Research ContributionArnold Mathematical Journal

Received: 7 November 2014 / Accepted: 2 May 2015

Abundance of 3-Planes on Real Projective Hypersurfaces

S. Finashin and V. Kharlamov

Abstract

We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2)={\left({{d+3}\atop{3}}\right)}$, contains “many” real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^{3}\log d$, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.

Everything you can imagine is real.

Pablo Picasso

Keywords
Real algebraic geometry, Enumerative geometry, Real Schubert calculus

1 Introduction

1.1 Phenomenon of Abundance

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

$n_{d}^{\mathbb{C}}\geqslant n_{d}^{\mathbb{R}}\geqslant n_{d}^{{\mathbb{R}}, \min}\geqslant n_{d}^{e},\quad\log n_{d}^{e}\sim\frac{1}{2}\log n_{d}^{\mathbb {C}},$

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

$\mathcal{N}_{d}^{\mathbb{C}}\geqslant\mathcal{N}_{d}^{\mathbb{R}}\geqslant \mathcal{N}_{d}^{{\mathbb{R}},\min}\geqslant\mathcal{N}_{d}^{e}\geqslant 0.$ (1.1.1)
Theorem 1.1.1.

The invariants $\mathcal{N}_{d}^{\mathbb{C}},\mathcal{N}_{d}^{e}$ are positive for each odd $d$ and satisfy the following asymptotic relations as (odd) $d\to\infty{\!}:$

$\log\mathcal{N}_{d}^{e}=\frac{1}{12}d^{3}\log d+O(d^{3})\leqslant\log N_{d}^{ \mathbb{C}}\leqslant\frac{1}{6}d^{3}\log{d}+O(d^{3}).$

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.

Corollary 1.1.2.

As odd degree $d$ increases, the invariants $\mathcal{N}_{d}^{\mathbb{C}}$ and $\mathcal{N}_{d}^{{\mathbb{R}},\min}$ have the same rate of growth in the logarithmic scale. More precisely,

$\frac{1}{12}d^{3}\log d+O(d^{3})\leqslant\log\mathcal{N}_{d}^{{\mathbb{R}}, \min}\leqslant\log\mathcal{N}_{d}^{\mathbb{C}}\leqslant\frac{1}{6}d^{3}\log{d} +O(d^{3}).$

(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 $n$.)

1.2 Examples, Applications, and Related Results

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).

1.3 The Content

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.

1.4 Conventions

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.

2 Elements of Complex Schubert Calculus

2.1 Schubert Basis

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

$\sigma_{\alpha}([C_{\beta}])=\begin{cases}1\quad\text{if }\alpha=\beta,\\ 0\quad\text{if }\alpha\neq\beta.\end{cases}$

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 $G_{[1]}({\mathbb{C}}^{k+m})$.

Proposition 2.1.1.

Schubert classes $[C_{\alpha}]$ and $[C_{\beta}]$ have intersection index 1 in $G_{[1]}({\mathbb{C}}^{k+m})$ if k-partitions $\alpha$ and $\beta$ are m-complementary, and index 0 if not. $\square$

2.2 Schur Polynomials

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

$\require{AMScd}{\begin{CD}BU_{1}^{k}=({\mathbb{C}}{\rm P}^{\infty})^{k}@>{\phi}>{}>G_{[1]}({ \mathbb{C}}^{\infty})=BU_{k},\end{CD}}$

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.

Theorem 2.2.1.

(e.g., (Bott and Tu1982)) The ring homomorphism $\phi^{*}$ is monomorphic and its image coincides with the subring ${\mathbb{Z}}^{S}[z]\subset{\mathbb{Z}}[z]$ formed by the symmetric polynomials. The Chern classes $c_{r}(\tau_{k}^{*})\in H^{*}(G_{[1]}({\mathbb{C}}^{\infty}))$, $1\leqslant r\leqslant k$, of the dual tautological vector bundle $\tau_{k}^{*}$ over $G_{[1]}({\mathbb{C}}^{\infty})$ are sent to the elementary symmetric polynomials

$\phi^{*}(c_{r})=\varepsilon_{r}(z_{1},\dots,z_{k})=\sum_{i_{1}<\dots<i_{k}}z_{ i_{1}}\dots z_{i_{k}}.$

These classes $c_{r}=c_{r}(\tau^{*}_{k})$ are multiplicative generators of the ring $H^{*}(G_{[1]}({\mathbb{C}}^{\infty}))$.

