2. Aspherical Complex Surfaces

In this section, we give a brief overview of those compact complex surfaces which are aspherical. These surfaces have contractible universal cover or equivalently, $\pi_k$ vanishes for $k \gt 1$ . By [Reference Albanese and Di CerboADC23, lemma 2], such surfaces are minimal. We will work through the Kodaira–Enriques classification by Kodaira dimension.

Kodaira dimension $-\infty$ : In the Kähler case, such surfaces are rational or ruled. The former consists of $\mathbb{CP}^2$ and Hirzebruch surfaces $\Sigma_n = \mathbb{P}_{\mathbb{CP}^1}(\mathcal{O}\oplus\mathcal{O}(n))$ . These are all simply connected, so they are their own universal covers. As they are not contractible, rational surfaces are not aspherical. Ruled surfaces are holomorphic fiber bundles with fiber $\mathbb{CP}^1$ and structure group $PGL(2, \mathbb{C})$ over a smooth connected curve C of positive genus. Every such surface is the projectivisation of a rank two holomorphic vector bundle over C. From the long exact sequence in homotopy, it follows that ruled surfaces have non-zero $\pi_2$ . In fact, if $\widetilde{C} \to C$ denotes the universal covering of C, pulling back the $\mathbb{CP}^1$ -bundle by this map exhibits ruled surfaces have universal cover $\mathbb{CP}^1\times\widetilde{C}$ – since $\widetilde{C}$ is Stein, we have $H^1(\widetilde{C}, \mathcal{PGL}(2, \mathbb{C})) = 0$ and hence the $\mathbb{CP}^1$ bundle over $\widetilde{C}$ is trivial.

A non-Kähler surface with Kodaira dimension $-\infty$ is called a Class VII surface. A minimal such surface is called a Class $\mathrm{VII}_0$ surface, and if furthermore the second Betti number is positive, then it is called a Class $\mathrm{VII}_0^+$ surface. A Class $\mathrm{VII}_0$ surface with second Betti number zero is biholomorphic to a Hopf surface or an Inoue–Bombieri surface, see [Reference BogomolovBog76, Reference BogomolovBog82, Reference Li, Yau and ZhengLYZ94 and Reference TelemanTel94]. Hopf surfaces have universal cover $\mathbb{C}^2\setminus\{0\}$ which is not contractible, while Inoue-Bombieri surfaces have universal cover $\mathbb{C}\times\mathbb{H}$ which is contractible, so they are aspherical.

A spherical shell in a complex surface is an open subset biholomorphic to a neighbourhood of $S^3$ in $\mathbb{C}^2\setminus\{0\}$ . If the complement is connected, then it is called a global spherical shell. A surface which admits a global spherical shell is a deformation of a primary Hopf surfaceFootnote ¹ blownup at finitely many points [Reference KatoKat78, theorem 1] – note that such surfaces are not aspherical. The global spherical shell conjecture asserts that all Class $\mathrm{VII}_0^+$ surfaces contains a global spherical shell. The conjecture remains open with some progress for small values of $b_2$ , see [Reference TelemanTel05, Reference TelemanTel10, Reference TelemanTel18]. It is not yet known if there exists an aspherical Class $\mathrm{VII}_0^+$ surface (it would necessarily violate the global spherical shell conjecture).

Since Class VII surfaces have first Betti number 1, such surfaces have $\chi_{top}(X) = b_2(X)$ . Furthermore, as they are non-Kähler, we see that $b^+(X) = 2h^{2,0}(X) = 0$ and hence $\sigma(X) = -b^-(X) = -b_2(X)$ . So Inoue-Bombieri surfaces have $\chi_{top}(X) = \sigma(X) = 0$ , while aspherical Class $\mathrm{VII}_0^+$ surfaces have $\chi_{top}(X) = -\sigma(X) = b_2(X) \gt 0$ .

Kodaira dimension 0: In the Kähler case, there are two families: tori and their quotients (bi-elliptic surfaces), and K3 surfaces and their quotients (Enriques surfaces). The former have universal cover $\mathbb{C}^2$ and are therefore aspherical, while the latter have K3 surfaces as their universal cover and hence are not aspherical.

