Normal bundle

For a smooth closed embedding $i:X\hookrightarrow Y$, the normal bundle fits into $0\to TX\to i^{\ast }TY\to N_{X/Y}\to 0$.

Embed a smooth plane conic in ${\mathbf{P}}^3$ through a plane. The plane equation and conic equation form a regular sequence of degrees one and two, so its normal sheaf is ${\mathcal{O}}_C(1)\oplus {\mathcal{O}}_C(2)$. Under $C\cong {\mathbf{P}}^1$, hyperplane degree is two, hence this normal bundle is ${\mathcal{O}}_{{\mathbf{P}}^1}(2)\oplus {\mathcal{O}}_{{\mathbf{P}}^1}(4)$. The normal exact sequence splits as well because $H^1({\mathbf{P}}^1,\mathcal{O}(2))=0$.

Near a projective subspace, transverse directions are linear maps from the tautological line to the complementary coordinate vector space. This describes the actual complex normal bundle. The conjugate line and the dual line are complex-isomorphic using a Hermitian metric; neither is the original tautological line as a complex bundle in general.

// Target: geometry-and-topology.bigb

For a transverse intersection, the quotient map $TM{|}_L\to {\nu }_1{|}_L\oplus {\nu }_2{|}_L$ is onto and has kernel $TL$. This gives the displayed isomorphism of actual normal bundles. Complex structures on the two normal bundles therefore induce a complex structure on the intersection's normal bundle.

// Target: geometry-and-topology.bigb

For an oriented submanifold of half the dimension of its ambient manifold, the self-intersection number is the intersection number with a transverse perturbation. It equals the Euler number of the normal bundle.

For a smooth projective algebraic curve $C$, the normal bundle of its diagonal in $C\times C$ is $TC$: on the diagonal the quotient of $TC\oplus TC$ by the diagonal subbundle is identified with $TC$ by taking the difference. The self-intersection formula therefore gives ${\Delta }_C^2=\mathrm{deg}TC=2-2g(C)$.

For a regular closed embedding $i:X\hookrightarrow Y$ of codimension $d$, the self-intersection formula is $i^{\ast }{i}_{\ast }(\alpha )=c_d(N_{X/Y})\cap \alpha$.

For an oriented closed surface smoothly embedded in an oriented four-manifold, a transverse section of the normal bundle gives a homologous small push-off. Its signed zeros are exactly the signed intersections with the original surface. This identifies the evaluation of the Euler class with the self-intersection number.

A framing of an embedding $S^{k-1}\hookrightarrow N^{n-1}$ is a trivialization of its rank-$(n-k)$ normal bundle, equivalently a continuously varying ordered basis of normal vectors. After fixing one framing, homotopy classes of orientation-compatible framings form a torsor for $[S^{k-1},SO(n-k)]$.