Hitchin map on even very stable upward flows

This paper is a write-up of the second author’s talk [12] at the conference ”Moduli spaces and geometric structures” in honour of Oscar García-Prada on the occasion of his 60th birthday at ICMAT Madrid in September 2022.

In [14], motivated by mirror symmetry, the notion of very stable Higgs bundle was introduced. Let $C$ be a smooth projective curve. Let ${\mathcal{M}}$ denote the moduli space of rank $n$ degree $d$ semistable Higgs bundles $(E,\Phi)$, where $E$ is a rank $n$ degree vector bundle and $\Phi\in H^{0}(C;\textnormal{End}(E)\otimes K)$ is a Higgs field. There is a ${\mathbb{T}}$-action on ${\mathcal{M}}$ by scaling the Higgs field, i.e. $\lambda\in{\mathbb{T}}$ acts by sending $(E,\Phi)$ to $(E,\lambda\Phi)$. A fixed point ${\mathcal{E}}\in{\mathcal{M}}^{s{\mathbb{T}}}$ is called very stable, if the upward flow

is closed. In Section 2 we recall the basic properties of very stable upward flows in general as well as for the the moduli space of Higgs bundles ${\mathcal{M}}$.

One of the main results of [14] is the classification of very stable Higgs bundles $(E,\Phi)\in{\mathcal{M}}^{{\mathbb{T}}}$ of type $(1,\dots,1)$. A fixed point is of type $(1,\dots,1)$ when the vector bundle $E=L_{0}\oplus\dots\oplus L_{n-1}$ is a direct sum of line bundles, and the Higgs field $\Phi|_{L_{i}}:L_{i}\to L_{i+1}K\subset EK$, which we denote by

Then we have

We recall this classification in Theorem 2.11 below, and a reformulation of it in Remark 2.12 in terms of minuscule dominant weights of $\mathrm{GL}(n,\mathbb{C})$.

Garcia-Prada and Ramanan in [9] study involutions on the moduli space of Higgs bundles. One important involution $\theta:{\mathcal{M}}\to{\mathcal{M}}$ is given by $\theta(E,\Phi):=(E,-\Phi)$. In [9] it is shown that the fixed points ${\mathcal{M}}^{\theta}$ correspond to $U(p,n-p)$-Higgs bundles (including the case $p=0$, where $U(0,n):=U(n)$). We recall these notions in Section 3.

In this paper we will be interested in the so-called even upward flows $W^{2+}_{\mathcal{E}}$ for any ${\mathcal{E}}\in{\mathcal{M}}^{s{\mathbb{T}}}\subset{\mathcal{M}}^{\theta}$ which are defined to be the upward flows ${\mathcal{E}}$ in the semi-projective ${\mathcal{M}}^{\theta}$, or equivalently, the intersection $W^{2+}_{\mathcal{E}}:=W^{+}_{\mathcal{E}}\cap{\mathcal{M}}^{\theta}$. Then we can define even very stable Higgs bundles ${\mathcal{E}}\in{\mathcal{M}}^{{\mathbb{T}}}$ for which the even upward flow $W^{2+}_{\mathcal{E}}\subset{\mathcal{M}}^{\theta}$ is closed. One of the main results of this paper is the following

To clarify the meaning of this complicated looking set of divisors, we reformulate this theorem in Theorem 3.13 in terms of so-called even minuscule dominant weights using positive weights of even height.

As the Hitchin map restricted to very stable upward flows is finite flat and ${\mathbb{T}}$-equivariant between affine spaces, with positive ${\mathbb{T}}$-action of the same dimension it is suspectible of explicit description. In the type $(1,\dots,1)$ very stable case, the second author found such an explicit description in [13] in terms of the spectrum of equivariant cohomology of the Grassmannian $\textnormal{Gr}_{k}(\mathbb{C}^{n})$. We will recall this in Section 4.2 below.

