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Chern character and charge normalization of a linear twistor monad

Statement

Let $k, N \in Z_{\geq 0}$ , and let $F$ be the locally free algebraic middle cohomology of a linear monad

O_{P^{3}} (- 1)^{\oplus k} ⟶ O_{P^{3}}^{\oplus (2 k + N)} ⟶ O_{P^{3}} (1)^{\oplus k}

on $P_{C}^{3}$ , with exactness at both ends.

For algebraic sheaf cohomology, write $H^{q} (P^{3}, G)$ for the right-derived global-sections group. If these groups are finite-dimensional and vanish for all sufficiently large $q$ , define the algebraic Euler characteristic by $χ (G) = \sum_{q \geq 0} (- 1)^{q} \dim_{C} H^{q} (P^{3}, G)$ . Let $H_{a l g} = c_{1}^{a l g} (O_{P^{3}} (1))$ in the Chow ring and let $H_{t o p} = c_{1}^{t o p} (O_{P^{3}} (1))$ in singular cohomology. The hyperplane cycle map $cl : A^{1} (P^{3}) \to H^{2} (P^{3}; Z)$ identifies $H_{a l g}$ with $H_{t o p}$ ; write $H$ for either class in the corresponding formula. The algebraic Chern character is $N - k H_{a l g}^{2}$ , the algebraic first Chern class is zero, the algebraic second Chern class is $k H_{a l g}^{2}$ , and the algebraic Euler characteristic is $N - 2 k$ . The analytification of $F$ has total topological Chern class $1 + k H_{t o p}^{2}$ through degree six and topological Chern character $N - k H_{t o p}^{2}$ .

Let $ȷ : C^{4} \to C^{4}$ be the conjugate-linear map $ȷ (λ_{1}, λ_{2}, μ_{1}, μ_{2}) = (- {\overset{ˉ}{λ}}_{2}, {\overset{ˉ}{λ}}_{1}, - {\overset{ˉ}{μ}}_{2}, {\overset{ˉ}{μ}}_{1}), ȷ^{2} = - 1.$ Use $H = C \oplus C j$ , where $j^{2} = - 1$ and $j a = \overset{ˉ}{a} j$ for $a \in C$ , and identify $C^{4}$ with the right quaternionic vector space $H^{2}$ by $(λ_{1}, λ_{2}, μ_{1}, μ_{2}) ⟼ (λ_{1} + {\overset{ˉ}{λ}}_{2} j, μ_{1} + {\overset{ˉ}{μ}}_{2} j) .$ Under this identification, right multiplication by $j$ is $ȷ$ . Give $H P^{1} = (H^{2} ∖ {0}) / H^{\times}$ its standard smooth quotient structure; it is the 4-sphere and its points are the one-dimensional right- $H$ subspaces of $H^{2}$ . The quaternionic quotient map is the smooth map $π : P (C^{4}) ⟶ H P^{1}, [λ_{1} : λ_{2} : μ_{1} : μ_{2}] ⟼ [(λ_{1} + {\overset{ˉ}{λ}}_{2} j, μ_{1} + {\overset{ˉ}{μ}}_{2} j)]_{H} .$ Equivalently, $π ([v]) = {span}_{C} (v, ȷ v)$ , a $ȷ$ -invariant complex two-plane. This is well-defined because $ȷ (c v) = \overset{ˉ}{c} ȷ v$ for $c \in C$ and $v, ȷ v$ are complex-linearly independent when $v \neq 0$ and $ȷ^{2} = - 1$ . Its fiber over a quaternionic line $L$ is $P_{C} (L) ≅ P^{1} (C)$ . In the finite chart with $λ = λ_{1} + {\overset{ˉ}{λ}}_{2} j \neq 0$ and $z = μ λ^{- 1} = z_{1} + {\overset{ˉ}{z}}_{2} j$ , the fiber equations are $μ_{1} = z_{1} λ_{1} - {\overset{ˉ}{z}}_{2} λ_{2}, μ_{2} = z_{2} λ_{1} + {\overset{ˉ}{z}}_{1} λ_{2},$ and the fiber over the remaining point is $ℓ_{\infty} = {λ_{1} = λ_{2} = 0}$ . Orient each fiber by its complex orientation. Write $z_{1} = x^{0} + i x^{1}$ and $z_{2} = x^{2} + i x^{3}$ , so the chart orientation is $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ ; orient $S^{4}$ so this positive stereographic chart followed by the fiber orientation induces the complex orientation of $P^{3}$ . Since the base and fiber dimensions are $4$ and $2$ , respectively, reversing these orientation factors has sign $(- 1)^{4 \cdot 2} = + 1$ , so this also agrees with the fiber-first convention. Let $u \in H^{4} (S^{4}; Z)$ be the positive point class for this orientation. Then $π^{*} u = H_{t o p}^{2}$ . Consequently, if the underlying topological bundle of $F$ is $π^{*} E$ for a rank- $N$ complex bundle $E \to S^{4}$ with $c_{1} (E) = 0$ , then $c_{2} (E) = k u$ and $⟨ c_{2} (E), [S^{4}] ⟩ = k$ .

