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Conditional twistor vanishing and normalized monad reconstruction

Statement

Let $A$ be a framed unitary ASD connection on the round oriented $S^{4}$ , let $E$ be its Ward bundle on $Z = C P^{3}$ , and let $k = - q_{N} (A)$ . Use the exact coupled cohomology-vanishing input stated below for both $E$ and $E^{\lor}$ . Assume also the exact projective GAGA and line-family base-change inputs needed to regard $E$ as an algebraic bundle, to make its restrictions trivial on a complex-open neighbourhood of at least one real twistor line, and to apply the diagonal resolution. Finally, use the Chern-character normalization $ch (E) = N - k H^{2}$ in $A^{*} (C P^{3})_{Q}$ , where $H = c_{1} (O (1))$ ; this normalization is conditional on the twistor pullback class convention matching the framed charge.

Under these inputs,

$H^{1} (Z, E (- 2)) = H^{1} (Z, E^{\lor} (- 2)) = 0.$

There is a canonical Beilinson three-term monad

$O_{Z} (- 1) \otimes U ⟶ O_{Z} \otimes K ⟶ O_{Z} (1) \otimes U^{'},$

whose middle cohomology is $E$ and whose terms are

$U = H^{2} (Z, E (- 3)), K = H^{2} (Z, E \otimes Ω_{Z}^{2}), U^{'} = H^{2} (Z, E \otimes Ω_{Z}^{1} (- 1)),$

with $\dim U = \dim U^{'} = k$ and $\dim K = 2 k + N$ . The construction is natural in framed bundle isomorphisms and unique up to changes of bases in $U, K, U^{'}$ . The values of all global parallel sections form its maximal trivial flat parallel summand, and the connection splits orthogonally into that summand and a summand with no nonzero parallel sections; arbitrary nonflat parallel summands are permitted. The monad construction is additive across this splitting and uses no hypothesis $H^{0} (Z, E) = 0$ .

Statement references

Conditional framed Penrose–Ward correspondence on the round four-sphere

Coupled Penrose vanishing on the round four-sphere

Smooth framed bundle of sections over real twistor lines

Projective GAGA for framed coherent and monad data

Donaldson, §5, printed pp. 8–10

Nakajima–Yoshioka, §3.1, printed pp. 17–19

Facts & Assumptions

Given: The Ward bundle and its framed ASD connection, the round twistor conventions, the exact coupled cohomology-vanishing input, the projective-family hypotheses, and the stated Chern-character normalization.

[A1]

For the Ward bundle, the underlying smooth bundle is the twistor pullback and its Dolbeault operator is the ordinary vertical derivative and the connection-horizontal alpha-plane operator, by the forward Ward construction. The exact coupled-vanishing theorem therefore gives $H^{1} (Z, E (- 2)) = H^{1} (Z, E^{\lor} (- 2)) = 0$ , including reducible and flat summands. Its approved interface asserts these two vanishings; no additional transform-map existence is inferred from that Statement. Coupled Penrose vanishing on the round four-sphere, Conditional framed Penrose–Ward correspondence on the round four-sphere

[A2]

Absolute projective GAGA and the universal-line direct-image lemma supply the needed local evaluation and degree-zero base-change maps; they are no longer extra assumed surjectivity inputs. Projective GAGA for framed coherent and monad data, Smooth framed bundle of sections over real twistor lines

[A3]

The twistor pullback of the framed Chern class has $c_{1} (E) = 0$ , $c_{2} (E) = k H^{2}$ , and $c_{3} (E) = 0$ , so $ch (E) = N - k H^{2}$ . The positive incidence convention and actual Thom/projective-cohomology calculation are supplied by Conditional framed Penrose–Ward correspondence on the round four-sphere; the $U (N)$ determinant and charge calculation is supplied by Determinant flatness and charge normalization for framed U(N) ASD data; the equality $π^{*} u = H^{2}$ is now checked for that exact convention; it is not inferred from a source sign convention.

[F1]

The exact round sphere, positive stereographic orientation and incidence are those of the Ward construction and the coupled-vanishing supplier. The latter has no GAGA, monad or inverse ADHM premise, so this application does not create a circular proof. Conditional framed Penrose–Ward correspondence on the round four-sphere, Coupled Penrose vanishing on the round four-sphere

[F2]

For a flat connection on the simply connected $S^{4}$ , parallel transport gives a global parallel frame; for a unitary connection the orthogonal complement of its parallel sections is preserved. $S^{n}$ is simply connected for every $n \geq 2$ , A flat connection admits local parallel frames, Flat connections have locally path-independent parallel transport on a coordinate ball

[F3]