Finally in Section 4.3 we study the problem of modelling the Hitchin map on certain even very stable upward flows, in terms of the equivariant cohomology of homogeneous spaces. We will find in Theorem 4.3 that for $\mathrm{GL}_{2n}$ the equivariant cohomology of quaternionic Grassmannians, for ${\mathrm{SO}}_{4n+2}$ the equivariant cohomology of the $4n$-sphere and finally for ${\rm E}_{6}$ the equivariant cohomology of the real Cayley plane should model the Hitchin map on some specific even very stable flows. The appearance of these symmetric spaces is interesting, partly because they are not of Hermitian type, and also because they are quotients of the Nadler group [22] of the quasi-split real form of Hodge type (see [9, Section 2.3] for the definition).

In this paper we are concentrating on type $(1,...,1)$ very stable and even very stable upward flows. By now there are many interesting results about other types of very stable or wobbly Higgs bundles see e.g. [20] for multiplicity algebras of type $(2)$ very stable Higgs bundles, [6] for many wobbly Higgs bundles - both papers in this conference proceedings - and [25] for a classification of all type $(n_{1},n_{2})$ very stable components.

In this section we first recall the definition of a semi-projective variety and then collect the basics of the Bialynicki-Birula decomposition associated to such a variety.

The latter is to be understood as the existence of a $\mathbb{T}$-equivariant morphism $f:\mathbb{A}^{1}\to X$ such that $f(1)=x$ and $f(0)=p$. Semi-projective varieties are endowed with a stratification in affine subvarieties known as the Bialynicki-Birula decomposition [3], which we now recall. We refer to [14, Section 2] for further details.

Given a smooth fixed point $\alpha\in X^{s\mathbb{T}}$, the $\mathbb{T}$-action on $X$ induces a representation of $\mathbb{T}$ on the tangent space $T_{\alpha}X$. We denote, for $k\in\mathbb{Z}$, the weight space $(T_{\alpha}X)_{k}\subset T_{\alpha}X$ where $\lambda\in\mathbb{T}$ acts via multiplication by $\lambda^{k}$. This leads to a decomposition $T_{\alpha}X=\bigoplus_{k\in\mathbb{Z}}(T_{\alpha}X)_{k}$ in weight spaces. We denote $T_{\alpha}^{+}X:=\bigoplus_{k>0}(T_{\alpha}X)_{k}$ the positive part and $T_{\alpha}^{-}X:=\bigoplus_{k<0}(T_{\alpha}X)_{k}$ the negative part. We have:

The proof was originally given in [3] for smooth complete $X$. A proof for the general case is given in [14, Proposition 2.1].

Finally, suppose further that $X^{s}$ is equipped with a symplectic form $\omega\in\Omega^{2}(X^{s})$ such that, for $\lambda\in\mathbb{T}$, we have $\lambda^{*}(\omega)=\lambda\omega$. This supposition is motivated by the fact that the semi-projective variety we will be studying, the moduli space of semistable Higgs bundles, is endowed with such a form. Then, we have:

The proof is given in [14, Proposition 2.10]. The main idea is that, for $v\in(T_{\alpha}X)_{k}$ and $w\in(T_{\alpha}X)_{l}$, we have

This definition was introduced in [14, Definition 4.1], where it was proven [14, Lemma 4.4] that $\alpha\in X^{s\mathbb{T}}$ is very stable if and only if $W^{+}_{\alpha}\subset X$ is closed.

In this section we extend the results summarized above to the subspace $\mathcal{M}^{\theta}\subset\mathcal{M}$ of the moduli space defined by the fixed points of the subgroup $C_{2}=\{1,-1\}\subseteq\mathbb{T}$, acting as the involution $\theta:(E,\Phi)\mapsto(E,-\Phi)$. Clearly, this subspace contains all the fixed points by the $\mathbb{T}$-action, so the previous concepts can be extended naturally. Moreover, by [9, Theorem 6.3], the space $\mathcal{M}^{\theta}$ contains the images of the maps $\mathcal{M}_{U(p,q)}\rightarrow\mathcal{M}$, given by extension of structure group, of the moduli spaces of $U(p,q)$-Higgs bundles (for the different $U(p,q)$ with $p+q=n$, $p\leq q$) into the moduli space of $\mathrm{GL}(n,\mathbb{C})$-Higgs bundles. The stable locus $\mathcal{M}^{\theta,s}$ is covered by these images, which are the Higgs bundles that we will consider.

We start by recalling the following definition from [4, Definition 3.3].

