Abundance of 3-Planes on Real Projective Hypersurfaces

Up to our knowledge, the phenomenon of abundance of real solutions in certain real enumerative problems was observed for the first time in (Itenberg et al.2003), where it was shown that the number of real rational curves of degree $d$ interpolating a generic collection of $3d-1$ real points in the real projective plane grows, in the logarithmic scale, as fast as the number of complex curves. Since then, similar abundance phenomena were observed in various other real enumerative problems (cf., (Itenberg et al.2013); (Georgieva and Zinger2013); (Kharlamov and Rasdeaconu2014)). In particular, in our previous paper (Finashin and Kharlamov2013) we proved that a generic real hypersurface of degree $2n-1$ in a real projective space of dimension $n+1$ contains at least $(2n-1)!!$ real lines, which is approximately the square root of the number of complex lines [the same bound was obtained by (Okonek and Teleman2014)]. This estimate was based on the signed counting of the real lines by means of the Euler number of an appropriate vector bundle, and as a result gave the following relations

where $n_{d}^{\mathbb{C}}$ and $n_{d}^{\mathbb{R}}$ denote respectively the numbers of complex and real lines on a generic real hypersurface of odd degree $d$ in a projective space of dimension $\frac{d+3}{2}$, the symbol $n_{d}^{{\mathbb{R}},\min}$ stands for the minimum of $n_{d}^{\mathbb{R}}$ taken over all such generic hypersurfaces and $n_{d}^{e}$ stands for the above mentioned signed count of real lines. (Here, the number $n_{d}^{\mathbb{R}}$ depends on the choice of such a hypersurface, while $n_{d}^{e}$, $n_{d}^{{\mathbb{R}},\min}$, and $n_{d}^{\mathbb{C}}$ depend only on $d$.)

The aim of the present paper is to show that a similar abundance phenomenon holds also as soon as we count the real 3-planes on generic real hypersurfaces of odd degree $d$ in a real projective space of an appropriate dimension, which we will still denote by $n+1$.

To achieve this goal we follow the same approach as in (Finashin and Kharlamov2013). Namely, the variety of (complex or real) 3-planes on a hypersurface that is defined by a homogeneous polynomial $f$ of degree $d$ in $n+2$ variables is viewed as the zero locus $\{s_{f}=0\}$ in the Grassmannian [$G_{[1]}({\mathbb{C}}^{n+2})$ or $G_{[1]}({\mathbb{R}}^{n+2})$, respectively], where $s_{f}$ is the determined by $f$ section of the symmetric power $\operatorname{Sym}^{d}\tau_{4,n+2}^{*}$ of the dual to the tautological (complex or real) 4-dimensional vector bundle $\tau_{4,n+2}$ on the corresponding Grassmannian. It is well-known [see, for example, (Debarre and Manivel1998), Theorem 1.2] that the section $s_{f}$ is transversal to the zero section for a generic choice of $f$. Thus, if $\dim\{s_{f}=0\}=4(n-2)-{\left({{d+3}\atop{3}}\right)}$ is zero, then, in the complex setting, the Chern number $c_{4(n-2)}(\operatorname{Sym}^{d}\tau_{4,n+2}^{*})[G_{[1]}({\mathbb{C}}^{n+2})]$ is equal to the number of complex 3-planes, while in the real setting, the Euler number of $\operatorname{Sym}^{d}\tau_{4,n+2}^{*}$ on $G_{[1]}({\mathbb{R}}^{n+2})$ counts the real 3-planes with signs.

The main feature of such a signed count is its invariance: dependence only on $d$ and not on the choice of a hypersurface (in particular, independence of the topology of the hypersurface). This count is well defined, if $d$ is odd (see Proposition 3.1.3), which is the only interesting case for invariant counting and lower bounds, since in the case of even $d$ the real locus of the hypersurface can be empty. Note also that the count of 3-planes makes sense only if $n=2+\frac{1}{4}{\left({{d+3}\atop{3}}\right)}$ (in higher dimension the 3-planes come in families, and in lower dimension their number is zero).