The standard projective diagonal has the Koszul resolution $0 \to O (- 3) ⊠ Ω^{3} (3) \to O (- 2) ⊠ Ω^{2} (2) \to O (- 1) ⊠ Ω^{1} (1) \to O \to O_{Δ} \to 0,$ because the diagonal is a regular embedding of codimension three and its three local coordinate differences form a regular sequence. Koszul resolutions and restriction to flat fibres

[F4]

The bounded hypercohomology spectral sequence has first page $E_{1}^{p, q} = H^{q} (K^{p})$ , differential $d_{r} : (p, q) \mapsto (p + r, q - r + 1)$ , and converges with a finite filtration to the hypercohomology of the diagonal complex. First hypercohomology spectral sequence

[F5]

On $P^{3}$ , Serre duality identifies $H^{q} (F)^{*}$ with ${Ext}^{3 - q} (F, O (- 4))$ ; for locally free $F$ this is $H^{3 - q} (F^{*} (- 4))$ . Serre duality for coherent sheaves on projective space

[F6]

On $P^{n}$ , $H^{0} (O (d)) = 0$ for $d < 0$ , and the only possible nonzero cohomology of $O (d)$ is in degrees $0$ and $n$ , with $H^{n} (O (d)) = 0$ for $d > - n - 1$ . Cohomology of O(d) on projective space

[F7]

Hirzebruch–Riemann–Roch on $P^{3}$ gives $χ (F) = \int_{P^{3}} ch (F) td (T P^{3})$ for the vector bundles and twists in this argument. Riemann-Roch for projective-space projections

[F8]

A holomorphic scalar function on a connected complex coordinate domain that vanishes on a nonempty open subset vanishes identically. Applying this in local holomorphic frames extends open-set vanishing to a bundle section on a connected complex manifold: the set where its germ is zero is open, and is closed by the scalar theorem in a connected coordinate chart. A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically

Proof

Given: [A1]–[A3] and [F1]–[F8].

1.1A1F1algebra

The Ward construction gives exactly the smooth pullback bundle and vertical/horizontal Dolbeault operator required in [A1], for the same sphere and incidence in [F1]. Thus its two cohomology vanishings apply directly. The dual is again unitary and ASD, since its curvature is $- F_{A}^{t}$ and Hodge star commutes with transpose, and its holomorphic operator is the induced dual one. Hence both $H^{1} (Z, E (- 2))$ and $H^{1} (Z, E^{\lor} (- 2))$ vanish. This uses the supplier's exact stated conclusion, not an unexposed injective-map premise.

1.2A2F5F6algebra

Let $L ≅ P^{1}$ be any real twistor line. Its normal bundle in $P^{3}$ is $O (1)^{\oplus 2}$ , and the conormal sequence splits because its extension group is $H^{1} (O (1))^{\oplus 2} = 0$ . Hence $Ω_{Z}^{1} ∣_{L} ≅ O (- 2) \oplus O (- 1)^{\oplus 2}$ and $T_{Z} ∣_{L} ≅ O (2) \oplus O (1)^{\oplus 2}$ . Since $Ω_{Z}^{2} ≅ T_{Z} (- 4)$ , each of $E (- 1) ∣_{L}$ , $E (- 3) ∣_{L}$ , $E (- 2) ∣_{L}$ , $(E \otimes Ω_{Z}^{1} (- 1)) ∣_{L}$ , and $(E \otimes Ω_{Z}^{2}) ∣_{L}$ is a sum of negative line bundles. The Serre-dual bundles restrict respectively to $E^{*} (- 3)$ , $E^{*} (- 1)$ , $E^{*} (- 2)$ , $E^{*} \otimes T_{Z} (- 3)$ , and $E^{*} \otimes Ω_{Z}^{1}$ , also sums of negative line bundles. Every point of $Z$ lies on a real line, so negativity on every such line makes every global section zero. Thus [F6] gives $H^{0} (Z, F) = 0$ and [F5] gives $H^{3} (Z, F) = 0$ for all the listed coefficient bundles $F$ .

2.1A2F6F8step 1.1algebra

Choose a plane $P ≅ P^{2}$ containing a real twistor line. By [A2] and the direct-image result, an open family of lines near that line has evaluation isomorphisms and hence trivial restrictions of $E$ . Those lines form a nonempty complex-open subset of the line family of $P$ . Choose affine coordinates $(x, y)$ with the original line given by $y = 0$ . This open family contains the projective lines $y = b$ for all sufficiently small complex $b$ , so their union contains a nonempty open subset of $P$ . A section of $E (- 2) ∣_{P}$ vanishes on each such line because $H^{0} (P^{1}, O (- 2)^{\oplus N}) = 0$ . It therefore vanishes on that open subset. In a connected coordinate chart and holomorphic bundle frame, [F8] makes its zero-germ locus closed as well as open. The plane is connected (any two points lie on a connected projective line), so the section vanishes everywhere; no density assertion for an arbitrary complex-open line family is needed. The divisor sequence $0 \to E (- 3) \to E (- 2) \to E (- 2) ∣_{P} \to 0$ and step 1.1 now give $H^{1} (Z, E (- 3)) = 0$ .

