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Thursday, September 22, 2016

The Grassmannian is a complex manifold

 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(nk). Thanks to Jinhua Xu and professor Mihai Păun for explaining the details.

To begin, take PGr(k,Cn) and an nk subspace Q of Cn, such that PQ={0}. Then PQ=Cn, so we have natural projections
A neighborhood of P, depending on Q may be described as UQ={SGr(k,Cn) : SQ={0}}. We claim that UQHom(P,Q). The isomorphism is described by
Hom(P,Q)UQHom(P,Q),A{v+Av : vP},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,Cnk)Ck(nk), so Gr(k,Cn) is locally of complex dimension k(nk). To show that there is a complex manifold structure, we need to show that the transition functions are holomorphic. Let P,PGr(k,Cn) and Q,QGr(nk,Cn) such that PQ=PQ={0}. Let XHom(P,Q) such that XφQ(UQUQ), with φQ(S)=X and φQ(S)=X for some SUQUQ. Define IX(v)=v+Xv, and note the transition map takes X to
X=φQφ1Q(X)(definition)=φQ(S)(assumption)=(πQ|S)(πP|S)1(definition)=(πQ|S)IXI1X(πP|S)1(creative identity)=(πQ|SIX)(πP|SIX)1(redistribution)=(πQ|P+πQ|QX)(πP|P+πP|QX).(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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