced morphism f id : U F U F). Everywhere in the paper V will denote a fixed vector space of dimension 4 over k and we set P 3 := P (V ). Also verywhere below we will reserve the letters u and v for denoting the two morphisms in the Euler exact sequence O P 3( 1) u V v O P 3 T P 3( 1). For any k-vector spaces U and W and any vector φ Hom(U, W 2 V ) Hom(U V, W V ) understood as a homomorphism φ : U V W V or, equivalently, as a homomorphism φ : U W 2 V, we will denote by φ the composition U O φ P 3 W 2 V ɛ O P 3 W Ω P 3(2), where ɛ is the induced morphism in the exact triple 2 Ω P 3(2) 2 v 2 V ɛ O P 3 Ω P 3(2) obtained by passing to the second wedge power in the dual Euler exact sequence. Also, shortening the notation, we will omit sometimes the subscript P 3 in the notation of sheaves on P 3, e.g., write O, Ω etc., instead of O P 3, Ω P 3 etc., respectively. Everywhere in the paper for m 1 we denote by S m the vector space S 2 (k m ) 2 V. Following W.Barth [B2], [B3] and A.Tyurin [T1], [T2] we call this space the space of nets of quadrics in the space k m. 3. Some generalities on instantons. Set MI n In this section we recall some well known facts about mathematical instanton bundles - see, e.g., [CTT].

3 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE3 For a given n-instanton E, the conditions (1), (2), Riemann-Roch and Serre duality imply (3) h 1 (E( 1)) = h 2 (E( 3)) = n, h 1 (E Ω 1 P 3) = h2 (E Ω 2 P3) = 2n + 2, h 1 (E) = h 2 (E( 4)) = 2n 2. Furthermore, the condition c 1 (E) = yields an isomorphism 2 E O P 3, hence a symplectic isomorphism j : E E. This symplectic structure j on E is unique up to a scalar, since E as a stable bundle is a simple bundle, i.e. Hom(E, E) = kid. Consider a triple (E, f, j) where E is an n-instanton, f is an isomorphism k n H 2 (E( 3)) and j : E E is a symplectic structure on E. We call two such triples (E, f, j) and (E f, j ) equivalent if there is an isomorphism g : E E such that g f = λf with λ {1, 1} and j = g j g, where g : H 2 (E( 3)) H 2 (E ( 3)) is the induced isomorphism. We denote by [E, f, j] the equivalence class of a triple (E, f, j). From this definition one easily deduces that the set F [E] of all equivalence classes [E, f, j] with given [E] is a homogeneous space of the group GL(k n )/{±id}. Each class [E, f, j] defines a point (4) A n = A n ([E, f, j]) S 2 (k n ) 2 V in the following way. Consider the exact sequences (5) Ω 1 P 3 i 1 V O P 3( 1) O P 3, Ω 2 P 3 2 V O P 3( 2) Ω 1 P, 3 4 V O P 3( 4) 3 V O P 3( 3) i 2 Ω 2 P, 3 induced by the Koszul complex of V O P 3( 1) ev O P 3. Twisting these sequences by E and passing to cohomoligy in view of (2) gives the diagram with exact rows (6) H 2 (E( 4)) 4 V H 2 (E( 3)) 3 V i 2 H 2 (E Ω 2 P ) 3 H 1 (E)) A H 1 (E( 1)) V H 1 (E Ω P 3) where A := i 1 1 i 2. The Euler exact sequence (5) yields the canonical isomorphism ω P 3 4 V O P 3( 4), and fixing an isomorphism τ : k 4 V induces the isomorphisms τ : V 3 V and ˆτ : ω P 3 O P 3( 4). Now the point A = A n in (4) is defined as the composition i 1 =, (7) A : k n V τ k n 3 V f H 2 (E( 3)) 3 V A H 1 (E( 1)) V j j H 1 (E ( 1)) V SD H 2 (E(1) ω P 3) V ˆτ H 2 (E( 3)) V f (k n ) V, where SD is the Serre duality isomorphism. One checks that A n is a skew symmetric map depending only on the class [E, f, j] and not depending on the choice of τ, and that this point A n 2 ((k n ) V ) lies in the direct summand S n = S 2 (k n ) 2 V of the canonical decomposition (8) 2 ((k n ) V ) = S 2 (k n ) 2 V 2 (k n ) S 2 V. Here S n is the space of nets of quadrics in k n. Following [B3], [T1] and [T2] we call A the n-instanton net of quadrics corresponding to the data [E, f, j].

4 4 TIKHOMIROV Denote W A := k n V/ ker A. Using the above chain of isomorphisms we can rewrite the diagram (6) as (9) ker A k n V c A W A ker A A c A = q A (k n ) V. W A Here dim W A = 2n + 2 and q A : W A WA is the induced skew-symmetric isomorphism. An important property of A = A n ([E, f, j]) is that the induced morphism of sheaves (1) a A : W A O P 3 c A (k n ) V O P 3 ev (k n ) O P 3(1) is an epimorphism such that the composition k n O P 3( 1) a A q W A O P 3 A W A O P 3 (k n ) O P 3(1) is zero, and E = ker(a A q A)/ Im a A. Thus A defines a monad a A (11) M A : k n O P 3( 1) a A W A O P 3 with the cohomology sheaf E, (12) E = E(A) := ker(a A q A )/ Im a A. a A q A (k n ) O P 3(1) Note that passing to cohomology in the monad M A twisted by O P 3( 3) and using (12) yields the isomorphism f : k n H 2 (E( 3)). Furthermore, the simplecticity of the form q A in the monad M A implies that there is a canonical isomorphism of M A with its dual which induces the symplectic isomorphism j : E E. Thus, the data [E, f, j] are recovered from the net A. This leads to the following description of the moduli space I n. Consider the set of n-instanton nets of quadrics (i) rk(a : k n V (k n ) V ) = 2n + 2, (ii) the morphism a A : W A O P 3 (kn ) O P 3(1) (13) MI n := A S n defined by A in (1) is surjective, (iii) h (E 2 (A)) =, where E 2 (A) := ker(a A q A)/ Im a A and q A : W A WA is a symplectic isomorphism defined by A in (9) The conditions (i)-(iii) here are called Barth s coditions. These conditions show that MI n is naturally supplied with a structure of a locally closed subscheme of the vector space S n. Moreover, the above description shows that there is defined a morphism π n : MI n I n : A [E(A)], and it is known that this morphism is a principal GL(k n )/{±id}-bundle in the étale topology - cf. [CTT]. Here by construction the fibre πn 1 ([E]) over an arbitrary point [E] I n coincides with the homogeneous space F [E] of the group GL(k n )/{±id} described above. Hence the irreducibility of (I n ) red is equivalent to the irreducibility of the scheme (MI n ) red. The definition (13) yields the following. Theorem 3.1. For each n 1, the space of n-instanton nets of quadrics MI n is a locally closed subscheme of the vector space S n given locally at any point A n MI n by ( ) 2n 2 (14) = 2n 2 5n equations obtained as the rank condition (i) in (13). Note that from (14) it follows that (15) dim [A] MI n dim S n (2n 2 5n + 3) = n 2 + 8n 3

