Proof. For (i), the map $c_1\colon \textrm {KR}^0(X)\to H^2(X, \mathbb {F}_2)$ is the composition

\begin{align*}\textrm {KR}^0(X)\longrightarrow \textrm {KU}^0(X)\xrightarrow {c_1} H^2(X, \mathbb {Z}) \to H^2(X, \mathbb {F}_2).\end{align*}

The first map is surjective by assumption, the second map is surjective since $K(\mathbb {Z}, 2)\cong \mathbb {C}\mathbb {P}^{\infty }$ classifies vector bundles of rank 1 and the third map is surjective since $H^3(X, \mathbb {Z})$ has no 2-torsion.

For (ii), since $c_1 \colon \textrm {KR}^0(X)\to H^2(X, \mathbb {F}_2)$ factors through the equivariant first Chern class map $c_1^G \colon \textrm {KR}^0(X)\to H^2_G(X, \mathbb {F}_2)$ , we get that $H^2_G(X, \mathbb {F}_2)\to H^2(X, \mathbb {F}_2)$ is surjective. Since $b_1(X)=0$ , $H^j_G(X, \mathbb {F}_2)\to H^j(X, \mathbb {F}_2)$ is surjective for $j=1, 3$ . Since $X(\mathbb {R})\neq \emptyset$ , $H^4_G(X, \mathbb {F}_2)\to H^4(X, \mathbb {F}_2)$ is also surjective. Therefore, $X$ is maximal.

Proof. The condition $H^1(X, \mathbb {F}_2)=0$ is equivalent to saying that $b_1(X)=b_3(X)=0$ and $H^*(X, \mathbb {Z})$ is 2-torsion-free.

For $({\rm i}) \Leftrightarrow {\rm (ii)}$ , by the Leray–Serre spectral sequence (3), together with the fact that $H^1(X, \mathbb {Z})=0$ (since $b_1=0$ ), we have an exact sequence

\begin{align*}0\to H^2(G, \mathbb {Z}(1))\to H^2_G(X, \mathbb {Z}(1))\to H^2(X, \mathbb {Z}(1))^G\to H^3(G, \mathbb {Z}(1))\to H^3_G(X, \mathbb {Z}(1)).\end{align*}

Since $H^2(G, \mathbb {Z}(1))=0$ and $H^3(G, \mathbb {Z}(1))\to H^3_G(X, \mathbb {Z}(1))$ is injective by Lemma 2.2 ( $X(\mathbb {R})\neq \emptyset$ since $X$ is maximal), the natural map $H^2_G(X, \mathbb {Z}(1))\to H^2(X, \mathbb {Z}(1))^G$ is an isomorphism. Hence the natural map $H^2_G(X, \mathbb {Z}(1))\to H^2(X, \mathbb {F}_2)$ factorizes as follows:

(17) \begin{align} H^2_G(X, \mathbb {Z}(1))\xrightarrow {\simeq } H^2(X, \mathbb {Z}(1))^G\hookrightarrow H^2(X, \mathbb {Z})\to H^2(X, \mathbb {F}_2). \end{align}

By Lemma 2.4,

\begin{align*}H^2(X, \mathbb {Z})=H^2(X, \mathbb {Z})^G\oplus H^2(X, \mathbb {Z}(1))^G.\end{align*}

Since $H^2(X, \mathbb {F}_2)=H^2(X, \mathbb {Z})\otimes \mathbb {F}_2$ , the surjectivity of (17) (i.e. the $c_1$ -maximality) is equivalent to the vanishing of $H^2(X, \mathbb {Z})^G\otimes \mathbb {F}_2$ or, equivalently, since $H^2(X, \mathbb {Z})^G$ is 2-torsion-free, $H^2(X, \mathbb {Z})^G$ is of rank zero.

For ${\rm (ii)}\Rightarrow {\rm (iii)}$ , supposing the contrary that $X(\mathbb {R})$ is disconnected, one can take two points $x, y$ belonging to distinct components of $X(\mathbb {R})$ . Let $\gamma$ be a path from $x$ to $y$ . Then $\gamma -\sigma _*(\gamma )$ is a 1-cycle in $X$ . Since $H_1(X, \mathbb {Z})$ is of odd torsion, there exists an odd number $N$ such that $N(\gamma -\sigma _*(\gamma ))$ is a boundary of a 2-chain $T$ , i.e.

\begin{align*}N(\gamma -\sigma _*(\gamma ))=\partial T.\end{align*}

Then $T+\sigma _*(T)$ is a 2-cycle, $\sigma$ -invariant. Moreover, its class in $H_2(X, \mathbb {Z})$ is non-torsion, since the intersection number of $T+\sigma _*(T)$ with the connected component of $S(\mathbb {R})$ containing $x$ is odd. We thus obtain a free class in $H_2(X, \mathbb {Z})^G$ , a contradiction.

For ${\rm (iii)}\Rightarrow {\rm (ii)}$ , assuming $X(\mathbb {R})$ is connected, the maximality of $X$ implies that

\begin{align*}\dim H^1(X(\mathbb {R}), \mathbb {F}_2)= \textrm {rk} H^2(X, \mathbb {Z})=\textrm {rk} H^2(X, \mathbb {Z})^G+\textrm {rk} H^2(X, \mathbb {Z}(1))^G.\end{align*}

Consider the Viro homomorphism (see [Reference Degtyarev, Itenberg and KharlamovDIK00, Section 1.5]),

\begin{align*}\operatorname {bv}\colon H_2(X, \mathbb {Z}(1))^G\to H_1(X(\mathbb {R}), \mathbb {F}_2),\end{align*}

which is, roughly speaking, defined as follows: for a two-dimensional $\sigma$ -invariant oriented real submanifold $\Sigma \subset X$ such that $\sigma$ reverses its orientation, $\operatorname {bv}$ sends the class $[\Sigma ]\in H_2(X, \mathbb {Z}(1))^G$ to the class of the smooth curve $\Sigma \cap X(\mathbb {R})$ in $H_2(X(\mathbb {R}), \mathbb {F}_2)$ .

We claim that the Viro homomorphism $\operatorname {bv}$ is surjective. Indeed, for any 1-cycle $\gamma$ representing a class $H_1(X(\mathbb {R}), \mathbb {F}_2)$ , its image inside $H_1(X, \mathbb {Z})$ is of odd torsion. Hence there is an odd number $N$ such that the class of $N\gamma$ vanishes, i.e. there exists a 2-chain $T$ in $X$ such that $\partial T=N\gamma$ . Then $T-\sigma _*(T)$ is a 2-cycle, $\sigma$ -anti-invariant, and hence defines an element in $H^2(X, \mathbb {Z}(1))^G$ , which is a preimage of $[\gamma ]$ . The surjectivity of the Viro homomorphism implies that $\textrm {rk} H^2(X, \mathbb {Z}(1))^G\geqslant \dim H^1(X(\mathbb {R}), \mathbb {F}_2)$ . Therefore, we must have $\textrm {rk} H^2(X, \mathbb {Z})^G=0$ .

Proof. The minimal model theory for surfaces has its real version: any smooth projective real surface can be obtained from a minimalFootnote ⁷ one by a sequence of successive blow-ups, where at each step the blow-up center is either a point in the real locus or a pair of complex conjugate points outside of the real locus; see [Reference KollárKol01].

Note that, given a maximal surface $S$ , (any of) its minimal model $S_0$ is again maximal by Proposition 3.8. Moreover, at each step of blow-up in $S=S_n\to \cdots \to S_1\to S_0$ , the blow-up center must be a point in the real locus (otherwise $b_2(S_{i+1}, \mathbb {F}_2)-b_2(S_i, \mathbb {F}_2)=2$ , but the real locus does not change). Therefore, any maximal real surface can be obtained from a maximal minimal real surface, via a sequence of blow-ups at points in the real loci. Since blowing up at a point in the real locus preserves the K-maximality, we only need to check that, for any minimal rational real surface, maximality implies K-maximality (which implies $c_1$ -maximality by Lemma 4.3).

