On singular generalizations of the Singer–Hopf conjecture

The Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara.


INTRODUCTION
The main purpose of this note is to overview and generalize some of the recent developments [2,23] around the Singer-Hopf conjecture in the complex algebraic context.In order to reach a wider audience, results are formulated in the convenient language of constructible functions (cf.Section 2 for basic definitions and properties).Along the way, we generalize a result of [1] on the non-negativity of Euler characteristics of an important class of constructible functions.
Recall that a connected CW complex is said to be aspherical if its universal cover is contractible.The homotopy type of an aspherical CW complex depends only of its fundamental group.Classical examples of aspherical spaces include (complex) tori, abelian varieties, ball quotients, etc.Moreover, by the Cartan-Hadamard theorem, closed Riemannian manifolds with non-positive sectional curvature are aspherical.See [25] for an overview.
The following conjecture on the topological Euler characteristic of a closed aspherical manifold was made by Hopf (and later on strengthened by Singer), for example, see [16,Conj. 25.1]: Conjecture 1.1 (Singer-Hopf).If  is a closed aspherical manifold of real dimension 2, then (−1)  ⋅ () ≥ 0. (1) In fact, Singer's formulation of Conjecture 1.1 also predicts the vanishing, except in middle degree, of Atiyah's  2 -Betti numbers [3] of the universal cover of  (e.g., see [16]).However, in this note, we focus only on the topological Euler characteristic.
Conjecture 1.1 is true for  = 1 (i.e., real dimension 2), since the only closed real surfaces with positive Euler characteristic are  2 and ℝ 2 , and they are the only non-aspherical ones.Since the topological Euler characteristic  is multiplicative, it follows immediately that the conjecture also holds for a finite product of closed aspherical real surfaces.This particular example explains the sign appearing in formula (1).
For closed Riemannian manifolds with non-positive sectional curvature, Conjecture 1.1 is due to Hopf and Chern, and it was mentioned in the list of problems [34] compiled by Yau in 1982 (cf. also [16]); it is true in this case for  = 2 since the Gauss-Bonnet integrand has the desired sign, cf.[7,Theorem 5] where the proof is attributed to Milnor.To our knowledge, Conjecture 1.1 is not known for all closed aspherical 4-manifolds, and it is open for  ≥ 3.
In [15], Gromov introduced the notion of Kähler hyperbolicity, including compact Kähler manifolds with negative sectional curvature, and he verified Conjecture 1.1 for Kähler hyperbolic manifolds.Cao-Xavier [6] and Jost-Zuo [17] independently introduced the concept of Kähler nonellipticity, including compact Kähler manifolds with non-positive sectional curvature, and settled the Singer-Hopf conjecture in this case (cf.also [11]).All these works proved a corresponding version of Conjecture 1.1 by means of vanishing  2 -cohomology; see also the overview [24,Ch. 11].
More recently, Liu-Maxim-Wang [23] proved the complex projective version of Conjecture 1.1 under the additional assumption that the (holomorphic) cotangent bundle  *  of  is numerically effective (nef, for short), and they conjectured that aspherical complex projective manifolds have nef cotangent bundles.Finally, Arapura-Wang gave in [2] a new proof of Conjecture 1.1 for compact Kähler manifolds with non-positive sectional curvature, using the fact that the cotangent bundle of such a manifold is nef.
If  = , then all statements in Conjecture 1.2 reduce to the complex projective version of the Singer-Hopf Conjecture 1.1.More generally, if  is a smooth closed irreducible subvariety of the aspherical complex projective manifold , all statements in Conjecture 1.2 become (−1) dim ℂ  ⋅ () ≥ 0.
In fact, we make the following more uniform conjecture (which, as explained in Proposition 2.2, turns out to be equivalent to Conjecture 1.2(i)).
Conjecture 1.3.Let  be an aspherical complex projective manifold and let  be a constructible function on  with an effective characteristic cycle.Then, the Euler characteristic of  is non-negative, that is, (, ) ≥ 0.