5 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE5 at any point A n MI n. On the other hand, by deformation theory for any n-instanton E we have dim [E] I n 8n 3. This agrees with (15), since MI n I n is a principal GL(k n )/{±id}- bundle in the étale topology. Let S n = {[E] I n there exists a line l P 3 of maximal jump for E, i.e. such a line l that h (E( n) l ) }. It is known [S] that S n is a closed subset of I n of dimension 6n + 2. Thus, since dim [E] I n 8n 3 at any [E] I n, it follows that (16) I n := I n S n is an open subset of I n and (I n) red is dense open in (I n ) red ; respectively, (17) MI n := π 1 n (I n) is an open subset of MI n and we have a dense open embedding (18) (MI n) red dense open (MI n ) red. For technical reasons we will below restrict ourselves to MI n instead of MI n. 4. A result of general position for (2m + 1)-instanton nets Definition 4.1. Let U and U be two vector spaces of dimensions respectively m and n, where m n. Consider the projective space P (U U ). We say that a point x P (U U ) has rank r (and denote this as rk(x) = r), if (i) there exist unique subspaces U r (x) U and U r(x) U of dimensions dim U k (x) = dim U k (x) = r such that x P (U r(x) U r(x)), and (ii) there do not exist subspaces Ũ U and Ũ U of dimension dim Ũ = dim Ũ < r such that x P (Ũ Ũ ). It is well known that each point x P (U U ) has a uniquely defined rank 1 rk(x) n. Fix a positive integer m 3 and a (2m + 1)-instanton vector bundle E such that [E] I 2m+1 and denote H 2m+1 = H 2 (E( 3)) and H 4m = H 2 (E( 4)). The Euler Exact sequence induces the exact triple E Ω P 3 V E( 1) E which gives a natural multiplication map in the first cohomology: (19) H 2m+1 V mult H 4m H 2 (E Ω P 3). Passing to cohomology of the exact triple E Ω 2 P 3 2 V E( 2) E Ω P 3 and using standard equalities = h 2 (E( 2)), h 3 (E Ω 2 P 3 ) = h (E Ω P 3) h (E( 1) V ) = for the instanton bundle E, we obtain: H 2 (E Ω P 3) =. Hence (19) gives the exact triple (2) W 4m+4 H 2m+1 V mult H 4m where (21) W 4m+4 := H 1 (E Ω P 3). We now prove the following main result of this section. Theorem 4.2. Let m 3 and let E be a (2m + 1)-instanton, [E] I 2m+1. Consider the spaces H 2m+1 = H 2 (E( 3)) and W 4m+4 = H 1 (E Ω P 3) together with the injection W 4m+4 H 2m+1 V defined in (2). Then for a generic m-dimensional subspace V m of H 2m+1 one has W 4m+4 V m V = {}.

6 6 TIKHOMIROV Доказательство. According to Definition 4.1 in which we put U = H 2m+1, U = V, each point x P (H 2m+1 V ) has rank 1 rk(x) dim V = 4. Thus (22) P (W 4m+4) = 4 r=1 Z r, where Z r := {x P (W 4m+4) rk(x) = r}, 1 r 4, are locally closed subsets of P (W 4m+4). Consider the Grassmannian and its locally closed subsets G := G(m, H 2m+1) (23) Σ r = {V m G V m U r (x) for some point x Z r }, 1 r 4. The condition that Z r P (V m V ) means that there exists a point x P (U r ) Z r for some r-dimensional subspace U r V m. This together with (22) implies that {V m G P (V m V ) P (W 4m+4) } = 4 r=1 Σ r. Thus, to prove the Theorem, it is enough to show that (24) dim Σ r < dim G, 1 r 4. We are starting now the proof of (24) for r = 4, 3, 2, 1. (i) r = 4. Set Γ 4 := {(x, U) P (W4m+4) G(4, H2m+1) rk(x) = 4 and U = U 4 (x)} and let P (W4m+4) p 4 q 4 Γ 4 G(4, H 2m+1 ) be the projections. By construction, p 4 (Γ 4 )) = Z 4 and the morphism p 4 : Γ 4 Z 4 is an isomorphism. Hence dim q 4 (Γ 4 ) dim Γ 4 = dim Z 4 dim P (W 4m+4) = 4m + 3. By construction we have the graph of incidence Π 4 = {(U, V m ) q 4 (Γ 4 ) Σ 4 U V m } with surjective projections q 4 (Γ 4 ) pr 1 pr 2 Π 4 Σ4 and a fibre over an arbitrary point U q 4 (Γ 4 ). Hence pr 1 1 (U) = G(m 4, H 2m+1/U) dim Σ 4 dim Π 4 = dim q 4 (Γ 4 )+dim G(m 4, H 2m+1/U) 4m+3+(m 4)(m+1) = m(m+1) 1 = = dim G 1 < dim G, i.e. (24) is true for r = 4. (ii) r = 3. Consider a morphism f 3 : Z 3 P (V ) = P 3 : x V 3 (x), where the pair of spaces (U 3 (x), V 3 (x)), U 3 (x) H2m+1 and V 3 (x) V, is determined uniquely by the point x via the condition x P (U 3 (x) V 3 (x)), since rk(x) = 3 (see Definition 4.1). Now for a given subspace V 3 V set (25) Σ 3 (V 3 ) = {V m G V m U 3 (x) for some point x f 1 3 (V 3 )}. Comparing this with (23) for r = 3 yields (26) Σ 3 = V 3 V Σ 3(V 3 ). Hence, (27) dim Σ 3 dim Σ 3 (V 3 ) + 3.

7 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE7 We are going to obtain an estimate for the dimension of Σ 3 (V 3 ) for an arbitrary 3-dimensional subspace V 3 in V. This subspace defines a commutative diagram (28) F Ω P 3 V 3 O P 3( 1) V O P 3( 1) I z O P 3 I x ( 1) O P 3( 1) k z, where z = P (ker : V V3 ) is a point in P 3 and the sheaf F has an O P 3-resolution O P 3( 2) 3O P 3( 1) F. Twisting this resolution by the vector bundle E and passing to cohomology we obtain the equalities H 1 (F E) H 2 (E( 3)) = H 2m+1, H 2 (F E) =. Respectively, passing to cohomology in diagram (28) twisted by E and using the above equalities and evident relations H (E k z ) k 2, H 1 (E k z ) = implies the diagram (29) k 2 H 2m+1 W 4m+4 H2m+1 V λ 3 H2m+1 V k 2 H 1 (E I z ) H 4m mult H 1 (E I z ( 1)) H 2m+1.

8 8 TIKHOMIROV In this diagram the composition ɛ := mult λ is surjective. Hence, setting W 2m+3 (V 3 ) := ker ɛ, where dim W 2m+3 (V 3 ) = 2m + 3, we obtain a commutative diagram (3) W 2m+3 (V 3 ) j W 4m+4 H 2m+1 H2m+1 V λ 3 H2m+1 V H 2m+1 ɛ H 4m H 4m mult Set. Z 3 (V 3 ) := {x P (W 2m+3 (V 3 )) rk(x) = 3}. The inclusion j in diagram (3) yields the bijection (31) Z 3 (V 3 ) f 1 3 (V 3 ). Consider the graph of incidence Γ 3 (V 3 ) := {(x, U) Z 3 (V 3 ) G(3, H2m+1) U = U 3 (x)} with projections Z 3 (V 3 ) p 3 Γ 3 (V 3 ) q 3 G(3, H2m+1). By construction, p 3 (Γ 3 (V 3 )) = Z 3 (V 3 ) and the morphism p 4 : Γ 3 (V 3 ) Z 3 (V 3 ) is an isomorphism. Hence (32) dim q 3 (Γ 3 (V 3 )) dim Γ 3 (V 3 ) = dim Z 3 (V 3 ) dim P (W 2m+3 (V 3 )) = 2m + 2. Consider the graph of incidence Π 3 (V 3 ) = {(U, V m ) q 3 (Γ 3 (V 3 )) Σ 3 (V 3 ) U V m } with projections q 3 (Γ 3 (V 3 )) pr 1 Π 3 (V 3 ) pr 2 Σ 3 (V 3 ) and a fibre pr 1 1 (U) = G(m 3, H 2m+1/U) over an arbitrary point U q 3 (Γ 3 (V 3 )). The projection Π 3 (V 3 ) pr 2 Σ 3 (V 3 ) is surjective in view of (31). Hence, using (32), we obtain dim Σ 3 (V 3 ) dim Π 3 (V 3 ) = dim q 3 (Γ 3 (V 3 ))+dim G(m 3, H 2m+1/U) 2m+2+(m 3)(m+1) = = m 2 1. This together with (27) and the assumption m 3 yields dim Σ 3 m = dim G + 2 m < dim G, i.e. (24) holds for r = 3. Before proceeding to the case r = 2 we need to make a small digression on jumping lines of E. Introduce some more notation. For a given line l P 3 we have E l O P 1(d) O P 1( d) for a welldefined nonnegative integer d called the jump of E l and is denoted d E (l); respectively, the line l is called a jumping line of jump d of E. Set G 2,4 := G(2, V ) and J k (E) := {l G 2,4 d E (l) k}, J k (E) := J k(e) J k+1 (E), k. From the semicontinuity of E l, l G 2,4, it follows that J k (E) (resp., J k (E)) is a closed (resp., locally closed) subset of G 2,4, k. Moreover, by Theorem of Grauert-Mülich, J (E) is a dense open subset of G 2,4. Next, since E I 2m+1, it follows that J 2m+1 (E) =, so that J 2m 1 (E) = J 2m 1(E) J 2m(E). We will use below the following lemma. Lemma 4.3. (1) dim J 2m 1 (E) 1. (ii) dim Jk (E) 3 for 1 k 2m 2.