By classification (see [Reference MangolteMan20, Theorem 4.3.23 and Theorem 4.4.11]), minimal rational real surfaces that are maximal are $\mathbb {P}^2$ , $\mathbb {P}^1\times _{\mathbb {R}} \mathbb {P}^1$ and $\mathbb {F}_n$ (real Hirzebruch surface) for $n\geqslant 2$ . For each surface $S$ in the list, $\textrm {KU}^1(S)=0$ and $\textrm {KU}^0(S)$ is generated by real algebraic line bundles:

- for $\mathbb {P}^2$ , the generators are $\mathcal {O}, \mathcal {O}(1), \mathcal {O}(2)$ ;

- for $\mathbb {P}^1\times \mathbb {P}^1$ , the generators are $\mathcal {O},p_1^* \mathcal {O}(1), p_2^* \mathcal {O}(1), p_1^* \mathcal {O}(1)\otimes p_2^* \mathcal {O}(1)$ ;

- for $\mathbb {F}_n=\mathbb {P}_{\mathbb {P}^1}(\mathcal {O}\oplus \mathcal {O}(n))\xrightarrow {\pi } \mathbb {P}^1$ , the generators are $\mathcal {O}, \pi ^*\mathcal {O}_{\mathbb {P}^1}(1), \mathcal {O}_\pi (1), \pi ^*\mathcal {O}_{\mathbb {P}^1}(1)\otimes \mathcal {O}_{\pi }(1)$ .

Hence $\textrm {KR}^0(S)\to \textrm {KU}^0(S)$ is surjective.

Proof. For simplicity, we drop the real structures from the notation. First, we assume that $C$ is maximal and prove the maximality of $M$ . Tensoring with an algebraic $\mathbb {R}$ -line bundle $\mathcal {L}$ on $C$ of positive degree induces an isomorphism of moduli spaces that is compatible with the real structures (in particular, the maximality is preserved):

\begin{align*} -\otimes \mathcal {L}\colon M_C(n, d)\xrightarrow {\simeq } M_C(n, d+n\deg (\mathcal {L})). \end{align*}

We can therefore assume that $d\gt (2g-2)n$ .

By passing to the $\mathbb {F}_2$ -coefficient in Theorem5.1, we get a set of generators for the ring $H^*_G(M, \mathbb {F}_2)$ . We will show that they are all in the image of the restriction map $H^*_G(M, \mathbb {F}_2)\to H^*(M, \mathbb {F}_2)$ . The key point is that the universal bundle $\mathcal {E}$ is a real algebraic/holomorphic vector bundle on the real variety $C\times M$ , and hence defines a class $[\mathcal {E}]\in \textrm {KR}^0(C\times M)$ and possesses equivariant Chern classes $c^G_r(\mathcal {E})\in H^{2r}_G(C\times M, \mathbb {F}_2)$ for any $r\in \mathbb {N}$ .

- For the generators in (i) and (ii), they are all of the form $[c_r(\mathcal {E})]_*(\alpha ):=\pi _*( p^*(\alpha )\smile c_r(\mathcal {E}))$ , where $\alpha \in H^*(C, \mathbb {F}_2)$ , upon which the Chern class $c_r(\mathcal {E})\in H^*(C\times M, \mathbb {F}_2)$ acts as cohomological correspondence. Since we have the commutative diagram

(18)

where the left vertical arrow is surjective by the maximality of $C$ (and Proposition 2.3), we obtain that $[c_r(\mathcal {E})]_*(\alpha )$ lies in the image of $H^*_G(M, \mathbb {F}_2)\to H^*(M, \mathbb {F}_2)$ .

- For the generators in (iii), consider the commutative diagram (for any $r\in \mathbb {N}$ )

(19)

where the left horizontal arrows are the K-theoretic push-forwards along the projection $\pi \colon C\times M\to M$ (which is a real map). Recall that $[\mathcal {E}]\in \textrm {KR}^0(C\times M)$ is mapped to $[\mathcal {E}]\in \textrm {KU}^0(C\times M)$ . Therefore, $c_r(\pi _*(\mathcal {E}))=c_r(\pi _![\mathcal {E}])$ (see Remark 5.2), which naturally lies in the image of $H^*_G(M, \mathbb {F}_2)\to H^*(M, \mathbb {F}_2)$ .

