The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

Abstract

By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by the toric structures one can put on the manifold. Our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of Lalonde–McDuff.

Appendices

Appendix A: Stability of symplectomorphism groups

This appendix is devoted to prove Proposition 3.3. The key tool in the proof is the Inflation Lemma of Lalonde–McDuff and a generalization of it by Olguta Buse. The Inflation Lemma states that in the presence of a $J$-holomorphic curve with nonnegative self-intersection, it is possible to deform a symplectic form $\tau _0$ through a family of forms $\tau _t$ that tame the almost complex structure $J$. Buse recently showed that, under some restrictions on the parameter $t$, actually the $J$-holomorphic curved used in the inflation procedure is allowed to have negative self-intersection.

Lemma 6.1

(Inflation Lemma, see [12] and [19]) Let $J$ be an $\tau _0$-tame almost complex structure on a symplectic 4-manifold $(M, \tau _0)$ that admits a $J$-holomorphic curve $Z$ with $Z\cdot Z \ge 0$. Then there is a family $\tau _t,\,t \ge 0$, of symplectic forms that all tame $J$ and have cohomology class $[\tau _t]=[\tau _0]+t \mathrm{PD}(Z)$, where $\mathrm{PD}(Z)$ is the Poincaré dual to the homology class $[Z]$.

Lemma 6.2

(Buse [5]) Let $J$ be an $\tau _0$-tame almost complex structure on a symplectic 4-manifold $(M, \tau _0)$ that admits a $J$-holomorphic curve $Z$ with $Z\cdot Z =-m, \, m \in \mathbb{N }$. Then for all $\epsilon >0$ there is a family $\tau _t$ of symplectic forms, all taming $J$, which satisfy $[\tau _t]=[\tau _0]+t \mathrm{PD}(Z)$ for all $0 \le t \le \frac{\tau _0(Z)}{m} - \epsilon$.

The proof of the proposition uses the same approach of the proof of Proposition 3.1 in [24], i.e., it is done in three steps. First we prove the following claim.

Claim The spaces $\mathcal{A }_{\mu ,c_1,c_2}$ and $\mathcal{A }_{\mu ^{\prime },c_1,c_2}$ are equal if:

• 1. $\lambda \le c_2 < c_1 < c_1+c_2$ and $\mu ^{\prime } \in (\ell , \mu ]$; or

• 2. $c_ 2 < \lambda \le c_1 < c_1+c_2$ and $\mu ^{\prime } \in (\ell + c_2, \mu ]$; or

• 3. $c_ 2 < c_1 <\lambda \le c_1+c_2$ and $\mu ^{\prime } \in (\ell + c_1, \mu ]$; or

• 4. $c_ 2 < c_1 < c_1+c_2 < \lambda$ and $\mu ^{\prime } \in (\ell + c_1+c_2, \mu ]$.

In the second step we show that the spaces $\mathcal{A }_{\mu ,c_1,c_2}$ and $\mathcal{A }_{\mu ,c_1^{\prime },c_2}$ are equal when the two parameters $c_1$ and $c_1^{\prime }$ satisfy $c_1 \le c_1^{\prime }< \lambda$ or $\lambda \le c_1 \le c_1^{\prime }$. The third step is similar to the second step and asserts that $\mathcal{A }_{\mu ,c_1,c_2}=\mathcal{A }_{\mu ,c_1,c_2^{\prime }}$ whenever the parameters $c_2$ and $c_2^{\prime }$ verify $c_2 \le c_2^{\prime }< \lambda$ or $\lambda \le c_2 \le c_2^{\prime }$.

Step 1 We begin by showing that for any $\mu \ge 1$ and any $\epsilon >0$ one has $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu +\epsilon , c_1,c_2}$. The proof of this fact goes as in the case of 1-blow-up done in [24]. The idea is that it is always possible to inflate the form $\omega _{\mu ,c_1,c_2}$ along some embedded $J$-holomorphic sphere representing the class $F$, to get a one parameter family of symplectic forms in classes $[\omega _{\mu ,c_1,c_2}]+\epsilon \, \mathrm PD (F)=\omega _{\mu +\epsilon ,c_1,c_2}$. All these forms tame $J$.

We next show that for $\mu =\ell +\lambda \ge 1$ and $\lambda \le c_2 < c_1 < c_1+c_2$ we have $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ^{\prime },c_1,c_2}$ whenever $\mu ^{\prime } \in (\ell , \mu ]$. Since $\lambda \le c_2$, there are almost complex structures giving configurations (7) and (9) in Fig. 2. Lemma 2.11 implies that the classes $D=E_1-E_2,\,D_{4\ell +1}=B+\ell F$ and $D_{4\ell -1}=B+\ell F-E_1$ are all represented by embedded $J$-holomorphic spheres. Therefore we can inflate $\omega _{\mu ,c_1,c_2}$ first along the curves $D_{4\ell +1}$ and $D_{4\ell -1}$ getting a two parameter family of symplectic forms

$$\begin{aligned} \omega _{be}=\frac{\omega _{\mu ,c_1,c_2}+bd_{4\ell +1}+ed_{4\ell -1}}{1+b+e} \end{aligned}$$

where $d_i$ is a 2-form representing the Poincaré dual of the homology class $D_i$, PD$(D_i)$ and $b,e \ge 0$. We then have $\omega _{be}(F)=1$,

$$\begin{aligned} \omega _{be}(B)=\frac{\mu +\ell (b+e)}{1+b+e}, \quad \omega _{be}(E_1)=\frac{c_1+e}{1+b+e} \quad \text{ and} \quad \omega _{be}(E_2)=\frac{c_2}{1+b+e}. \end{aligned}$$