9 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE9 Proof of Lemma. (1) Suppose the contrary, i.e. dim J 2m (E) 2. Take any irreducible surface S J 2m (E) and let D be the degree of S with respect to the sheaf O G2,4 (1). Fix an integer r 5 and take any irreducible curve C belonging to the linear series OG2,4 (r) S. Then the degree deg C w.r.t. O G2,4 (1) equals to Dr, hence deg C 5. Hence by [C, Lemma 6] there exist two distinct lines, say, l 1, l 2 C, which intersect in P 3. Let the plane P 2 be the span of l 1 and l 2 in P 3. Now the exact triple E( 2) P 2 E P 2 E l1 l 2 implies (33) H (E P 2) H (E l1 l 2 ) H 1 (E( 2) P 2). Next, as [E] I 2m+1, we have h (E( 1)) = h 1 (E( 2)) =, hence the exact triple E( 2) E( 1) E( 1) P 2 implies (34) H (E( 1) P 2) =. Now assume h (E P 2) >. Then a section s H s (E P 2) defines an injection O P 2 E P 2. This injection and (34) show that the zero-set Z of section s is -dimensional and the injection s s extends to a triple O P 2 E P 2 I Z,P 2. Whence (35) h (E P 2) 1. Furthermore, equality together with Riemann-Roch and Serre duality for the vector bundle E( 1) P 2 shows that h 1 (E( 2) P 2) = 2m + 1. Whence in view of (33) and (34) we obtain (36) h (E l1 l 2 ) 2m + 2. On the other hand, let x := l 1 l 2. Since by construction l 1, l 2 J 2m 1 (E), it follows that either E li O P 2(2m 1) O P 2(1 2m), or E li O P 2(2m) O P 2( 2m), hence h (E I x,li ) 2m 1, i = 1, 2. This clearly implies h (E l1 l 2 ) h (E I x,l1 l 2 ) h (E I x,l1 )+h (E I x,l2 ) = 4m 2. Comparing this with (36) we obtain the inequality 2m + 2 4m 2, i.e. m 2. This contradicts to the assumption m 3. Hence, the assertion (1) follows. (2) This is an immediate corollary of Theorem of Grauert-Mülich. Lemma is proved. (iii) r = 2. Our notation and argument is completely parallel to that in the case r = 3. Consider a morphism f 2 : Z 2 G 2,4 : x V 2 (x), where the pair of spaces (U 2 (x), V 2 (x)), U 2 (x) H2m+1 and V 2 (x) V, is determined uniquely by the point x via the condition x P (U 2 (x) V 2 (x)), since rk(x) = 2 (see Definition 4.1). According to the above remarks on jumping lines of E we may assume that l Jk (E) for some k 2m, i.e. respectively, h (E l) = 2, h 1 (E l) =, if l J (E), h (E l) = k + 1, h 1 (E l) = k 1, if l J k(e), 1 k 2m. Now for 1 k 2m and a given subspace V 2 J k set (37) Σ 2,k (V 2 ) = {V m G V m U 2 (x) for some point x f 1 2 (V 2 )}. Then similarly to (26) we have Hence, in view of Lemma 4.3 Σ 2 = 2m Σ 2,k (V 2 ). k= V 2 Jk (38) dim Σ 2 max V 2 J k (dim Σ 2,k (V 2 ) + dim Jk). k 2m

10 1 TIKHOMIROV We are going to obtain an estimate for the dimension of Σ 2,k (V 2 ) for an arbitrary 2-dimensional subspace V 2 in Jk, k 2m. This subspace defines a commutative diagram (39) O P 3( 2) s Ω P 3 F V 2 O P 3( 1) V O P 3( 1) V 2 O P 3( 1) I l O P 3 O l, where l = P (ker V V 2 ) is a line in P 3, V 2 := V /V 2, and F := coker s. Passing to cohomology in diagram (39) twisted by E, we obtain the diagram (4) W 4m+4 H (E l) H 1 (E F ) H 2m+1 V 2 H 2m+1 V H 2m+1 V 2 H (E l) H 1 (E I l ) H 4m mult H 1 (E l) Assume for definiteness that 1 k 2m. (The case k = is treated in a similar way.) In this case diagram (4) leads to a diagram. (41) W k+1 (V 2 ) j W 4m+4 H 4m k+3 H2m+1 V 2 H2m+1 V V 4m k+1 H 4m mult H 2m+1 V 2 W k 1. where we set W k+1 (V 2 ) := H (E l), W k 1 := H 1 (E l), V 4m k+1 := H 2m+1 V 2 /W k+1 (V 2 ).

11 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE11 Set Z 2,k (V 2 ) := {x P (W k+1 (V 2 )) rk(x) = 2}. The inclusion j in diagram (41) yields the bijection (42) Z 2,k (V 2 ) f 1 2 (V 2 ). Consider the graph of incidence Γ 2,k (V 2 ) := {(x, U) Z 2,k (V 2 ) G(2, H2m+1) U = U 2 (x)} with projections Z 2,k (V 2 ) p 2 Γ 2,k (V 2 ) q 2 G(2, H2m+1). By construction, p 2 (Γ 2,k (V 2 )) = Z 2,k (V 2 ) and the morphism p 4 : Γ 2,k (V 2 ) Z 2,k (V 2 ) is an isomorphism. Hence (43) dim q 2 (Γ 2,k (V 2 )) dim Γ 2,k (V 2 ) = dim Z 2,k (V 2 ) dim P (W k+1 (V 2 )) = k. Consider the graph of incidence Π 2,k (V 2 ) = {(U, V m ) q 2 (Γ 2,k (V 2 )) Σ 2,k (V 2 ) U V m } with projections q 2 (Γ 2,k (V 2 )) pr 1 Π 2,k (V 2 ) pr 2 Σ 2,k (V 2 ) and a fibre pr 1 1 (U) = G(m 2, H 2m+1/U) over an arbitrary point U q 2 (Γ 2,k (V 2 )). The projection Π 2,k (V 2 ) pr 2 Σ 2,k (V 2 ) is surjective in view of (42). Hence using (43) we obtain dim Σ 2,k (V 2 ) dim Π 2,k (V 2 ) = dim q 2 (Γ 2,k (V 2 ))+dim G(m 2, H 2m+1/U) k+(m 2)(m+1) = = m 2 m 2 + k = dim G (2m k + 2), 1 k 2m. In a similar way we obtain for k = dim Σ 2, (V 2 ) 1 + (m 2)(m + 1) = m 2 m 1 = dim G (2m + 1). The last two inequalities together with (38), Lemma 4.3 and the assumption m 3 yield dim Σ 2 < dim G, i.e. (24) is true for r = 2. (ii) r = 1. Consider a morphism f 1 : Z 1 P (V ) = (P 3 ) : x V 1 (x), where the pair of spaces (U 1 (x), V 1 (x)), U 1 (x) H2m+1 and V 1 (x) V, is determined uniquely by the point x via the condition x P (U 1 (x) V 1 (x)), since rk(x) = 1 (see Definition 4.1). Now for a given subspace V 1 (P 3 ) set Then similar to (26) we have Σ 1 (V 1 ) := {V m G V m U 1 (x) for some point x f 1 1 (V 1 )}. (44) Σ 1 = V 1 (P 3 ) Σ 1(V 1 ). Hence, (45) dim Σ 1 dim Σ 1 (V 1 ) + 3.

12 12 TIKHOMIROV We are going to obtain an estimate for the dimension of Σ 1 (V 1 ) for an arbitrary 1-dimensional subspace V 1 in V. This subspace defines a commutative diagram (46) Ω P 3 Ω P 3 V 1 O P 3( 1) V O P 3( 1) V 3 O P 3( 1) O P 3( 1) O P 3 O P 2. Note that to the point V 1 (P 3 ) there clearly corresponds a projective plane P (V 1 ) in P 3. Set B(E) := {V 1 (P 3 ) h (E P (V1 )) }. It is known that, for m 1, dim B(E) 2. (see [B1]). Moreover, in view of (35) h (E P (V1 )) = 1, V 1 B(E). Passing to cohomology in diagram (46) twisted by E and using the equality h (E) = for [E] I 2m+1 we obtain the diagram (47) W 4m+4 H (E P (V1 )) W 4m+4 H2m+1 V λ 1 H2m+1 V H2m+1 V 3 H (E P (V1 )) H 2m+1 H 4m mult H 1 (E P (V1 )).

13 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE13 Let V 1 B(E). Setting ɛ := mult λ and W 1 (V 1 ) := ker ɛ = H (E P (V1 )), where dim W 1 (V 1 ) = 1, we obtain from (47) a commutative diagram (48) W 1 (V 1 ) j W 4m+4 W 4m+4/W 1 (V 1 ) H2m+1 λ V 1 H2m+1 V ɛ H 2m+1/W 1 (V 1 ) H 4m mult H 2m+1 V 3 H 1 (E P 2 (V 1 )). Set Z 1 (V 1 ) := if V 1 B(E), resp., Z 1 (V 1 ) := j(w 1 (V 1 )) if V 1 B(E). The diagrams (47) and (48) yield the bijection (49) Z 1 (V 1 ) f 1 1 (V 1 ), V 1 (P 3 ). The rest argument is completely the same as in cases r = 3 and r = 2 above. Consider the graph of incidence Γ 1 (V 1 ) := {(x, U) Z 1 (V 1 ) P (H2m+1) U = U 1 (x)} with projections Z 1 (V 1 ) p 1 Γ 1 (V 1 ) q 1 P (H2m+1). By construction, p 1 (Γ 1 (V 1 )) = Z 1 (V 1 ) and the morphism p 4 : Γ 1 (V 1 ) Z 1 (V 1 ) is an isomorphism. Hence (5) dim q 1 (Γ 1 (V 1 )) dim Γ 1 (V 1 ) = dim Z 1 (V 1 ). Consider the graph of incidence Π 1 (V 1 ) = {(U, V m ) q 1 (Γ 1 (V 1 )) Σ 1 (V 1 ) U V m } with projections q 1 (Γ 1 (V 1 )) pr 1 Π 1 (V 1 ) pr 2 Σ 1 (V 1 ) and a fibre pr 1 1 (U) = G(m 1, H 2m+1/U) over an arbitrary point U q 1 (Γ 1 (V 1 )). The projection Π 1 (V 1 ) pr 2 Σ 1 (V 1 ) is surjective in view of (49). Hence in view of (5) we have dim Σ 1 (V 1 ) dim Π 1 (V 1 ) = dim q 1 (Γ 1 (V 1 )) + dim G(m 1, H 2m+1/U) + (m 1)(m + 1) = = m 2 1. This together with (45) and the assumption m 3 yields dim Σ m = dim G + 2 m < dim G, i.e. (24) holds for r = 1. Theorem is proved. 5. Decomposition k 2m+1 k m+1 k m and related constructions 5.1. Decomposition k 2m+1 k m+1 k m. Fix an isomorphism (51) ξ : k m+1 k m k 2m+1 and let (52) k m+1 i m+1 k m+1 k m im k m

14 14 TIKHOMIROV be the injections of direct summands. For a given (2m + 1)-instanton vector bundle E, [E] I 2m+1, fix an isomorphism f : k 2m+1 H 2 (E( 3)) = H 2m+1 and a symplectic structure j : E E. The data [E, f, j] define a net of quadrics A MI 2m+1 (see section 3), and the exact triple (2) is naturally identified with the dual to the triple ker A k 2m+1 V W A and fits in diagram (9) for n = 2m + 1 (53) ker A k 2m+1 V c A W A Consider the composition ker A (54) j ξ,a : k m+1 V i m+1 k m+1 V k m V A c A = q A (k 2m+1 ) V. W A ξ k 2m+1 V c A W A. Under these notations Theorem 4.2 can be reformulated in the following way: (*) Assume m 3 and let A be an arbitrary (2m + 1)-net from MI 2m+1. Then for a generic isomorphism ξ : k 2m+1 k m+1 k m one has (55) ker A ξ i m+1 (k m+1 V ) = {}. Equivalently, j ξ,a : k m+1 V W A is an isomorphism. Consider the direct sum decomposition corresponding to the isomorphism (51) (56) ξ : Sm+1 (k m ) (k m+1 ) 2 V S m S 2m+1 and let (57) ξ 1 : S 2m+1 S m+1, ξ 2 : S 2m+1 (k m ) (k m+1 ) 2 V, ξ 3 : S 2m+1 S m be projections onto summands. By definition, ξ 1 (A) considered as a skew-symmetric homomorphism k m+1 V (k m+1 ) V coincides with the composition (58) ξ 1 (A) : k m+1 V j ξ,a q W A A W A j ξ,a (k m+1 ) V. This means that assertion (*) can be reformulated as: (**) Assume m 3 and let A be an arbitrary (2m + 1)-net from MI 2m+1. Then for a generic isomorphism ξ in (51) the skew-symmetric homomorphism ξ 1 (A) : k m+1 V (k m+1 ) V is invertible. For A and ξ from (**) we have the commutative diagram (59) k m+1 V i m+1 k m+1 V k m V ξ j ξ,a k 2m+1 V c A W A ξ 1 (A) q A ξ(a) A (k m+1 ) V i m+1 (k m+1 ) V (k m ) V j ξ,a (k 2m+1 ) V c WA A, ξ