In order to state the results, let us introduce the following notation: assuming that $d$ is odd, let us denote by $\mathcal{N}_{d}^{\mathbb{C}}$, $\mathcal{N}_{d}^{\mathbb{R}}$ and $\mathcal{N}_{d}^{{\mathbb{R}},\min}$ the number of complex 3-planes on a generic hypersurface of degree $d$ in a complex projective space of dimension $3+\frac{1}{4}{\left({{d+3}\atop{3}}\right)}$, the number of real 3-planes on a generic real hypersurface of degree $d$ in a real projective space of dimension $3+\frac{1}{4}{\left({{d+3}\atop{3}}\right)}$, and the minimum of $\mathcal{N}_{d}^{\mathbb{R}}$ taken over all generic real hypersurfaces as above (the number $\mathcal{N}_{d}^{\mathbb{C}}$ does not depend on the choice of a generic hypersurface). To avoid cumbersome discussions of explicit orientation conventions needed to fix the sign of the Euler number in question, we take into account only its absolute value and denote the latter by $\mathcal{N}_{d}^{e}$. Note that the numbers introduced are linked by the following trivial relations

Our conjecture is that, in fact, $\log\mathcal{N}_{d}^{\mathbb{C}}\sim 2\log\mathcal{N}_{d}^{e}$ which would imply that $\log\mathcal{N}_{d}^{\mathbb{C}}\sim\frac{1}{6}d^{3}\log d.$ It seems to us that even the positivity of $\mathcal{N}_{d}^{\mathbb{R}}$ (which follows from $\mathcal{N}_{d}^{e}\neq 0$) was not acknowledged in the literature before.

Amazingly, the answers that we obtain in the real setting look more simple than those in the complex setting. Similar phenomena are observed in other enumerative problems, see (Finashin and Kharlamov2013) and Sect. 5.

(For us the same rate of growth for two sequences $f(n),g(n)$, means existence of a constant $C$ such that $|f(n)|\leqslant C|g(n)|$ and $|g(n)|\leqslant C|f(n)|$ for all sufficiently big $n$.)

In the case of even $d$, we still have $\mathcal{N}_{d}^{\mathbb{C}}>0$ [see, for example, (Debarre and Manivel1998),Theorem 2.1] as well as $\log N_{d}^{\mathbb{C}}\leqslant\frac{1}{6}d^{3}\log{d}+O(d^{3})$ (see Proposition 2.5.1). By contrary, if $d$ is even, then $\mathcal{N}_{d}^{e}$ either vanishes or is defined only modulo 2, see the explanation in Remark $(1)$ after Proposition 3.1.3.

Note that the positivity $\mathcal{N}_{d}^{\mathbb{R}}\geqslant\mathcal{N}^{e}_{d}>0$ implies that a generic real projective $n$-dimensional hypersurface of odd degree $d$ with $4(n-2)>{\left({{d+3}\atop{3}}\right)}$ contains an infinite number of real 3-planes, and then due to (Debarre and Manivel1998), Theorem 2.1 the variety of these real 3-planes is of (pure) dimension $4(n-2)-{\left({{d+3}\atop{3}}\right)}$ if $n>6$. (In fact, Debarre and Manivel have proved in (Debarre and Manivel2000) such positivity and pure dimension results for the variety of real $r$-planes on odd degree real complete intersections, but only under the assumption that the dimension of the ambient projective space is large enough. Apparently, for hypersurfaces, due to this dimension assumption their result applies only to $d\leqslant 3$.)

The approach that we develop in this paper can be applied to counting real $(2r-1)$-planes on real projective $n$-dimensional hypersurfaces of any odd degree $d$, under an appropriate dimension condition, that is $2r(n-2r+2)={\left({{d+2r-1}\atop{2r-1}}\right)}$. (Counting of even dimensional planes gives a trivial result, since the dimension of the corresponding vector bundle is odd, and the Euler count, whenever it gives an integer rather than a modulo 2 residue, would give zero.) The result of such counting still gives an invariant, and hence provides, like in the cases $r=1$ [considered in (Finashin and Kharlamov2013)] and $r=2$ (considered in this paper), an effective universal lower bound for the number of real $(2r-1)$-planes on a hypersurface. We restricted ourselves here to the case of 3-planes, since for the moment in the higher dimensional cases we can not provide an explicit answer, but can only set up an upper bound (see Sect. 2.5), give an implicit formula in the form of multivariate Cauchy integral (see Theorem 5.3.1), and suggest a conjecture (see Conjecture 2.6).

In Sect. 2, we recall some facts from the complex Schubert calculus that are related to counting the number of projective subspaces in hypersurfaces. In Sect. 3, we discuss real Schur polynomials and the modifications required to make similar counting in the real setting. The techniques developed in these sections are applied in Sect. 4 to prove the main results. In Sect. 5 we discuss a few other real enumerative problems that can be solved by using the same methods.

If in a homology or cohomology notation the coefficients are not specified, then they are supposed to be integer. The notation $p_{i}$ for the Pontryagin classes may refer to $p_{i}(\tau_{k,\infty})\in H^{*}(G_{[1]}({\mathbb{R}}^{\infty}))$ as well as for their pull-backs in $H^{*}(G_{[1]}({\mathbb{R}}^{k+n}))$, in $H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))$, and in $H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{k+n}))$ ($\tilde{G}$ stands for the Grassmannians of oriented supspaces). This ambiguity should be resolved by the context.

Our decision not to fix explicit orientations results in a number of identities valid up to sign, and we write $x={\scriptscriptstyle{}^{\pm}}\,{y}$, which means that $x=y$, or $x=-y$. The symbol $\square$ marks the end of a proof, or signifies that the corresponding statement is either a citation or an immediate consequence of previous claims.

By a $k$-partition of $n\in{\mathbb{Z}}_{\geqslant 0}$ we mean a decreasing integer sequence $\alpha=(\alpha_{1},\dots,\alpha_{k})$, $\alpha_{1}\geqslant\dots\geqslant\alpha_{k}\geqslant 0$, $|\alpha|=\alpha_{1}+\dots+\alpha_{k}=n$. Graphically $\alpha$ is presented as a Young–Ferrers diagram of size $n$. For example, the constant $k$-partition $\mathbf{m}=(m,\dots,m)$ is presented by the $k\times m$ rectangle. In what follows we assume that some $k>0$ is fixed throughout the whole section, and omit sometimes the vanishing components of $\alpha$.

A filtration $0\subset{\mathbb{C}}^{1}\subset{\mathbb{C}}^{2}\subset\dots$ of ${\mathbb{C}}^{\infty}$ yields a CW-decomposition of Grassmannian $G_{[1]}({\mathbb{C}}^{\infty})$ into open Schubert cells $C_{\alpha}$ indexed with $k$-partitions, namely, a $k$-subspace $L\subset{\mathbb{C}}^{\infty}$ belongs to $C_{\alpha}$ if and only if $\alpha_{k+1-s}=\min\{j\,|\,\dim(L\cap{\mathbb{C}}^{j+s})=s\}$, for each $1\leqslant s\leqslant k$.

With any k-partition $\alpha$ we associate a homology and a cohomology classes of Grassmannians as follows. The closure ${\text{Cl}}(C_{\alpha})$ is the so-called Schubert variety; being equipped with the complex orientation it yields the Schubert class $[C_{\alpha}]\in H_{2n}(G_{[1]}({\mathbb{C}}^{\infty}))$, where $n=|\alpha|$. The cohomology class $\sigma_{\alpha}\in H^{2n}(G_{[1]}({\mathbb{C}}^{\infty}))$ associated to $\alpha$ is characterized by

In other words, the classes $\sigma_{\alpha}$ taken over all $k$-partitions of size $n$ form an additive basis in $H^{2n}(G_{[1]}({\mathbb{C}}^{\infty}))$ such that $h=\sum_{\alpha}({h([C_{\alpha}])}\sigma_{\alpha}$ for any $h\in H^{*}(G_{[1]}({\mathbb{C}}^{\infty}))$.

Note that the Schubert cell $C_{\alpha}\subset G_{[1]}({\mathbb{C}}^{\infty})$ is contained in a finite dimensional Grassmannian $G_{[1]}({\mathbb{C}}^{k+m})$ if and only if $\alpha_{1}\leqslant m$, that is if the Young–Ferrers diagram of $\alpha$ lies inside the $(k\times m)$-rectangle diagram. It follows that the additive bases of $H_{*}(G_{[1]}({\mathbb{C}}^{k+m}))$ and $H^{*}(G_{[1]}({\mathbb{C}}^{k+m}))$ are given respectively by $[C_{\alpha}]$ and $\sigma_{\alpha}$ such that $\alpha_{1}\leqslant m$ (here and below, to simplify notation, we denote by $\sigma_{\alpha}$ not only the class in $H^{*}(G_{[1]}({\mathbb{C}}^{\infty}))$ but also its pull-back in $H^{*}(G_{[1]}({\mathbb{C}}^{k+m}))$).

We say that $k$-partitions $\alpha$ and $\beta$ are m-complementary to each other if $\alpha_{i}+\beta_{k+1-i}=m$ for $i=1,\dots,k$ (so that, in particular, $\alpha_{1},\beta_{1}\leqslant m$). It is well-known (and easy to check) that the Schubert cycles of $m$-complementary $k$-partitions are Poincare-dual in $G_{[1]}({\mathbb{C}}^{k+m})$.

Denote by $U_{k}$ the unitary group and by $U_{1}^{k}$ its maximal torus formed by the diagonal matrices. The inclusion map $U_{1}^{k}\subset U_{k}$ induces a map of the classifying spaces

and the induced cohomology map $\phi^{*}:H^{*}(G_{[1]}({\mathbb{C}}^{\infty}))\to H^{*}({\mathbb{C}}{\rm P}^{ \infty})^{k}\cong{\mathbb{Z}}[z_{1},\dots,z_{k}]$ is independent of the choice of the maximal torus, since such tori form a single conjugacy class.

For the unitary groups, the so called splitting principle can be expressed formally as follows.

$\square$

We call $z_{i}$, $1\leqslant i\leqslant k$, the Chern roots as they are the formal roots of $t^{k}-c_{1}t^{k-1}+\cdots+(-1)^{k}c_{k}$. For each $h\in H^{*}(G_{[1]}({\mathbb{C}}^{\infty}))$, we say that $\phi^{*}(h)\in{\mathbb{Z}}^{S}[z]$ is the root polynomial representing class $h$.

The root polynomials $s_{\alpha}\in{\mathbb{Z}}^{S}[z]$ that represent classes $\sigma_{\alpha}\in H^{2n}(G_{[1]}({\mathbb{C}}^{\infty}))$, $n=|\alpha|$, are called Schur polynomials. The relation $h=\sum_{\alpha}({h([C_{\alpha}])}\sigma_{\alpha}$ implies

where $\lambda_{\alpha}(h)=h[C_{\alpha}]\in{\mathbb{Z}}$ will be called the Schur coefficients for $h$ (or for $\phi^{*}(h)$).

Recall that the Schur polynomials can be calculated using the following generalized Vandermonde polynomial

where ${\rm sign}({{\rm\tau}})$ is the sign of a permutation ${{\rm\tau}}\in S_{k}$, and $\delta=(k-1,k-2,\dots,1,0)$. Recall that the usual Vandermonde polynomial is

We denote by $c_{{\rm top}}\in H^{*}(G_{k}({\mathbb{C}}^{k+m}))$ the top Chern class of the symmetric power $\operatorname{Sym}^{d}(\tau^{*}_{k,m})$ of the dual to the tautological bundle $\tau^{*}_{k,m}$ on $G_{k}({\mathbb{C}}^{k+m})$, and by $f_{d}(z)=\phi^{*}(c_{{\rm top}})\in{\mathbb{Z}}^{S}[z]$ the root polynomial of $c_{{\rm top}}$. The number of $(k-1)$-planes in a projective hypersurface can be evaluated as the following Chern number.

Let us denote the above Chern number $c_{{\rm top}}([G_{k}({\mathbb{C}}^{k+m})])$ by $\mathcal{N}_{d,k}^{\mathbb{C}}$. Proposition 2.4.1 together with Corollary 2.3.2 provides the following integral formula.

To estimate this integral, we need the following well-known root factorization formula for $f_{d}$.

We call the factors in the right hand side of (2.4.1) the root factors.

In this section we study the growth rate of the sequence $\mathcal{N}_{d,k}^{\mathbb{C}}$ in the logarithmic scale.

Some heuristic arguments make plausible to conjecture that the sequence $\mathcal{N}_{d,k}^{\mathbb{C}}$ has the following logarithmic asymptotics:

We denote the tangent bundle of a real Grassmannian $G_{k}({\mathbb{R}}^{k+m})$ by $T_{k,m}$ and the tautological $k$-dimensional vector bundle over $G_{k}({\mathbb{R}}^{k+m})$ by $\tau_{k,m}$.

As an immediate consequence we obtain the following result.

Let $\widetilde{G}_{[1]}({\mathbb{R}}^{\infty})$ be the Grassmannian of oriented $2k$-planes in ${\mathbb{R}}^{\infty}$, and $G_{[1]}({\mathbb{R}}^{\infty})$ be that of non oriented $2k$-planes. Denote by ${\widetilde{\vartheta}}:\widetilde{G}_{[1]}({\mathbb{R}}^{\infty})\to G_{[1]}( {\mathbb{C}}^{\infty})$ the composition of the double covering $\pi:\widetilde{G}_{[1]}({\mathbb{R}}^{\infty})\to G_{[1]}({\mathbb{R}}^{\infty})$ and of the inclusion $\vartheta:G_{[1]}({\mathbb{R}}^{\infty})\subset G_{[1]}({\mathbb{C}}^{\infty})$. The following description of the integer cohomology ring $H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))/{\rm Tors}$ is classical.