We can then inflate $\omega _{be}$ along the curve $E_1-E_2$ getting a family of symplectic forms

$$\begin{aligned} \omega _a=\omega _{be}+a\mathrm{PD}(E_1-E_2), \end{aligned}$$

for all $0\le a < \frac{\omega _{be}(E_1-E_2)}{2}$, where $\mathrm{PD}(E_1-E_2)$ is a 2-form representing the Poincaré dual of the homology class $E_1-E_2$. The area of the exceptional classes $E_i,\,i=1,2$ can be made equal to $c_i$ by setting $e=(c_1+c_2)b/(1-c_1-c_2) \ge 0$ and $a=c_2b/(1-c_1-c_2+b)$. Note that

$$\begin{aligned} 2a < \omega _{be}(E_1-E_2) = \frac{(c_1-c_2)(1-c_1-c_2)+ b(c_1+c_2)}{1-c_1-c_2+b}, \end{aligned}$$

for all $b \ge 0$. This gives a one parameter family of forms $\omega _b$ verifying $\omega _b(F)=1,\,\omega _b(E_1)=c_1,\,\omega _b(E_2)=c_2$ and

$$\begin{aligned} \omega _b(B)=\frac{\mu (1-c_1-c_2)+b\ell }{1-c_1-c_2 +b}. \end{aligned}$$

Letting $b \rightarrow \infty$, this shows that the almost complex structure $J$ is tamed by a symplectic form in class $[\omega _{\ell +\epsilon ,c_1,c_2}]$ for all $0 <\epsilon \le \lambda$. Therefore $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ^{\prime },c_1,c_2}$, for all $\mu ^{\prime } \in (\ell , \mu ]$.

In order to finish the proof of the claim, that is, to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ^{\prime },c_1,c_2}$, we need to consider all other possible cases which includes all configurations up to number (18). For configurations (13) and (15) we use the same inflation process, while for the remaining cases the process is simpler since we do not need to use negative inflation. Nevertheless one needs a different choice of curves, since it has to satisfy Lemma 2.11. The Tables 3, 4 and 5 summarize these choices as well as the values of the parameters.

Table 3 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ^{\prime },c_1,c_2}$ when $\mu ^{\prime } \le \mu$

Table 4 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ^{\prime },c_1,c_2}$ when $\mu ^{\prime } \le \mu$

Table 5 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ^{\prime },c_1,c_2}$ when $\mu ^{\prime } \le \mu$

Step 2 First we show that $\mathcal{A }_{\mu ,c_1^{\prime },c_2} \subset \mathcal{A }_{\mu ,c_1,c_2}$ whenever $c_1 \le c_1^{\prime }$. We consider the case $\lambda \le c_2$ and almost complex structures $J$ such that the embedded $J$-holomorphic spheres satisfy configurations (8) or (10). Given $J\in \mathcal{A }_{\mu ,c_1^{\prime },c_2}$ we need to show that there is a symplectic form that tames $J$ and for which $B,F,E_1$ and $E_2$ have area $\mu ,1,c_1$ and $c_2$ respectively. By Lemmas 2.8 and 2.11 there exist embedded $J$-holomorphic spheres representing the classes $D_{4\ell +1}=B+\ell F,\,D_{4\ell }=B+\ell F-E_2$ and $F$. The Inflation Lemma implies that the Poincaré duals of $D_{4\ell +1},\,D_{4\ell }$ and $F$ are represented by 2-forms $d_{4\ell +1},\,d_{4\ell }$ and $f$ such that the form

$$\begin{aligned} \omega _{abe}=\frac{\omega _{\mu ,c_1^{\prime },c_2}+ ad_{4\ell }+bd_{4\ell +1}+ef}{1+a+b} \end{aligned}$$

(11)

is symplectic for all $a,b,e \ge 0$ and all tame $J$. We then have $\omega _{abe}(F)=1$,

$$\begin{aligned} \omega _{abe}(B)=\frac{\mu + (a+b)\ell +e}{1+a+b} \quad \text{ and} \quad \omega _{abe}(E_2)=\frac{c_2+a}{1+a+b}. \end{aligned}$$

Setting $e=\lambda b/(1-c_2) \ge 0$ and $a=c_2b/(1-c_2) \ge 0$, the resulting family of symplectic forms $\omega _b$ satisfies $\omega _b(F)=1,\,\omega _b(B)=\mu ,\,\omega _b(E_2)=c_2$ and $\omega _b(E_1)=c_1^{\prime }(1-c_2)/(1-c_2+b)$. Since $b$ can take any non negative value, this implies that $J \in \mathcal{A }_{\mu ,c_1,c_2}$ for all $c_1 \le c_1^{\prime }$ in this particular case. For all other cases the choice of curves to use in the inflation process is shown in Tables 6, 7 and 8. Note that for almost complex structures $J$ such as in configurations (7), (9), (13) and (15) one again uses negative inflation.

Table 6 Inflation process to show that $\mathcal{A }_{\mu ,c_1^{\prime },c_2} \subset \mathcal{A }_{\mu ,c_1,c_2}$ when $c_1 \le c_1^{\prime }$

Table 7 Inflation process to show that $\mathcal{A }_{\mu ,c_1^{\prime },c_2} \subset \mathcal{A }_{\mu ,c_1,c_2}$ when $c_1 \le c_1^{\prime }$

Table 8 Inflation process to show that $\mathcal{A }_{\mu ,c_1^{\prime },c_2} \subset \mathcal{A }_{\mu ,c_1,c_2}$ when $c_1 \le c_1^{\prime }$

In the first column of Table 4 the parameter $b$ satisfies $0 \le b \le \frac{c_1^{\prime }-c_2}{2c_2}$. This implies that $2a \le \omega _{be}(E_1-E_2)$ as required by negative inflation and $c_2 \le \omega _b(E_1) \le c_1^{\prime }$ as desired.