In the non-Kähler realm, such surfaces are primary Kodaira surfaces and their quotients (secondary Kodaira surfaces). Primary Kodaira surfaces are holomorphic principal elliptic curve bundles over a smooth connected genus one curve. It follows from the long exact sequence in homotopy that $\pi_k = 0$ for $k \gt 1$ . Just as in the case of ruled surfaces, we can also identify the universal cover of Kodaira surfaces as $\mathbb{C}^2$ by pulling back such a bundle by the universal covering of the base. A description of primary Kodaira surfaces as quotients of $\mathbb{C}^2$ by a group of affine transformations was given by Suwa [Reference SuwaSuw75, theorem 2].

Kodaira dimension 1: A compact surface X is called an elliptic surface if there is a smooth connected curve C and a holomorphic map $\pi \;:\; X \to C$ such that the generic fiber is a smooth genus one curve; the map $\pi$ is called an elliptic fibration. We call an elliptic surface X relatively minimal if there are no $-1$ curves in the fibers of $\pi$ – every elliptic surface is an iterated blowup of a relatively minimal elliptic surface. Every surface of Kodaira dimension 1 is elliptic (see [Reference WallWal86, lemma 7·2(a)]), but there are also elliptic surfaces of Kodaira dimension $-\infty$ and Kodaira dimension 0. An elliptic surface with Kodaira dimension 1 is called a properly elliptic surface.

The non-generic fibers of a relatively minimal elliptic fibration $\pi \;:\; X \to C$ , called exceptional fibers, were classified by Kodaira, see [Reference KodairaKod63, theorem 6·2]. Aside from multiples of a smooth genus one curve (known as a multiple fibers with smooth reduction), every other possibility is a configuration of (possibly singular) rational curves. The elliptic fibration induces an orbifold structure on C by declaring images of multiple fibers as cone points whose order is the multiplicity of the fiber. We denote the orbifold Euler characteristic and orbifold fundamental group of C by $\chi^{\text{orb}}(C)$ and $\pi_1^{\text{orb}}(C)$ respectively.

Conversely, if $X \to C$ is relatively minimal with no exceptional fibers other than multiple fibers with smooth reduction, and X has Kodaira dimension 0 or 1, then $\chi^{\text{orb}}(C) \leq \chi(\mathcal{O}_X) = 0$ . Therefore C is a good orbifold, i.e. there is a finite orbifold covering $C' \to C$ where C ^′ is a manifold. Pulling back $X \to C$ by this map induces an elliptic fibration $X' \to C'$ with no multiple fibers such that X ^′ is a finite unramified cover of X, see [Reference Barth, Hulek, Peters and Van de VenBHPV04, proposition III·9·1]. Since $\chi(\mathcal{O}_{X'}) = 0$ , all the fibres of $X' \to C'$ are isomorphic by [Reference Barth, Hulek, Peters and Van de VenBHPV04, proposition V·12·2] and the remark which precedes it, and hence $X' \to C'$ is locally trivial by [Reference Fischer and GrauertFG65]. As the orbifold Euler characteristic is multiplicative under orbifold coverings, we have $\chi^{\text{orb}}(C') \leq 0$ and hence C ^′ has positive genus. Applying the long exact sequence of homotopy groups, we see that X ^′ is aspherical.

There are examples of elliptic surfaces which contain finitely many rational curves, and examples with infinitely many, see [Reference Barbaro, Fagiolo and OrtizBFO22, section 5].

Note, there are elliptic surfaces with Kodaira dimension $-\infty$ , but none of them are aspherical (they are either rational, ruled, or Hopf). As for Kodaira dimension 2, none of them are elliptic.

There are non-aspherical elliptic surfaces in Kodaira dimensions 0 and 1. By combining Proposition 2·1 with [Reference WallWal86, lemma 7·2(b)], such surfaces must be Kähler. In Kodaira dimension 0, such surfaces are the elliptic K3 surfaces and all Enriques surfaces, while for Kodaira dimension 1, homotopy K3 surfaces and Dolgachev surfaces provide examples. One can construct many more Kodaira dimension 1 examples as follows (the stated examples arise this way). Choose an elliptic surface with an exceptional fiber which is not a multiple fiber with smooth reduction (equivalently, has positive Euler characteristic). Applying logarithmic transformations decreases the value of $\chi^{\text{orb}}(C)$ , so by [Reference WallWal86, lemma 7·1], the result will eventually have Kodaira dimension 1.