In the rest of the paper, we allow ambiguity and keep traditional notation $p_{i}$ for the classes $\tilde{p}_{i}$, and moreover, use the same notation for the Pontryagin classes induced in the Grassmannians $H^{*}(G_{[1]}({\mathbb{R}}^{n}))/{\rm Tors}$ and $H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{n}))/{\rm Tors}$.

The embedding ${\rm SO}_{2}^{k}\subset{\rm SO}_{2k}$ given by $2\times 2$ special orthogonal matrices along the diagonal, gives a maximal torus in ${\rm SO}_{2k}$ and induces a map between the classifying spaces

This map associates to a k-tuple $(\xi_{1},\dots,\xi_{k})$ of ${\rm SO}_{2}$-bundles their Whitney sum $\xi_{1}\oplus\dots\oplus\xi_{k}$. We denote by $x_{1},\dots,x_{k}$ the standard (Euler class) generators of $H^{*}((B{\rm SO}_{2})^{k})$ and, for a given a class $h\in H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))$, by a real root polynomial of $h$ we mean the polynomial $\phi_{\mathbb{R}}^{*}(h)\in{\mathbb{Z}}[x_{1},\dots,x_{k}]=H^{*}((B{\rm SO}_{2 })^{k})$.

We consider two subrings of the ring ${\mathbb{Z}}[x]$, $x=(x_{1},\dots,x_{k})$. The Pontryagin ring ${\mathbb{Z}}^{P}[x]={\mathbb{Z}}[x_{1}^{2},\dots,x_{k}^{2}]\cap{\mathbb{Z}}^{S }[x]$ is formed by the symmetric polynomials in $x_{i}^{2}$, $1\leqslant i\leqslant k$, and the Euler–Pontryagin ring ${\mathbb{Z}}^{EP}[x]={\mathbb{Z}}[x_{1}^{2},\dots,x_{k}^{2},x_{1}\dots x_{k}] \cap{\mathbb{Z}}^{S}[x]$, where ${\mathbb{Z}}[x_{1}^{2},\dots,x_{k}^{2},x_{1}\dots x_{k}]$ is generated by the squares $x_{i}^{2}$ and the product $x_{1}\dots x_{k}$.

As in the complex case, the $k$-partitions $\alpha$ determine a CW-decomposition of $G_{[1]}({\mathbb{R}}^{\infty})$ into real Schubert cells $C_{\alpha,{\mathbb{R}}}=C_{a}\cap G_{[1]}({\mathbb{R}}^{\infty})$ with the property $C_{\alpha,{\mathbb{R}}}\subset G_{[1]}({\mathbb{R}}^{k+m})$ if and only if $\alpha_{1}\leqslant m$. The cells $C_{\alpha,{\mathbb{R}}}$ lead in general only to ${\mathbb{Z}}/2$-homology classes

(here and further on, the same notation is used for the classes in Grassmannians for different $m$, including $m=\infty$).

For a $k$-partition $\beta=(\beta_{1},\dots,\beta_{k})$, denote by $2\beta$ the $k$-partition $(2\beta_{1},\dots,2\beta_{k})$ and by $\beta(2)$ the $2k$-partition $(\beta_{1},\beta_{1},\dots,\beta_{k},\beta_{k})$, where each of the $\beta_{i}$ is repeated twice. We say that $2k$-partition $\alpha$ is an even partition if it can be presented as $\alpha=2\beta(2)$, and that $\alpha$ is an odd partition if it has form $\alpha=2\beta(2)+\mathbf{1}=(2\beta_{1}+1,\dots,2\beta_{k}+1)$ for some $k$-partition $\beta$.

If $\alpha$ is an even $2k$-partition, then the real Schubert cell $C_{\alpha,{\mathbb{R}}}$ defines an integral homology class, which we denote

(concerning the sign of $[C_{\alpha,{\mathbb{R}}}]$, in what follows we do not need to make any specific choice of the orientation).