15 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE15 ). As j ξ,a in this diagram is invertible, the compo- ( ξ1 (A) ξ where ξ(a) is the matrix 2 (A) ξ 2 (A) ξ 3 (A) sition g ξ,a = j 1 ξ,a c A ξ i m is well-defined, and we obtain a commutative diagram (6) k m V (k m ) V g ξ ξ 2 (A) g ξ ξ 2 (A) k m+1 ξ 1 (A) V (k m+1 ) V. In particular, ξ 3 (A) (61) ξ 3 (A) = ξ 2 (A) ξ 1 (A) 1 ξ 2 (A). For m 1 let Isom 2m+1 be the set of all isomorphisms ξ in (51). Consider the open subset MI 2m+1 of MI 2m+1 defined in (17) and set (62) MI 2m+1 (ξ) := {A MI 2m+1 the skew symmetric homomorphism ξ 1 (A) in (58) is invertible}, ξ Isom 2m+1. The relation (61) together with (**) implies the following corollary of Theorem 4.2. Theorem 5.1. Fom m 3 the following statements hold. (i) The sets MI 2m+1 (ξ), ξ Isom 2m+1, are dense open subsets of the set MI 2m+1 constituting its open cover. (ii) For any ξ Isom 2m+1 and any A MI 2m+1 (ξ) the relation (61) is true. We will need below the following lemma. Lemma 5.2. Let ξ and A MI 2m+1 (ξ) be as in Theorem 5.1 and set (63) B := ξ 1 (A), C := ξ 2 (A). Then the following statements hold. (i) Consider a subbundle morphism (64) α ξ,a := j 1 ξ a A ξ : (k m+1 k m ) O P 3( 1) k m+1 V O P 3. Then there exists an epimorphism (65) λ ξ,a : coker(b α ξ,a ) (k m+1 ) O P 3(1). making commutative the diagram (66) (k m+1 ) V O P 3 coker(b α ξ,a ) can u λ ξ,a (k m+1 ) O P 3(1), where can is a canonical surjection.

16 16 TIKHOMIROV (ii) Consider the commutative diagram (67) k m O P 3( 1) (k m+1 k m ) O P 3( 1) B αξ,a (k m+1 ) V O P 3 can coker(b α ξ,a ) i m+1 k m+1 O P 3( 1) B u (k m+1 ) V O P 3 ɛ ξ,a v B 1 k m+1 T P 3( 1) τ ξ,a k m O P 3( 1), where τ ξ,a and ɛ ξ,a are the induced morphisms. Then the morphism τ ξ,a is a subbundle morphism fitting in a commutative diagram (68) (k m+1 ) V O P 3 v B 1 k m+1 T P 3( 1) C u k m O P 3( 1) Доказательство. (i) Consider the commutative diagram (69) k 2m+1 O( 1) ξ a A W A O j ξ,a q A τ ξ,a k m O P 3( 1). WA O a A (k 2m+1 ) O(1) (k m+1 k m ) O( 1) α ξ,a (k m+1 ) V O B (k m+1 ) V O α ξ,a (k m+1 k m ) O(1) i i m+1 u m+1 u k m+1 O( 1) (k m+1 ) O(1) j ξ,a ξ Here the upper triple is the monad (11) for n = 2m + 1. Whence the statement (i) follows. (ii) Standard diagram chasing using (63) and diagrams (59) and (67) Remarks on t Hooft instantons. Consider the set I th 2m+1 := {[E] I 2m+1 h (E(1)) }, of t Hooft instanton bundles and the corresponding set of t Hooft instanton nets MI th 2m+1 := πn 1 (I th 2m+1). We collect some well-known facts about I th 2m+1 in the following lemma - see [BT], [NT], [T2, Prop. 2.2]. Lemma 5.3. Let m 1. Then the following statements hold. (i) I2m+1 th is an irreducible (1m + 9)-dimensional subvariety of I 2m+1. Respectively, MI2m+1 th is an irreducible (4m m + 1)-dimensional subvariety of I 2m+1. (ii) I2m+1 th := I2m+1 th I 2m+1 is a smooth dense open subset of I2m+1 th and (7) h (E(1)) = 1, [E] I th 2m+1.

17 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE17 (iii) MI th 2m+1 is a smooth dense open subset of the set (71) T H 2m+1 := {A S 2m+1 A = 2m+2 i=1 h 2 w, where h (k 2m+1 ), w 2 V, w w = }. We are going to extend the statement of Theorem 5.1 to the cases m = 1 and 2. To this end, for m = 1, 2 and ξ Isom 2m+1 consider the sets MI 2m+1 (ξ) defined in (62) and set (72) MI 2m+1 := ξ Isom 2m+1 MI 2m+1 (ξ), m = 1, 2. For m 1 let ξ Isom 2m+1 be the standard isomorphism k m+1 k m k m+1 : ((a 1,..., a m+1 ), (a m+2,..., a 2m+1 )) (a 1,..., a 2m+1 ). Let {h 1 = (1,,..., ),..., h 2m+1 (,...,, 1) be the standard basis in (k 2m+1 ) and let e 1,..., e 4 be some fixed basis in V. Consider the nets A (m) T H 2m+1, m = 1, 2, defined as follows (73) A (1) = h 2 1 (e 1 e 2 + e 3 e 4 ) + h 2 2 (e 1 e 3 + e 4 e 2 ), A (2) = h 2 1 (e 1 e 2 + e 3 e 4 ) + h 2 2 (e 1 e 3 + e 4 e 2 ) + h 2 3 (e 1 e 4 + e 2 e 3 ). It is an exercise to show that the homomorphisms ξ 1(A (m) ) : k m+1 V (k m+1 ) V, m = 1, 2, are invertible. On the other hand, for a given ξ Isom 2m+1 the condition that a homomorphism ξ 1 (A) : k m+1 V (k m+1 ) V is invertible is an open condition on the net A T H 2m+1. Hence, since the sets MI 2m+1, m = 1, 2, are irreducible, Lemma 5.3 yields the following corollary. Corollary 5.4. Let 1 m 2. (i) For m = 1, 2 the set MI 2m+1 is a dense open subset of MI 2m+1 and of MI 2m+1, and the statement of Theorem 5.1 extends to the cases m = 1 and 2, with MI 2m+1 being substituted by MI 2m+1. (ii) Let m 1. The set { MI MI2m+1 th th := 2m+1, m 3, MI 2m+1 MI2m+1, th m = 1, 2, is a dense open subset of MI th 2m+1 and of MI th 2m+1 covered by dense open subsets (74) MI th 2m+1(ξ) := MI th 2m+1 MI 2m+1 (ξ), ξ Isom 2m+1. Note that (18), Theorem 5.1 and Corollary 5.4 yield Corollary 5.5. Let m 1. Then for any ξ Isom 2m+1 the scheme (MI 2m+1 (ξ)) red is dense open in (MI 2m+1 ) red. In particular, (75) dim MI 2m+1 (ξ) = dim MI 2m Invertible nets of quadrics from S m+1 and symplectic rank-(2m + 2) bundles. Introduce more notations. Set (76) N m+1 := {B S m+1 B : k m+1 V (k m+1 ) V is an invertible homomorphism}. The set N m+1 is a dense open subset of the vector space S m+1, and it is easy to see that for any B N m+1 the following conditions are satisfied. (1) The morphism B : k m+1 O P 3( 1) (k m+1 ) Ω P 3(1) induced by the homomorphism B : k m+1 V (k m+1 ) V is a subbundle morphism, i.e. (77) E 2m+2 (B) := coker( B)

18 18 TIKHOMIROV is a vector bundle of rank 2m + 2 на P 3. This follows from the diagram (78) k m+1 O P 3( 1) B (k m+1 ) Ω P 3(1) e E 2m+2 (B) u k m+1 V O P 3 B v (k m+1 ) V O P 3 E 2m+2 (B) k m+1 T P 3( 1) v B u (k m+1 ) O P 3(1) (2) The homomorphism B : k m+1 (k m+1 ) 2 V induced by B : k m+1 V (k m+1 ) V is injective. This follows from the commutative diagram extending the upper horizontal triple in (78) (79) (k m+1 ) T P 3( 2) (k m+1 ) T P 3( 2) k m+1 O P 3 B (k m+1 ) 2 V O P 3 can H (E 2m+2 (B)(1)) O P 3 k m+1 O P 3 w B (k m+1 ) Ω P 3(2) e ev E 2m+2 (B)(1), where w is the morphism induced by the morphism v from the Euler exact sequence in (78). From this diagram we obtain the isomorphism (8) coker( B) H (E 2m+2 (B)(1)). (3) Diagram (78) and the Five-Lemma yield an isomorphism (81) θ : E 2m+2 (B) E 2m+2 (B) which is in fact symplectic, θ = θ,

