nLab homotopy of rational maps

Idea

On the stable homotopy type of mapping spaces of (regular) rational maps.

Under good conditions, the subspace inclusion of the space of rational maps (regular, see Rem. ), between given projective complex manifolds or algebraic varieties, into the mapping space of all continuous maps
induces an isomorphism in integral ordinary homology in low degrees, while for maps out of the Riemann sphere this is even an isomorphism on homotopy groups in low degrees:

$$Maps_{rat}(X_1, X_2) \xhookrightarrow{ {homology\; iso} \atop {in\; low\; degree} } Maps_{cts}(X_1, X_2) \,.$$

This phenomenon originates in results of Segal 1979 and is commonly referred to by Segal’s name (e.g. “theorems of Segal-type” in Friedlander & Lawson 1997, Sec. 5.C.

Whenever this holds it provides

1. from left to right: homotopy theoretic tools for analyzing moduli spaces of rational hypersurfaces;

2. from right to left: small algebraic models for stable homotopy types of mapping spaces

at least up to some dimension.

Details

Some remarks on the terminology being used:

Remark

(“degree”)
Most or all of the following statement invoke an integer “degree” of continuous functions. Beware that this is not the degree of a continuous function (see there) in the usual sense of algebraic topology, except in special cases (such as the archetypical example ).

Remark

(“rational maps”)
It it tradition (starting with Segal) to speak of rational maps in the following, but in the end the focus on regular rational maps (“morphisms”: e.g. Friedlander & Lawson 1997, p. 27)), as is necessary to regard them as continuous functions defined everywhere on the given domain $X_1$.

In many cases of interest, such as when the domain $X_1$ is a non-singular complex curve/Riemann surface and the codomain $X_2$ a complex projective space, then all rational maps from $X_1$ to $X_2$ are automatically regular (e.g. Shafarevich Vol1, Cor. 2.3).

For review of more details see Havlicek 92, §1.

Maps from a Riemann surface to a projective space

• For $n \in \mathbb{N}_+$, consider complex projective n-space $\mathbb{C}P^n$.

• Say that a continuous map $f \;\colon\; \Sigma_2 \to \mathbb{C}P^n$ out of a 2-dimensional manifold has degree $d \in \mathbb{N}$ (Rem. ) if the pullback of the generator $1 \in \mathbb{Z} \simeq H^2\big( \mathbb{C}P^n;\, \mathbb{Z}\big)$ (see here) is

  (1)$$f^\ast(1) \,=\, d \,\in\, \mathbb{Z} \,\simeq\, H^2(\Sigma_2;\, \mathbb{Z}) \,.$$

• For $\Sigma$ a compact connected Riemann surface write $g_\Sigma \in \mathbb{N}$ for its genus.

Proposition

(Segal’s theorem)
For $\Sigma$ a compact connected Riemann surface, the inclusion

(2)$$Maps_{ {rat} }^{deg = d} \big( \Sigma ,\, \mathbb{C}P^n \big) \xhookrightarrow{ \;\; i \;\; } Maps_{ {cts} }^{deg = d} \big( \Sigma ,\, \mathbb{C}P^n \big)$$

of

• the topological subspace of rational maps to complex projective n-space

  (the moduli spaces of rational complex curves in $\mathbb{C}P^n$)

into

• the connected component of maps of degree $d$ of the mapping space (with the compact open topology) of all continuous maps

is

• for $g_\Sigma = 0$ (i.e. $\Sigma$ the Riemann sphere):

  a $d (2 n -1)$-equivalence, hence

  an isomorphism on homotopy groups in degrees $\lt d (2 n -1)$

  and a surjection on homotopy groups in degree $d (2 n - 1)$

• for $g_\Sigma \geq 1$:

  an isomorphism on ordinary homology groups in degrees $\lt (d - 2g) (2 n -1)$

  and a surjection on homology groups in degree $(d - 2g) (2 n -1)$.

This is due to Segal 1979, Prop. 1.2, 1.3 (bewaring the Note on terminology on p. 44). The analogous statement for rational curves in real projective spaces is in Mostovoy 01.

Example

(the archetypical case)
In the special case that $n = 1$ and $\Sigma = S^2$ is the 2-sphere with its complex structure, so that both domain and codomain are the Riemann sphere $\mathbb{C}P^1$, Prop. says that

$$Maps_{ {rat} }^{deg = d} \big( \mathbb{C}P^1 ,\, \mathbb{C}P^1 \big) \xhookrightarrow{ \;\; i \;\; } Maps_{ {top} }^{deg = d} \big( S^2 ,\, S^2 \big)$$

is an isomorphism on homotopy groups up to degree $\leq d$.

(Segal 1979, Prop. 1.1’)

Remark

(relation to Yang-Mills monopoles)
Example controls the classification of Yang-Mills monopoles. See there for more

Remark

(relation to Gromov-Witten theory) A compactification and quotient stack of the space of rational maps in (2) is considered in Gromov-Witten theory, e.g. Bertram 2002, p. 9.

Remark

(relation to twistor string theory)
In the context of twistor string theory, the spaces of rational maps $\Sigma \to \mathbb{C}P^3$ (2) are interpreted as moduli spaces of D1-brane-instantons in the twistor space $\mathbb{C}P^3$ (Witten 2004, Sec. 3).

Such rational maps are also argued to encode scattering amplitudes in D=4 N=8 supergravity (Cachazo & Skinner 2012, Adamo 2015).

Here the number of poles in the rational function is the number $n$ of particles in the n-point function, and the genus and degree encode the particle’s helicity and the loop order of the scattering amplitude.

Remark

(comparison to the homotopical Oka principle)
Prop. may be compared to the homotopical Oka principle, which applies (since $\mathbb{C}P^n$ is an Oka manifold by this Prop.) to the complementary case of connected non-compact (“open”) Riemann surfaces $\Sigma$ (which are Stein manifolds by this Example), in which case it says that the corresponding inclusion of the space of holomorphic maps

$$Maps_{ {hol} } \big( \Sigma ,\, \mathbb{C}P^n \big) \xhookrightarrow{ \;\; i \;\; } Maps_{ {top} } \big( \Sigma ,\, \mathbb{C}P^n \big)$$

induces an isomorphism on all homotopy groups, hence is a weak homotopy equivalence – reflecting the fact that non-compactness of the Riemann surfaces and absence of any asymptotic boundary condition provides a large supply of holomorphic functions.

In fact:

Proposition

The full homotopy type of the space of pointed rational maps from the Riemann sphere to complex projective $n$-space $\mathbb{C}P^n$ of algebraic degree $d$ is that of the configuration space of at most $d$ points in $\mathbb{R}^2$ with labels in $S^{2n-1}$:

$$Maps_{ {rat} }^{deg = d} \big( \Sigma ,\, \mathbb{C}P^n \big) \;\simeq_{htpy}\; \underset{ \leq d}{Conf} \big( \mathbb{R}^2; S^{2k+1} \big)$$

(Cohen & Shimamoto 91, Theorem 1)

Maps between projective spaces

Generalization to higher dimensional domain spaces:

• Say that the degree of a rational map $f \;\colon\; \mathbb{C}P^{n_1} \to \mathbb{C}P^{n_2}$ between two complex projective spaces is the degree of the polynomials that define it.