Statement references

The Chern character and the Todd class

Cohomology of O(d) on projective space

Integral cohomology ring of complex projective space

The normal Thom class realizes the Poincare dual of a closed submanifold

Donaldson, §§4–6

Nakajima–Yoshioka, §3.1

Facts & Assumptions

Given: The locally free exact monad, the positive twistor orientation, and the conditional pullback bundle $E$ in the Statement.

[A1]

AC is inherited from the projective cohomology and characteristic-class suppliers. The Axiom of Choice

[F1]

The Chern character is additive on exact complexes, $ch (L) = e^{c_{1} (L)}$ for a line bundle, and ${ch}_{2} = \frac{1}{2} (c_{1}^{2} - 2 c_{2})$ . The Chern character and the Todd class

[F2]

On $P^{3}$ , $H^{*} (P^{3}; Z) = Z [x] / (x^{4})$ for the tautological line generator $x$ ; $O (1)$ is its dual, so $H = - x$ and $H^{2} = x^{2}$ is the positive degree-four generator. Integral cohomology ring of complex projective space

[F3]

For a closed oriented $z$ -submanifold of an oriented $n$ -manifold, the absolute image of its normalized normal Thom class caps the ambient fundamental class to $(- 1)^{(n - z) z}$ times the submanifold class, under the supplier's tangent-first orientation convention. The normal Thom class realizes the Poincare dual of a closed submanifold

[F4]

The algebraic cohomology of $O (d)$ on projective space is given by the twisting-sheaf theorem; in particular on $P^{3}$ , $χ (O (- 1)) = 0$ , $χ (O) = 1$ , and $χ (O (1)) = 4$ . Cohomology of O(d) on projective space

[F5]

For numerable finite-rank complex bundles over a path-connected CW base, integral Chern classes are natural, satisfy $c (E \oplus G) = c (E) c (G)$ , and obey $c_{1} (L) = e (L_{R})$ for a complex line, with $c (0) = 1$ and $c (E \oplus ε^{r}) = c (E)$ . Naturality, normalization, and Whitney sum for Chern classes

[F6]

The algebraic first Chern class of $O (1)$ is the hyperplane divisor class; its cycle class is the Poincaré dual of the complex hyperplane, which is the Euler class of its complex normal line. Thus its cycle class is $H_{t o p}$ . The Chern character and the Todd class, The normal Thom class realizes the Poincare dual of a closed submanifold

[F7]

For a short exact sequence of finite locally free sheaves, total Chern classes satisfy the Whitney product formula, and pullback to a composition of projective bundles is injective and splits the bundles into line-bundle filtrations. On that pullback the Chern roots of the middle bundle are the union of the roots of the subbundle and quotient; since $ch (E) = \sum_{i} e^{x_{i}}$ , the Chern character is additive in the short exact sequence. Additivity, naturality and the splitting principle for Chern classes, The Chern character and the Todd class

[F8]

A short exact sequence of sheaves has the natural long exact sequence in the same sheaf cohomology groups. If the cohomology groups in the sequence are finite-dimensional and vanish in sufficiently high degree, alternating sums of their dimensions are additive. Long exact sequence of sheaf cohomology

Proof

Given: The two exact sequences obtained by factoring the locally free monad through $K = \ker (O (1)^{\oplus k} \leftarrow O^{\oplus (2 k + N)})$ .

1.1A1F4F7F8algebra

Exactness gives $0 \to K \to O^{2 k + N} \to O (1)^{k} \to 0$ and $0 \to O (- 1)^{k} \to K \to F \to 0$ . By [F7], additivity of the Chern character in these short exact sequences gives

ch (F) = (2 k + N) ch (O) - k ch (O (- 1)) - k ch (O (1)) .

By [F4], the cohomology groups of $O (- 1)$ , $O$ , and $O (1)$ are finite-dimensional and vanish above degree three. The long exact sequences [F8] then give the same finiteness and vanishing for $K$ and $F$ , so [F8] also gives additivity of $χ$ . Since $χ (O (- 1)) = 0$ , $χ (O) = 1$ , and $χ (O (1)) = 4$ by [F4], we obtain $χ (F) = (2 k + N) - 4 k - 0 = N - 2 k$ .