Kodaira dimension 2: Aspherical surfaces with Kodaira dimension 2 exist, but as with most problems regarding general type surfaces, we have nothing even close to a classification. Indeed, the list of known aspherical surfaces of general type is not particularly rich even if there are reasons to expect such surfaces exist in great profusion. The list includes ball quotients (e.g., fake projective planes), surfaces isogeneous to product of curves, Kodaira fibrations, Mostow–Siu surfaces, and certain branched covers of ball quotients due to Domingo–Stover [Reference Stover and ToledoST22, theorem 1·5]. We refer to the paper of Bauer–Catanese [Reference Bauer and CataneseBC18] for more details. The list of aspherical surfaces of general type also includes the vast majority of smooth minimal toroidal compactifications of ball quotients, see [Reference Di CerboDiC12, theorem A]. In all of these examples, when the signature is explicitly computed one has that $\sigma\geq 0$ . It seems currently unknown whether or not an aspherical complex surface of general type can have negative signature. In conclusion, we can summarise this discussion into a table, see Table 1.

Table 1.

Aspherical complex surfaces via the Kodaira-Enriques classification

3. Singer Conjecture for Surfaces with Residually Finite Fundamental Group

In this section, we show that the Singer conjecture holds true for closed aspherical complex surfaces with residually finite fundamental group. The proof we present here does not rely on Gromov’s characterisation of Kähler manifolds with non-vanishing first $L^2$ -Betti number [Reference GromovGro89]. We rely upon the study of the Albanese map given in [Reference Di Cerbo and LombardiDL24a] and [Reference Di Cerbo and LombardiDL24b], and Lück’s approximation theorem which we now briefly recall.

Let X be a manifold with $\Gamma\stackrel{{\rm def}}{=}\pi_1(X)$ residually finite. We consider a sequence of nested, normal, finite index subgroups $\{\Gamma_i\}$ of $\Gamma$ such that $\cap_i\Gamma_i$ is the identity element. This sequence is usually called a cofinal filtration of $\Gamma$ . Let $\pi_i\colon X_i\rightarrow X$ be the finite regular cover of X associated to $\Gamma_i$ . Lück’s approximation theorem [Reference LückLuc94a] ensures that

(3·1) \begin{align}\lim_{i \to \infty}\frac{b_{k}(X_i)}{\deg (\pi_i) } = b^{(2)}_k (X;\; \widetilde{X}),\end{align}

where $b_k(X_i)$ denotes the $k{\text{th}}$ Betti number of $X_i$ , and $b^{(2)}_k (X;\; \widetilde{X})$ is the $L^2$ -Betti number of X computed with respect to the universal cover $\widetilde{X}$ . Thus, the limit in (3·1) always exists and it is independent of the cofinal filtration. We refer to the ratio $b_k(X_i)/\deg (\pi_i)$ as the normalised $k{\text{th}}$ -Betti number of the cover $\pi_i\colon X_i \rightarrow X$ . Conjecture 1·2 is then equivalent to the sub-degree growth of Betti numbers along a tower of covers associated to a cofinal filtration.

We start with the following proposition that is not limited to complex dimension two.

Proposition 3·1. Let X be an aspherical smooth projective variety. Assume that $\pi_1(X)$ is residually finite and there exists a cofinal tower of coverings $\pi_i\colon X_i\to X$ such that the images $a_{X_i}(X_i)$ of the Albanese maps are either points or curves in $\mathrm{Alb}(X_i)$ . We then have

\[ {}\lim_{i \to \infty}\frac{b_{1}(X_i)}{\deg (\pi_i) } = 0. {}\]

Proof. Clearly, we just need to study the case where $b_{1}(X_i)\neq 0$ from some point on in the cofinal tower. Recall that if $a_{X_i}(X_i)$ is a curve, it must be smooth, connected, and its genus equals $({1}/{2})b_{1}(X_i)$ . For simplicity sake, from now on we assume that $a_{X_i}(X_i)$ is a curve for any $i\geq 0$ in the cofinal tower. Moreover, we set $S:=a_{X}(X)=a_{X_0}(X_0)$ and $S_i:=a_{X_i}(X_i)$ . Due to the universal property of the Albanese variety, there is a map $a_{\pi_i} \;:\; S_i \to S$ such that the following diagram commutes:

Since $\pi_i$ is unramified, for any $i\geq 1$ , the branching locus $B_i$ of $a_{\pi_i}$ is contained in the (finite) set of critical values of $a_{X}$ . In particular, there exists a positive constant $C\gt0$ such that for all i we have $ \#( B_i ) \leq C$ . Thus the degree of the ramification divisor $R_i$ of $a_{\pi_i}$ is bounded by $\deg R_i \leq C\cdot \deg \big( a_{\pi_i} \big) $ for any i. By using the Riemann–Hurwitz formula, we have

\[ {}b_1(S_i) = 2 \deg \big( a_{\pi_i} \big) \cdot \chi (\omega_S) + \deg R_i +2 {}\leq \deg \big( a_{\pi_i} \big) \cdot ( 2\chi(\omega_S) + C) +2. {}\]

Since $b_1(X_i)=b_1(S_i)$ , dividing by $\deg \big( a_{\pi_i} \big) \gt 0$ yields

(3·2) \begin{equation} {}\frac{b_1(X_i)}{\deg(a_{\pi_i})} \leq 2\chi(\omega_S) + C + \frac{2}{\deg (a_{\pi_i})} \leq 2\chi(\omega_S) + C + 2. {}\end{equation}

Next, let $k_i$ be the minimal degree of the restriction of $\pi_i$ to a general fiber of $a_{X_i}$ . Note that $\{k_i\}_{i\in\mathbb N}$ is a sequence of non-decreasing positive integers and

(3·3) \begin{equation}\deg \pi_i \geq k_i \, \cdot \, \deg \big( a_{\pi_i} \big) \quad \forall i.\end{equation}

We claim that

(3·4) \begin{equation} {}\lim_{i\to\infty}k_i = \infty. {}\end{equation}

By contradiction, as the tower of coverings $\pi_i \colon X_i\to X$ is cofinal, the $X_i$ ’s converge to the topological universal cover $\widetilde{X}$ (cf. [Reference Di Cerbo and Di CerboDD19, section 3]). Now equip the covers $X_i$ with the metrics induced by a fixed Kähler metric on the base, given by an ample line bundle L on X, via pullback. Moreover, let $G_i$ be a general fiber of $a_{X_i}$ such that $k_i = \deg \big( \pi_i|G_i \big)$ . In this way, denoting by $F$ the general fiber of $a_X$ , the volume of $G_i$ is computed as

\[( \pi_i^*L \cdot G_i ) = ( L \cdot \pi_{i*}G_i ) = k_i \cdot ( L \cdot F) \quad \forall i.\]

If the $k_i$ ’s were bounded, there would exist an integer $N\gt0$ such that $( \pi_i^*L \cdot G_i ) \lt N$ for all i; but this contradicts [Reference Di Cerbo and Di CerboDD19, proposition 3·3] (note that if X is aspherical, then $\pi_1(X)$ is large, see [Reference Liu, Maxim and WangLMW21, proposition 6·7] and [Reference KollárKol93, proposition 2·12·1]). In conclusion, by (3·2), (3·3) and (3·4) it follows that

\[ {}0 \leq \frac{b_{1}(X_i)}{\deg (\pi_i) } \leq \frac{b_1(X_i)}{k_i\deg(a_{\pi_i})} \leq \frac{2\chi(\omega_S) + C + 2}{k_i }. {}\]

Taking the limit as $i \to \infty$ concludes the proof.

We can now give a proof of Theorem 1·5 stated in the introduction.

Proof of Theorem 1·5. Since $b_0^{(2)}(X;\; \widetilde{X}) = b_4^{(2)}(X;\; \widetilde{X}) = 0$ and the alternating sum of $L^2$ -Betti numbers is equal to $\chi_{top}(X)$ , it is enough to show that $b_1^{(2)}(X;\; \widetilde{X}) =0$ by Poincaré duality. Moreover, we can assume that X is minimal by the asphericity assumption (see [Reference Albanese and Di CerboADC23, lemma 2]). We divide the proof into several steps according to the Kodaira dimension (using the results of Section 2).