Letting $m\to\infty$ in Theorem 3.3.1, we can conclude that the classes $[C_{\alpha,{\mathbb{R}}}]$ for all even $2k$-partitions $\alpha$ form an additive basis in $H_{*}(G_{[1]}({\mathbb{R}}^{\infty}))/{\rm Tors}$. Let us introduce the dual additive basis, $\{\sigma_{\alpha,{\mathbb{R}}}\}_{\text{even }\alpha}$, in $H^{*}(G_{[1]}({\mathbb{R}}^{\infty}))/{\rm Tors}$; namely, let $\sigma_{\alpha,{\mathbb{R}}}$ be the integral cohomology class taking value 1 on $[C_{\alpha,{\mathbb{R}}}]$ and vanishing on the other integral Schubert classes. Then, we take the pull-back of $\sigma_{\alpha,{\mathbb{R}}}$ by the double covering $\pi:\widetilde{G}_{[1]}({\mathbb{R}}^{\infty})\to G_{[1]}({\mathbb{R}}^{\infty})$, that is $\widetilde{\sigma}_{\alpha,{\mathbb{R}}}=\pi^{*}(\sigma_{\alpha,{\mathbb{R}}}) \in H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))$, and define the real Schur polynomial of an even $2k$-partition$\alpha$ as the root polynomial

(cf., Corollary 3.2.4).

The relation between the real and the complex Schur polynomials can be described as follows. Consider the ring isomorphism $T$ between the Chern ring

and the Pontryagin ring

sending $z_{i}$ to $x_{i}^{2}$, in terms of polynomial rings, and equivalently, the Chern classes $c_{i}$ to the Pontryagin classes $p_{i}$, $i=1,\dots,k$, in terms of cohomology rings.

Now we extend the definition of the real root polynomials to odd $2k$-partitions, $\alpha^{\prime}=\alpha+\mathbf{1}$ (where $\alpha$ is an even partition) by letting $\widetilde{\sigma}_{\alpha^{\prime},{\mathbb{R}}}$ be the product of $\widetilde{\sigma}_{\alpha,{\mathbb{R}}}$ by the Euler class $e_{2k}\in H^{2k}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))$ and define similarly the real Schur polynomial of $\alpha^{\prime}$ as the root polynomial

In $\widetilde{G}_{[1]}({\mathbb{R}}^{\infty})$, we have a pair of cells, $C_{\alpha,{\mathbb{R}}}^{\pm}$, that form the pull-back of $C_{\alpha,{\mathbb{R}}}$ with respect to the double covering $\pi:\widetilde{G}_{[1]}({\mathbb{R}}^{\infty})\to G_{[1]}({\mathbb{R}}^{\infty})$. The closure of the union $\widetilde{C}_{\alpha,{\mathbb{R}}}=C_{\alpha,{\mathbb{R}}}^{+}\cup C_{\alpha, {\mathbb{R}}}^{-}$ yields an integral class of infinite order $[\widetilde{C}_{\alpha,{\mathbb{R}}}]\in H_{|\alpha|}(\widetilde{G}_{[1]}({ \mathbb{R}}^{\infty}))/{\rm Tors}$ if $\alpha$ is an even or an odd $2k$-partition. Here, the orientations of $C_{\alpha,{\mathbb{R}}}^{\pm}$ are chosen to be respected by the covering $\pi$ (and, therefore, by the deck transformation), and for even $2k$-partitions $\alpha$ we just take the sum of cells $C_{\alpha,{\mathbb{R}}}^{\pm}$, that is the pull-back $[\widetilde{C}_{\alpha,{\mathbb{R}}}]=\pi^{!}([C_{\alpha,{\mathbb{R}}}])$, while for odd $2k$-partitions $\alpha$ we take the difference $[\widetilde{C}_{\alpha,{\mathbb{R}}}]=[C_{\alpha,{\mathbb{R}}}^{+}-C_{\alpha,{ \mathbb{R}}}^{-}]$, where the sign “$-$” signifies reversion of the orientation of $C_{\alpha,{\mathbb{R}}}^{-}$.

By Lemma 3.4.4, each class $h\in H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))/{\rm Tors}$ can be decomposed into an integer linear combination

This gives the corresponding decomposition of the root polynomial $f=\phi_{\mathbb{R}}^{*}h\in{\mathbb{Z}}^{EP}[z]:$

The coefficients $\lambda_{\alpha}=\lambda_{a}(h)\in{\mathbb{Z}}$ will be called real Schur coefficients of $h$, and of its root polynomial $f$. If $h\in H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))$, we put $\lambda_{\alpha}(h)=\lambda_{\alpha}(h/{\rm Tors})$.