$\square$

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

$\phi^{*}(h)=\sum_{\alpha}\lambda_{\alpha}(h)s_{\alpha},$

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

$V_{\alpha+\delta}(z)=\sum_{{{\rm\tau}}\in S_{k}}{\rm sign}({{\rm\tau}})z_{{{ \rm\tau}}(1)}^{\alpha_{1}+k-1}z_{{{\rm\tau}}(2)}^{\alpha_{2}+k-2}\dots z_{{{ \rm\tau}}(k)}^{\alpha_{k}}$

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

$V_{\delta}(z)=\sum_{{{\rm\tau}}\in S_{k}}{\rm sign}({{\rm\tau}})z_{{{\rm\tau}} (1)}^{k-1}z_{{{\rm\tau}}(2)}^{k-2}\dots z_{{{\rm\tau}}(k-1)}=\prod_{1\leqslant i <j\leqslant k}(z_{i}-z_{j}).$
Proposition 2.2.2.

(e.g., (Stanley1999), Theorem 7.15.1) For any $k$-partition $\alpha$ we have

$\hskip{144.54pt}s_{\alpha}=\frac{V_{\alpha+\delta}}{V_{\delta}}.\hskip{144.54pt}$

$\square$

Example.
  1. (1)

    The Schur polynomial $s_{1,\dots,1}$ with $r\leqslant k$ components “ 1 ” is the elementary symmetric polynomial

    $\varepsilon_{r}(z_{1},\dots,z_{k})=\sum_{i_{1}<\dots<i_{r}}z_{i_{1}}\dots z_{i _{r}}=\phi^{*}(c_{r}),$

    it is the root polynomial of $c_{r}=\sigma_{1,\dots,1}$ (cf., Theorem 2.2.1 ).

  2. (2)

    The Schur polynomial $s_{m,\dots,m}$ with $k$ components “ $m$ ” equals $(z_{1}\dots z_{k})^{m}$ , it is the root polynomial of $c_{k}^{m}$ .

2.3 Multivariate Cauchy Integral Formula for the Coefficients $\lambda_{\alpha}$

Lemma 2.3.1.

For any $f\in{\mathbb{Z}}^{S}[z]$, $z=(z_{1},\dots,z_{k})$, and any $k$-partition $\alpha$, we have

$\lambda_{\alpha}(f)=\frac{1}{k!(2\pi i)^{k}}\int_{T^{k}}f(z)\overline{s_{ \alpha}}(z)V_{\delta}(z)\overline{V_{\delta}}(z)\,\frac{{\rm d}z}{z},$

where $T^{k}=\{z\in{\mathbb{C}}^{k}\,:\,|z_{1}|=\dots=|z_{k}|=1\}$ and $\frac{{\rm d}z}{z}=\frac{{\rm d}z_{1}}{z_{1}}\dots\frac{{\rm d}z_{k}}{z_{k}}$.

Proof.

The monomials $z^{\alpha}$ with $\alpha=(\alpha_{1},\dots,\alpha_{k})$, $\alpha_{i}\geqslant 0$, form an orthonormal basis in ${\mathbb{C}}[z]$ with respect to the inner product

$\langle f_{1},f_{2}\rangle=\frac{1}{(2\pi i)^{k}}\int_{T^{k}}f_{1}(z)\overline {f_{2}(z)}\frac{{\rm d}z}{z}.$

It follows that for each pair of $k$-partitions $\alpha$ and $\beta$, $\langle V_{\alpha+\delta},V_{\beta+\delta}\rangle=k!\langle z^{\alpha+\delta}, z^{\beta+\delta}\rangle$. Thus, according to Proposition 2.2.2, the Schur polynomials $s_{\alpha}$ form an orthonormal basis in the vector space ${\mathbb{Z}}^{S}[z]\otimes{\mathbb{C}}$, with respect to the modified inner product