We now suppose that $\lambda \le c_2$ and prove that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1^{\prime },c_2}$ whenever $c_1 \le c_1^{\prime }$. From Lemma 2.11 we know that, for the configurations (7) and (9) we can inflate along the classes $D_{4\ell -1}=B+\ell F-E_1,\, F$ and $E_1-E_2$. Then, setting first

$$\begin{aligned} \omega _{be}=\frac{\omega _{\mu ,c_1,c_2}+bd_{4\ell -1}+ef}{1+b} \end{aligned}$$

and then

$$\begin{aligned} \omega _a=\omega _{b,e}+a\mathrm{PD}(E_1-E_2), \end{aligned}$$

where $a=c_2b$ and $e=\lambda b$, the resulting one parameter family of symplectic forms satisfy $\omega _b(F)=1,\,\omega _b(B)=\mu ,\,\omega _b(E_2)=c_2$ and $\omega _b(E_1)=(c_1+(1-c_2)b)/(1+b)$. Letting $b\rightarrow \infty$, this yields the desired result. Note that since $2c_2<1$ one has $2a < \omega _{be}(E_1-E_2)$, for all $b \ge 0$, as required by negative inflation. All remaining cases that show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1^{\prime },c_2}$ when $c_1 \le c_1^{\prime }$, are summarized in the Tables 9, 10 and 11.

Table 9 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1^{\prime },c_2}$ when $c_1 \le c_1^{\prime }$

Table 10 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1^{\prime },c_2}$ when $c_1 \le c_1^{\prime }$

Table 11 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1^{\prime },c_2}$ when $c_1 \le c_1^{\prime }$

Step 3 The proof of this step is similar to the last one, although it is simpler when one shows that $\mathcal{A }_{\mu ,c_1,c_2^{\prime }} \subset \mathcal{A }_{\mu ,c_1,c_2}$ whenever $c_2 \le c_2^{\prime }$, since we can inflate along the curve $E_2$. More precisely, in this case it is always possible to inflate the form $\omega _{\mu ,c_1,c_2^{\prime }}$ along an embedded $J$-holomorphic sphere representing the class $E_2$, to get a one parameter family of symplectic forms in classes $[\omega _{\mu ,c_1,c_2^{\prime }}]+\epsilon \mathrm{PD}(E_2)=\omega _{\mu , c_1,c_2^{\prime }-\epsilon }$, where $0 \le \epsilon < c_2^{\prime }$. All these forms tame $J$.

Next, in Tables 12, 13, we show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1,c_2^{\prime }}$ whenever $c_2 \le c_2^{\prime }$.

Table 12 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1,c_2^{\prime }}$ when $c_2 \le c_2^{\prime }$

Table 13 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1,c_2^{\prime }}$ when $c_2 \le c_2^{\prime }$

Note that in the first column of Table 12 if $c_1 \ge 1/2 \Leftrightarrow 1-c_1 \le c_1$ then the negative inflation requirement $2a < \omega _{be}(E_1-E_2)$ is satisfied for all $b \ge 0$, and this gives $c_2 \le \omega _b(E_2)< 1-c_1$. On the other hand, if $c_1 \le 1/2 \Leftrightarrow 1-c_1 \ge c_1$ then the condition $2a < \omega _{be}(E_1-E_2)$ implies that the parameter $b$ satisfies $0 \le b \le \frac{c_1-c_2}{1-2c_1}$ and therefore one has $c_2 \le \omega _{be}(E_2) \le c_1$ as desired. There is a similar remark regarding the parameter $b$ in the second column.

For configurations (11) and (12) we need to consider two distinct cases: $\lambda \ge 1-c_1$ and $\lambda < 1-c_1$ (see Table 14).

Table 14 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1,c_2^{\prime }}$ when $c_2 \le c_2^{\prime }$

And finally for configurations (17) and (18) we also need to consider two distinct cases: $\lambda -c_1 \le c_1 \Leftrightarrow \lambda \le 2c_1$ and $c_1 < \lambda -c_1 \Leftrightarrow \lambda > 2c_1$ (see Table 15).

Table 15 Inflation process to show that $\mathcal{A }_{\mu ,c_1,c_2}\subset \mathcal{A }_{\mu ,c_1,c_2^{\prime }}$ when $c_2 \le c_2^{\prime }$

Appendix B: Compatible integrable complex structures

The purpose of this Appendix is to briefly explain some differential and topological results relative to the stratification of the space $\mathcal{J }_{\omega }:=\widetilde{\mathcal{J }}_{\mu ,c_1,c_2}$ of compatible almost complex structures on $\widetilde{M}_{\mu ,c_1,c_2}$, and to the stratification it induces on the subspace $\mathcal{J }_{\omega }^{int}\subset \mathcal{J }_{\omega }$ of compatible, integrable, complex structures.

Recall from Sect. 2.2 that the space $\mathcal{J }_{\omega }$ is the disjoint union of finitely many strata, $\mathcal{J }_{\omega }=\sqcup U_{i},\,i=0,\ldots ,k$. Each stratum is characterized by the existence of a unique chain of holomorphic embedded spheres of one of the types (1)–(18) and containing a curve in class $D_{-m}$, for some $m\ge 0$ (see Figs. 2, 3, 4, 5, 6 and the definition of the classes $D_i$ in Sect. 2.2).

Proposition 7.1

Suppose the stratum $U_{\mathcal{A }}\subset \mathcal{J }_{\omega }$ is characterized by the existence of a configuration of $J$-holomorphic embedded spheres $C_{1}\cup C_{2}\cup \cdots \cup C_{N}$ representing a given set of distinct homology classes $A_{1},\ldots , A_{N}$ of negative self-intersection. Then $U_{\mathcal{A }}$ is a cooriented Fréchet submanifold of $\mathcal{J }_{\omega }$ of (real) codimension $\mathrm{codim}_{\mathbb{R }}(U_{\mathcal{A }})= 2N-2c_{1}(A_{1}+\cdots +A_{N})$.