If  is an abelian variety, Conjecture 1.3 can be deduced from [13,Theorem 1.3], see also [1] (where the triviality of the tangent bundle is essential) and [10].When  is obtained (by taking the stalkwise Euler characteristic) from a perverse sheaf on , Conjecture 1.3 reduces to [23,Conjecture 6.2].
In this note, we prove Conjecture 1.3 under the additional assumption that the cotangent bundle  *  of  is nef (e.g., globally generated), or, more generally, if  admits a finite morphism to a complex projective manifold with nef cotangent bundle.Moreover, the inequalities become strict if "nef" is replaced by "ample."Theorem 1.4.Let  be a complex projective manifold and let  be a constructible function on  with effective characteristic cycle.Assume  admits a finite morphism  ∶  →  to a complex projective manifold  with nef cotangent bundle (e.g.,  has non-positive sectional curvature).Then, (, ) ≥ 0.Moreover, the inequality is strict if  *  is ample (e.g.,  has negative sectional curvature).
A weaker version of Conjecture 1.3, for  coming from a perverse sheaf on  and  *  nef, was proved in [23,Proposition 3.6].While the proof of Theorem 1.4 follows the same lines as that of loc.cit.(see also Theorem 3.5 in Section 3 for a more general statement), what we want to emphasize here are the various facets of Conjecture 1.3, reflected in the statement of Conjecture 1.2, if more general coefficients are allowed.Note that asking for the characteristic cycle of a constructible function to be effective is much weaker that asking for that function to come (by taking the stalkwise Euler characteristic) from a perverse sheaf.Conjecture 1.2 is obtained directly from Conjecture 1.3 by letting  be one of the following constructible functions supported on : (−1) dim ℂ    ,   and   , respectively, all of which are known to have effective characteristic cycles, for example, see the discussion in [1, Section 7].Note that, in the notations of Section 2, only   comes from a perverse sheaf.
It was also proved in [2] that Conjecture 1.3 is true if  is an aspherical complex projective manifold (or, more generally, if  has a large fundamental group [20]) which admits a cohomologically rigid almost faithful semi-simple representation, provided that  comes from a perverse sheaf on .In Section 3, we note that the proof of this result in loc.cit.extends to all constructible functions with effective characteristic cycles, since the only operations involved in the proof preserve the effectivity of characteristic cycles of constructible functions.
The paper is organized as follows.In Section 2, we review the relevant background about constructible complexes, constructible functions, characteristic cycles, and prove (in Proposition 2.2) that Conjecture 1.3 is equivalent to Conjecture 1.2(i).In Section 3, we review (semi-)positivity results for nef and, resp., ample vector bundles on complex projective manifolds, and prove Theorem 1.4 (as a consequence of the more general statement of Theorem 3.5).We conclude with a discussion around another conjecture of Hopf, whose proof in the projective/Kähler context is known to follow from classical results in the complex algebraic geometry.

PRELIMINARIES: CONSTRUCTIBLE COMPLEXES AND CHARACTERISTIC CYCLES
Let  be a complex algebraic manifold.We denote by    () the bounded derived category of ℂ-constructible complexes on .The reader may consult [9] or [27] for an overview of the theory of constructible complexes.
Consider One then has a group isomorphism to the group () of algebraic cycles on , defined on generators by:  *   ↦ (−1) dim ℂ  .
Let () be the group of constructible functions on a complex algebraic variety  (e.g., a closed subvariety of ), that is, the free abelian group generated by indicator functions 1  of closed irreducible subvarieties  of .There is a unique linear map  ∶ () ⟶ ℤ called the Euler characteristic, defined on generators by (1  ) ∶= ().In order to keep track of spaces, for any  ∈ () we will use the notation (, ) ∶= ().
To any bounded constructible complex  • ∈    () on a complex algebraic variety , one associates a constructible function   ( • ) ∈ () by taking the stalkwise Euler characteristic, that is, for any  ∈ .For example,   (ℂ  ) = 1  , and if  is pure-dimensional we let where   is the intersection cohomology (-)complex of .Note that if  =   ( • ), then For a closed subvariety  in , a constructible function (or complex) on  can be regarded as a constructible function (or complex) on  by extending it by 0 on  ⧵ , hence for  ∈ () we have (, ) = (, ).For the purpose of this note, we may, without any loss of generality, work with constructible functions on  (with support in a closed subvariety) and their corresponding Euler characteristics.