• In generalization of (1), say that a continuous map $f \;\colon\; \mathbb{C}P^{n_1} \to \mathbb{C}P^{n_2}$ between two complex projective spaces has degree $d \in \mathbb{N}$ (Rem. ) if this is the induced factor for pullback in their second integral ordinary cohomology (see here)

  (3)$$\array{ \mathbb{Z} \simeq H^2\big( \mathbb{C}P^{n_2};\, \mathbb{Z}\big) &\xrightarrow{ f^\ast }& H^2\big( \mathbb{C}P^{n_1};\, \mathbb{Z}\big) \simeq \mathbb{Z} \\ 1 &\mapsto& d } \,.$$

Proposition

For $1 \leq n_1 \leq n_2$, and $d \in \mathbb{N}$, the inclusion

$$Maps^{deg = d}_{rat} \big( \mathbb{C}P^{n_1} ,\, \mathbb{C}P^{n_2} \big) \xhookrightarrow{\;\;\;\;} Maps^{deg = d}_{cts} \big( \mathbb{C}P^{n_1} ,\, \mathbb{C}P^{n_2} \big)$$

of the topological subspace of rational maps of algebraic degree $d$ into the mapping space of continuous functions of degree $d$ in the sense of (3) induces an isomorphism on integral ordinary homology in degrees

$$\leq\, \big( 2 n_1 - 2 n_2 + 1 \big) \big( \lfloor(d+2)/2\rfloor + 1 \big) \,,$$

where $\lfloor - \rfloor$ denotes the integer floor of a rational number.

(Mostovoy 2006, Theorem 2, with corrected proof in Mostovoy 2012)

An analogous result for real projective spaces is in Adamaszek, Kozlowski & Yamaguchi 2008.

Prop. generalizes to the case that the codomain $\mathbb{C}P^{n_2}$ is allowed to be any smooth toric variety (Mostovoy & Munguia-Villanueva 2012, Kozlowski & Yamaguchi 2018).

Related concepts

• Yang-Mills monopole

• Oka principle

• Cohomotopy

References

Maps out of Riemann surfaces

The original theorem for rational maps/continuous maps from compact Riemann surfaces to complex projective spaces:

• Graeme Segal, The topology of spaces of rational functions, Acta Math. Volume 143 (1979), 39-72 (euclid:1485890033)

Further discussion:

• Fred Cohen, Ralph Cohen, B. M. Mann, R. J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math (1991) 166: 163 (doi:10.1007/BF02398886)

• Eric M. Friedlander, H. Blaine Lawson, Section 5.C of: Duality Relating Spaces of Algebraic Cocycles and Cycles, Topology Volume 36, Issue 2, March 1997, Pages 533-565 (pdf)

• Ralph L. Cohen, John D. S. Jones, Graeme B. Segal, Stability for holomorphic spheres and Morse theory, in: K. Grove, I. H. Madsen, E. K. Pedersen (eds.) Geometry and Topology: Aarhus, Contemporary Mathematics

  Volume: 258 (2000) (arXiv:math/9904185, ISBN:978-0-8218-2158-9)

• Yasuhiko Kamiyama, Remarks on spaces of real rational functions, The Rocky Mountain Journal of Mathematics Vol. 37, No. 1 (2007), pp. 247-257 (jstor:44239357)

The analog for rational curves into real projective spaces:

• Jacob Mostovoy, Spaces of Rational Loops on a Real Projective Space, Transactions of the American Mathematical Society, Vol. 353, No. 5 (May, 2001), pp. 1959-1970 (jstor:221802)

Identification of the higher homotopy groups of $Maps_{rat}(\mathbb{C}P^1, \mathbb{C}P^1)$:

• Martin A. Guest, Andrzej Kozlowski, M. Murayama, Kohhei Yamaguchi, The homotopy type of the space of rational functions, J. Math. Kyoto Univ. 35(4): 631-638 (1995) (doi:10.1215/kjm/1250518652)

Identification of the full homotopy type of $Maps_{rat}(\mathbb{C}P^1, \mathbb{C}P^n)$ with a configuration space of points:

• Ralph L. Cohen, Don H. Shimamoto, Rational Functions, Labelled Configurations, and Hilbert Schemes, Journal of the London Mathematical Socienty 43 2 (1991) 509-528 (doi:10.1112/jlms/s2-43.3.509)

Generalization to the case that the codomain is

… a Grassmannian:

• Frances Kirwan, On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Mat. 24(1-2): 221-275 (1985) (doi:10.1007/BF02384399)