Like in Lemma 2.3.1, we can find the coefficients $\lambda_{\alpha}$ using a residue formula.

In the case of $\lambda_{\mathbf{m}}$ that we are interested in, we obtain a formula analogous to the one in Corollary 2.3.2.

Throughout this section $d\geqslant 1$ is odd, as it is in Theorem 1.1.1. We start with a real analogue of the root factorization formula (2.4.1) that takes the following form, in which we call it the real root factorization formula.

The factors in the above product formula will be called real root factors.

Theorem 1.1.1 concerns the case $k=2$, and in this case Proposition 4.1.1 reads as follows

The polynomial $g_{d}(x)=f^{2}_{d}(x)\in{\mathbb{Z}}^{EP}[x]$ can be written as

where $g_{d,i}$, $0\leqslant i\leqslant\frac{d-1}{2}$, is the product of those real root factors that give $|\ell_{1}-\ell_{\bar{1}}|+|\ell_{2}-\ell_{\bar{2}}|=d-2i$ (or, equivalently, $\min(\ell_{1},\ell_{\bar{1}})+\min(\ell_{2},\ell_{\bar{2}})=i$).

It is convenient to group the real root factors of $g_{d}$ by letting

and

As in Lemma 4.2.2, the root factors $(\ell_{1}x_{1}+\ell_{2}x_{2})(\ell_{2}x_{1}+\ell_{1}x_{2})$ in $h_{d-2i}(x_{1},x_{2})$ can be grouped together with the factors $(\ell_{1}x_{1}-\ell_{2}x_{2})(\ell_{2}x_{1}-\ell_{1}x_{2})$ in $h_{d-2i}(x_{1},-x_{2})$ to give us the product

This product enters in $f_{d}$ in the power $(i+1)$. Therefore, we can put

where

Note that in the initial definition of $f_{d}$ it was defined only up to sign (see the remark after Proposition 4.1.1), and from now on we eliminate this ambiguity by prescribing to $f_{d}$ the sign given by formula (4.2).

We can also summarize the above formulae as follows:

where $C_{d}=[d!!(d-2)!!\dots]^{2}$ and $N=\frac{1}{2}{\left({{d+3}\atop{3}}\right)}$.

Consider a sequence of functions

Assume that $\phi_{\ell,r}(x)>0$ for all $x\in T^{2}$, $d$ and $(\ell,r)$, and that the maximum of $\phi_{\ell,r}$ over $T^{2}$ is equal to $M_{\ell,r}$ and is achieved on a submanifold $L\subset T^{2}$, which is common for all $d$ and $\ell$. This is the case in our setting, where $\ell=(\ell_{1},\ell_{2})$ and

and $M_{\ell,r}=(\ell_{1}^{2}+\ell_{2}^{2})^{r+1}$. Namely, if $x_{j}=e^{i\phi_{j}}$, $\phi_{j}\in[0,2\pi]$, $j=1,2$, then

where $\phi=\frac{\pi}{2}-\phi_{1}+\phi_{2}$. Thus, $(x_{1},x_{2})=(e^{i\phi_{1}},e^{i\phi_{2}})$ is a point of maximum if and only if $\phi=0$ or $\phi=\pi$, so that in our case $L$ consists of two disjoint circles $\phi_{1}-\phi_{2}=\pm\frac{\pi}{2}$.

We are concerned about the asymptotics, in the logarithmic scale, of the integral

for a certain function $W(x)\geqslant 0$ on $T^{2}$, namely, for $W(x)=V_{2\delta}(x)\overline{V}_{2\delta}(x)$

The asymptotic upper bound for $\mathcal{N}_{d}^{\mathbb{C}}$ is established in Proposition 2.5.1. The positivity of $\mathcal{N}_{d}^{e}$ follows from Theorem 4.1.2 and the positivity of $F_{d}$ (see Proposition 4.3.1). The asymptotic expression for $\mathcal{N}_{d}^{e}$ follows from Theorem 4.1.2 and Proposition 4.4.1.