Proof

(Sketch of proof) Given a symplectic $4$-manifold $(M,\omega )$ and a set of $N$ distinct spherical homology classes

$$\begin{aligned} \mathcal{A }=\{A_{1},\ldots ,A_{N}\}\subset H_{2}(M;\mathbb{Z }) \end{aligned}$$

let denote by $\mathcal{M }(\mathcal{A },\mathcal{J }_{\omega })$ the space of $(N+1)$-tuples

$$\begin{aligned} (u,J):=(u_{1},\ldots ,u_{N},J)\in C^{\infty }\left((\mathbb{C }P^{1})^{N},M\right)\times \mathcal{J }_{\omega } \end{aligned}$$

such that $u_{i}:\mathbb{C }P^{1}\rightarrow M$ is a somewhere injective $J$-holomorphic map whose image represents the homology class $A_{i}$. By Proposition 6.2.7 in [21], this space is always a smooth infinite dimensional Fréchet manifold. Its tangent space at $(u,J)$ can be identified with the vector space of $(N+1)$-tuples

$$\begin{aligned} (\xi ,Y):=(\xi _{1},\ldots ,\xi _{N},Y)\in \Omega ^{0}(u_{1}^{*}(TM))\times \cdots \times \Omega ^{0}(u_{N}^{*}(TM))\times S\Omega _{J}^{0,1}(TM) \end{aligned}$$

such that

$$\begin{aligned} \bar{\partial }_{J}\xi + u_{i}^{*}Y = 0,~\forall i \end{aligned}$$

where $S\Omega _{J}^{0,1}(TM)$ is the image of the complex symmetric $(0,2)$-tensors $S^{0,2}_{J}(M)\subset T^{(0,2)}_{J}(M)$ under the natural identification $T^{(0,2)}_{J}(M)\simeq \Omega ^{(0,1)}_{J}(TM)$ defined by the Hermitian metric $h_{J}(v,w):=\omega (v,Jw)-i\omega (v,w)$, and where $\bar{\partial }_{J}:\Omega ^{0}(u^{*}(TM))\rightarrow \Omega ^{0,1}_{J}(u^{*}(TM))$ is a $0$-order perturbation of a genuine Dolbeault operator. The image under the projection $\pi :\mathcal{M }(A,\mathcal{J }_{\omega })\rightarrow \mathcal{J }_{\omega }$ is the set $U_{\mathcal{A }}$ of all $J$ such that each class in $\mathcal{A }$ is represented by an irreducible $J$-holomorphic sphere. The differential is given by the projection

$$\begin{aligned} \pi _{*}: T_{(u,J)}\mathcal{M }(\mathcal{A },\mathcal{J }_{\omega })&\rightarrow T_{J}\mathcal{J }_{\omega }\\ (\xi ,Y)&\mapsto Y \end{aligned}$$

and is Fredholm of index $2c_{1}(A_{1}+\cdots +A_{N})+4N$. Its image at $(u,J)$ is

$$\begin{aligned} \mathrm{Im}(\pi _{*})&= \left\{ Y~|~u_{i}^{*}(Y)\in \mathrm{Im}\left(\bar{\partial }_{J}:\Omega ^{0}(u_{i}^{*}(TM))\rightarrow \Omega ^{0,1}_{J}(u_{i}^{*}(TM)\right),~\forall i\right\} \\&\simeq \mathrm{Im}\left(\oplus \bar{\partial }_{J}:\bigoplus \Omega ^{0}(u_{i}^{*}(TM))\rightarrow \bigoplus \Omega ^{0,1}_{J}(u_{i}^{*}(TM))\right) \end{aligned}$$

while its kernel is isomorphic to $\ker (\oplus \bar{\partial }_{J})$. Moreover, because the classes $A_{i}$ have negative self-intersection, each element in the preimage $\pi ^{-1}(J)$ is obtained from a single $J$-holomorphic map $u:(\mathbb{C }P^{1})^{N}\rightarrow M$ by reparametrization of each component under the free action of $\mathrm{PSL}(2,\mathbb{C })$. It follows that $\ker (\pi _{*})\simeq \ker (\oplus \bar{\partial }_{J})$ is isomorphic to $(\mathrm{sl}(2,\mathbb{C }))^{N}$. Consequently, $\mathrm{coker}(\pi _{*})$ has constant dimension $6N - \left(2c_{1}(A_{1}+\cdots A_{N})+4N\right)$ and the projection $\pi$ factors through a smooth embedding

$$\begin{aligned} \mathcal{M }(\mathcal{A }, \mathcal{J }_{\omega }) / (\mathrm{PSL}(2,\mathbb{C }))^{N} \hookrightarrow \mathcal{J }_{\omega } \end{aligned}$$

To show that the bundle $\mathrm{coker}(\pi _{*})\rightarrow \mathrm{U}_{i}$ is oriented, we first note that $\pi _{*}$ lies in a component of Fredholm operators that contains a complex linear operator (namely the projection $\pi _{*}$ at an integrable $J$). Therefore, the determinant bundle $\Lambda ^{\max }\ker (\pi _{*})\otimes \Lambda ^{\max }\mathrm{coker}(\pi _{*})$ is oriented. Since $\ker (\pi _{*})\simeq \mathrm{sl}(2,\mathbb{C })$ is also oriented, this defines an orientation on the cokernel bundle. For more details, see Appendix A in [1], Proposition 2.8 in [2], and Proposition 6.2.7 in [21]. $\square$

We now turn our attention to the subspace $\mathcal{J }_{\omega }^{int}\subset \mathcal{J }_{\omega }$ of compatible, integrable, complex structures. Given a stratum $\mathrm{U}_{i}\subset \mathcal{J }_{\omega }$, let write $V_{i}=\mathrm{U}_{i}\cap \mathcal{J }_{\omega }^{int}$.

