Lecture topic
Let Gr(k,Cn) be the space of k-dimensional complex subspaces of Cn, also known as the complex Grassmannian. We will show that it is a complex manifold of dimension k(n−k). Thanks to Jinhua Xu and professor Mihai Păun for explaining the details.
To begin, take P∈Gr(k,Cn) and an n−k subspace Q of Cn, such that P∩Q={0}. Then P⊕Q=Cn, so we have natural projections
A neighborhood of P, depending on Q may be described as UQ={S∈Gr(k,Cn) : S∩Q={0}}. We claim that UQ≅Hom(P,Q). The isomorphism is described by
Hom(P,Q)→UQ→Hom(P,Q),A↦{v+Av : v∈P},S↦(πQ|S)∘(πP|S)−1.
The map on the right, call it φQ, is also the chart for the manifold structure. The idea of decomposing Cn into P and Q and constructing a homomorphism from P to Q may be visualized in the following diagram.
Then Hom(P,Q)≅Hom(Ck,Cn−k)≅Ck(n−k), so Gr(k,Cn) is locally of complex dimension k(n−k). To show that there is a complex manifold structure, we need to show that the transition functions are holomorphic. Let P,P′∈Gr(k,Cn) and Q,Q′∈Gr(n−k,Cn) such that P∩Q=P′∩Q′={0}. Let X∈Hom(P,Q) such that X∈φQ(UQ∩UQ′), with φQ(S)=X and φQ′(S)=X′ for some S∈UQ∩UQ′. Define IX(v)=v+Xv, and note the transition map takes X to
X′=φQ′∘φ−1Q(X)(definition)=φQ′(S)(assumption)=(πQ′|S)∘(πP′|S)−1(definition)=(πQ′|S)∘IX∘I−1X∘(πP′|S)−1(creative identity)=(πQ′|S∘IX)∘(πP′|S∘IX)−1(redistribution)=(πQ′|P+πQ′|Q∘X)∘(πP′|P+πP′|Q∘X).(definition)
At this last step we have compositions and sums of homomorphisms and linear maps, which are all holomorphic. Hence the transition functions of Gr(k,Cn) are holomorphic, so it is a complex manifold.
To begin, take P∈Gr(k,Cn) and an n−k subspace Q of Cn, such that P∩Q={0}. Then P⊕Q=Cn, so we have natural projections
A neighborhood of P, depending on Q may be described as UQ={S∈Gr(k,Cn) : S∩Q={0}}. We claim that UQ≅Hom(P,Q). The isomorphism is described by
Hom(P,Q)→UQ→Hom(P,Q),A↦{v+Av : v∈P},S↦(πQ|S)∘(πP|S)−1.
The map on the right, call it φQ, is also the chart for the manifold structure. The idea of decomposing Cn into P and Q and constructing a homomorphism from P to Q may be visualized in the following diagram.
Then Hom(P,Q)≅Hom(Ck,Cn−k)≅Ck(n−k), so Gr(k,Cn) is locally of complex dimension k(n−k). To show that there is a complex manifold structure, we need to show that the transition functions are holomorphic. Let P,P′∈Gr(k,Cn) and Q,Q′∈Gr(n−k,Cn) such that P∩Q=P′∩Q′={0}. Let X∈Hom(P,Q) such that X∈φQ(UQ∩UQ′), with φQ(S)=X and φQ′(S)=X′ for some S∈UQ∩UQ′. Define IX(v)=v+Xv, and note the transition map takes X to
X′=φQ′∘φ−1Q(X)(definition)=φQ′(S)(assumption)=(πQ′|S)∘(πP′|S)−1(definition)=(πQ′|S)∘IX∘I−1X∘(πP′|S)−1(creative identity)=(πQ′|S∘IX)∘(πP′|S∘IX)−1(redistribution)=(πQ′|P+πQ′|Q∘X)∘(πP′|P+πP′|Q∘X).(definition)
At this last step we have compositions and sums of homomorphisms and linear maps, which are all holomorphic. Hence the transition functions of Gr(k,Cn) are holomorphic, so it is a complex manifold.
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