• Benjamin M. Mann, R. James Milgram, Some spaces of holomorphic maps to complex Grassmann manifolds, J. Differential Geom. 33(2): 301-324 (1991) (doi:10.4310/jdg/1214446318)

• John W. Havlicek, On spaces of holomorphic maps from two copies of the riemann sphere to complex grassmannians, Stanford University 1992 (proquest:304009877)

… a toric variety:

• Martin A. Guest, Configuration spaces and the space of rational curves on a toric variety, Bull. Amer. Math. Soc. 31 (1994), 191-196 (doi:10.1090/S0273-0979-1994-00515-4)

• Martin A. Guest, The topology of the space of rational curves on a toric variety, Acta Math. 174(1): 119-145 (1995) (doi:10.1007/BF02392803, arXiv:alg-geom/9301005)

… a flag manifold:

• C. P. Boyer, B. M. Mann, J. C. Hurtubise, R. J. Milgram, The topology of the space of rational maps into generalized flag manifolds, Acta Mathematica. 1994 Mar 1;173(1):61-101 (doi:10.1007/BF02392569)

• J. C. Hurtubise, Holomorphic maps of a Riemann surface into a flag manifold, J. Differential Geom. 43(1): 99-118 (1996) (doi:10.4310/jdg/1214457899)

Application to the quantization of Skyrmions (via their rational map Ansatz, see the references there):

• Steffen Krusch, Homotopy of rational maps and the quantization of Skyrmions, Annals of Physics Volume 304, Issue 2, April 2003, Pages 103-127 (doi:10.1016/S0003-4916(03)00014-9)

• Steffen Krusch, Skyrmions and Rational Maps, talk at KIAS 2004 (pdf, pdf)

• Steffen Krusch, Quantization of Skyrmions (arXiv:hep-th/0610176)

Maps between projective spaces

On maps between complex projective spaces:

• Jacob Mostovoy, Spaces of rational maps and the Stone–Weierstrass theorem, Topology Volume 45, Issue 2, March 2006, Pages 281-293 (doi:10.1016/j.top.2005.08.003)

corrected proof in:

• Jacob Mostovoy, Truncated Simplicial Resolutions and Spaces of Rational Maps, The Quarterly Journal of Mathematics, Volume 63, Issue 1, March 2012, Pages 181–187 (doi:10.1093/qmath/haq031)

On maps between real projective spaces:

• Michal Adamaszek, Andrzej Kozlowski, Kohhei Yamaguchi, Spaces of algebraic and continuous maps between real algebraic varieties, Quart. J. Math. 62 (2011), 771–790 (arXiv:0809.4893, doi:10.1093/qmath/haq029)

On maps from real projective space to complex projective space:

• Andrzej Kozlowski, Kohhei Yamaguchi, Spaces of algebraic maps from real projective spaces into complex projective spaces (arXiv:0812.3954), in: Yves Félix et al. (eds.) Homotopy Theory of Function Spaces and Related Topics, Contemporary Mathematics Volume: 519; 2010 (ISBN:978-0-8218-4929-3)

and equivariantly:

• Andrzej Kozlowski, Kohhei Yamaguchi, Spaces of equivariant algebraic maps from real projective spaces into complex projective spaces, RIMS Kôkyûroku Bessatsu B39 (2013), 051−061 (arXiv:1109.0353, published pdf)

Maps from projective spaces to toric varieties

On maps from complex projective space to smooth toric varieties:

• Jacob Mostovoy, Erendira Munguia-Villanueva, Spaces of morphisms from a projective space to a toric variety, Revista Colombiana de Matematicas 48 1 (2014) (arXiv:1210.2795, published pdf)

• Andrzej Kozlowski, Kohhei Yamaguchi, The homotopy type of spaces of rational curves on a toric variety, Topology and its Applications Volume 249, 1 November 2018, Pages 19-42 (doi:10.1016/j.topol.2018.06.006)