Moreover, since the class map    () →  0 (   ()) is onto,   is already an epimorphism on    ().The usual functors in the sheaf theory, which respect the corresponding category of bounded constructible complexes, induce via   welldefined group homomorphisms on the level of constructible functions (see, e.g., [30, Section 2.3]).
Another important example of a constructible function on a complex algebraic variety  is the MacPherson local Euler obstruction function   , see [26].When  is a closed irreducible subvariety  of ,   can be seen as a function defined on all of  by setting   () = 0 for  ∈  ⧵ .In particular, one may consider the group homomorphism In particular, one can associate a characteristic cycle to any constructible function.For example, if  is a closed irreducible subvariety of , one has: Note also that for any constructible complex  • ∈    ().Kashiwara's global index theorem [18] computes the Euler characteristic of any bounded constructible complex  • on  with supp( • ) compact, or, equivalently, that of the constructible function  =   ( • ) ∈ (), by the formula: that is, the intersection index in the complex manifold  * , of the characteristic cycle of  with the zero section of  * . ).In particular, CC(  ) is effective.Finally, the Behrend function   appearing in the Donaldson-Thomas theory has an effective characteristic cycle (see [1,4]).
We can now prove the following.
for uniquely determined closed irreducible subvarieties  of  and positive integers   .By Equation (4), the coefficients   are determined by the following equality of constructible functions: Hence,

𝜒(𝑋, 𝜑) = ∑
⊆   ⋅ (−1) dim ℂ  ⋅ (,   ) ≥ 0, with the last inequality following by applying Conjecture 1.2(i) to each subvariety  in the support of CC().□ Several basic operations of constructible functions preserve the property of having an effective characteristic cycle.In particular, one has the following.

Proposition 2.3. [1, Proposition 7.2(2)]
Let  be a closed reduced subscheme of a smooth complex algebraic variety , and assume that  is a constructible function on  with () effective.
As already mentioned, the above functors of constructible functions coincide with those induced via   from the corresponding functors of constructible complexes of sheaves.

POSITIVITY RESULTS FOR NEF BUNDLES: APPLICATIONS
In this section, we recall (semi-)positivity results for ample (resp., nef) vector bundles on complex projective manifolds.We use such results to deduce (semi-)positivity statements for the Euler characteristics of constructible functions with effective characteristic cycles, thus proving Theorem 1.4.
Definition 3.1.If  is a vector bundle on a complex projective manifold , denote by () the projective bundle of hyperplanes in the fibers of .A vector bundle  on  is called ample (resp.nef) if the line bundle (1) on () is ample (resp.nef) in the classical sense.
The nef condition is a degenerate ampleness condition.Globally generated bundles are nef.Properties of nef/ample bundles are studied, for example, in [8,12,22].
The following semi-positivity result was proved by Fulton-Lazarsfeld [12] for ample bundles, and extended to nef bundles by Demailly-Peternell-Schneider [8] (cf.e.g., [22,Section 8.1.B] for the definition of the intersection number).[12, Theorem II]) Let  be a complex projective manifold and let  be a rank  nef (resp., ample) vector bundle on .For any -dimensional conic subvariety  of , one has  ⋅   ≥ 0 (resp., > 0), where   is the zero section of , and  ⋅   denotes the intersection number of cycles in .

Theorem 3.2. ([8, Proposition 2.3],
Remark 3.3.The notion of nef vector bundle can be extended to the Kähler manifold context, with nefness of a line bundle understood in the sense of [8,Definition 1.2]; this coincides with the usual definition in the projective case.Then, the above theorem holds more generally, for nef vector bundles on compact Kähler manifolds (cf.[8,Proposition 2.3]).
The first result of this section involves nef/ample cotangent bundles.As already mentioned in Section 1, complex projective manifolds with non-positive (resp., negative) sectional curvature have nef (resp., ample) cotangent bundle.
Example 3.4.The class of complex projective manifolds whose cotangent bundles are nef is closed under taking finite (unramified) covers, products, and subvarieties, and it includes smooth subvarieties of abelian varieties.However, if  is an abelian variety of dimension  and  is a smooth subvariety of  of dimension  and codimension  −  < , then the cotangent bundle of  is not ample (e.g., see [22,Example 7.2.3]).On the other hand, for an arbitrary -dimensional complex projective manifold  and each  ≤ ∕2, there are plenty of smooth -dimensional subvarieties  of  with ample cotangent bundle (e.g., complete intersections of sections of  by general hypersurfaces of sufficiently high degrees in the ambient projective space, see [5,33]).Finally, Kratz [21,Theorem 2] showed that if  is a complex projective manifold, whose universal cover is a bounded domain in ℂ  or in a Stein manifold, then  *  is nef.This result prompted Liu-Maxim-Wang [23] to conjecture that the nefness of the cotangent bundle should hold more generally, if the universal cover is Stein, a claim refuted recently in [32].
We can now prove the following result.Theorem 3.5.Let  be a complex algebraic variety, and let  ∶  →  be a morphism to a complex projective manifold  with  *  nef.Let  ∈ () be a constructible function on  such that the characteristic cycle of the constructible function  * () ∈ () is effective.Then, (, ) ≥ 0, and the inequality is strict if  *  is ample.with strict inequality if  *  is ample.Kashiwara's global index formula (5) then yields: with strict inequality if  *  is ample.The assertion of the theorem follows now by combining Equations ( 8) and ( 9).□ Theorem 1.4 is an immediate consequence of Theorem 3.5, as we now show.We next notice that the main result of [2] admits a natural generalization from (constructible functions coming from) perverse sheaves to the case of coefficients (constructible complexes or constructible functions) with effective characteristic cycles.We formulate this generalized version below (see loc. cit.for the relevant definitions and constructions).Theorem 3.7.Let  be a smooth complex projective variety with large algebraic fundamental group (e.g.,  is aspherical).Suppose that there exists a cohomologically rigid almost faithful semi-simple representation  ∶  1 () → (, ℂ).Let  be a constructible function on  with () effective.Then, (, ) ≥ 0.
Proof.The proof of this result in [2] for perverse sheaves coefficients can be readily extended to the more general setup of effective characteristic cycles, since the only operations involved in the proof preserve the effectivity of characteristic cycles of constructible functions.For the benefit of the reader, we summarize the main steps in the proof, by adapting [2, Theorems 1.8 and 1.6] to the setup of this note.
Let   be the local system on  corresponding to the representation .As noted in [2],   underlies a complex variation of Hodge structure (CVHS)  with discrete monodromy.
Step 1. Upon passing to a finite unramified cover  ∶  ′ → , one can moreover assume that the monodromy group Γ = ( 1 ()) of   is torsion free (cf.[2, Lemma 5.1]).After performing this step, the Euler characteristic (, ) gets multiplied by the degree of the cover  and hence it preserves its sign, and the characteristic cycle of  * () is still effective (cf.Proposition 2.3(ii)).
The effect of performing Step 1 is that the quotient  = Γ∖ of the Griffiths period domain by the monodromy group Γ is now a manifold, and the period map of the CVHS  induces a horizontal map  ∶  →  (i.e., its derivative lies in the subbundle  ℎ  ⊂  induced from the horizontal subbundle of ).
Step 2. Since (, ) = (,  * ()), it suffices to show that (,  * ()) ≥ 0. Conditions of the theorem force  ∶  →  to be quasi-finite, and hence a finite morphism (since  is compact).Hence, by Proposition 2.3(i), the characteristic cycle of the constructible function  * () (supported on the compact subvariety ()) is effective in  * .We conclude this note with the following discussion about complex projective (or compact Kähler) manifolds with nef tangent bundles.Let  be a complex projective manifold (or even a Kähler manifold) whose tangent bundle is nef.Examples include rational homogeneous manifolds (e.g., complex projective spaces, flag manifolds, or quotients ∕ of a semisimple complex Lie group  by a parabolic subgroup), abelian varieties (or complex tori), etc. (cf.[8, Section 3.A]).Moreover, as we will argue below, complex projective (or compact Kähler) manifolds with non-negative sectional curvature have nef tangent bundles.By applying Theorem 3.2 to the tangent bundle , one gets via the Gauss-Bonnet formula that () = ∫    () =    ⋅    ≥ 0, where    is the zero section of .Hence, one gets the following result from [8].
Note that Theorem 3.2 also shows that the inequality in Equation ( 11) is strict if  is ample.In fact, Mori [29] proved that a complex projective manifold with ample tangent bundle is isomorphic to a complex projective space.Proposition 3.8 is a generalization in the complex projective (or Kähler) context of another conjecture of Hopf (also appearing on Yau's problem list [34]): Conjecture 3.9 (Hopf).A compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic.A compact, even-dimensional Riemannian manifold with non-negative sectional curvature has nonnegative Euler characteristic.
Let us finally note that in the Kähler context, Siu-Yau [31] showed that if  has positive bisectional curvature then  is biholomorphic to a complex projective space.Furthermore, a classification of all compact Kähler manifolds with nonnegative bisectional curvature was obtained by Mok [28], and this can be used along with results of [8] to give a direct proof of Conjecture 3.9 in this case.
defined on an irreducible cycle  by the assignment  ↦   , and then extended by ℤ-linearity.It is well known (e.g., see[9, Theorem 4.1.38],and references therein), that the homomorphism  ∶ () → () is an isomorphism.The local Euler obstruction function appears in the formulation of the local index theorem, which in the above notations asserts the existence of the following commutative diagram (e.g., see [30, Section 5.0.3], and references therein):

Proof.
First note that (, ) = (,  * ()), (8) so it suffices to show that (,  * ()) ≥ 0, with strict inequality if  *  is ample.Let CC( * ()) = ∑ ⊆   ⋅  *  , for uniquely determined closed irreducible subvarieties  of , and   > 0 by the effectivity assumption.Since  *  is nef, one gets by Theorem 3.2 that  *   ⋅  *   ≥ 0, ) denote the free abelian group spanned by the irreducible conic Lagrangian cycles in the cotangent bundle  * .Its elements are of the form ∑    ⋅  *  , for some   ∈ ℤ, and  closed irreducible subvarieties of .Recall that, if  is a closed irreducible subvariety of  with smooth locus  reg , its conormal bundle  *   is the closure in  *  of [19,which associates characteristic cycles to (Grothendieck classes of) constructible complexes on  (e.g., see[19, Chapter  IX]).Here, we let LCZ( * Definition 2.1.If  ≠ 0 ∈ () with CC() = ∑    ⋅  *  , we say that CC() is effective if all coefficients   are positive.It is also well known that if 0 ≠  =   ( • ) is a nontrivial constructible function associated with a perverse sheaf  • on , then CC() is effective (e.g., see [27, Corollary 4.7] [1,,f of Theorem 1.4.Let  ∶  →  be a finite morphism between complex projective manifolds, with  *  nef.Since  ∈ () has, by assumption, an effective characteristic cycle, it follows from Proposition 2.3(i) that CC( * ()) is effective in  * .The assertion follows now from Theorem 3.5.□Besides the situation considered in Theorem 1.4, there are other interesting classes of morphisms  ∶  →  (and constructible functions  ∈ () defined on their domain) satisfying the assumptions of Theorem 3.5.For instance, if  ∈ () comes from a perverse sheaf, let  ∶  →  be a morphism such that  * preserves perverse sheaves (e.g., a closed embedding or a quasi-finite affine morphism); then  * () ∈ () also comes from a perverse sheaf and hence, since  is assumed smooth, CC( * ()) is effective in  * .Another example is provided by the projection  ∶  →  from a semi-abelian variety  onto its abelian part .In this case, it was shown in [1, Section 8] that if  ∈ () comes from a perverse sheaf, then  * () ∈ () also comes from a perverse sheaf.Hence, CC( * ()) is effective, and Theorem 3.5 applies to give (, ) ≥ 0, an inequality initially proved in[13, Corollary 1.4].For more examples in this direction, the interested reader may consult[1, Proposition 8.4, Example 8.5].