19 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE19 since the homomorphism B : k m V (k m ) V is skew-symmetric. The isomorphism θ together with the upper triple from (78) and its dual fits in the commutative diagram (82) k m+1 O P 3( 1) B (k m+1 ) Ω P 3(1) e E 2m+2 (B) k m+1 O P 3( 1) B u (k m+1 ) V O P 3 v v B 1 e θ k m+1 T P 3( 1) u (k m+1 ) O P 3(1) B (k m+1 ) O P 3(1) Note that this diagram immediately implies that. (83) h (E 2m+2 (B)) = h i (E 2m+2 (B)( 2)) =, i. Let ξ and A MI 2m+1 (ξ) be as in Theorem 5.1 for m 3, respectively, in Corollary 5.4 for m = 1, 2. Then the homomorphism B : k m+1 V (k m+1 ) V defined in (63) by definition lies in N m+1. Hence by Lemma 5.2 diagrams (66) and (66) hold. These diagrams together with (82) imply B τ ξ,a =, so that there exists a morphism (84) ρ ξ,a : k m O( 1) E 2m+2 (B) such that τ ξ,a = e θ ρ ξ,a. Since τ ξ,a is a subbundle morphism, ρ ξ,a is also a subbundle morphism. Moreover, diagrams (68) and (82) yield the commutative diagram (85) (k m+1 ) Ω P 3(1) E 2m+2 (B) C ρ ξ,a v k m O( 1) e θ C τ ξ,a (k m+1 ) V v B O 1 k m+1 T P 3( 1). Diagrams (82) and (85) yield the commutative diagram e (86) k m O( 1) (k C ρ ξ,a m+1 ) V O v E 2m+2 (B) Ω P 3(1) D C θ C e e θ e E 2m+2 (B) e T P 3( 1) ρ ξ,a v C (k m ) O(1) k m+1 V O, C B 1

20 2 TIKHOMIROV where D C := C B 1 C = u (C B 1 C) u is the zero map. In fact, by (61) and (63) we have D C = p 2 (ξ 3 (A)), where p 2 : 2 ((k n ) V ) 2 (k n ) S 2 V is the projection onto the second direct summand of the decomposition (8). Since by (57) ξ 3 (A) lies in the first direct summand of (8) it follows that D C =. We thus obtain the monad (87) k m O( 1) ρ ξ,a E 2m+2 (B) θ ρ ξ,a (k m ) O(1) with the cohomology sheaf (88) E 2 (ξ, A) := ker(θ ρ ξ,a)/ Im ρ ξ,a which is a vector bundle since ρ ξ,a is a subbundle morphism. Furthermore, by (83) it follows from the monad (87) that E 2 (ξ, A) is a (2m + 1)-instanton, (89) [E 2 (ξ, A)] I 2m+1. Lemma 5.6. E 2 (ξ, A) E(A), where the sheaf E(A) is defined in (12). Доказательство. Diagram chasing using (59), (6), (67)-(69), (78)-(79) and (82). 6. Scheme X m. An isomorphism between X m and an open subset of the space MI 2m Space X m. Consider the vector space S m+1, respectively, its dual space S m+1 and set (9) (S m+1) := {B S m+1 D : (k m+1 ) V k m+1 V is an invertible homomorphism}, (91) Σ m+1 := Hom(k m, (k m+1 ) 2 V ) According to our convention on notations we will understand an arbitrary point C Σ m+1 either as a homomorphism C : k m V (k m+1 ) V, or as a homomorphism C : k m (k m+1 ) 2 V, or as an induced morphism C : k m O( 1) (k m+1 ) Ω(1). Note also that the set (S m+1) is a dense open subset of the vector space S m+1. Consider the set (92) (i) (C D C : k m V (k m ) V ) S m, (ii) the map (k m+1 k m ) O (D 1,C) u (k m+1 ) V O(1) is a subbundle morphism, X m := (D, C) (S m+1) Σ m+1 (iii) the composition Ĉ : km C (k m+1 ) 2 V can (k m+1 ) 2 V / Im( D 1 ) H (E 2m+2 (D 1 )(1)) yields a subbundle morphism k m O P 3( 1) ρ D,C E 2m+2 (D 1 ), i.e. ρ D,C is surjective and E 2(D, C) := Ker( t ρ D,C )/ Im(ρ D,C ) is locally free.

21 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE21 By definition X m is a locally closed subset of (S m+1) Σ m+1. Hence it is naturally supplied with the structure of a reduced scheme. Note that in the condition (iii) of the definition of X m we set t ρ D,C := θ ρ D,C, where θ : E 2m+2 (D 1 ) E 2m+2(D 1 ) is a natural symplectic structure on E 2m+2 (D 1 ) defined in (81). Theorem 6.1. Let m 1 and let ξ be as in Theorem 5.1 and Corollary 5.4. (i) There is an isomorphism of reduced schemes (93) f m : (MI 2m+1 (ξ)) red X m : A (ξ 1 (A) 1, ξ 2 (A)). (ii) The inverse isomorphism is given by the formula (94) g m : X m (MI 2m+1 (ξ)) red : (D, C) ξ(d 1, C, C D C). 1 Доказательство. (i) We first show that the image of the map f m : (MI 2m+1 (ξ)) red (S m+1) Σ in m,m+1 lies in X m, i.e. satisfies the conditions (i)-(iii) in the definition of X m. Indeed, the condition (i) is automatically satisfied, since (57) and (61) give C D C = ξ 2 (A) ξ 1 (A) 1 ξ 2 (A) = ξ 3 (A) S 2 (k m ) 2 V. Next, the morphism ρ D,C defined in (iii) above coincides by its definition with the morphism ρ ξ,a defined in (84). In fact, the upper triangle of the diagram (85) twisted by O(1) and the lower part of the diagram (79) in which we put (95) B = D 1 (note that D is invertible) fit in the diagram (96) k m+1 O D 1 (k m+1 ) 2 V O C k m+1 O D 1 (k m+1 ) Ω(2) w C can k m O H (E 2m+2 (D 1 )(1)) O Ĉ ρ ξ,a e E 2m+2 (D 1 )(1), where the composition Ĉ = can C is defined in the condition (iii) of the definition of X m. Whence (97) ρ D,C = ρ ξ,a. Since ρ ξ,a is a subbundle morphism, the condition (iii) is satisfied and, moreover, Ĉ is a subbundle morphism as well. Thus, the lower part of the diagram (96) shows that the morphism ( D 1, C) : (k m+1 k m ) O (k m+1 ) Ω(2) is a subbundle morphism. Hence its composition with the subbundle morphism v : (k m+1 ) Ω(2) (k m+1 ) V O(1) is a subbundle morphism as well. By definition, this composition coincides with (D 1, C) u. Hence the condition (ii) in the definition of X m is satisfied. This shows that f m ((MI 2m+1 (ξ)) red ) lies in X m. Last, the equality g m f m = id follows directly from (57) and (61). (ii) We first prove that the image of the map (98) g m : X m S 2m+1 : (D, C) (D 1, C, C D C) 2 1 Here we use the decomposition (56) fixed by the choice of ξ. 2 We identify here the triple (D 1, C, C D C) with a point in S 2 (k 2m+1 ) 2 V via the decomposition (56). ev

22 22 TIKHOMIROV lies in (MI 2m+1 (ξ)) red. In fact, the subbundle morphism A := (D 1, C) u : (k m+1 k m ) O (k m+1 ) V O(1) and its dual extend to the right and left exact sequence (99) (k m+1 k m ) O( 1) A (k m+1 ) V O A D (k m+1 k m ) O(1). ( ) D Furthermore, by definition A D A = u 1 C A u, where A is the matrix C C. D C Since the condition (i) is satisfied, under the direct sum decomposition (56) this matrix A can be treated an element of S 2m+1. Hence u A u =, i.e. (99) is a monad. Show that its cohomology bundle E(D, C) := ker(a D)/ Im A is an (2m + 1)-instanton, this giving the desired inclusion g(x m ) (MI 2m+1 (ξ)) red. For this, consider the diagram (67) in which we substitute B α ξ,a by A, respectively, B by D 1, denote G := coker A, and change the notation for τ ξ,a and ɛ ξ,a, respectively, to τ D,C and ɛ D,C (1) k m O P 3( 1) (k m+1 k m ) O P 3( 1) i m+1 k m+1 O P 3( 1) A D 1 u (k m+1 ) V O P 3 (k m+1 ) V O P 3 can G ɛ D,C v D k m+1 T P 3( 1) τ D,C k m O P 3( 1). In these notations the diagram (82) becomes the display of the antiselfdual monad (11) k m+1 O( 1) D 1 u (k m+1 ) V O u (k m+1 ) O(1) with the symplectic cohomology sheaf E 2m+2 (D 1 ): (12) E 2m+2 (D 1 ) = ker(u )/ Im(D 1 u). Moreover, as in (84) and (85) we obtain a subbundle morphism (13) ρ D,C : k m O( 1) E 2m+2 (D 1 ) such that (14) τ D,C = e θ ρ D,C, where θ : E 2m+2 (D 1 ) E 2m+2 (D 1 ) is a symplectic structure on E 2m+2 (D 1 ). Besides, as in (83) we have (15) h (E 2m+2 (D 1 )) = h i (E 2m+2 (D 1 )( 2)) =, i. Furthermore, as before, the antiselfdual monads (99) and (11) imply the (antiselfdual) monad (87) (16) k m O( 1) ρ D,C E 2m+2 (D 1 ) θ ρ D,C (k m ) O(1) with the cohomology sheaf E(D, C), (17) E(D, C) = ker(θ ρ D,C)/ Im(ρ D,C ). Now (15) and (16) yield h (E(D, C)) = h i (E(D, C)( 2)) =, i, i.e. E(D, C) is an (2m + 1)-instanton. Thus Im g m I 2m+1 (ξ). The fact that f m g m = id follows directly from (93) and (94).

23 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE Scheme Z m. Set 7. Variety Z m (18) Λ m := 2 (k m ) S 2 V, Φ m := Hom(k m, (k m ) ) 2 V, and consider the set (19) Z m := (D, φ) S m Φ m Θ m (D, φ) := φ D φ : k m V (k m ) V satisfies the condition. Θ m (D, φ) S m (Here, as in (9), we understand a point D S m as a homomorphism (k m ) V k m V.) Consider the standard decomposition with the induced projections 2 ((k m ) V ) = S m Λ m pr 1 S m 2 ((k m ) V ) pr 2 Λ m. We have a morphism h m : S m Φ m Λ m : (A m, φ m ) pr 2 (Θ(A m, φ m )). By the definition Z m we have (11) Z m = h 1 m (). Convention: If Z m is nonempty, we supply Z m with a scheme structure of a scheme-theoretic fibre h 1 m () of the morphism h m. Assume that (111) Z m. Then from the definition of Z m we obtain the estimate for the dimension of Z m at each point z Z m (112) dim z Z m = dim h 1 m () dim(s m Φ m ) dim 2 (k m ) S 2 V = = 3m(m + 1) + 6m 2 5m(m 1) = 4m(m + 2). Consider the open dense subset Φ m := {φ Φ m φ : k m (k m ) 2 V ) is injective} of Φ m and set (113) Z m := { (D, φ) Z m (S m) Φ m The set Z m is by definition an open subset in Z m. Assume Z m. Pick a point z = (D, φ) Z m and set Im( φ) Im( (D 1 ) : k m (k m ) 2 V ) = {} } W 5m := (k m ) 2 V / Im( (D 1 )), dim W 5m = 5m. Let i(z) be the composition in the diagram (114) k m φ i(z) k m (D 1 ) (k m ) 2 V can W 5m The lower horizontal triple in (114) yields the diagram (115) k m O P 3 k m O P 3 (D 1 ) (k m ) 2 V O P 3 ev D 1 (k m ) Ω P 3(2) can W 5m O P 3 ev can E 2m (D 1 )(1),

24 24 TIKHOMIROV where E 2m (D 1 ) is a symplectic bundle (see (81)). From this diagram we deduce the equalities (116) h i (E 2m (D 1 )( 2)) =, i, and the isomorphism (117) h (ev) : W 5m H (E 2m (D 1 )), i, Moreover, the diagrams (114) and (115) define the composition (118) i z : k m O P 3( 1) i(z) W 5m O P 3( 1) ev E 2m (D 1 ). Note that from the definition of the set Z m it follows that (119) t i z i z =, where t i z := i z θ and θ : E 2m ((D 1 )) E 2m ((D 1 )) is the symplectic structure on E 2m ((D 1 )) mentioned above, i.e. we have an antiselfdual complex (12) k m O P 3( 1) iz E 2m (D 1 ) t i z (k m ) O P 3(1). (Warning: this complex is not right exact.) Twisting the sequence (118) by O P 3(1) and passing to sections, we obtain in view of Furthermore, the standard embedding (121) j : k m 1 k m : (a 1,..., a m 1 ) (a 1,..., a m 1, ) and the morphism i z from (118) define the composition (122) j z : k m 1 O P 3( 1) j k m O P 3( 1) iz E 2m (D 1 ) 7.2. Varieties Z m and N th 2m 1. Assume, as above, that Z m and set (123) Z m = {z = (D, φ) Z m j z : k m 1 O P 3( 1) E 2m (D 1 ) is a subbundle morphism}. By definition, Z m is an open subset of Z m, hence also of Z m. If Z m, then for any point z = (D, φ) Z m we obtain from (119) that t j z j z =, where t j z := j z θ. Thus j z defines a monad (124) k m 1 O P 3( 1) j z E 2m (D 1 ) t j z (k m 1 ) O P 3(1), and in view of (116) the cohomology sheaf of this monad is an instanton bundle (125) E 2 (z) := Ker( t j z )/ Im(j z ), [E 2 (z)] I(2m 1). Consider the subvariety I th 2m 1 I 2m 1 of t Hooft instanton bundles I th 2m 1 := {[E] I 2m 1 h (E(1)) }. Lemma 7.1. Assume Z m. Then for any z = (D, φ) Z m the bundle E 2 (z) is a t Hooft instanton bundle, i.e. [E 2 (z)] I th 2m 1. Proof. Consider the complexes (12) and (124) and set H m 1 := k m 1 O P 3( 1), H m := k m O P 3( 1), K m+1 := coker j z, K m := coker i z.

25 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE25 The complexes (12) and (124) are antiselfdual, hence they extend to a commutative diagram (126) H m 1 E 2m (D 1 ) j i H m z E 2m (D 1 ) t j z O P 3( 1) O P 3(1) t i z j z Hm 1 j Hm O P 3(1), K m γ H m E 2 (z) τ K m+1 β δ Hm 1 j s α O P 3( 1) in which α, β, γ, δ and τ are the induced morphisms. In this diagram we have β α = and j γ β = δ. Hence δ α =. This implies that α factors through the morphism τ, i.e. there exists an injection s : O P 3( 1) E 2 (z) such that α = τ s. This injection s is a nonzero section s H (E 2 (z)(1)). Hence E 2 (z) is a t Hooft bundle. We will show that Zm is an irreducible variety of dimension 4m(m+2), hence it is nonempty. For this, fix an isomorphism (127) ξ : k m k m 1 k 2m 1 and consider the variety MI th 2m 1(ξ) defined in (74). Take an arbitrary point A MI th 2m 1(ξ). The point A defines a point B = ξ 1 (A) and a monad k m 1 O P 3( 1) ρ ξ,a E 2m (B) t ρ ξ,b (k m 1 ) O P 3(1) with the cohomology bundle [E 2 (A)] = π 2m 1 (A) (see subsection 5.3). The display of this monad twisted by O P 3(1) is (128) E 2 (A)(1) k m 1 ρ O ξ,a P 3 E 2m (B)(1) K m+1 (A)(1) t ρ ξ,a (k m 1 ) O P 3(2), where K m+1 (A) := coker ρ ξ,a. Note that from (7) and the definition of MI2m 1(ξ) th it follows that h (E 2 (A)(1)) = 1. Hence, passing to sections in the diagram (128) we obtain a well defined epimorphism (129) b(ξ, A) : H (E 2m (B)(1)) h (ɛ) H (K m+1 (A)(1)) can H (K m+1 (A)(1))/H (E 2 (A)(1)) k 4m. ɛ

26 26 TIKHOMIROV On the other hand, similar to (115) and (117) we obtain the exact triple (13) k m B 1 (k m ) 2 V c(a) H (E 2m (B)(1)). Denote by c(a) the epimorphism (k m ) 2 V H (E 2m (B)(1)) in this triple and set (131) V 2m (ξ, A) := c(a) 1 (ker b(ξ, A)) k 2m, V 2m(ξ, A) := {v V 2m (ξ, A) Span(Im (ξ 1 (A) 1 ), Im (ξ 2 (A)), kv) = V 2m (ξ, A)}, (132) V 2m (ξ) := {(A, v) A MI th 2m 1(ξ), v V 2m (ξ, A)}. Here the projection V 2m (ξ) MI2m 1(ξ) th : (A, v) A is a k 2m -bundle over MI2m 1(ξ), th hence by Lemma 5.3 and Corollary 5.4 V 2m (ξ) is irreducible of dimension (133) dim V 2m (ξ) = dim MI th 2m 1(ξ) + 2m = 4m(m + 2). Besides, V 2m(ξ, A) is a dense open subset of V 2m (ξ, A) for each A MI th 2m 1(ξ), (134) V 2m(ξ, A) dense open V 2m (ξ, A) k 2m. Next, set Π m := Hom(k m, (k m ) 2 V) and (135) N(ξ, A) := (φ : km V (i) Span(Im (ξ 1 (A) 1 ), Im φ) = V 2m (ξ, A), (k m ) V ) Π m (ii) φ j = ξ 2 (A), (iii) φ (ξ 1 (A) 1 ) φ S m, (136) N th 2m 1(ξ) := {(A, φ) A MI th 2m 1(ξ), φ N(ξ, A)}. Consider the standard decomposition k m = k m 1 k, so that the injection j in (121) is an embedding of the left direct summand of this decomposition. Then each monomorphism ( φ : k m (k m ) 2 V ) N(ξ, A) in view of the conditions (i)-(iii) of (135) is uniquely determined by its restriction onto the right direct summand k of the standard decomposition, satisfying the conditions and φ k : k V 2m (ξ, A) (k m ) 2 V : 1 v Span(Im (ξ 1 (A) 1 ), Im φ) = Span(Im (ξ 1 (A) 1 ), Im (ξ 2 (A)), kv) = V 2m (ξ, A). (ξ 2 (A) + φ k V ) (ξ 1 (A) 1 ) (ξ 2 (A) + φ k V ) S m. These conditions and the definition of V2m(ξ, A) mean that N(ξ, ) is a closed subset of V2m(ξ, A), hence by (134) it is a locally closed subset of V 2m (ξ, A). As a result, we have (137) N th 2m 1(ξ) In particular, locally closed V 2m (ξ). (138) dim N th 2m 1(ξ) dim V 2m (ξ) = 4m(m + 2). Now consider the map (139) h m : N th 2m 1(ξ) Z m : (A, φ) (D := ξ 1 (A) 1, φ). This map is well defined. In fact, take any point (A, φ) N2m 1(ξ). th Since A MI2m 1(ξ), th we have D (S m), so that the vector bundle E 2m (D 1 ) is well-defined. Next, since φ j = ξ 2 (A) (see condition (ii) in (135)), it follows from Theorem 6.1 that the morphism j z : k m 1 O( 1) E 2m (D 1 )

27 MODULI OF MATHEMATICAL INSTANTON VECTOR BUNDLES WITH ODD c 2 ON PROJECTIVE SPACE27 for z = (D, φ) coincides with the subbundle morphism ρ ξ,a satisfying diagram (96). Note that in view of (97) we can rewrite this also as (14) j z = ρ D,C, C = φ j. The diagram (96), in turn, implies that the condition Im( D) Im( φ) = {} is satisfied. This together with the injectivity of j z and the condition (iii) in (135) precisely means that z Zm. As a result, it follows that Zm and, respectively, Z m is nonempty. Moreover, since Zm is supplied with the structure of a reduced scheme and N2m 1(ξ) th is smooth (hence reduced) it follows that the map h m given by formula (139) is a morphism of reduced schemes. Next, consider the set where Z m(ξ) := {z Z m z = (D, φ) satisfies the condition ( )} (D 1, φ j) u : (k m k m 1 ) O( 1) (k m ) V O is a subbundle morphism. ( ) Since the condition (*) is open and Z m(ξ) contains a subset h m (N th 2m 1(ξ)), it follows that Z m(ξ) is a nonempty open subset of Z m. Consider the map (141) λ m : Z m(ξ) S 2m 1 : z = (D, φ) A := ξ(d 1, φ j, (φ j) D (φ j)). Since (φ D φ) S m by the definition of Z m, it follows that (142) (φ j) D (φ j) S m 1, i.e. the map λ m in (141) is well-defined. Moreover, since Z m(ξ) is a reduced scheme, the map λ m is a morphism of reduced schemes. Theorem 7.2. Let m 1 and ξ be a fixed isomorphism (127). Then Z m(ξ) is a smooth irreducible variety of dimension 4m(m + 2) and there is an isomorphism of smooth varieties (143) ν m : Z m(ξ) N th 2m 1(ξ) : (D, φ) (A, φ), where A is given by (141). Proof. Consider the set X m 1 defined in (92) and the morphism of reduced schemes (144) η m : Z m(ξ) X m 1 : z = (D, φ) (D, φ j). This morphism is well-defined since (142), (*) and (14) are precisely the conditions (i), (ii) and (iii) of the definition of X m 1. Next, comparing (94), (141) and (144) we obtain that λ m = g m 1 η m for m 1. Whence Im λ m MI 2m 1 (ξ). Moreover, for any point z = (D, φ) the diagram (126) defines a section s E 2 (A)(1) for A = λ m (z), so that [E 2 (A)] I2m 1, th i.e. A MI2m 1(ξ). th Hence (A, φ) N2m 1(ξ), th and the morphism ν m in (143) is well-defined. Comparing now (139) and (143), we obtain that h m = νm 1, i.e. ν m is an isomorphism of reduced schemes. Next, since by definition Zm(ξ) is an open subset of Z m, it follows from (112) that dim Zm(ξ) 4m(m + 2). This together with (138) and the isomophism ν m shows that dim Z m(ξ) = dim N th 2m 1(ξ) = dim V 2m (ξ) = 4m(m + 2). Whence by (137) and the irreducibility and smoothness of V 2m (ξ) we obtain that Zm(ξ) N2m 1(ξ) th is a dense open subset of V 2m (ξ), so that Zm(ξ) is smooth and irreducible of dimension 4m(m + 2).