Consequently, $H^*_G(M, \mathbb {F}_2)\to H^*(M, \mathbb {F}_2)$ is surjective. We conclude the maximality of $M$ by Proposition 2.3.

Conversely, assuming that $M$ is maximal, then by Proposition 3.10 and the fact that $H^*(M, \mathbb {Z})$ is torsion-free, we get the maximality of the Albanese variety of $M$ , which is identified with $J^d(C)$ . Since $C(\mathbb {R})\neq \emptyset$ by our hypothesis, tensoring with a degree-1 real line bundle shows that $J(C)$ is isomorphic to $J^d(C)$ , and hence is also maximal. By Corollary 3.12, $C$ is maximal.

Proof. By [Reference Boden and HuBH95], [Reference Boden and YokogawaBY99] and [Reference ThaddeusTha96], there is a wall-and-chamber structure on the space of weights and, by varying the weight, the moduli space stays the same within the chamber and undergoes a standard flip when crossing a wall with flipping centers being projective bundles over products of moduli spaces of stable parabolic bundles with smaller invariants (see [Reference Boden and YokogawaBY99, Theorem 4.1] or [Reference Fu, Hoskins and LehalleurFHPL23, Theorem 5.11]). Therefore, by wall-crossing, induction and Remark 3.6, we only need to prove the maximality of $M^{\alpha }_{C, D}(n, d, \underline {1})$ for $\alpha$ generic in the minimal chamber, where the stability of a parabolic bundle reduces to the stability of the underlying vector bundle. To this end, one uses natural morphisms which forget the flags. By [Reference Boden and YokogawaBY99, Theorem 4.2], such morphisms identify the moduli space of parabolic bundles as iterated flag bundles (hence iterated Grassmannian bundles) over $M_C(n,d)$ . By Proposition 3.3 (or rather Remark 3.4) and Theorem5.3, we can conclude the maximality of $M^{\alpha }_{C, D}(n, d, \underline {1})$ .

Proof. As is mentioned above, $H^*(H, \mathbb {Z})$ is torsion-free thanks to [Reference García-Prada and HeinlothGPH13, Corollary 9].

The Atiyah–Hirzebruch spectral sequence $E_2^{p,q}=H^p(C, \textrm {KU}^q(\textrm {pt}))\Rightarrow \textrm {KU}^{p+q}(C)$ degenerates at the $E_2$ -page for dimension reasons. We obtain that:

1. – the Chern character map $(\textrm {rk}, c_1)\colon \textrm {KU}^0(C)\xrightarrow {\cong } H^0(C, \mathbb {Z})\oplus H^2(C, \mathbb {Z})$ is an isomorphism;

2. – the higher first Chern classFootnote ⁹ map $c_{\frac {1}{2}}\colon \textrm {KU}^1(C) \xrightarrow {\cong } H^1(C, \mathbb {Z})$ is an isomorphism.

As in the statement of the theorem, let $\{\alpha _i\}_{i=1}^{2g}$ be a basis of $H^1(C, \mathbb {Z})$ . We consider the following basis of $H^0(C, \mathbb {Z})\oplus H^2(C, \mathbb {Z})$ :

\begin{align*} \alpha _{2g+1}:=1_C+(g-1) [\textrm {pt}] ; \quad \quad \alpha _{2g+2}:= [\textrm {pt}].\end{align*}

Via the above Chern class isomorphisms, we lift $\{\alpha _i\}_{i=1}^{2g+2}$ to a basis $\{x_i\}_{i=1}^{2g+2}$ of $\textrm {KU}^*(C)$ . In particular, $c_{\frac {1}{2}}(x_i)=\alpha _i$ for $1\leqslant i\leqslant 2g$ , and $x_{2g+1}$ (respectively, $x_{2g+2}$ ) is represented by a degree $g-1$ line bundle (respectively, by $[\mathcal {O}_C]-[\mathcal {O}_C(-x)]$ for a point $x\in C$ ).

We can write the decomposition of the class $[\mathcal {E}]\in \textrm {KU}^0(C\times H)$ via the Künneth formula $\textrm {KU}^0(C\times H)=\textrm {KU}^1(C)\otimes \textrm {KU}^1(H)\oplus \textrm {KU}^0(C)\otimes \textrm {KU}^0(H)$ as follows:

(20) \begin{align} [\mathcal {E}]=\sum _{i=1}^{2g+2}p_1^*(x_i)\otimes p_2^*(e_i),\ \end{align}

where $e_i\in \textrm {KU}^*(H)$ . By Markman [Reference MarkmanMar07, Theorem 3], the (usual) Chern classes $c_r(e_i)$ for $i=2g+1$ and $2g+2$ , and the higher Chern classes $c_{r-\frac {1}{2}}(e_i)$ for $i\leqslant 2g$ , where $r\in \mathbb {N}^*$ , generate the cohomology ring $H^*(H, \mathbb {Z})$ . We will show that they all can be written in the form of generators in (i), (ii) and (iii) in the statement of the theorem:

- For the Chern classes of $e_{2g+2}$ , it is easy to check that when applying $p_{2,!}$ to (20), on the right-hand side, the only contribution is the term $p_{2,!}(p_1^*(x_{2g+2})\otimes p_2^*(e_{2g+2}))=\chi (x_{2g+2})e_{2g+2}=e_{2g+2}$ . Indeed, $\chi (x_{2g+1})=0$ and $\chi (x_{2g+2})=1$ by the Riemann–Roch theorem and the choices for $x_{2g+1}$ and $x_{2g+2}$ . Therefore, $e_{2g+2}=p_{2, !}([\mathcal {E}])$ , and its Chern classes are included in (iii).

- For the Chern classes of $e_{2g+1}$ , it is easy to check that when restricting both sides of (20) to a fiber of $p_1$ , on the right-hand side, the only contribution comes from the term $p_1^*(x_{2g+1})\otimes p_2^*(e_{2g+1})$ , which restricts to $e_{2g+1}$ . Indeed, when restricting to a point of $C$ , $x_{2g+1}$ becomes $1_{\textrm {pt}}$ and $x_{2g+2}$ becomes zero. Therefore, $e_{2g+1}=[\mathcal {E}]|_{H}$ , and its Chern classes are included in (i).

- For the higher Chern classes of $\{e_i\}_{i=1}^{2g}$ , take the $(1, 2r-1)$ -Künneth component of the $r$ th Chern classes of both sides of (20):

\begin{align*} &[c_r(\mathcal {E})]_{(1, 2r-1)}\\ =&\left [c_r\left (\sum _{i=1}^{2g+2}p_1^*(x_i)\otimes p_2^*(e_i)\right )\right ]_{(1, 2r-1)}\\ =&\left [c_r\left (\sum _{i=1}^{2g}p_1^*(x_i)\otimes p_2^*(e_i)\right )\right ]_{(1, 2r-1)}\\ =& \sum _{i=1}^{2g}\left [c_r\left (p_1^*(x_i)\otimes p_2^*(e_i)\right )\right ]_{(1, 2r-1)}\\ =& \sum _{i=1}^{2g} p_1^*c_{\frac {1}{2}}(x_i)\smile p_2^*c_{r-\frac {1}{2}}(e_i)\\ =& \sum _{i=1}^{2g} p_1^*\alpha _i\smile p_2^*c_{r-\frac {1}{2}}(e_i), \end{align*}

where the second equality follows from the fact that $x_{2g+1}, x_{2g+2}, e_{2g+1}, e_{2g+2}$ are in $\textrm {KU}^0$ and hence do not have higher Chern classes in odd cohomology; the third equality is because $c_r\left (\sum _{i=1}^{2g}p_1^*(x_i)\otimes p_2^*(e_i)\right )$ is a polynomial in higher Chern classes of $x_i$ and $e_i$ , but each $x_i$ can only contribute to $c_{\frac {1}{2}}(x_i)\in H^1(C, \mathbb {Z})$ for the $(1, 2r-1)$ -Künneth component, so only one of the $x_i$ can contribute each time; the fourth equality follows from the explicit computation in [Reference MarkmanMar07], as is explained in the next lemma; the last equality is by the definition of the $x_i$ . Therefore, $c_{r-\frac {1}{2}}(e_i)$ is a Künneth component of $c_r(\mathcal {E})$ , which is included in (ii).

To summarize, we have showed the Markman’s generators are indeed the ones in (i), (ii) and (iii) in the statement of the theorem.

Proof. Combining Markman [Reference MarkmanMar07, Lemma 22 (3)] and [Reference MarkmanMar07, the last formula in Section 2], we get

\begin{align*}&c_r(p_1^*x\otimes p_2^*y)=\\& \sum _{\lambda ={(1^{m_1}\cdots s^{m_s})}\dashv r}(-1)^{r-\ell (\lambda )}\prod _{i\geqslant 1}(-1)^{m_i(i-1)}\sum _{0\leqslant k_1\lt \cdots \lt k_{m_i}\leqslant i-1}\prod _{j=1}^{m_i}\left(\begin{array}{c}{i-1}\\{k_j}\end{array}\right)p_1^*c_{k_j+\frac {1}{2}}(x)\smile p_2^*c_{i-k_j-\frac {1}{2}}(y).\end{align*}

On the right-hand side, $c_{k_j+\frac {1}{2}}(x)$ is non-zero only if $k_j=0$ . Therefore, all $k_j=0$ , and hence $m_i=0$ or 1 (i.e. $\lambda$ has no repeated parts), and the above formula simplifies to

\begin{align*}c_r(p_1^*x\otimes p_2^*y)=\sum _{\lambda =(\lambda _1\gt \cdots \gt \lambda _{\ell })\dashv r}(-1)^{r-\ell }\prod _{j\geqslant 1}^{\ell }(-1)^{(\lambda _j-1)}p_1^*c_{\frac {1}{2}}(x)\smile p_2^*c_{i-\frac {1}{2}}(y).\end{align*}

Now if we take the $(1, 2r-1)$ -Künneth component, for each term there can be at most one $c_{\frac {1}{2}}(x)$ in the product, so only the trivial partition $\lambda =(r)$ contributes to this Künneth component, and we get the desired formula.

Proof. The proof is completely analogous to the proof of Theorem5.3. We wrote that proof in the way that it applies here verbatim, with $M$ replaced by $H$ and we use Theorem6.1 in place of Theorem5.1.

Proof. (i) It is easy to see that $X^{[1,2]}$ is isomorphic to $\textrm {Bl}_{\Delta }(X\times X)$ , the blow-up of $X\times X$ along its diagonal, whose maximality follows from Lemma 3.1 and Proposition 3.5.

(ii) Denote by $\tau \colon \textrm {Bl}_{\Delta }(X\times X)\to X\times X$ and $g\colon \textrm {Bl}_{\Delta }(X\times X)\to X^{[2]}$ the natural morphisms, both defined over $\mathbb {R}$ . For any class $\gamma \neq [X]\in H^*(X, \mathbb {F}_2)$ , we first remark that

\begin{align*}\tau _*g^*g_*\tau ^*(\gamma \otimes [X])=\gamma \otimes [X]+[X]\otimes \gamma \in H^*(X\times X, \mathbb {F}_2).\end{align*}

Since $g_*\tau ^*(\gamma \otimes [X])\in H^*(X^{[2]}, \mathbb {F}_2)$ , which is the image of some $\alpha \in H^*_G(X^{[2]}, \mathbb {F}_2)$ by the maximality assumption, we obtain that

\begin{align*}\gamma \otimes [X]+[X]\otimes \gamma =\tau _*g^*(\alpha )\in \textrm {Im} \left (H^*_G(X\times X, \mathbb {F}_2)\to H^*(X\times X, \mathbb {F}_2)\right ).\end{align*}

Taking any real point $p\in X(\mathbb {R})$ , define the $\mathbb {R}$ -morphism $i_p\colon X\to X\times X$ sending $x$ to $(x, p)$ . It follows that $i_p^*(\gamma \otimes [X]+[X]\otimes \gamma )=\gamma$ is in the image of $H^*_G(X, \mathbb {F}_2)\to H^*(X, \mathbb {F}_2)$ . Therefore, $X$ is maximal by Proposition 2.3. (The maximality of $X$ in the surface case is also shown by Kharlamov and Răsdeaconu in [Reference Kharlamov and RăsdeaconuKR23, Theorem 1.1] with a different argument and without the assumption $X(\mathbb {R})\neq \emptyset$ .)

Using (i), $X^{[1,2]}$ is maximal as well.

For $X^{[2,3]}$ , we first note that the universal subscheme in $X\times X^{[2]}$ is canonically identified with the nested Hilbert scheme $X^{[1,2]}$ , which is in turn isomorphic to $\textrm {Bl}_{\Delta }(X\times X)$ as mentioned above. It is known that $X^{[2,3]}$ is isomorphic to the blow-up of $X\times X^{[2]}$ along its universal subscheme. Hence

\begin{align*}X^{[2,3]}\cong \textrm {Bl}_{X^{[1,2]}}(X\times X^{[2]}).\end{align*}

All the involved embeddings are real morphisms between smooth real varieties. Therefore, we can conclude the maximality of $X^{[2,3]}$ by invoking Lemma 3.1 and Proposition 3.5 together with the maximalities of $X$ , $X^{[2]}$ and $X^{[1,2]}$ .

Finally, for $X^{[3]}$ , note that there is a natural surjection $\pi \colon X^{[2,3]}\to X^{[3]}$ by forgetting the length 2 subscheme. It is easy to see that $\pi$ is generically finite of degree $3$ (an odd number). Proposition 3.8 allows us to deduce the maximality of $X^{[3]}$ from that of $X^{[2,3]}$ .

Proof. By Krasnov [Reference KrasnovKra05], the maximality of $X$ implies that its Fano surface of lines $F:=F(X)$ is also maximal (see Proposition 3.14). Applying the realization functors $H^*(-, \mathbb {F}_2)$ and $H^*(-(\mathbb {R}), \mathbb {F}_2)$ to the motivic Galkin–Shinder relation [Reference Galkin and ShinderGS14] established in [Reference Belmans, Fu and RaedscheldersBFR22, Corollary 18], we obtain the following relations between the total $\mathbb {F}_2$ -Betti numbers of $X$ , $F$ and $X^{[2]}$ :

\begin{align*} b_*(F, \mathbb {F}_2)+4 b_*(X, \mathbb {F}_2)&=b_*(X^{[2]}, \mathbb {F}_2)\\ b_*(F(\mathbb {R}), \mathbb {F}_2)+4 b_*(X(\mathbb {R}), \mathbb {F}_2)&=b_*(X^{[2]}(\mathbb {R}), \mathbb {F}_2). \end{align*}

Therefore, $X^{[2]}$ is also maximal. We can conclude by applying Theorem7.3.

Proof of Theorem. By Remark 8.2, the condition $H^1(S, \mathbb {F}_2)=0$ implies that $b_{\operatorname {odd}}(S^{[n]})=0$ and $H^*(S^{[n]}, \mathbb {Z})$ has no 2-torsion. By the universal coefficient theorem, the cohomology ring $H^*(S^{[n]}, \mathbb {F}_2)$ is generated by the same collection of generators in Theorem8.4 by passing to $\mathbb {F}_2$ -coefficients. Like in Theorem5.3, we only need to show that all the generators are in the image of the restriction map $H^*_G(S^{[n]}, \mathbb {F}_2)\to H^*(S^{[n]}, \mathbb {F}_2)$ .

- For generators in (i), since the tautological vector bundle $\mathcal {O}_S^{[n]}$ is a holomorphic real vector bundle on $S^{[n]}$ by construction, it hence defines a class $[\mathcal {O}_S^{[n]}]\in \textrm {KR}^0(S^{[n]})$ . As $c^G_r(\mathcal {O}_S^{[n]})\in H^*_G(S^{[n]}, \mathbb {F}_2)$ is mapped to $c_r(\mathcal {O}_S^{[n]}) \in H^*(S^{[n]}, \mathbb {F}_2)$ , the latter is in the image of the restriction map.

- For generators in (ii), given a class $\alpha \in H^2(S, \mathbb {F}_2)$ , by Proposition 4.4, $S$ is $c_1$ -maximal, i.e. $c_1\colon \textrm {KR}^0(S)\to H^2(S, \mathbb {F}_2)$ is surjective, and there exists an element $L\in \textrm {KR}^0(S)$ such that $c_1(L)=\alpha$ . Up to replacing $L$ by its determinant, one can in fact assume that $L$ is a real vector bundle of rank 1.

We denote by $\widetilde {\alpha }:=c_1^G(L)\in H^2_G(S, \mathbb {F}_2)$ the equivariant first Chern class of $L$ . Clearly, $\widetilde {\alpha }$ is mapped to $\alpha$ under the restriction map. Let $\mathcal {Z}_n\subset S^{[r]}\times S$ be the universal subscheme. Let $p_1\ \text{and}\ p_2$ be the two natural projections from $\mathcal {Z}_r$ to $S^{[r]}$ and $S$ , which are real morphisms since $\mathcal {Z}_n$ is a subscheme defined over $\mathbb {R}$ . We thus obtain an element $p_{1,!}(p_2^*(L))\in \textrm {KR}^0(S^{[n]})$ . By the proof of [Reference Li and QinLQ08, Lemma 3.5], we have the following equality (which holds integrally):

\begin{align*}c_r(p_{1,!}(p_2^*(L)))=\mathfrak {m}_{(1^r), \alpha }\in H^*(S^{[r]}, \mathbb {F}_2).\end{align*}

Therefore, $c^G_r(p_{1,!}(p_2^*(L)))\in H_G^*(S^{[r]}, \mathbb {F}_2)$ is mapped to $\mathfrak {m}_{(1^r), \alpha }$ via the restriction map.

Now consider the commutative diagram

(21)

where the vertical arrows are restriction maps, and the horizontal maps are given by the correspondences induced by the nested Hilbert scheme $S^{[r,n]}:=\{(Z, Z^{\prime})\in S^{[r]}\times S^{[n]}\ |\ Z \subset Z^{\prime}\}$ , viewed as algebraic cycles (defined over $\mathbb {R})$ . By Remark 8.5, the bottom arrow sends $\mathfrak {m}_{(1^r), \alpha }$ to $\unicode {x1D7D9}_{-(n-r)}\mathfrak {m}_{(1^r), \alpha }$ . Since we just proved that $\mathfrak {m}_{(1^r), \alpha }$ is in the image of the left vertical arrow, we conclude by the commutativity of the diagram that $\unicode {x1D7D9}_{-(n-r)}\mathfrak {m}_{(1^r), \alpha }$ is in the image of the right vertical arrow.

- For generators in (iii), by definition, $\unicode {x1D7D9}_{-(n-r)}\mathfrak {p}_{-r}(\textrm {pt})|0\rangle$ is the image of the class $[\textrm {pt}]\in H^4(S)$ under the following composition of correspondences:

\begin{align*}H^*(S)\xrightarrow {[\Gamma _r]_*} H^*(S^{[r]})\xrightarrow {[S^{[r,n]}]_*} H^*(S^{[n]}),\end{align*}

where $\Gamma _r:=\{(x, Z)\in S\times S^{[r]}\ |\ \operatorname {supp}(Z)=\{x\}\}$ and $S^{[r,n]}$ is as above, both viewed as algebraic cycles (see Remark 8.5). However, $\Gamma _n$ and $S^{[r,n]}$ (although singular) are defined over $\mathbb {R}$ , so they induce equivariant correspondences; we have the commutative diagram

(22)

where all the vertical arrows are restriction maps and the left one is surjective since $S$ is maximal (Lemma 4.3). Therefore, $\unicode {x1D7D9}_{-(n-r)}\mathfrak {p}_{-r}(\textrm {pt})|0\rangle =[S^{[r,n]}]_*\circ [\Gamma _r]_*([\textrm {pt}])$ is in the image of the restriction map.

To summarize, all types of generators lie in the image of the restriction map $H^*_G(S^{[n]}, \mathbb {F}_2)\to H^*(S^{[n]}, \mathbb {F}_2)$ , and hence the latter is surjective, and $S^{[n]}$ is maximal by Proposition 2.3.