To begin with, suppose ${\rm Kod}(X)=-\infty$ . In the non-Kähler case, X is of Class VII. All such surfaces have $b_1 (X) =1$ . Moreover any finite covering of a surface of Class VII is again of Class VII (see for example [Reference Friedman and MorganFM94, proposition II·1·21] or [Reference DürrDur05, lemma 5·1]). Thus, the vanishing of $b^{(2)}_{1}(X;\; \widetilde{X})$ follows immediately from Lück’s approximation [Reference LückLuc94a]. In the Kähler case, no surface with ${\rm Kod}(X)=-\infty$ is aspherical.

As discussed in Section 2, aspherical complex surfaces of Kodaira dimension 0 are finitely covered by either a torus or a primary Kodaira surface. It follows that the fundamental groups of such surfaces contain a normal subgroup isomorphic to $\mathbb{Z}^2$ . Hence, the $L^2$ -Betti numbers vanish by a classical result of Cheeger–Gromov [Reference Cheeger and GromovCG86, corollary 0·6], see also [Reference LückLuc02, theorem 1·44].

The aspherical complex surfaces of Kodaira dimension 1 are the properly elliptic surfaces with no exceptional fibers other than multiple fibers with smooth reduction. By Remark 2·2, such surfaces are finitely covered by holomorphic elliptic curve bundle, and hence their fundamental groups also contain a normal subgroup isomorphic to $\mathbb{Z}^2$ . Again, this implies that the $L^2$ -Betti numbers vanish.

For surfaces of general type, we first recall that they have to be projective (see [Reference Barth, Hulek, Peters and Van de VenBHPV04, p.189]), and we can use the Albanese map if the surface is irregular (i.e. $b_1(X) \neq 0$ ). Then, we proceed by using Lück’s approximation [Reference LückLuc94a] on a cofinal tower. If none of the covers in the tower is irregular, then the vanishing of $b^{(2)}_{1}(X;\; \widetilde{X})$ is immediate and the result follows. In the other cases, we use either Proposition 3·1, or Theorem [Reference Di Cerbo and LombardiDL24b, theorem 1·3] specialised to complex dimension two. Recall that in complex dimension two, $a_{X}$ is semismall if and only if it is generically finite onto its image.

4. On the Singer Conjecture for Complex Surfaces

In [Reference GromovGro89], Gromov shows that if $(X, \omega)$ is a closed, Kähler manifold with $b^{(2)}_{1}(X;\widetilde{X})\neq 0$ , then $\pi_1(X)$ is commensurable to the fundamental group of a compact surface of genus $g\geq 2$ . For more details about this important result, we refer to [Reference Arapura, Bressler and RamachandranABR92] and the nice book [Reference Amorós, Burger, Corlette, Kotschick and ToledoABCKT96, chapter 4] on Kähler groups.

Gromov’s theorem implies that no aspherical Kähler surface $(X^2, \omega)$ can have non-vanishing $b^{(2)}_1$ . Indeed, if this was the case then a finite cover of X, say $X^\prime$ , would have the same fundamental group as a hyperbolic Riemann surface, say C. Since both $X^\prime$ and C are aspherical with isomorphic fundamental groups, they are homotopy equivalent [Reference LückLuc12, theorem 2·1], which is clearly not possible as $H_{4}(X';{\mathbb Z})\neq H_{4}(C;\;{\mathbb Z})=0$ . Let’s summarise this discussion into a theorem.

We can now combine some parts of the proof of Theorem 4·1 with Theorem 1·5 to prove the following.

Note, if one could show that the fundamental group of a Class $\mathrm{VII}_0^+$ surface was residually finite, then we could apply the argument in the proof of Theorem 1·5 to extend Theorem 4·2 to all complex surfaces.

The big elephant in the room. Aspherical Class $\mathrm{VII}_0^+$ surfaces conjecturally do not exist. That said, their cohomological structure seems somewhat simple. This motivates the following.

5. Reid’s Conjecture and Gromov-Lück Inequality

In this section, we prove Theorem 1·4.