2.2F5F6step 1.1step 1.2algebra

Tensor $0 \to Ω_{Z}^{1} (- 1) \to O (- 2)^{\oplus 4} \to O (- 1) \to 0$ by $E$ . Step 1.2 gives $H^{0} (E (- 2)) = H^{0} (E (- 1)) = 0$ , and step 1.1 gives $H^{1} (E (- 2)) = 0$ . The long exact sequence yields $H^{1} (E \otimes Ω_{Z}^{1} (- 1)) = 0$ . Serre duality and step 1.1 also give $H^{2} (E (- 2)) = 0$ .

3.1F3F4step 1.2step 2.1step 2.2algebra

Apply [F4] to the diagonal Koszul resolution [F3], tensoring the second factor by $E (- 2)$ , pushing to the first factor, and twisting that output factor by $O (2)$ . Its first page is $E_{1}^{p, q} = H^{q} (Z, E (- 2) \otimes Ω_{Z}^{- p} (- p)) \otimes O_{Z} (p + 2)$ for $p = - 3, - 2, - 1, 0$ . Steps 1.2, 2.1 and 2.2 and Serre duality make every term in rows $q = 0, 1, 3$ vanish except possibly $H^{1} (E \otimes Ω_{Z}^{2})$ at $(- 2, 1)$ . No differential can enter this term; its only possible outgoing $d_{2}$ lands in $H^{0} (E (- 2)) = 0$ . It therefore survives to $E_{\infty}$ , but the diagonal construction resolves $E (- 2)$ before the output twist and $E$ after that twist; it is concentrated in degree zero and has no degree $- 1$ hypercohomology. Since $(- 2) + 1 = - 1$ , the term is zero. Thus the remaining spectral-sequence row is $q = 2$ .

4.1F3F4step 3.1algebra

The row $q = 2$ gives the complex $H^{2} (E (- 3)) \otimes O (- 1) \to H^{2} (E \otimes Ω_{Z}^{2}) \otimes O \to H^{2} (E \otimes Ω_{Z}^{1} (- 1)) \otimes O (1)$ . All higher differentials vanish because all other rows are zero. Its only cohomology is the degree-zero hypercohomology of the diagonal resolution, namely $E$ ; the first and last terms are exact because the other total degrees vanish. The maps are linear in the homogeneous output coordinates, since each Koszul differential is contraction with the universal linear section. Thus this is the stated monad; its cohomology vector spaces are canonical, and changing bases acts by $G L (U) \times G L (K) \times G L (U^{'})$ .

4.2A3F6F7step 1.2step 2.1step 2.2step 3.1algebra

From [A3], $ch (E (t)) = N e^{t H} - k H^{2} e^{t H}$ and $td (T_{Z}) = 1 + 2 H + \frac{11}{6} H^{2} + H^{3}$ . Hirzebruch–Riemann–Roch [F7] gives $χ (E (t)) = N (\binom{t + 3}{3}) - k (t + 2)$ , with values $k, - N + 2 k, 0, - k$ for $t = - 3, - 4, - 2, - 1$ . The Euler sequences $0 \to O (- 4) \to O (- 3)^{\oplus 4} \to Ω_{Z}^{2} \to 0$ and $0 \to Ω_{Z}^{1} (- 1) \to O (- 2)^{\oplus 4} \to O (- 1) \to 0$ give Euler characteristics $k, 2 k + N, k$ for the three monad coefficient bundles. Steps 1.2, 2.1, 2.2 and 3.1 leave only $H^{2}$ in each, so these values are the dimensions.

5.1F2F6step 4.1algebra∎

The space of parallel sections of $\hat{E}$ evaluates to a trivial subbundle, and its orthogonal complement is parallel because the connection is unitary. No nonzero parallel section lies in the complement. Every flat summand is trivial since $S^{4}$ is simply connected [F2]. The Ward transform and diagonal construction preserve direct sums. For a trivial rank- $m$ summand $O_{Z}^{\oplus m}$ , the two Euler sequences and [F6] give coefficient groups $H^{2} (O (- 3)) = 0$ , $H^{2} (Ω_{Z}^{2})^{\oplus m} ≅ C^{m}$ , and $H^{2} (Ω_{Z}^{1} (- 1)) = 0$ , so its monad is $0 \to O_{Z}^{\oplus m} \to 0$ . Flat summands are therefore included without imposing $H^{0} (Z, E) = 0$ or $H^{0} (Z, E^{\lor}) = 0$ . No Dirac index or kernel-dimension formula is used.

Conditional twistor vanishing and normalized monad reconstruction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources