STABLE MAPS OF GENUS ZERO TO FLAG SPACES

Yu.I.Manin

Max–Planck–Institut für Mathematik, Bonn

Abstract

We calculate a generating series for the virtual Euler–Poincaré characteristics of the spaces of stable maps of genus zero to flag spaces using the summation over trees technique.

§0. Introduction

Topological quantum field theory led recently to a spectacular progress in numerical algebraic geometry. It was shown that generating functions of certain charactertistic numbers of modular spaces of stable algebraic curves with labelled points satisfy remarkable differential equations of KP–type (E. Witten, M. Kontsevich). In a later series of developments, this was generalized, partly conjecturally, to the spaces of maps of curves into algebraic varieties leading to the Mirror Conjecture and the construction of quantum cohomology.

The key technical notion in the context of algebraic geometry is that of a stable map introduced by M. Kontsevich (cf. [K] and [BM]) following the earlier work by M. Gromov in symplectic geometry. It provides a natural compactification of spaces of maps, in the same way as stable curves compactify moduli spaces.

We will be working over a ground field. Let W be an algebraic variety.

0.1. Definition. A stable map (to W) is a structure (C; x_{1}, …, x_{n}; f) consisting of the following data.

a). (C; x_{1}, …, x_{n}) is a connected complete reduced curve with n ≥ 0 labelled pairwise distinct non–singular points x_{i} and at most ordinary double singular points.

b). f : C → W is a morphism having no non–trivial first order infinitesimal automorphisms identical on W and x_{i}’s (stability). This means that every irreducible component of C of genus zero (resp. 1) has at least three (resp. one) special points (inverse images of singular and labelled points) on its normalization.

A family of stable maps parametrized by a noetherian scheme S is a structure consisting of a flat proper morphism π : C → S, n sections x_{i} : S → C, and a morphism f : C → W whose restriction to each geometric fiber of π is a stable map in the sense of the previous Definition.

Families of stable maps form an algebraic stack in the sense of [DM].

For fixed n ≥ 0, g ≥ 0 and an algebraic homology class of dimension two β, denote by \overline{M}_{g,n}(W, β) the substack of maps for which g is the arithmetic genus of C and β = f_{∗}([C]). For the proof of the following theorem see [K], [BM] and [FP].

0.2. Theorem. a). If W is projective, then \overline{M}_{g,n}(W, β) is a proper separated algebraic stack of finite type.

b). Assume that W is convex in the following sense: H^{1}(C, f^{∗}(T_{W})) = 0 for any stable map f : C → W of genus zero. Then the stacks \overline{M}_{0,n}(W, β) are smooth.

Denote by M_{0,n}(W, β) the big cell of this stack over which C is smooth. The complement of this cell is a divisor with normal crossings.

In addition, the spaces of geometric points of these stacks are represented by algebraic schemes, crude moduli spaces of stable maps, \overline{M}_{g,n}(W, β) and M_{g,n}(W, β) respectively. If (C; x_{1}, …, x_{n}; f) is a stable map, we denote by [(C; x_{1}, …, x_{n}; f)], or simply [f], the corresponding point. Two maps [f^{(i)}], i = 1, 2, define the same point iff there is an isomorphism g : C^{(1)} → C^{(2)} such that g(x_{i}^{(1)}) = x_{i}^{(2)} for all i and g ∘ f^{(2)} = f^{(1)}.

This paper is a sequel to [M].

We will consider mostly the case g = 0, W = G/P (generalized flag spaces). Although flag spaces are convex, the respective spaces of stable maps are not smooth but only orbifolds in general. For a recent account of these spaces see [Th]. Our goal is to calculate their virtual Poincaré polynomials, or rather an appropriate generating function for these polynomials. This calculation generalizes the one made for \overline{M}_{0,n} in [M] and [G]. Remarkably, up to a change of variables, this function satisfies the same universal differential equation as that in [M], the dependence on W being reflected only in the initial condition which involves the Eisenstein series of W = G/P: see 2.3 below.

§1. Stratification of the space of stable maps

1.1. Virtual Euler–Poincaré maps. Let R be a commutative associative Q– algebra, V ar the category of algebraic varieties, not necessarily complete, smooth, or irreducible, over a fixed ground field.

We will call a map Ob V ar → R : V ↦ [V] a virtual Euler–Poincaré map if the following conditions are satisfied:

i). [V] depends only on the isomorphism class of V.

ii). Additivity: if V = ∐_{i} V_{i} is a finite disjoint union of locally closed subsets (“strata”), then [V] = ∑_{i}[V_{i}].

iii). Multiplicativity: [∏_{j} V_{j}] = ∏_{j}[V_{j}].

¿From ii) and iii) it follows that if V → B is a Zariski locally trivial fiber space with fiber F, then [V] = [B][F].

If we work over C, and R = Q[q] (resp. R = Q[r, s]) the main examples are virtual Poincaré (resp. Hodge) polynomials:

P_{V} (q) := ∑_{i,j}(−1)^{i+j}dim(gr_{W}^{j} H_{c}^{i}(V))q^{j},  (1.1)

H_{V} (r, s) := ∑_{i,j,k}(−1)^{i+j+k}h^{j,k}(H_{c}^{i}(V))r^{j}s^{k},  (1.2)

and also the virtual Euler characteristics χ(V) = P_{V} (−1). These properties were for the first time systematically used by Danilov and Khovanskii in the toric geometry.

Recently C. Soulé and H. Gillet ([GS]) established that the map V ↦ class of the motive h^{∗}(V) of V in the K_{0}–ring of Grothendieck’s motives extends from projective smooth varieties to V ar and becomes an Euler–Poincaré map. For a somewhat different construction see [GuNA].

On the subcategory generated by the (dual) Tate motive one can identify the Gillet–Soulé map with virtual Poincaré map putting h^{∗}(P^{1}) = q^{2} +1, and calculate the latter via point count over F_{q^{2}}. Most of our calculations are in fact restricted to this situation.

We will have also to localize R, most notably by inverting [P GL(2)] = q^{2}(q^{4}−1). Since the Euler characteristic of this manifold vanishes, we have to apply a limiting procedure q^{2} → 1 producing logarithms and infinite ramification in our formulas for generating functions of χ, as already happened in [M].

1.2. Poincaré polynomials of \underline{Map}_{β}(P^{1}, W). If W is a flag manifold, the scheme of maps of P^{1} to W landing in β is a smooth (generally non–complete) manifold. Its Gillet–Soulé motive lies in the Tate’s subring, and the generating function

E(W, z) := ∑_{β∈B}[\underline{Map}_{β}(P^{1}, W)]z^{β}  (1.3)

is rational in z. Here B ⊂ N means the effective subsemigroup of the lattice N of algebraic homology classes of curves, and z^{β} are formal monomials, elements of the dual lattice M written multiplicatively.

In fact, in the F_{q^{2}}–avatar, (1.3) essentially coincides with an appropriate Eisenstein series discussed e.g. in [H]. If we introduce a complex vector variable s ∈ M_{C} and replace z^{β} by q^{−2(β.s)}, the poles of Eisenstein series will lie on the complexified walls of the ample cone B_{R}^{t} in N shifted by −K_{W} , and reflections with respect to the walls will generate the functional equations.

1.2.1. Example. We have

E(P^{n}, z) = \frac{1 − q^{2n+2}}{1 − q^{2}} \frac{1 − q^{2}z}{1 − q^{2n+2}z} .  (1.4)

In fact, denote by N_{d} the number of non–zero (n+1)–tuples (f_{0}(t_{0}, t_{1}), …, f_{n}(t_{0}, t_{1})) of coprime forms of degree d over F_{q^{2}} divided by |F_{q^{2}}^{∗}| = q^{2} − 1. Since the number of all such (n + 1)–tuples is q^{2(n+1)(d+1)} − 1 and the map ((f_{0}, …, f_{n}), g) ↦ (f_{0}g, …, f_{n}g) with fixed g ≠ 0 and degrees of f_{i}, g has all fibers of cardinality q^{2}−1, we have

∑_{k=0}^{d} N_{d−k}(q^{2(k+1)} − 1) = q^{2(k+1)(d+1)} − 1,

from which (1.4) easily follows.

Identifying M = Pic P^{n} with Z via O(1) ↦ 1 and putting z = q^{−2s} one sees that the real pole of (1.4) is s = n + 1 = −K_{P^{n}}.

1.3. Big cells of stable map spaces. We will say that a stable map (C; x_{1}, …, x_{n}; f) to W belongs to the big cell M_{0,k}(W, β) if C ≅ P^{1}, and f_{∗}([C]) = β. We will explain how these big cells are related to \underline{Map}_{β}(P^{1}, W).

a). β = 0. In this case, k ≥ 3, and the map

(P^{1}; x_{1}, …, x_{k}; f) ↦ (f(x_{1}), [(P^{1}; x_{1}, …, x_{k})])

induces the identification

M_{0,k}(W, 0) ≅ W × M_{0,k}.  (1.5)

In particular, when W is a point, we get simply M_{0,k}.

b). β ≠ 0. In this case, a choice of three points x_{1}, x_{2}, x_{3} in P^{1} defines an isomorphism

M_{0,3}(W, β) ≅ \underline{Map}_{β}(P^{1}, W).  (1.6)

Let G^{(i)} ⊂ P GL(2) be the subgroup fixing x_{1}, …, x_{i}, i = 0, …, 3. Then G^{(i)} freely acts upon M_{0,3}(W, β), and we have

M_{0,i}(W, β) ≅ \underline{Map}_{β}(P^{1}, W)/G^{(i)}, i ≤ 3.  (1.7)

Finally, for k ≥ 4 the forgetful map M_{0,k}(W, β) → M_{0,3}(W, β) identifies M_{0,k}(W, β) with a locally trivial fibration over \underline{Map}_{β}(P^{1}, W) with fiber

M_{0,k} = ((P^{1})^{k} \ ∩_{i<j}Δ_{ij})/P GL(2).

To summarize, we have the following formula for the virtual Euler–Poincaré class of M_{0,k}(W, β) valid for β = 0 as well:

1.3.1. Proposition. We have

[M_{0,k}(W, β)] = [\underline{Map}_{β}(P^{1}, W)]\binom{[P^{1}]}{k}k!\frac{1}{[P GL(2)]}.  (1.8)

In fact,

[M_{0,k}] = \binom{[P^{1}] − 3}{k − 3}(k − 3)!, [P GL(2)] = \binom{[P^{1}]}{3}3!.

We will now construct strata of the space of stable maps numbered by marked trees. Their virtual Euler–Poincaré classes will be expressed via products of those of big cells.

1.4. Trees. In this paper, a tree τ is a finite connected simply connected CW– complex of dimension 1 or 0 (one–vertex tree). A flag of a tree is a pair consisting of a vertex and an adjoining edge. The valency |v| of a vertex v is the number of flags containing v. The sets of vertices (resp. edges, flags) of τ are denoted V_{τ} (resp. E_{τ} , F_{τ}).

For a set S, P(S) denotes the set of its subsets. As above, B is the semigroup of classes of effective algebraic curves in a flag manifold W.

1.4.1. Definition. A (k, W)–marking of a tree τ is a map

μ : V_{τ} → B × P(1, …, k) : v ↦ (β_{v}, S_{v}),

satisfying the following conditions:

a). The family {S_{v}|v ∈ V_{τ} , S_{v} ≠ ∅} forms a partition of {1, …, k} into pairwise disjoint subsets. (For k = 0, we interpret P as one–element set, and this condition as empty).

b). If β_{v} = 0, then |v| + |S_{v}| ≥ 3.

There is an obvious notion of isomorphism of marked trees.

1.4.1. Type of a stable map. Let (C; x_{1}, …, x_{k}; f) be a stable map of genus 0 to W. Its combinatorial type is, by definition, the isomorphism class of the dual tree of C with the marking depending on f. We can describe it as follows. First of all, put

V_{τ} = {irreducible components C_{v} of C},
E_{τ} = {intersection points of irreducible components}.

Thus a flag of τ is a pair (intersection point of two components, one of these components). Furthermore,

β_{v} := f_{∗}([C_{v}]), S_{v} = {i|x_{i} ∈ C_{v}}.

Obviously, β_{v} = 0 means that f(C_{v}) is a point so that condition 1.4.1 b) means stability.

1.5. Maps of fixed type. Consider a combinatorial type (τ, μ). For any vertex v ∈ V_{τ}, construct the stack M_{0,S_{v}}(W, β_{v}) parametrizing stable maps of P^{1} with points labelled by S_{v}, of class β_{v}. For any t ∈ S_{v}, there is a canonical evaluation map ev_{t} : M_{0,S_{v}}(W, β_{v}) → W : [f] ↦ f(x_{t}) where x_{t} is the point marked by t. Put

M_{0,k}(W, β, τ, μ) = ∏_{W} M_{0,S_{v}}(W, β_{v}),

where ∏_{W} means the partial fibered product over W which leaves in the total product only those families of stable maps (f_{v} | v ∈ V_{τ}) for which f_{v}(s) = f_{w}(t) whenever s, t are halves of the same edge of τ.

1.5.1. Theorem. Let W be a generalized flag variety. Put

N(W, β) := \frac{[M_{0,3}(W, β)]}{[W][P GL(2)]} = \frac{[\underline{Map}_{β}(P^{1}, W)]}{[W][P GL(2)]}.  (1.9)

Then

[M_{0,k}(W, β, τ, μ)] =
[W] ∏_{v∈V_{τ}} ε(β_{v}, |v| + k_{v})N(W, β_{v})\binom{[P^{1}]}{|v| + k_{v}}(|v| + k_{v})!  (1.10)

where μ : v ↦ (β_{v}, S_{v}), k_{v} = |S_{v}|, β = ∑ β_{v}, and ε(β, n) = 0 for β = 0, n = 0, 1, 2, and 1 otherwise.

Proof (sketch). For β = 0, we have a projection M_{0,k}(W, 0, τ, μ) → W : (C; x_{1}, …, x_{k}; f) ↦ f(C). Its fiber is the stratum of the moduli space of stable curves with labelled points of the given combinatorial type (τ, μ); notice that μ now is just a map v ↦ S_{v}. This fiber is isomorphic to ∏_{v∈V_{τ}} M_{0,|v|}. The fibration is locally trivial over [W]. Hence

[M_{0,k}(W, 0, τ, μ)] = [W] ∏_{v∈V_{τ}} \binom{[P^{1}] − 3}{|v + k_{v}| − 3}(|v + k_{v}| − 3)!.

One easily sees that this coincides with (1.10).

For β ≠ 0, τ one–vertex that is, C = P^{1}, (1.10) follows from (1.8).

Finally, in the remaining cases we represent a stable map f : C → W as a vector of stable maps of irreducible components f_{v} : C_{v} → W. If we fix one component, the rest will be constrained by incidence conditions. This accounts for the necessity to divide by [W] in each v–factor; we incorporated this division in the definition (1.9) of N(W, β). The exterior factor [W] in (1.10) appears because one component can be moved freely.

§2. Generating function of the space of stable maps

2.1. The problem. In this section, we set to calculate the following series in R[[t, z]] (we remind that monomials in z belong to a semigroup ring):

Φ_{W} (t, z) := ∑_{τ/(iso)} \frac{1}{|Aut τ|} ∑_{{Vτ →B×N \atop v↦(βv ,kv)}}[W]×

∏_{v∈V_{τ}} \frac{t^{k_{v}}}{k_{v}!}z^{β_{v}}ε(β_{v}, |v| + k_{v})N(W, β_{v})\binom{[P^{1}]}{|v| + k_{v}}(|v| + k_{v})!  (2.1)

We retain the notation of §1; in particular, ε is introduced in order to exclude inadmissible markings.

To motivate this definition, consider the following more natural generating function:

Φ̃_{W} (t, z) := ∑_{β,k}[\overline{M}_{0,k}(W, β)]\frac{t^{k}}{k!} z^{β}.  (2.2)

Assume in addition that [\overline{M}_{0,k}(W, β)] in (2.2) means an “orbifold” virtual class of [\overline{M}_{0,k}(W, β)] which is defined on the category of small algebraic stacks and besides the usual additivity and multiplicativity properties postulated in 1.1 satisfies the following condition: if V is smooth, G a finite group freely acting upon V, then

[V/G] = \frac{[V]}{|G|}.

In this case (2.2) coincides with (2.1). To see this, one has to consider the stratified covering of \overline{M}_{0,k}(W, β) by M_{0,k}(W, β, τ, μ) where (τ, μ) runs over all admissible marked trees with ∑_{v} β_{v} = β, ∑_{v} k_{v} = k, and M_{0,k}(W, β.τ, μ) are moduli spaces of rigidified stable maps. The claim follows from the fact that Aut τ acts freely on ∐_{μ} M_{0,k}(W, β, τ, μ).

2.2. Theorem. Denote by φ^{0} = φ_{W} (t, z) the unique root in R[[t, z]] of the following equation:

\frac{[P^{1}]}{[W]}E(W, z)(1 + t + φ^{0})^{[P^{1}]−1} =

φ^{0} \frac{[P^{1}] − 1}{[P^{1}] − 2} + \frac{t}{[P^{1}] − 2} + \frac{1}{([P^{1}] − 1)([P^{1}] − 2)}  (2.3)

satisfying the condition

φ^{0} ≡ ∑_{β∈ind} N(W, β)z^{β}mod (z, t)^{2},

where the summation is taken over indecomposable elements of B. Then we have

Φ_{W} (t, z) = −\frac{[P^{1}] − 1}{2[P^{1}]}(φ^{0})^{2} + \frac{φ^{0}}{[P^{1}]} − \frac{t^{2}}{2[P^{1}]}  (2.4)

\frac{∂Φ_{W}}{∂t} (t, z) = φ^{0}.  (2.5)

Proof. We first rearrange the inner sum and product in (2.1):

Φ_{W} (t, z) = ∑_{τ/(iso)} \frac{[W]}{|Aut τ|} ∏_{v∈V_{τ}} ∑_{β,k≥0} \frac{t^{k}}{k!} z^{β}×

ε(β, |v| + k)N(W, β)\binom{[P^{1}]}{|v| + k}(|v| + k)!  (2.6)

Furthermore, for a fixed v ∈ V_{τ},

∑_{β,k} \frac{t^{k}}{k!} z^{β}ε(β, |v| + k)N(W, β)\binom{[P^{1}]}{|v| + k}(|v| + k)! =

(∑_{β} N(W, β)z^{β}) (∑_{k≥0} \frac{t^{k}}{k!} \binom{[P^{1}]}{|v| + k}(|v| + k)!) − ε_{|v|} =

\frac{E(W, z)}{[P GL(2)][W]} ∑_{k≥0} \frac{t^{k}}{k!} \binom{[P^{1}]}{|v| + k}(|v| + k)! − ε_{|v|},  (2.7)

where ε_{n} = 0 for n ≥ 3, and

ε_{n} = ∑_{k=0}^{2−n} N(W, 0)\frac{t^{k}}{k!} \binom{[P^{1}]}{n + k}(n + k)!  for n ≤ 2.  (2.8)

To calculate the resulting sum over marked trees we can now apply the formalism of [K], [M]. Introduce one more formal variable φ and consider the formal potential

S(φ) := −\frac{φ^{2}}{2} + ∑_{n=0}^{∞} \frac{C_{n}}{n!} φ^{n},  (2.9)
C_{n} := \frac{E(W, z)}{[P GL(2)][W]} ∑_{k=0}^{∞} \frac{t^{k}}{k!} (n + k)!\binom{[P^{1}]}{n + k} − ε_{n}.  (2.10)

We have

S(φ) = −\frac{φ^{2}}{2} + \frac{E(W, z)}{[P GL(2)][W]} ∑_{n,k=0}^{∞} φ^{n}t^{k}\binom{n + k}{k}\binom{[P^{1}]}{n + k} − ∑_{n=0}^{2} \frac{ε_{n}}{n!} φ^{n} =

\frac{E(W, z)}{[P GL(2)][W]}(1 + t + φ)^{[P^{1}]} − φ^{2} \frac{[P^{1}] − 1}{2([P^{1}] − 2)}

−φ (\frac{1}{([P^{1}] − 1)([P^{1}] − 2)} + \frac{t}{[P^{1}] − 2})

− (\frac{1}{[P^{1}]([P^{1}] − 1)([P^{1}] − 2)} + \frac{t}{([P^{1}] − 1)([P^{1}] − 2)} + \frac{t^{2}}{2([P^{1}] − 2)}) . (2.11)

According to a general formula of perturbation theory (cf. [K], [M]) we have

Φ_{W} (t, z) = S^{crit} := S(φ^{0}),

where φ^{0} is an appropriate formal solution of \frac{∂}{∂φ}S(φ) = 0 which we will now

identify. Differentiating (2.11) in φ we obtain:

\frac{E(W, z)}{[W]}[P^{1}](1 + t + φ^{0})^{[P^{1}]−1} =

φ^{0} \frac{[P^{1}] − 1}{[P^{1}] − 2} + \frac{t}{[P^{1}] − 2} + \frac{1}{([P^{1}] − 1)([P^{1}] − 2)}

which is (2.3).

Rewriting this equation as

φ^{0} = C_{1} + C_{2}φ^{0} + C_{3} \frac{(φ^{0})^{2}}{2} + …

and taking into account that C_{1} modulo (t, z)^{2} starts with terms linear in z we see that there is a unique solution φ^{0} ∈ R[[t, z]] such that φ^{0} ≡ C_{1}mod (t, z)^{2}. In view of (2.10), this coincides with the congruence in the statement of the Theorem.

It remains to calculate S(φ^{0}). Multiplying (2.3) by \frac{1 + t + φ^{0}}{[P^{1}]} and simplifying,

we get (2.4).

Finally, derivating (2.3) and (2.4) in t, we can obtain (2.5): we reproduce this calculation below in simplified notation.

2.3. Comments and supplements. We will now consider the case R = Q[q], [V] = P_{V} (q) (the virtual Poincaré polynomial of V). Equations (2.3) and (2.4) take form

\frac{q^{2} + 1}{P_{W} (q)}E(W, z)(1 + t + φ^{0})^{q^{2}} = φ^{0} \frac{q^{2}}{q^{2} − 1} + \frac{t}{q^{2} − 1} + \frac{1}{q^{2}(q^{2} − 1)},  (2.12)

Φ_{W} (t, z) = −\frac{q^{2}}{2(q^{2} + 1)}(φ^{0})^{2} + \frac{1}{q^{2} + 1}φ^{0} − \frac{1}{2(q^{2} + 1)}t^{2}.  (2.13)

a). A differential equation for φ^{0}. Derivating (2.12) in t and multiplying the

result by \frac{1 + t + φ^{0}}{q^{2}} we get the differential equation

(1 − q^{2}φ^{0})φ_{t}^{0} = (q^{2} + 1)φ^{0} + t.  (2.14)

Up to a simple variable change φ^{0} = ψ − t, this is the same equation as (0.7) (and (0.15)) in [M]:

(1 + q^{2} − q^{2}ψ)ψ_{t} = 1 + ψ.  (2.15)

Its universality is remarkable: in (2.14) there is no dependence on W and z (below it will be encoded in the choice of constant in a general solution of (2.14)), and in [M] it emerged also in a calculation of the generating function for arbitrary configuration spaces X[n].

b). Euler characteristics: q^{2} = 1. A formal substitution q^{2} = 1 into (2.11) is impossible, however a limiting procedure gives:

(1 + t + φ^{0}) log (1 + t + φ^{0}) = 2φ^{0} + t,  (2.16)

and Φ_{W} (t, z) becomes

Φ_{W} (t, z) = −\frac{1}{4}(φ^{0})^{2} + \frac{1}{2}φ^{0} − \frac{1}{4}t^{2}.  (2.17)

c). General solution of (2.14). We can apply to (2.14) Proposition 1.5.1 in [M]. Put

x = t + \frac{q^{2} + 1}{q^{2}}, y = q^{2}φ^{0} − 1, w = \frac{y}{x}.

The general solution of (2.14) in implicit form is given by

Cx = (w + 1)^{\frac{1}{q^{2}−1}} (w + q^{2})^{\frac{q^{2}}{1−q^{2}}} .  (2.18)

This makes evident the ramification structure of φ for q^{2} ≠ 1. The constant C in our case is a function of z which can be calculated by considering (2.18) at t = 0

that is, x = \frac{q^{2} + 1}{q^{2}}. The function w(z) = \frac{q^{4}φ^{0}(0, z) − q^{2}}{q^{2} + 1} is found from (2.12) at

t = 0.

d). As in [M], the derivative of Φ_{W} in t is simpler than Φ_{W} itself. Indeed, from (2.13) and (2.14) we have:

\frac{∂Φ_{W}}{∂t} = −\frac{q^{2}}{q^{2} + 1}φ^{0}φ_{t}^{0} + \frac{1}{q^{2} + 1}φ_{t}^{0} − \frac{t}{q^{2} + 1}

= −\frac{1}{q^{2} + 1} (φ_{t}^{0} − (q^{2} + 1)φ^{0} − t) + \frac{1}{q^{2} + 1}φ_{t}^{0} − \frac{t}{q^{2} + 1} = φ^{0}.

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