We denote by $\mathcal{M}^{s}_{U(p,q)}$ the moduli space of stable $U(p,q)$-Higgs bundles, where stability is defined as for $\mathrm{GL}(n,\mathbb{C})$-Higgs bundles. As explained before, this space sits inside $\mathcal{M}$ as the fixed point locus of the involution $\theta$, that is, a stable Higgs bundle $(E,\Phi)$ is a $U(p,q)$-Higgs bundle (for some $p$ and $q$) if and only if $(E,\Phi)\simeq(E,-\Phi)$ [9, Theorem 6.3]. We will often use interchangeably the moduli space $\mathcal{M}_{U(p,q)}$ and its image inside $\mathcal{M}$.

Note that every fixed point $\mathcal{E}=(E,\Phi)$ of the $\mathbb{T}$-action is in particular fixed by $-1$ and hence a $U(p,q)$-Higgs bundle for some $p$ and $q$. This can be seen more explicitly by observing that, in the decomposition $E=E_{0}\oplus\dots\oplus E_{k-1}$ with $\Phi(E_{i})\subset E_{i+1}\otimes K$, the Higgs field $\Phi$ interchanges the summands with odd indices by those of even indices. In particular, type $(1,1,\dots,1)$ fixed points are Higgs bundles for the quasi-split group $U(p,p)$ or $U(p,p+1)$.

Moreover, recall from Białynicki-Birula theory in Section 2 that the upward flow $W^{+}_{\mathcal{E}}$ is an affine space isomorphic to the subspace of positive weights $T^{+}_{\mathcal{E}}\mathcal{M}\subset T_{\mathcal{E}}\mathcal{M}$. It is easy to identify the subspace of $U(p,q)$-Higgs bundles.

Note that this coincides with the standard upward flow when defined in $\mathcal{M}^{\theta}$ instead of $\mathcal{M}$. Hence, we also have the following natural definition of very stable points:

We remark, as one of the main interests for this study, that the subspaces of even weights $T^{2}_{\mathcal{E}}\mathcal{M}\subset T_{\mathcal{E}}\mathcal{M}$ that we are considering are Lagrangian, since the symplectic form $\omega$ pairs the subspace of weight $k$ with that of $1-k$, as explained at the end of Section 2, so that the subspaces of even weights are paired with those of odd weights. In fact, the subvariety $\mathcal{M}^{\theta}\subset\mathcal{M}$ itself is Lagrangian, as explained in [9, Theorem 8.10].

Obviously, a very stable fixed point is also even very stable. However, a wobbly fixed point $\mathcal{E}$ can either remain even wobbly or instead be even very stable, depending on whether the nontrivial intersection $(W^{+}_{\mathcal{E}}\cap\mathcal{C})\setminus\{\mathcal{E}\}$ happens at even weights or not. We will classify the even very stable Higgs bundles of type $(1,1,\dots,1)$, revealing that both situations already arise in this case. We follow the same notation of Subsection 2.2.

This is another $\Phi^{\prime}_{c}$-invariant subspace, since $\Phi^{\prime}_{c}(v_{k-1}+v_{k+l})\in\left<v_{k},v_{k+l+1}\right>$, $\Phi^{\prime}_{c}(v_{m})\in\left<v_{m+1}\right>$ for $m\in\{k,\dots,k+l-2\}\cup\{k+l+1,\dots,n-1\}$, and $\Phi^{\prime}_{c}(v_{k+l-1})=0$ (in this analysis we take $v_{j}=0$ for any $j>n-1$). Now, recall that there is an induced $\mathbb{T}$-action on $Gr_{n-k}(E^{\prime}|_{c})$ given by the $\mathbb{T}$-action on $E^{\prime}$ with weight $-i$ on the $i$-th summand of $E^{\prime}$. With this, given $\lambda\in\mathbb{T}$ we define $V_{\lambda}:=\lambda V$, that is:

Note that this subspace is always $n-k$ dimensional. This yields a curve within the connected subvariety $S_{n-k}(\Phi^{\prime}|_{c})\subset Gr_{n-k}(E^{\prime}|_{c})$ of vector subspaces of the fiber $E^{\prime}|_{c}$ which are invariant by $\Phi^{\prime}|_{c}$. As argued in the proof of [14, Theorem 4.16], this translates into a curve in the moduli space of Higgs bundles, defined by $\mathcal{H}_{V_{\lambda}}(E^{\prime},\Phi^{\prime})\simeq\lambda\mathcal{H}_{V}(E^{\prime},\Phi^{\prime})$. This curve connects $\mathcal{H}_{V_{0}}(E^{\prime},\Phi^{\prime})=(E(-c),\Phi)$ with a different fixed point, given by $\mathcal{H}_{V_{\infty}}(E^{\prime},\Phi^{\prime})$, where $V_{\infty}=\left<v_{k-1},v_{k},v_{k+1},\dots,v_{k+l-1},v_{k+l+1},v_{n-1}\right>$. This fixed point is also stable [14, Lemma 4.18].

Moreover, since $l$ is odd, then $l+1$ is even and we have that $V_{\lambda}=V_{-\lambda}$, hence $\lambda\mathcal{H}_{V}(E^{\prime},\Phi^{\prime})\simeq-\lambda\mathcal{H}_{V}(E^{\prime},\Phi^{\prime})$ so that this curve is fixed by the action of $-1$ and hence it lies in $W^{2+}_{(E(-c),\Phi)}$. This shows that $(E(-c),\Phi)$ is even wobbly thus $(E,\Phi)$ as well. ∎

There are still more examples of even wobbly stable Higgs bundles of type $(1,1,\dots,1)$, which are covered by the following proposition.

Let $1<i_{1}<i_{2}<\dots<i_{m}<n-1$ be all the indices other than $1$ and $n-1$ such that $b_{i_{k}}(c)=0$. We can assume that all of them are of the same parity since otherwise we apply the previous proposition. We start again with a Hecke transform at $(L_{0}\oplus L_{1}\dots\oplus L_{i_{m}-1})|_{c}$. This yields $(E^{\prime},\Phi^{\prime})$ with the same form as in the proof of Proposition 3.5.

Now we will construct a basis $\{v_{0},v_{1},\dots,v_{n-1}\}$ of $E^{\prime}|_{c}$ as follows. We start with any nonzero $v_{0}\in L_{0}^{\prime}|_{c}$, then apply $\Phi^{\prime}|_{c}$ until a zero vector is obtained. This will produce a string: $v_{1}=\Phi^{\prime}|_{c}(v_{0})$, $v_{2}=\Phi^{\prime}|_{c}(v_{1})\dots$ all the way up to $v_{i_{1}-1}$, if $b_{1}(c)\neq 0$, or just $v_{0}$ if $b_{1}(c)=0$. The process then iterates: we pick $v_{i_{1}}$ (or $v_{1}$ if $b_{1}(c)=0$) nonzero in the corresponding summand and iterate, repeating until $v_{n-1}$. By construction, the basis is partitioned in strings inside of which each element maps to the next one and the last one maps to zero. Also, each $v_{i}$ is in the corresponding summand $L^{\prime}_{i}|_{c}$. Let $V_{0}=\left<v_{i_{m}},\dots,v_{n-1}\right>$.

Now recall that $i_{m}<n-1$ and $i_{1}\geq 2$. Note that if $m=1$, it is possible that $i_{1}=i_{m}$, meaning a multiple zero of $b_{i_{1}}$ at $c$. In any case, we define the following $(n-i_{m})$-dimensional subspace:

It is $\Phi^{\prime}|_{c}$-invariant: each generator is taken to the next, and the last one to zero. Notice that it is important that $i_{m}<n-1$ since otherwise we get a $1$-dimensional space that does not work, for $\left<v_{i_{m}}+v_{i_{1}-2}\right>\mapsto\left<v_{i_{1}-1}\right>$. Also notice that if $b_{n-1}(c)=0$ then the second generator might map to zero instead of $v_{i_{m}+2}$, but this is not a problem for the invariance. We compute $\lambda V$:

Once again this gives a curve in $S_{n-k}(\Phi^{\prime}|_{c})$ connecting $V_{0}$ with a different $V_{\infty}$ such that the Hecke transform is a new fixed point and, since $i_{m}-i_{1}+2$ is even, it follows that $\lambda V=-\lambda V$, as desired. ∎

We now prove that these constitute all even wobbly cases.