The above approach can be also applied to counting 3-planes in the complete intersections, except that the formulas become more cumbersome and the asymptotics is difficult to disclose. For simplicity, let us restrict ourselves to the intersections of cubic hypersurfaces. Recall that for one cubic hypersurface (see Example 4.2.4) the Euler class is given by

An intersection $X$ of $r$ generic real cubic hypersurfaces in $P^{m+3}$ contains a finite number of 3-planes if the dimension $20r$ of $e_{20}^{r}$ is equal to $4m=\dim\widetilde{G}_{[1]}({\mathbb{R}}^{m+4})$ [see, for example, (Debarre and Manivel2000)], that is if $m=5r$. The signed count of real 3-planes in $X$ gives a number $\mathcal{N}_{3,\dots,3}^{e}$, which is the half of the corresponding count of oriented 3-planes that is $e_{20}^{r}[\widetilde{G}_{[1]}({\mathbb{R}}^{m+4})]$, so we obtain

Due to the multiplicative structure of the Euler–Pontryagin ring, the right-hand side is equal to the result of substitution of the Catalan numbers $C_{k}$ instead of $t^{k}$ in the polynomial $9^{r}(25-4t)^{r}$. Standard manipulation with generating functions and radii of convergence shows that the rate of growth of this sequence is linear in the logarithmic scale:

To count the number $\mathcal{N}_{3,\dots,3}^{\mathbb{C}}$ of complex 3-planes on a generic intersection of cubic hypersurfaces we can use the multivariate integral Cauchy formula (cf., Corollary 2.4.2):

Applying to this integral the saddle point version of the Laplace method, namely, by deforming the integration cycle locally keeping the points of the maximal absolute value of the product (this value is equal to $3^{20}$) but making the values of the product real at each point of a small neighborhood of the locus of maxima, we obtain

Note that the Schubert cell $C_{2,2}\subset G_{[1]}({\mathbb{C}}^{2n+4})$ is formed by the projective 3-planes in $P^{2n+3}$ which intersect $P^{2n-1}\subset P^{2n+3}$ along a line. Therefore, if we choose a generic set $S=\{S_{1},\dots,S_{2n}\}$ of projective $(2n-1)$-dimensional subspaces $S_{i}\subset P^{2n+3}$, then the number $\mathcal{N}^{\mathbb{C}}_{\boxplus^{2n}}$ of 3-planes $L\subset P^{2n+3}$ such that the intersections $L\cap S_{i}$, $1\leqslant i\leqslant 2n$, are lines, can be found by evaluation of the power of the Schubert class $\sigma_{2,2}$:

In the real setting, we define the number $\mathcal{N}^{\mathbb{R}}_{\boxplus^{2n}}$ similarly, by counting the real 3-planes intersecting a generic set $S$ of real subspaces $S_{i}\subset P^{2n+3}$ along lines. This number depends on the choice of a generic set $S$, and we denote by $\mathcal{N}^{\min}_{\boxplus^{2n}}$ the minimum of $\mathcal{N}^{\mathbb{R}}_{\boxplus^{2n}}$ with respect to all generic choices of such $S$. The number $\sigma_{2,2,{\mathbb{R}}}^{2n}[G_{[1]}({\mathbb{R}}^{2n+4})]$ can be interpreted as the signed count of the real 3-planes, and thus its absolute value, which we denote by $\mathcal{N}^{e}_{\boxplus^{2n}}$, provides an estimate

Here, $\sigma_{2,2,{\mathbb{R}}}\in H^{4}(G_{[1]}({\mathbb{R}}^{2n+4}))$ is nothing but the Pontryagin class $p_{1}$. The following result shows that for this enumerative problem we get once more a fixed proportion in the logarithmic scale between the number of real solutions and the number of complex ones.

Now, let us look for the asymptotical behavior of the number of complex solutions in the same Schubert problem as that in Proposition 5.2.2.

Passing to the polar coordinates, $x=\exp(i\theta)$, we obtain

where $R=[0,2\pi]^{4}$, and the maximal value of $g(x)=|s_{2,2}(x)|^{2}$ is $20^{2}$, since the sum of the coefficients of $s_{2,2}=x_{1}^{2}x_{2}^{2}+\dots+x_{1}^{2}x_{3}x_{4}+\dots+2x_{1}x_{2}x_{3}x_{4}$ is 20. $\square$

Theorem 4.1.2 and its proof extends easily to the case of counting $2k-1$-planes with any $k\in{\mathbb{N}}$.

Some heuristic arguments applied to the above integral formula suggest a conjecture that $\mathcal{N}_{d}^{e}$ is non zero for each odd $d$ and any $k$, and that, for $k$ fixed and odd $d$ growing to infinity, the following asymptotic relation holds (cf., Conjecture 2.6)