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homotopy of rational maps

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Complex geometry

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

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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)homologyisoinlowdegreeMaps cts(X 1,X 2). 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 1X_1.

In many cases of interest, such as when the domain X 1X_1 is a non-singular complex curve/Riemann surface and the codomain X 2X_2 a complex projective space, then all rational maps from X 1X_1 to X 2X_2 are automatically regular (e.g. Shafarevich Vol1, Cor. 2.3).

Maps from a Riemann surface to a projective space

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

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

    (1)f *(1)=dH 2(Σ 2;). f^\ast(1) \,=\, d \,\in\, \mathbb{Z} \,\simeq\, H^2(\Sigma_2;\, \mathbb{Z}) \,.
  • For Σ\Sigma a compact connected Riemann surface write g Σg_\Sigma \in \mathbb{N} for its genus.

Proposition

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

(2)Maps rat deg=d(Σ,P n)iMaps cts deg=d(Σ,P n) 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

into

is

(Segal 1979, Prop. 1.2, 1.3)

Example

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

Maps rat deg=d(P 1,P 1)iMaps top deg=d(S 2,S 2) 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 d\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 ΣP 3\Sigma \to \mathbb{C}P^3 (2) are interpreted as moduli spaces of D1-brane-instantons in the twistor space P 3 \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 nn 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 P n\mathbb{C}P^n is an Oka manifold by this Prop.) to the complementary case of connected non-compact (“open”) Riemann surfaces Σ\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(Σ,P n)iMaps top(Σ,P n) 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.

Maps between projective spaces

Generalization to higher dimensional domain spaces:

  • Say that the degree of a rational map f:P n 1P n 2f \;\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:P n 1P n 2f \;\colon\; \mathbb{C}P^{n_1} \to \mathbb{C}P^{n_2} between two complex projective spaces has degree dd \in \mathbb{N} (Rem. ) if this is the induced factor for pullback in their second integral ordinary cohomology (see here)

    (3)H 2(P n 2;) f * H 2(P n 1;) 1 d. \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 1n 1n 21 \leq n_1 \leq n_2, and dd \in \mathbb{N}, the inclusion

Maps rat deg=d(P n 1,P n 2)Maps cts deg=d(P n 1,P n 2) 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 dd into the mapping space of continuous functions of degree dd in the sense of (3) induces an isomorphism on integral ordinary homology in degrees

(2n 12n 2+1)((d+2)/2+1), \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 P n 2\mathbb{C}P^{n_2} is allowed to be any smooth toric variety (Mostovoy & Munguia-Villanueva 2012, Kozlowski & Yamaguchi 2018).

References

Maps out of Riemann surfaces

The original theorem for maps from compact Riemann surfaces to projective spaces:

Further discussion:

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)

… a toric variety:

… a flag manifold:

Application to Skyrmions:

  • 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)

Maps between projective spaces

On maps between complex projective spaces:

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:

On maps from real projective space to complex projective space:

and equivariantly:

Maps from projective spaces to toric varieties

On maps from complex projective space to smooth toric varieties:

Last revised on July 29, 2021 at 12:52:04. See the history of this page for a list of all contributions to it.