$\langle f_{1},f_{2}\rangle_{\operatorname{Sym}}=\frac{1}{k!(2\pi i)^{k}}\int_{ T^{k}}f_{1}(z)\overline{f_{2}(z)}V_{\delta}(z)\overline{V_{\delta}(z)}\frac{{ \rm d}z}{z},$ (2.3.1)

namely,

$\begin{align} \langle s_{\alpha},s_{\beta}\rangle_{\operatorname{Sym}}=&\left\langle\frac{V _{\alpha+\delta}}{V_{\delta}},\frac{V_{\beta+\delta}}{V_{\delta}}\right\rangle _{\operatorname{Sym}}=\frac{1}{k!(2\pi i)^{k}}\int_{T^{k}}V_{\alpha+\delta}(z) \overline{V_{\alpha+\delta}(z)}\frac{{\rm d}z}{z}\\ =&\frac{1}{k!}\langle V_{\alpha+\delta},V_{\beta+\delta}\rangle=\langle z^{ \alpha+\delta},z^{\beta+\delta}\rangle. \end{align}$

The claim of the Lemma follows now from $\lambda_{\alpha}(f)=\langle f,s_{\alpha}\rangle_{\operatorname{Sym}}$ and (2.3.1). $\square$

Corollary 2.3.2.

For each $h\in H^{2km}(G_{k}({\mathbb{C}}^{\infty}))$ its value $h([G_{k}({\mathbb{C}}^{k+m})])$ on the fundamental class $[G_{k}({\mathbb{C}}^{k+m})]\in H_{2km}(G_{k}({\mathbb{C}}^{\infty}))$ is given by

$h([G_{k}({\mathbb{C}}^{k+m})])=\lambda_{m,\dots,m}(\phi^{*}(h))=\frac{1}{k!(2 \pi i)^{k}}\int_{T^{k}}\frac{\phi^{*}(h)(z)}{z^{\mathbf{m}}}V_{\delta}(z) \overline{V_{\delta}}(z)\,\frac{{\rm d}z}{z},$

where $\phi^{*}(h)\in{\mathbb{Z}}^{S}[z]$ is the root polynomial of $h$ and $z^{\mathbf{m}}$ in the denominator stands for $(z_{1}\dots z_{k})^{m}$.

Proof.

We apply Lemma 2.3.1 to $\alpha=(m,\dots,m)$ and use Example 2.2.2(2). $\square$

2.4 Counting $(k-1)$-Planes on Projective Hypersurfaces

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.

Proposition 2.4.1.

(e.g., (Debarre and Manivel1998)) Assume that $X\subset P^{m+k-1}$ is a generic hypersurface of degree $d\geqslant 1$, and that $mk={\left({{d+k-1}\atop{k-1}}\right)}$. Then, $X$ contains a finite number of projective $(k-1)$-planes and this number is equal to $c_{{\rm top}}$ evaluated on the fundamental class $[G_{k}({\mathbb{C}}^{k+m})]$. $\square$

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.

Corollary 2.4.2.

If $mk={\left({{d+k-1}\atop{k-1}}\right)}$, then

$\mathcal{N}_{d,k}^{\mathbb{C}}=\lambda_{m,\dots,m}(f_{d})=\frac{1}{k!(2\pi i)^ {k}}\int_{T^{k}}\frac{f_{d}(z)}{z^{\mathbf{m}}}V_{\delta}(z)\overline{V_{ \delta}}(z)\,\frac{{\rm d}z}{z}.$

$\square$

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

Proposition 2.4.3.

(e.g., (Debarre and Manivel1998)) For any $d\geqslant 1$,

$f_{d}(z)=\prod_{\ell_{1}+\dots+\ell_{k}=d}(\ell_{1}z_{1}+\dots+\ell_{k}z_{k}).$ (2.4.1)

$\square$

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

2.5 An Upper Bound

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

Proposition 2.5.1.