By the discussion in Section 2, the only aspherical surfaces with Kodaira dimension $-\infty$ are Inoue-Bombieri surfaces and potential aspherical class $\mathrm{VII}_0^+$ surfaces. As we have seen, the former satisfy $\chi_{top}(X) = \sigma(X) = 0$ , while the latter satisfy $\chi_{top}(X) = -\sigma(X) = b_2(X) \gt 0$ . On the other hand, the aspherical surfaces of Kodaira dimension 0 or 1 all have $\chi_{top}(X) = \sigma(X) = 0$ . This only leaves surfaces of general type.

Note, the Bogomolov–Miyaoka–Yau inequality states that for a general type surface X we have $\chi_{top}(X) \geq 3\sigma(X)$ . However, it is not true that $\chi_{top}(X) \geq 3|\sigma(X)|$ for every such X. For example, let $X_d$ be a smooth degree d hypersurface of $\mathbb{CP}^3$ . Note that $X_d$ is a surface of general type for $d \geq 5$ , and a simple characteristic class argument shows that $\chi_{top}(X_d) = d^3 - 4d^2 + 6d$ and $\sigma(X_d) = -\frac{1}{3}(d-2)d(d+2)$ . So, for example, we have $\chi_{top}(X_5) = 55$ and $\sigma(X_5) = -35$ so $3|\sigma(X_5)| = 105 \gt 55 = \chi_{top}(X_5)$ . In fact, the proposed inequality is violated by $X_d$ for all $d \geq 5$ (also $d = 3, 4$ , but these are not surfaces of general type). Of course, none of these examples are aspherical since they are simply connected by the Lefschetz hyperplane theorem (see for example [Reference LazarsfeldLaz04, theorem 3·1·17]).

If the signature is non-negative, then of course $\chi_{top}(X) \geq 3\sigma(X)$ is equivalent to $\chi_{top}(X) \geq 3|\sigma(X)|$ . The discrepancy occurs, as in the examples above, when the signature is negative.

This question is yet to be answered, so we continue on our quest to find an inequality relating $\chi_{top}(X)$ and $|\sigma(X)|$ . To do so, we need to recall the circle of ideas related to Reid’s conjecture, see for example [Reference Barth, Hulek, Peters and Van de VenBHPV04, chapter VII]. We also refer to the beautiful survey [Reference Mendes Lopes and PardiniMLP12] of Mendes Lopes-Pardini on the geography of irregular surfaces for much more on this fascinating topic.

As shown by Horikawa [Reference HorikawaHor76] and Reid [Reference ReidRei79], Conjecture 5·2 holds true under the stronger assumption $K^{2}_{X}\lt3\chi_{hol}$ . We therefore can observe the following.

Proof. By [Reference Albanese and Di CerboADC23, lemma 2], X must be minimal. Now an aspherical surface must have infinite $\pi_1$ , see for example [Reference LückLuc12, lemma 4·1]. As it was observed at the beginning of Section 4, $\pi_1(X)$ cannot be commensurable with the fundamental group of a curve. Since Conjecture 5·2 holds true for minimal surfaces of general type satisfying $K^{2}_{X}\lt3\chi_{hol}$ , we conclude that

\[K^{2}_{X}\geq 3\chi_{hol},\]

for any aspherical surface of general type.

We can now prove the desired inequality relating $\chi_{top}(X)$ and $|\sigma(X)|$ for aspherical general type surfaces.

Lemma 5·4. Let X be an aspherical surface of general type. We then have:

\[\chi_{top}(X)\geq \frac{9}{5}|\sigma|.\]

Proof. Recall that

\[K^{2}_{X}=2\chi_{top}(X)+3\sigma(X), \quad \chi_{hol}(X)=\frac{\chi_{top}(X)+\sigma(X)}{4}.\]

By using Proposition 5·3, we obtain

\[\chi_{top}(X)\geq \frac{9}{5}(-\sigma(X)),\]

which, combined with the Bogomolov–Miyaoka–Yau inequality, gives $\chi_{top}(X)\geq ({9}/{5})|\sigma(X)|$ .

This completes the proof of Theorem 1·4.

Note that for a minimal surface of general type X, we have $c_1^2(X) \gt 0$ from which it follows that $\chi_{top}(X) \gt ({3}/{2})(-\sigma(X))$ . If the signature is negative, the inequality $\chi_{top}(X) \geq ({9}/{5})$ $(-\sigma(X))$ is stronger. For example, if $\sigma(X) = -3$ , the former inequality implies $\chi_{top}(X) \geq 5$ while the latter implies $\chi_{top}(X) \geq 6$ .

Note that we actually have $\chi_{top}(X) \geq ({9}/{5})|\sigma(X)|$ for all aspherical complex surfaces, except any potential Class $\mathrm{VII}_0^+$ examples (in all other cases, $\chi_{top}(X) = \sigma(X) = 0$ ).

Remark 5·6. Very recently, in [Reference Arapura, Maxim and WangAMW25, conjecture 1·2], Arapura, Maxim and Wang stated a Hodge-theoretic version of the Singer–Hopf conjecture: if X is a compact Kähler manifold of dimension n which is either aspherical or it has nef cotangent bundle, then

(·5) \begin{equation}(\!-\!1)^{n-p} \chi(\Omega_X^p) \geq 0\quad \quad {for\,all} \quad p=0, \ldots ,n.\end{equation}

Here $\Omega_X^p$ denotes the bundle of holomorphic p-forms, and

$$ \chi(\Omega_X^p)=\sum_{i=0}^{n} (\!-\!1)^i \dim H^i(X, \Omega_X^p)$$

is the associated Euler characteristic. The conjecture is verified by the same authors in the case of surfaces with nef cotangent bundle (cf. loc. cit. Proposition 2·4). Moreover, by following [Reference Johnson and KotschickJK93], it also holds for aspherical complex surfaces as

$$ \chi(\omega_X) =\chi({{\mathcal O}}_X) = \frac{\chi_{ top}(X) + \sigma(X)}{4} \quad \mbox{and}\quad\chi(\Omega^1_X)=\frac{\sigma(X)-\chi_{top}(X)}{2}.$$

In higher dimension, the conjecture holds for Kähler hyperbolic and Kähler nonelliptic manifolds (see [Reference GromovGro91] and [Reference Jost and ZuoJZ00]). As an application of the inequality $\chi_{top}(X) \geq ({9}/{5}) |\sigma(X)|$ of Theorem 1·4, we observe that the inequalities in (5·5) are actually strict for all complex aspherical surfaces of general type. More precisely, as aspherical surfaces are minimal and $\chi_{top}(X)\gt0$ for all minimal surfaces of general type, if $\sigma(X)\neq 0$ we have

\begin{align}\notag\chi(\omega_X) =\chi({{\mathcal O}}_X) \geq \frac{1}{5}|\sigma(X)|\gt0 \quad \mbox{and}\quad-\chi(\Omega^1_X)\geq \frac{2}{5}|\sigma(X)|\gt0,\end{align}

while if $\sigma(X)=0$ we clearly obtain

\begin{align*}\chi(\omega_X) =\chi({{\mathcal O}}_X) = \frac{\chi_{ top}(X)}{4}\gt0 \quad \mbox{and}\quad-\chi(\Omega^1_X)=\frac{\chi_{top}(X)}{2}\gt0.\end{align*}

Finally, it is tantalising to ask what is the optimal constant $a\gt0$ , such that $\chi_{top}(X)\geq a|\sigma(X)|$ for all aspherical surfaces of general type. As remarked above, we currently seem not to know any example of aspherical surfaces of general type with negative signature. If this is not an accident simply due to our lack of good examples, but a true fact of nature, by using the Bogomolov–Miyaoka–Yau inequality we would have

\[\chi_{top}(X)\geq 3\sigma(X)\geq 0,\]

where the first inequality is saturated if and only if X is a ball quotient. Notice that given a minimal surface of general type X with $\sigma(X)\gt 0$ , the reversed oriented 4-manifold $\overline{X}$ can never admit a complex structure compatible with the orientation. This follows from Seiberg–Witten theory, see [Reference KotschickKot97, theorem 2]. Thus, in order to give a positive answer to Question 5.1, a genuinely new example of a surface of general type would need to be constructed, or alternatively one would need to provide an aspherical counterexample to the global spherical shell conjecture!