Intersection theory

In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form.

There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov–Witten theory and the extension of intersection theory from schemes to stacks.

Topological intersection form

For a connected oriented manifold

M

{\displaystyle M}

of dimension

2

n

{\displaystyle 2n}

the intersection form is defined on the

n

{\displaystyle n}

-th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class

[

M

]

{\displaystyle [M]}

in

H

2

n

(

M

,

∂

M

)

{\displaystyle H_{2n}(M,\partial M)}

. Stated precisely, there is a bilinear form

λ

M

:

H

n

(

M

,

∂

M

)

×

H

n

(

M

,

∂

M

)

→

Z

{\displaystyle \lambda _{M}\colon H^{n}(M,\partial M)\times H^{n}(M,\partial M)\to \mathbf {Z} }

given by

λ

M

(

a

,

b

)

=

⟨

a

⌣

b

,

[

M

]

⟩

∈

Z

{\displaystyle \lambda _{M}(a,b)=\langle a\smile b,[M]\rangle \in \mathbf {Z} }

with

λ

M

(

a

,

b

)

=

(

−

1

)

n

λ

M

(

b

,

a

)

∈

Z

.

{\displaystyle \lambda _{M}(a,b)=(-1)^{n}\lambda _{M}(b,a)\in \mathbf {Z} .}

This is a symmetric form for n even (so 2n = 4k doubly even), in which case the signature of M is defined to be the signature of the form, and an alternating form for n odd (so 2n = 4k + 2 is singly even). These can be referred to uniformly as ε-symmetric forms, where ε = (−1)n = ±1 respectively for symmetric and skew-symmetric forms. It is possible in some circumstances to refine this form to an ε-quadratic form, though this requires additional data such as a framing of the tangent bundle.