If we fix $k\geqslant 1$ and vary $d\geqslant 1$ so that $\frac{1}{k}{\left({{d+k-1}\atop{k-1}}\right)}$ is integer, then the invariant $\mathcal{N}_{d,k}^{\mathbb{C}}$ defined in Sect. 2.4 has the following asymptotics as $d\to\infty{\!}:$

$\log(\mathcal{N}_{d,k}^{\mathbb{C}})\leqslant\frac{1}{(k-1)!}d^{k-1}\log d+O(d ^{k-2}\log d).$
Proof.

Since $|l_{1}x_{1}+\dots+l_{k}x_{k}|\leqslant\ell_{1}+\dots+\ell_{k}=d$ at each point of $T^{k}$, the integral formula given in Corollary 2.4.2 implies that there exists a constant $C$ such that $\mathcal{N}_{d,k}^{\mathbb{C}}\leqslant Cd^{b(d)}$, where $b(d)={\left({{d+k-1}\atop{k-1}}\right)}=\frac{1}{(k-1)!}d^{k-1}+O(d^{k-2}).$ $\square$

2.6 Conjecture

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

$\log(\mathcal{N}_{d,k}^{\mathbb{C}})\sim\frac{1}{(k-1)!}d^{k-1}\log d\quad \text{for }k\text{ fixed and }d\to\infty.$

3 Elements of Real Schubert Calculus

3.1 Orientability and the Euler Class for the Symmetric Powers

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 $\tau_{k,m}$.

Lemma 3.1.1.

For any $k,m\geqslant 0$, we have

$w_{1}(T_{k,m})=(k+m)w_{1}(\tau_{k,m}).$

In particular, $G_{k}({\mathbb{R}}^{k+m})$ is orientable if and only if $k+m$ is even.

Proof.

Note that $T_{k,m}=\operatorname{Hom}(\tau_{k,m},\tau_{k,m}^{\perp})$, where $\tau_{k,m}^{\perp}$ is the $m$-dimensional vector bundle orthogonal to $\tau_{k,m}$, so that $\tau_{k,m}+\tau_{k,m}^{\perp}$ is a trivial bundle and, in particular, $w_{1}(\tau_{k,m})=w_{1}(\tau_{k,m}^{\perp})$. Following the splitting principle, we write $w_{1}(\tau_{k,m})=\sum_{i=1}^{k}a_{i}$ and $w_{1}(\tau_{k,m}^{\perp})=\sum_{j=1}^{m}b_{j}$, where $a_{i},b_{j}\in H^{1}(G_{k}({\mathbb{R}}^{k+m};{\mathbb{Z}}/2))$, which gives

$w_{1}(T_{k,m})=\sum_{i,j}(a_{i}+b_{j})=mw_{1}(\tau_{k,m})+kw_{1}(\tau_{k,m}^{ \perp})=(k+m)w_{1}(\tau_{k,m}).\quad$

$\square$

Lemma 3.1.2.
  1. (1)

    The vector bundle $\operatorname{Sym}^{d}(\tau^{*}_{k,m})$ is orientable if and only if ${\left({{d+k-1}\atop{k}}\right)}$ is even.

  2. (2)

    If $\dim\operatorname{Sym}^{d}(\tau^{*}_{k,m})=\dim G_{k}({\mathbb{R}}^{k+m})$ , then $w_{1}(\operatorname{Sym}^{d}(\tau^{*}_{k,m}))={dm}\,w_{1}(\tau_{k,m})$ . In particular, under the assumption that $\dim\operatorname{Sym}^{d}(\tau^{*}_{k,m})=\dim G_{k}({\mathbb{R}}^{k+m})$ the bundle $\operatorname{Sym}^{d}(\tau^{*}_{k,m})$ is orientable if and only if $dm$ is even.

Proof.

For proving (1), we use the splitting principle and obtain the following expression for the total Stiefel-Whitney class, $W_{*}$, of the symmetric power $\operatorname{Sym}^{d}(\tau_{k,m})$:

$W_{*}(\operatorname{Sym}^{d}(\tau_{k,m}))=\Pi_{\ell_{1}+\dots+\ell_{k}=d}(1+ \ell_{1}a_{1}+\dots+\ell_{k}a_{k})$

with respect to $w_{1}(\tau_{k,m})=\sum_{i=1}^{k}a_{i}$. Hence,

$w_{1}(\operatorname{Sym}^{d}(\tau_{k,m}))=\sum_{\ell_{1}+\dots+\ell_{k}=d}( \ell_{1}a_{1}+\dots+\ell_{k}a_{k})=n(a_{1}+\dots+a_{k})$

where

$n=\frac{d}{k}{\left({{d+k-1}\atop{k-1}}\right)}={\left({{d+k-1}\atop{k}}\right )},$

from where $w_{1}(\operatorname{Sym}^{d}(\tau_{k,m}))=0$ if and only if $n={\left({{d+k-1}\atop{k}}\right)}$ is even.

To deduce (2), note that ${\left({{d+k-1}\atop{k-1}}\right)}=\dim\operatorname{Sym}^{d}(\tau_{k,m})=\dim G _{k}({\mathbb{R}}^{k+m})=km$ implies $n=\frac{d}{k}km=dm$. $\square$

As an immediate consequence we obtain the following result.

Proposition 3.1.3.

Assume that the dimension of the vector bundle $\operatorname{Sym}^{d}(\tau^{*}_{k,m})$, that is ${\left({{d+k-1}\atop{k-1}}\right)}$, is equal to the dimension $km$ of the Grassmannian $G_{k}({\mathbb{R}}^{k+m})$. Assume also that $k+m=dm\mod 2$. Then, the Euler number $e(\operatorname{Sym}^{d}(\tau^{*}_{k,m}))[G_{k}({\mathbb{R}}^{k+m})]$ with respect to the local coefficient system twisted by $w_{1}(G_{k}({\mathbb{R}}^{k+m}))$ is an integer well-defined up to sign. $\square$

Remarks.
  1. (1)

    The first assumption of Proposition 3.1.3 is always satisfied in what follows, since it simply signifies that the virtual dimension of the variety $F_{k-1}(X)$ of $(k-1)$ -planes contained in a hypersurface $X\subset P^{k+m-1}$ of degree $d$ is 0 . Note that for even $d$ there exist real hypersurfaces $X$ with $X({\mathbb{R}})=\varnothing$ , for instance, the Fermat hypersurface, and thus, a signed count of the real $(k-1)$ -planes on such hypersurfaces (if invariant) gives 0 . Note also that if $km$ is odd, then even when the Euler number is well defined, it is equal to zero, as it happens for any real odd dimensional vector bundle.

    For odd $d$ the second assumption just means that $k$ is even, so, the case of even $k$ and odd $d$ is the only one which makes sense to study. As the case $k=2$ was analyzed in ( (Finashin and Kharlamov2013) ) , we are concerned in what follows with the case $k=4$ .

  2. (2)

    Calculation of the Euler number $e(\operatorname{Sym}^{d}(\tau^{*}_{k,m}))[G_{k}({\mathbb{R}}^{k+m})]$ can (and will) be done by passing to the Grassmannian $\widetilde{G}_{k}({\mathbb{R}}^{k+m})$ of orientable $k$ -planes and its (orientable) dual tautological bundle $\widetilde{\tau}^{*}_{k,m}$ , because of the relation

    $e(\operatorname{Sym}^{d}(\tau^{*}_{k,m}))[G_{k}({\mathbb{R}}^{k+m})]={ \scriptscriptstyle{}^{\pm}}\,\frac{1}{2}e(\operatorname{Sym}^{d}(\widetilde{ \tau}^{*}_{k,m}))[\widetilde{G}_{k}({\mathbb{R}}^{k+m})].$ (3.1.1)

3.2 Pontryagin Classes and the Real Root Polynomials

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.

Theorem 3.2.1.

(e.g., (Brown1982))

  1. (1)

    The ring $H^{*}(G_{[1]}({\mathbb{R}}^{\infty}))/{\rm Tors}$ is freely generated by the Pontryagin classes $p_{i}=(-1)^{i}\vartheta^{*}(c_{2i})\in H^{4i}(G_{[1]}({\mathbb{R}}^{\infty})), 1\leqslant i\leqslant k.$

  2. (2)

    The ring $H^{*}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))/{\rm Tors}$ is generated by the Pontryagin classes

    $\tilde{p}_{i}=\pi^{*}(p_{i})=(-1)^{i}\widetilde{\vartheta}^{*}(c_{2i})\in H^{4 i}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty})),\quad 1\leqslant i\leqslant k$

    and the Euler class $e_{2k}\in H^{2k}(\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}))$ , the only defining relation in these generators is $\tilde{p}_{k}=e_{2k}^{2}$ .

