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For a real or complex vector bundle $E$, choose a fiber metric. Its Thom space is the pointed quotientcollapsing the sphere bundle to one basepoint. When the base is compact, this is the one-point compactification of the total space: fiberwise radial compression identifies the total space with the interior of $D(E)$, and all directions approaching its boundary become the single point at infinity.

Decompose ${\mathbb{C}}^{n+m+2}=V\oplus W$ with $V={\mathbb{C}}^{n+1}$, $W={\mathbb{C}}^{m+1}$, so the subspace being collapsed is $\mathbb{P}(V)={\mathbb{CP}}^n$. Every line outside it has a representative $(v,w)$ with $w\ne 0$. Its projection to $W$ gives the line $\ell =\mathbb{C}w\in \mathbb{P}(W)={\mathbb{CP}}^m$, and there is a unique complex-linear map $A:\ell \to V$ such that $A(w)=v$. The line in $V\oplus W$ is precisely the graph of $A$. This construction is unaffected by replacing $(v,w)$ with $(\lambda v,\lambda w)$.

Let $\gamma ={\eta }_1$ be the tautological line over $\mathbb{P}(W)$. The graph construction and its inverse are continuous, indeed smooth in the usual affine charts, and identifyA Hermitian metric gives a complex-linear bundle isomorphism $\bar{\gamma }\cong {\gamma }^{\ast }$. With the Hermitian form linear in its first argument, it sends $\bar{v}$ to the functional $w\mapsto ⟨w,v⟩$. Linearity in the conjugate vector is exactly what makes this a complex, rather than merely real, isomorphism. Thus the bundle is $({\bar{\eta }}_1{)}^{\oplus (n+1)}$, with the conjugation bars present in the original PDF.

Collapsing the closed subspace ${\mathbb{CP}}^n$ in compact Hausdorff space ${\mathbb{CP}}^{n+m+1}$ gives the one-point compactification of its open complement: neighborhoods of the collapsed point have compact complements, and conversely every compact subset of the complement is closed in the ambient space. The graph homeomorphism therefore extends over the added point. This proves the projective-space quotient as a Thom space of conjugate tautological lines:

The graph coordinates also determine the normal complex structures requested in the second part. The copy $\mathbb{P}(W)$ corresponds to the zero section of the graph bundle, whose normal bundle is that bundle itself. For a coordinate embedding ${\mathbb{CP}}^k\subset {\mathbb{CP}}^n$, use $W={\mathbb{C}}^{k+1}$ and its complementary coordinate space $V={\mathbb{C}}^{n-k}$. Then the normal bundle of a linear complex-projective embedding isThe complex structure is the linear structure in these graph coordinates. Equivalently, applying the preceding homeomorphism with its parameters $n-k-1$ and $k$ identifies the Thom space of this normal bundle with ${\mathbb{CP}}^n/{\mathbb{CP}}^{n-k-1}$, after reordering coordinates. Define ${\nu }_m$ in the identical way. The Hermitian-metric identification with the conjugate line does not change the complex normal orientation.

The original coordinate embeddings may be nested, so first move one into transverse position. Keep ${\mathbb{CP}}^k=\mathbb{P}(A)$ for the first $k+1$ coordinates, and move ${\mathbb{CP}}^m$ to $\mathbb{P}(B)$ for the last $m+1$ coordinates. A unitary coordinate permutation accomplishes this. Every unitary matrix can be joined to the identity by diagonalizing it and varying its eigenvalue phases continuously, so this move is an ambient isotopy. Its trace gives a cobordism carrying the transported normal complex structure, and hence leaves the input cobordism class unchanged.

If $k+m\ge n$, thenTheir projectivizations are transverse and meet in ${\mathbb{CP}}^r$, where $r=k+m-n$. The direct-sum normal structure from question 1 restricts toThe quotient map defining the transverse normal-bundle isomorphism is complex-linear here, so it agrees with the standard normal complex structure and contributes no orientation sign. If $k+m<n$, the first and last coordinate subspaces have no nonzero vector in common, so their projectivizations are disjoint.

Consequently the product of linear projective subspaces in complex cobordism isThe product degree is $2(2n-k-m)$, equal to the real codimension of the stated intersection when it exists. In the equality case $k+m=n$, the representative is one point with its complex-oriented normal space.