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Spaces of holomorphic maps from \(\mathbb{C} P^1\) to complex Grassmann manifolds. (English) Zbl 1025.58002

Banyaga, A. (ed.) et al., Topics in low-dimensional topology. In honor of Steve Armentrout. Proceedings of the conference on low dimensional topology, University Park, PA, USA, May 1996. Singapore: World Scientific. 99-111 (1999).
The author provides a detailed proof of a well-known ‘folk theorem’. This theorem gives a description of the space of holomorphic maps from \(\mathbb CP^1\) to the complex Grassmann manifold \(G_{n,n+k}(\mathbb C)\) in terms of equivalence classes of \(\lambda\)-matrices \(M_{n,n+k}(\mathbb C[z])\), i.e. \(n\times (n + k)\) matrices with entries in the polynomial ring \(\mathbb C[z]\). The equivalence relation is given by the action of the topological group \(GL_n(\mathbb C[z])\) consisting of those \(n\times n\) \(\lambda\)-matrices whose determinant is a non-zero constant. This group acts on the space of \(n\times (n + k)\) \(\lambda\)-matrices by matrix multiplication on the left.
The author shows that the action \(GL_n(\mathbb C[z])\times M_{n,n+k}(\mathbb C[z])\to M_{n,n+k}(\mathbb C[z])\) restricts to an action \(GL_n(\mathbb C[z])\times P_{n,n+k}(\mathbb C[z])\to P_{n,n+k}(\mathbb C([z])\), where \(P_{n,n+k}(\mathbb C[z])\) is the space of polynomial maps from \(\mathbb C\) to the Stiefel manifold \(V_{n,n+k}(\mathbb C)\). The quotient space is in bijective correspondence with the space of holomorphic maps \[ f :\mathbb CP^1\to G_{n,n+k}(\mathbb C),\quad \text{Hol}(\mathbb CP^1,G_{n,n+k}(\mathbb C)) \longleftrightarrow P_{n,n+k}(\mathbb C[z])/GL_n(\mathbb C[z]). \] The space of holomorphic maps \(f :\mathbb CP^1\to G_{n,n+k}(\mathbb C)\) of degree \(d\) corresponds to the subspace of \(P_{n,n+k}(\mathbb C[z])/(GL_n(\mathbb C[z])\) consisting of those matrices such that the determinants of the minors are all polynomials of degree at most \(d\) (with at least one determinant having degree \(d\)). It is shown that when restricted to the space of holomorphic maps of degree \(d\) the above bijection is a homeomorphism.
It should be noted that the fact that a holomorphic map from \(\mathbb CP^1\) to \(G_{n,n+k}(\mathbb C)\) is locally given by a matrix of polynomials follows quickly from Chow’s Theorem and the GAGA principal. The theorem proved in this note (without reference to Chow’s Theorem or the GAGA principal) improves the local result given by Chow’s Theorem. First, it is shown that a holomorphic map \(f:\mathbb CP^1\to G_{n,n+k}(\mathbb C)\) can be represented by a single global matrix of polynomials. Second, the compact open topology on \(\text{Hol}(\mathbb CP^1, G_{n,n+k}(\mathbb C))\) agrees with the quotient topology on \(P_{n,n+k}(\mathbb C[z])/GL_n(\mathbb C[z])\) when one restricts to elements of degree \(d\).
For the entire collection see [Zbl 0988.00070].

MSC:

58D15 Manifolds of mappings
46T25 Holomorphic maps in nonlinear functional analysis
58C10 Holomorphic maps on manifolds
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