Lemma 7.2

Given any $J\in \mathcal{J }_{\omega }^{int}$, the complex surface $(\mathbb{X }_{3}, J)$ is the twofold blow-up of a Hirzebruch surface $(\mathbb{F }_{m},J_{m})$.

Proof

This follows from the fact that given any $J\in \mathcal{J }_{\omega }^{int}$, the class $E_{2}$ is always represented by an exceptional complex curve that we can blow-down, and that on the resulting surface $(\mathbb{X }_{2},J)$, the exceptional classes $E_{1}$ and $F-E_{1}$ are both represented. The blow-down along the class $E_{1}$ yields an even Hirzebruch surface $\mathbb{F }_{2k}$ while the blow-down along the class $F-E_{1}$ yields an odd Hirzebruch surface $\mathbb{F }_{2k-1}$. $\square$

From a complex analytic (or algebraic) point of view, the 18 types of compatible complex structures on $\mathbb{X }_{3}$ can be constructed as follows:

• Type (1) Twofold blow-up of $\mathbb{F }_{0}$ at two generic points (not lying on the same fiber $F$ nor on the same section $B$).

• Type (2) Twofold blow-up of $\mathbb{F }_{0}$ at two distinct points on the same fiber $F$.

• Type (3) Twofold blow-up of $\mathbb{F }_{0}$ at two distinct points on the same section $B$.

• Type (4) Twofold blow-up of $\mathbb{F }_{0}$ at two “infinitely near” points on a fiber, that is, the blow-up of $\mathbb{F }_{0}$ at $p$ followed by the blow-up of $\widetilde{\mathbb{F }}_{0}$ at the line $\ell _{p}=T_{p}F\subset T_{p}\mathbb{F }_{0}$ on the exceptional divisor.

• Type (5) Twofold blow-up of $\mathbb{F }_{0}$ at two “infinitely near” points on a flat section $B$, that is, at $(p, \ell _{p}=T_{p}B)$.

• Type (6) Twofold blow-up of $\mathbb{F }_{0}$ at two “infinitely near” points $(p, \ell _{p})$ with the direction $\ell _{p}$ transverse to $T_{p}F$ and $T_{p}B$.

• Type (7) Twofold blow-up of $\mathbb{F }_{2m-1}$ at two distinct points on the same fiber, one of which lying on the section $s_{0}$ of self-intersection $-2m+1$.

• Type (8) Twofold blow-up of $\mathbb{F }_{2m-1}$ at two “infinitely near” points $(p,\ell _{p})$, where $p\in s_{0}$ and $\ell _{p}$ is tranverse to $s_{0}$ and to the fiber $F_{p}$.

• Type (9) Twofold blow-up of $\mathbb{F }_{2m-1}$ at two “infinitely near” points $(p,\ell _{p})$, where $p\in s_{0}$ and $\ell _{p}=T_{p}F$.

• Type (10) Twofold blow-up of $\mathbb{F }_{2m}$ at two generic points, that is, at two points on two different fibers and away from $s_{0}$.

• Type (11) Twofold blow-up of $\mathbb{F }_{2m}$ at two distinct points on the same fiber, one of which belonging to $s_{0}$.

• Type (12) Twofold blow-up of $\mathbb{F }_{2m}$ at two points on different fibers, one points belonging to $s_{0}$.

• Type (13) Twofold blow-up of $\mathbb{F }_{2m}$ at two distinct points on the same fiber, one of which belonging to $s_{0}$ (note that only the order of the two blow-ups distinguishes (13) from (11)).

• Type (14) Twofold blow-up of $\mathbb{F }_{2m}$ at two “infinitely near” points $(p,\ell _{p})$, where $p\in s_{0}$ and $\ell _{p}$ is tranverse to $s_{0}$ and to the fiber $F_{p}$.

• Type (15) Twofold blow-up of $\mathbb{F }_{2m}$ at two “infinitely near” points $(p,\ell _{p})$, where $p\in s_{0}$ and $\ell _{p}=T_{p}F$.

• Type (16) Twofold blow-up of $\mathbb{F }_{2m}$ at two points on different fibers, one points belonging to $s_{0}$ (note that only the order of the two blow-ups distinguishes (16) from (12)).

• Type (17) Twofold blow-up of $\mathbb{F }_{2m}$ at two “infinitely near” points $(p,\ell _{p})$, where $p\in s_{0}$ and $\ell _{p}=T_{p}s_{0}$.

• Type (18) Twofold blow-up of $\mathbb{F }_{2m}$ at two distinct points on $s_{0}$.

Let $\mathrm{Diff}_{h}$ denote the group of diffeomorphisms of $\mathbb{X }_{3}$ acting trivially on homology.

Proposition 7.3

Given any two compatible complex structures $J_{1}, J_{2}$ in $V_{i}$, there exists a diffeomorphism $\phi$ acting trivially on homology such that $J_{2}=\phi _{*}J_{1}$. Consequently, given any $J_{i}\in V_{i}$, we have

$$\begin{aligned} V_{i} = U_{i}\cap \mathcal{J }_{\omega }^{int}=(\mathrm{Diff}_{h}\cdot J_{i})\cap \mathcal{J }_{\omega } \end{aligned}$$