1.2F2F3F6algebra

The quaternionic coordinates in the Statement give local product charts for $π$ : on the finite chart, with $v = λ_{2} / λ_{1}$ , the base coordinates $z_{1}, z_{2}$ recover the remaining affine coordinates by $μ_{1} / λ_{1} = z_{1} - {\overset{ˉ}{z}}_{2} v$ and $μ_{2} / λ_{1} = z_{2} + {\overset{ˉ}{z}}_{1} v$ ; the charts with $λ_{2} \neq 0$ and the reciprocal quaternionic chart at infinity cover the omitted points. Hence $π$ is a smooth submersion, and each fiber $L_{x} = π^{- 1} (x)$ is a closed complex linear $P^{1} \subset P^{3}$ . Let $ω_{x} \in H^{4} (S^{4}, S^{4} ∖ {x}; Z)$ be the positive local orientation class, whose image in absolute cohomology is $u$ . In a local product chart over an oriented ball around $x$ , the pulled-back relative class $π^{*} ω_{x}$ is the positive generator on every normal four-disk to $L_{x}$ ; it is therefore the normalized Thom class of that fiber, with normal orientation induced from the oriented base. Its absolute image is $π^{*} u$ . The fiber has real dimension $2$ and codimension $4$ , so [F3] gives the sign $(- 1)^{4 \cdot 2} = + 1$ and identifies $π^{*} u$ with the Poincaré dual of $[L_{x}]$ . To identify this dual class, choose two complex hyperplanes meeting transversely in $L_{x}$ . Their positive normal Thom classes multiply to the positive normal Thom class of $L_{x}$ (in local holomorphic coordinates this is the product of the two complex-disk generators); by [F6] each hyperplane class is $H_{t o p}$ . Thus the Poincaré dual of $[L_{x}]$ is $H_{t o p}^{2}$ , proving $π^{*} u = H_{t o p}^{2}$ .

2.1A1F1F2F5F7step 1.1algebra

In the algebraic Chow ring, [F1] and $H = c_{1} (O (1))$ give

ch (F) = (2 k + N) - k e^{- H} - k e^{H} = N - k H^{2},

because $H^{4} = 0$ on the complex threefold and the odd powers cancel. The integral Whitney formula [F7] applied to the same two sequences gives $c^{a l g} (F) = (1 - H^{2})^{- k} = 1 + k H^{2}$ through codimension three, so $c_{1}^{a l g} (F) = 0$ and $c_{2}^{a l g} (F) = k H^{2}$ integrally. Analytification preserves these locally split exact bundle sequences. Their underlying topological sequences split: a Hermitian metric on each middle bundle identifies the orthogonal complement of the subbundle with the quotient. Such metrics exist by patching local metrics with a partition of unity on compact $P^{3}$ . This base is a path-connected finite CW complex, and its finite-rank bundles are numerable. Thus the integral Whitney formula [F5], applied to these splittings with $c_{1} (O (1)) = H$ and $c_{1} (O (- 1)) = - H$ from [F2], gives

c (F) = \frac{1}{(1 - H)^{k} (1 + H)^{k}} = (1 - H^{2})^{- k} = 1 + k H^{2}

through degree six, since $H^{4} = 0$ . Hence the topological classes are $c_{1} (F) = 0$ , $c_{2} (F) = k H^{2}$ , and $c_{3} (F) = 0$ . The universal formulas ${ch}_{1} = c_{1}$ , ${ch}_{2} = \frac{1}{2} (c_{1}^{2} - 2 c_{2})$ , and ${ch}_{3} = \frac{1}{6} (c_{1}^{3} - 3 c_{1} c_{2} + 3 c_{3})$ then give the topological $ch (F) = N - k H^{2}$ . This compares only the identical finite line-bundle resolution and its line class, not arbitrary analytic and algebraic cohomology.

3.1F2F5step 2.1step 1.2algebra∎

Write $c_{2} (E) = m u$ for $m \in Z$ . Naturality of Chern classes [F5] and the pullback hypothesis give $m H^{2} = π^{*} c_{2} (E) = c_{2} (F) = k H^{2}$ by steps 2.1 and 1.2. Since $H^{2}$ is a nonzero generator by [F2], $m = k$ , proving both the stated Chern class and its evaluation on $[S^{4}]$ .

Chern character and charge normalization of a linear twistor monad

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources