Definitions Kähler_manifold

1 definitions

1.1 symplectic viewpoint
1.2 complex viewpoint
1.3 riemannian viewpoint

definitions

since kähler manifolds equipped several compatible structures, can described different points of view:

symplectic viewpoint

a kähler manifold symplectic manifold (x,ω) equipped integrable almost-complex structure j compatible symplectic form ω, meaning bilinear form

g
(
u
,
v
)
=
ω
(
u
,
j
v
)


{\displaystyle g(u,v)=\omega (u,jv)}

on tangent space of x @ each point symmetric , positive definite (and hence riemannian metric on x).

complex viewpoint

a kähler manifold complex manifold x hermitian metric h associated 2-form ω closed. in more detail, h gives positive definite hermitian form on tangent space tx @ each point of x, , 2-form ω defined by

ω
(
u
,
v
)
=

r
e


h
(
i
u
,
v
)


{\displaystyle \omega (u,v)=\mathrm {re} \;h(iu,v)}

for tangent vectors u , v (where complex number





−
1




{\displaystyle {\sqrt {-1}}}

). kähler manifold x, kähler form ω real closed (1,1)-form. kähler manifold can viewed riemannian manifold, riemannian metric g defined by

g
(
u
,
v
)
=

r
e


h
(
u
,
v
)
.


{\displaystyle g(u,v)=\mathrm {re} \;h(u,v).}

equivalently, kähler manifold x hermitian manifold of complex dimension n such every point p of x, there holomorphic coordinate chart around p in metric agrees standard metric on c order 2 near p. is, if chart takes p 0 in c, , metric written in these coordinates hab=⟨∂/∂za,∂/∂zb), then

h

a
b


=

δ

a
b


+
o
(
∥
z

∥

2


)


{\displaystyle h_{ab}=\delta _{ab}+o(\|z\|^{2})}

for a,b in {1,...,n}.

since 2-form ω closed, determines element in de rham cohomology h(x,r), known kähler class.

riemannian viewpoint

a kähler manifold riemannian manifold x of dimension 2n holonomy group contained in unitary group u(n). equivalently, there complex structure j on tangent space of x @ each point (that is, real linear map tx j = −1) such j preserves metric g (meaning g(ju,jv) = g(u,v)) , j preserved parallel transport.