Proof

First recall that the complex automorphism group of $\mathbb{F }_{0}$ is isomorphic to

$$\begin{aligned} \mathrm{Aut}(\mathbb{F }_{0})\simeq (\mathrm{PSL}(2,\mathbb{C })\times \mathrm{PSL}(2,\mathbb{C }))\ltimes \mathbb{Z }_{2} \end{aligned}$$

while the automorphism group of $\mathbb{F }_{m},\,m\ge 1$, is isomorphic to the semi-direct product

$$\begin{aligned} \mathrm{Aut}(\mathbb{F }_{m}) \simeq GL(2,\mathbb{C })/\mu _{m}\ltimes H^{0}(\mathbb{C }P^{1}, (\Lambda ^{m})^{*}) \end{aligned}$$

where $\mu _{m}$ is the group of $m$th roots of unity. The action of $GL(2,\mathbb{C })/\mu _{m}$ lifts the action of $\mathrm{PSL}(2,\mathbb{C })$ on $\mathbb{C }P^{1}$ and thus acts triply transitively on the set of fibers of $\mathbb{F }_{m}$. This action preserves the zero section $s_{0}$ (of self-intersection $-m$) and the section at infinity $s_{\infty }$ (of self-intersection $+m$) and is transitive on their complement $\mathbb{F }_{m}\setminus \{s_{0},s_{\infty }\}$. The group of sections of the dual line bundle $(\Lambda ^{m})^{*}$ is isomorphic to the space $\mathrm{Sym}^{m}(\mathbb{C }^{2})$ of symmetric $m$-linear forms on $\mathbb{C }^{2}$—that is, homogenous polynomials of degree $m$ in two variables—and acts fiberwise by

$$\begin{aligned} \alpha \cdot [z:u^{\otimes m}] = [z+\alpha (u^{\otimes m}):u^{\otimes m}] \end{aligned}$$

Once restricted to a single fiber $F_{p}\simeq \mathbb{C }P^{2}$, the action of $\mathrm{Sym}^{m}(\mathbb{C }^{2})$ fixes the point $s_{0}\cap F_{p}=[1:0]$ while it is simply transitive on $F_{p}\setminus [1:0]$. We note that for $m\ge 1$, these automorphism groups are all connected and, consequently, they act trivially on homology.

Now, the proof of the statement reduces to showing that the appropriate automorphism group acts transitively on the pairs of points $(p_{1},p_{2})$ or $(p,\ell _{p})$ that define the compatible complex structures of a given type (1)–(18) as blow-ups of Hirzebruch structures. $\square$

In their paper [2], Abreu-Granja-Kitchloo proved that, under some cohomological conditions, $\mathcal{J }^{int}$ is a genuine Fréchet submanifold of $\mathcal{J }$ whose tangent bundle may be described using standard deformation theory. In order to state their result, let write $H^{0,q}_{J}(M)$ for the $q$th Dolbeault cohomology group with coefficients in the sheaf of germs of holomorphic functions, and $H^{0,q}_{J}(TM)$ for the $q$th Dolbeault cohomology group with coefficients in the sheaf of germs of holomorphic vector fields. Then,

Theorem 7.4

([2] , Theorem 2.3) If $(M,\omega )$ is a symplectic $4$-manifold, $J\in \mathcal{J }^{int}_{\omega }$ is a compatible integrable complex structure, and the cohomology groups $H^{0,2}_{J}(M)$ and $H^{0,2}_{J}(TM)$ are zero, then $\mathcal{J }^{int}_{\omega }$ is a submanifold of $\mathcal{J }_{\omega }$ in the neighborhood of $J$. Moreover, the moduli space of infinitesimal compatible deformations of $J$ in $\mathcal{J }^{int}_{\omega }$ coincides with the moduli space of infinitesimal deformations of $J$ in the set of all integrable structures, that is, it is given by $H^{0,1}_{J}(TM)$. Finally, the tangent space of $\mathcal{J }^{int}_{\omega }$ at $J$ is naturally identified with

$T_{J}((\mathrm{Diff}\cdot J)\cap \mathcal{J }^{int}_{\omega })\oplus H^{0,1}_{J}(TM)$.

Lemma 7.5

For any $J\in \mathcal{J }_{\omega }^{int}$, the cohomology groups $H^{0,2}_{J}(\mathbb{X }_{3})$ and $H^{0,2}_{J}(T\mathbb{X }_{3})$ are zero.

Proof

(See also [11] Sect. 5.2(a)(iv) p.220.) By Lemma 7.2, we know that any compatible complex structure $J\in \mathcal{J }_{\omega }^{int}$ is obtained by blowing-up a Hirzebruch structure on $S^2\times S^2$. Now, for any complex surface $(X,J)$, the geometric genus $p_{g}:=\mathrm{rk}H^{0,2}_{J}(X)$ is a birational invariant. Consequently, the first assertion follows from the classical fact that $p_{g}(\mathbb{F }_{m})=0$, for all $m\ge 0$.

As for $H^{0,2}_{J}(T\mathbb{X }_{3}):=\check{H}^{2}_{J}(T\mathbb{X }_{3})$, Serre duality implies that $\check{H}^{2}_{J}(T\mathbb{X }_{3})^{\vee }\simeq \check{H}^{0}(\mathcal K _{J}\otimes \Omega ^{1}_{J})$. Now, by Lemma 2.5, $(\mathbb{X }_{3},J)$ contains a two-dimensional family of embedded rational curves of zero self-intersection (the fibers in class $[F]$) which cover a dense open set, and the restriction of the rank $2$ bundle $\mathcal K _{J}\otimes \Omega ^{1}_{J}$ to any of those curves is isomorphic to $\mathcal{O }(-4)\oplus \mathcal{O }(-2)$. Hence, $\mathcal K _{J}\otimes \Omega ^{1}_{J}$ cannot have any nontrivial holomorphic sections. $\square$

Let $\mathrm{Aut}_{h}(J)\subset \mathrm{Diff}_{h}$ for the subgroup of complex automorphism of $(M,J)$. Let $\mathrm{Iso}_{h}(\omega ,J)\subset \mathrm{Aut}_{h}(J)$ denote the Kähler isometry group of $(M,\omega ,J)$. The next result shows that in some cases the part of the $\mathrm{Diff}_{h}$-orbit of $J$ which lies in $\mathcal{J }^{int}_{\omega }$ may be identified with the $\mathrm{Symp}_{h}(M,\omega )$-orbit:

Theorem 7.6

([2], Corollary 2.6) If $J\in \mathcal{J }^{int}_{\omega }$ is such that the inclusion $\mathrm{Iso}_{h}(\omega ,J)\hookrightarrow \mathrm{Aut}_{h}(J)$ is a weak homotopy equivalence, then the inclusion of the $\mathrm{Symp}_{h}(M,\omega )$-orbit of $J$ in $(\mathrm{Diff}_{h}\cdot J)\cap \mathcal{J }^{int}_{\omega }$

$$\begin{aligned} \mathrm{Symp}_{h}(M,\omega )/\mathrm{Iso}_{h}(\omega ,J)\hookrightarrow (\mathrm{Diff}_{h}\cdot J)\cap \mathcal{J }^{int}_{\omega } \end{aligned}$$

is also a weak homotopy equivalence.

Remark 7.7

The actual statement of [2] Corollary 2.6 gives a condition for the full $\mathrm{Symp}(M,\omega )$ orbit of $J\in \mathcal{J }_{\omega }^{int}$ to be homotopy equivalent to its $\mathrm{Diff}_{[\omega ]}$ orbit, where $\mathrm{Diff}_{[\omega ]}$ is the group of diffeomorphisms preserving the class $[\omega ]$. However, the same arguments apply to the orbits of $J$ under $\mathrm{Symp}_{h}$ and $\mathrm{Diff}_{h}$.

Lemma 7.8

Given any $J\in \mathcal{J }^{int}(\widetilde{M}_{\mu ,c_1,c_2})$, the inclusion $\mathrm{Iso}_{h}(\omega _{\mu ,c_1,c_2},J)\hookrightarrow \mathrm{Hol}_{h}(J)$ is a weak homotopy equivalence.

Proof

First observe that if $(\widetilde{M},\widetilde{J})$ is the blow-up of $(M,J)$ at a point $p$, then the automorphism group $\mathrm{Hol}_{h}(\widetilde{J})$ is isomorphic to the stabilizer subgroup of $p$ in $\mathrm{Hol}_{h}(J)$. Looking at the 18 possible types of compatible complex structures defined on $\mathbb{X }_{3}\simeq \widetilde{M}_{\mu ,c_1,c_2}$, it is easy, but tedious, to check that the complex automorphism groups $\mathrm{Hol}_{h}(\mathbb{X }_{3},J)$ are homotopy equivalent to

$$\begin{aligned} \mathrm{Aut}_{h}(\mathbb{X }_{3},J)\simeq {\left\{ \begin{array}{ll} S^{1}&\text{ if}\,J \text{ is} \text{ of} \text{ type} \text{6,} \text{8} \text{ or} \text{14},\\ T^{2}&\text{ otherwise.} \end{array}\right.} \end{aligned}$$

On the other hand, the isometry groups of the Hirzebruch surfaces (with respect to any Kähler form $\omega$) are the maximal compact Lie subgroups of their complex automorphism groups, namely

$$\begin{aligned} \mathrm{Iso}(\mathbb{F }_{m},\omega ) \simeq {\left\{ \begin{array}{ll} (\mathrm{SO}(3)\times \mathrm{SO}(3))\ltimes \mathbb{Z }_{2}&\text{ if} \, m=0,\\ S^{1}\times \mathrm{SO}(3)&\text{ if}\, m\,\text{ is} \text{ even},\\ \mathrm{U}(2)&\text{ if}\,m\,\text{ is} \text{ odd}. \end{array}\right.} \end{aligned}$$

In particular, they are deformation retracts of the automorphism groups $\mathrm{Aut}(\mathbb{F }_{m})$. After blow-up, they induce isometry groups $\mathrm{Iso}_{h}(\mathbb{X }_{3},\omega ,J)$ isomorphic to

$$\begin{aligned} \mathrm{Iso}_{h}(\mathbb{X }_{3},\omega ,J)\simeq {\left\{ \begin{array}{ll} S^{1}&\text{ if}\,J\,\text{ is} \text{ of} \text{ type} \text{6,} \text{8,} \text{ or} \text{14},\\ T^{2}&\text{ otherwise.} \end{array}\right.} \end{aligned}$$

$\square$

Together with Proposition 7.3, this gives

Corollary 7.9

Given $J\in V_{i}\subset \mathcal{J }_{\omega }^{int}$, there is a weak homotopy equivalence

$$\begin{aligned} \mathrm{Symp}_{h}(\widetilde{M}_{\mu ,c_1,c_2},\omega )/\mathrm{Iso}_{h}(\omega ,J) \simeq V_{i} \end{aligned}$$

Let $\mathcal{A }=\{A_{1},\ldots ,A_{N}\}$ be a set of distinct spherical homology classes of negative self-intersections. Let $U_{\mathcal{A }}$ be the stratum it defines in $\mathcal{J }_{\omega }$. The next proposition gives conditions ensuring that $U_{\mathcal{A }}$ is tranversal to $\mathcal{J }_{\omega }^{int}$ and that its normal bundle at $J\in \mathcal{J }_{\omega }^{int}$ may be described in terms of deformation theory.

Theorem 7.10

([2], Theorem 2.9) Let $(M,\omega ,J)$ be a Kähler $4$-manifold such that $J\in V_{\mathcal{A }}:=U_{\mathcal{A }}\cap \mathcal{J }_{\omega }^{int}$ and that $V_{\mathcal{A }}=(\mathrm{Diff}_{[\omega ]}\cdot J)\cap \mathcal{J }_{\omega }$. Suppose that the cohomology groups $H^{0,2}_{J}(M)$ and $H^{0,2}_{J}(TM)$ are zero. Suppose also that $(u,J):=(u_{1},\ldots ,u_{N},J)\in \mathcal{M }(\mathcal{A },\mathcal{J }_{\omega })$ is such that $u^{*}:H^{0,1}_{J}(TM)\rightarrow \bigoplus H^{0,1}(u_{i}^{*}(TM))$ is surjective. Then the projection $\pi :\mathcal{M }(\mathcal{A },\mathcal{J }_{\omega })\rightarrow \mathcal{J }_{\omega }$ is tranversal at $(u,J)$ to $\mathcal{J }_{\omega }^{int}\subset \mathcal{J }_{\omega }$ and the infinitesimal complement to the image $U_{\mathcal{A }}$ of $\pi$ in a neighborhood of $J$ can be identified with the moduli space of infinitesimal deformations $H^{0,1}_{J}(TM)$.

Lemma 7.11

Given $J\in V_{\mathcal{A }}\subset \mathcal{J }^{int}(\mathbb{X }_{3},\omega )$, let denote by $C$ the unique $J$-holomorphic configuration of type $\mathcal{A }$, and let $u=(u_{1},\ldots ,u_{N})$ be some $J$-holomorphic parametrization of $C$. Then the induced map $u^{*}:H^{0,1}(T\mathbb{X }_{3})\rightarrow \bigoplus H^{0,1}(u^{*}(T\mathbb{X }_{3}))$ is surjective.

Proof

Consider the exact sequence of sheaves associated to the inclusion $f:C\hookrightarrow \mathbb{X }_{3}$ of $C$ viewed as a nodal curve:

$$\begin{aligned} 0\rightarrow \mathcal{O }_{\mathbb{X }_{3}}(-C)\rightarrow \mathcal{O }_{\mathbb{X }_{3}}\rightarrow \mathcal{O }_{C}\rightarrow 0 \end{aligned}$$

Tensoring with $T\mathbb{X }_{3}$ we get the short exact sequence

$$\begin{aligned} 0\rightarrow \mathcal{O }_{\mathbb{X }_{3}}(-C)\otimes T\mathbb{X }_{3}\rightarrow T\mathbb{X }_{3}\rightarrow \mathcal{O }_{C}\otimes T\mathbb{X }_{3}\simeq f_{*}f^{*}T\mathbb{X }_{3}\rightarrow 0 \end{aligned}$$

whose associated cohomology sequence is

$$\begin{aligned} \cdots \rightarrow H^{1}(\mathbb{X }_{3};TX_{3})\stackrel{f^{*}}{\rightarrow } H^{1}(C;f^{*}T\mathbb{X }_{3})\rightarrow H^{2}(\mathbb{X }_{3};\mathcal{O }_{\mathbb{X }_{3}}(-C)\otimes T\mathbb{X }_{3})\rightarrow \cdots \end{aligned}$$

The sheaf $\mathcal{O }_{\mathbb{X }_{3}}(-C)\otimes T\mathbb{X }_{3}$ being locally free, Serre duality implies that $H^{2}(\mathbb{X }_{3};\mathcal{O }_{\mathbb{X }_{3}}(-C)\otimes T\mathbb{X }_{3})\simeq H^{0}(\mathbb{X }_{3};\mathcal{O }_{\mathbb{X }_{3}}(C)\otimes T\mathbb{X }_{3}^{\vee }\otimes K_{\mathbb{X }_{3}}$. But, since $C\cdot F = 1$, the restriction of $\mathcal{O }_{\mathbb{X }_{3}}(C)\otimes T\mathbb{X }_{3}^{\vee }\otimes K_{\mathbb{X }_{3}}$ to any fiber $F$ is isomorphic to $\mathcal{O }(1)\otimes (\mathcal{O }(-4)\oplus \mathcal{O }(-2))$. Since the fibers cover an open dense subset of $\mathbb{X }_{3}$, it follows that $\mathcal{O }_{\mathbb{X }_{3}}(C)\otimes T\mathbb{X }_{3}^{\vee }\otimes K_{\mathbb{X }_{3}}$ has no nontrivial sections and, by duality, that $H^{2}(\mathbb{X }_{3};\mathcal{O }_{\mathbb{X }_{3}}(-C)\otimes T\mathbb{X }_{3})=0$.

Let $\nu :\widetilde{C}\rightarrow C$ be the normalization defined by $u=f\circ \nu$. We claim that

$$\begin{aligned} \nu ^{*}: H^{1}(C;f^{*}T\mathbb{X }_{3})\rightarrow H^{1}(\widetilde{C};\nu ^{*}f^{*}T\mathbb{X }_{3}) \end{aligned}$$

is surjective. To see this, we first observe that $H^{1}(\widetilde{C};\nu ^{*}f^{*}T\mathbb{X }_{3})\simeq H^{1}(C;\nu _{*}\nu ^{*}f^{*}T\mathbb{X }_{3})$ since $\nu$ is a finite-to-one proper map. Now consider the short exact sequence

$$\begin{aligned} 0\rightarrow f^{*}T\mathbb{X }_{3}\rightarrow \nu _{*}\nu ^{*}f^{*}T\mathbb{X }_{3}\rightarrow S \rightarrow 0 \end{aligned}$$

Since the cokernel $S$ is supported on a finite number of points of $C$, it follows that $H^{1}(C,S)=0$, so that $H^{1}(C,f^{*}T\mathbb{X }_{3})\rightarrow H^{1}(C,\nu _{*}\nu ^{*}f^{*}T\mathbb{X }_{3})$ is surjective. $\square$

Corollary 7.12

The action of $\mathrm{Symp}(\widetilde{M}_{\mu ,c_1,c_2})$ on $\mathcal{J }_{\omega }$ is homotopy equivalent to its restriction to $\mathcal{J }_{\omega }^{int}$.

Corollary 7.13

The space $\mathcal{J }_{\omega }^{int}$ of compatible integrable complex structures on $\widetilde{M}_{\mu ,c_1,c_2}$ is contractible.