$\square$

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

$\require{AMScd}{\begin{CD}({\mathbb{C}}{\rm P}^{\infty})^{k}=(B{\rm SO}_{2})^{k}@>{\phi_{ \mathbb{R}}}>>B{\rm SO}_{2k}=\widetilde{G}_{[1]}({\mathbb{R}}^{\infty}).\end{CD}}$

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 $\phi_{\mathbb{R}}^{*}(h)\in{\mathbb{Z}}[x_{1},\dots,x_{k}]=H^{*}((B{\rm SO}_{2 })^{k})$.

Lemma 3.2.2.

Consider any class $h\in H^{*}(G_{[1]}({\mathbb{C}}^{\infty}))$ and let $h_{\mathbb{R}}=\widetilde{\vartheta}^{*}(h)\in H^{*}(\widetilde{G}_{[1]}({ \mathbb{R}}^{\infty}))$ denote its pull-back. Then the real root polynomial $\phi_{\mathbb{R}}^{*}(h_{\mathbb{R}})(x)$ is obtained from the complex one, $\phi^{*}(h)(z)$, by letting $z_{2k-1}=-z_{2k}=x_{k}$, that is

$\phi_{\mathbb{R}}^{*}(h_{\mathbb{R}})(x_{1},\dots,x_{k})=\phi^{*}(h)(x_{1},-x_ {1},\dots,x_{k},-x_{k}).$
Proof.

The tautological embedding ${\rm SO}_{2}\to{\rm SU}_{2}\subset{\rm U}_{2}$ is conjugate to the embedding ${\rm SO}_{2}\to{\rm SU}_{2}\subset{\rm U}_{2}$ defined as the composition of the antidiagonal homomorphism ${\rm SO}_{2}\to{\rm SO}_{2}\times{\rm SO}_{2}$, $g\mapsto(g,g^{-1})$, the isomorphism ${\rm SO}_{2}\times{\rm SO}_{2}\cong{\rm U}_{1}\times{\rm U}_{1}$ and the maximal torus inclusion ${\rm U}_{1}\times{\rm U}_{1}\subset U_{2}$. This leads to a commutative up to conjugation diagram

$\require{AMScd}{\begin{CD}({\rm SO}_{2})^{k}@>{}>{}>({\rm U}_{1})^{2k}\\ @V{}V{}V@V{}V{}V\\ {\rm SO}_{2k}@>{}>{}>{\rm U}_{2k}\end{CD}}$

with the “vertical” block-diagonal inclusion maps, the k-th power of the antidiagonal homomorphism ${\rm SO}_{2}\to{\rm SO}_{2}\times{\rm SO}_{2}\cong{\rm U}_{1}\times{\rm U}_{1}$ at the “top”, and the tautological homomorphism at the “bottom”. Passing to the corresponding classifying spaces we obtain a commutative up to homotopy diagram

$\require{AMScd}{\begin{CD}(B{\rm SO}_{2})^{k}=({\mathbb{C}}{\rm P}^{\infty})^{k}@>{(\Delta_{a })^{k}}>{}>({\mathbb{C}}{\rm P}^{\infty})^{2k}=(B{\rm U}_{1})^{2k}\\ @V{\phi_{\mathbb{R}}}V{}V@V{}V{\phi_{\mathbb{C}}}V\\ B{\rm SO}_{2k}=\widetilde{G}_{[1]}({\mathbb{R}}^{\infty})@>{\widetilde{ \vartheta}}>{}>G_{[1]}({\mathbb{C}}^{\infty})=B{\rm U}_{2k}\end{CD}}$

where $\Delta_{a}:{\mathbb{C}}{\rm P}^{\infty}\to({\mathbb{C}}{\rm P}^{\infty})^{2}$ is the antidiagonal map $z\mapsto(z,\bar{z