Vanishing and Chern-class constraints for compact complex threefolds with $b_2=0$ and holomorphic algebraic reduction in full generality

For a compact complex threefold $X$ with vanishing second Betti number ($b_2(X)=0$) and positive algebraic dimension, Campana–Demailly–Peternell (1998) claimed a generic vanishing theorem forcing $c_3(X)=0$; the proof rested on a lemma (Lemma 1.5 of the 1998 paper) asserting that higher direct images under a fibration over a curve are locally free, which is false in general. Their 2020 corrigendum repairs the case where the algebraic reduction is non-holomorphic in full generality, but in the remaining case — algebraic dimension $1$ with holomorphic algebraic reduction $f:X\to C$ — the key vanishing and negativity theorem (Theorem 2.2: $H^i(X,T_X\otimes M)=0$ for $i\neq1$, $\chi(X,T_X\otimes M)\le0$, $c_3(X)\le0$) is proven only under the auxiliary hypotheses $H^1(X,\mathbb Z)=H^2(X,\mathbb Z)=0$, adopted 'for simplicity'. The open problem is to establish these conclusions, together with the accompanying fiber-classification statements, for all threefolds with $b_2(X)=0$ alone, or to refute them by an explicit counterexample; an auxiliary question asks for the exact scope of validity of the original Lemma 1.5 within the geometric contexts of the 1998 paper. The famous motivating question — whether the six-sphere admits a complex structure — is logically independent of this gap and remains open.

Background

A compact complex threefold is a three-dimensional complex manifold that is compact without boundary; such a manifold with $b_2(X)=0$ (second Betti number, i.e. $\dim_{\mathbb R}H^2(X;\mathbb R)=0$) can be neither projective nor Kähler, so it is studied through its meromorphic functions. The algebraic dimension $a(X)$ is the transcendence degree over $\mathbb C$ of the field of meromorphic functions on $X$; $a(X)>0$ exactly when $X$ carries a non-constant meromorphic function, and (after resolving indeterminacies) such functions organize into the algebraic reduction, a fibration $f:X\to C$ onto a curve or surface, unique up to bimeromorphism; the reduction may or may not be holomorphic on $X$ itself. For bundles, $\mathrm{Pic}^\circ(X)$ denotes the identity component of the Picard group of holomorphic line bundles (those topologically trivial along the connected component), and 'generic $M$' means outside a proper analytic subset of it. The third Chern class equals the topological Euler characteristic, $c_3(X)=\chi_{\mathrm{top}}(X)$, a theorem of Hopf, so Chern-class statements are topological statements. Campana–Demailly–Peternell (Compositio Math. 112 (1998), 77–91) proved, as motivation, that a compact complex manifold homeomorphic to the six-sphere $S^6$ has $a(X)=0$, excluding any complex structure on $S^6$ with a non-constant meromorphic function. Their argument for the general main theorem (a generic vanishing statement implying $c_3(X)=0$ for all threefolds with $b_2(X)=0$ and $a(X)>0$) used Lemma 1.5: for a surjective holomorphic map $f:X\to C$ from a smooth compact threefold to a smooth curve and a locally free sheaf $F$, all higher direct image sheaves $R^if_*(F)$ (the sheaves on $C$ collecting the cohomology of the fibers) are locally free, i.e. vector bundles. That lemma is false in general: Lehn–Rollenske–Schinko (Diff. Geom. Appl. 57 (2018), 121–137, Remark 3.7) gave the counterexample of the second projection $p:E\times E\to E$ with $D=\{0\}\times E-\Delta$ on the product of an elliptic curve with itself, for which $R^1p_*\mathcal O(D)$ is a skyscraper sheaf (a sheaf supported at a single point, hence not locally free), pulled back to $E\times E\times\mathbb P^1$ to reach dimension three. The 2020 corrigendum (Compositio Math. 156, 679–696) therefore re-proved the theory: the case where $X$ admits a non-holomorphic meromorphic map to $\mathbb P^1$ — which includes all $a(X)=2$ threefolds, since a holomorphic algebraic reduction onto a projective surface would force $b_2(X)>0$, and which is also covered by Lehn–Rollenske–Schinko's independent proof — is settled in full generality under $b_2(X)=0$ alone. In the remaining case, $a(X)=1$ with holomorphic algebraic reduction $f:X\to C$, the corrigendum classifies the possible general fibers — by Kodaira's classification of compact complex surfaces these are surfaces of special type (of Kodaira dimension $\le0$): tori (complex quotients of $\mathbb C^2$ by a lattice), primary Hopf surfaces (quotients of $\mathbb C^2\setminus\{0\}$ by a contraction), Inoue surfaces (quotients of $\mathbb C\times H$ built from indefinite binary quadratic forms), Kodaira surfaces (non-Kähler elliptic fibrations with trivial canonical bundle), and hyperelliptic (bi-elliptic) surfaces — and rules out Kodaira and hyperelliptic fibers, but carries out the whole analysis under the extra assumptions $H^1(X,\mathbb Z)=H^2(X,\mathbb Z)=0$ (which force $C\simeq\mathbb P^1$ and are used, e.g., to conclude $c_1(\omega_F)=0$ on fibers). These hypotheses are automatic for manifolds homeomorphic to $S^6$, so the six-sphere corollary $a(S^6)=0$ is unconditional. The corrigendum states the hypotheses were adopted 'for simplicity', indicating the authors expect the general statement; the restriction is a genuine gap in the literature, and the status of the original Lemma 1.5 within the specific fibrations arising from algebraic reductions of $b_2=0$ threefolds is likewise left undetermined. The broader context is the Hopf problem — whether $S^6$ carries any complex structure at all — which remains open (see the 2018 historical survey by Agricola, Bazzoni, Goertsches, Konstantis, Rollenske) and has attracted only necessary conditions and unaccepted claimed solutions since.

Problem Statement

Let $X$ be a compact complex threefold with $b_2(X)=0$, algebraic dimension $a(X)=1$, and holomorphic algebraic reduction $f:X\to C$ onto a curve. Prove, in this generality (that is, without the auxiliary hypotheses $H^1(X,\mathbb Z)=H^2(X,\mathbb Z)=0$ used in the corrigendum), that:

(1) $H^i(X,T_X\otimes M)=0$ for all $i\neq1$ and generic $M\in\mathrm{Pic}^\circ(X)$;

(2) $\chi(X,T_X\otimes M)\le0$ (equivalently, for one and hence all $M\in\mathrm{Pic}^\circ(X)$, noting $\chi$ is constant on the connected group $\mathrm{Pic}^\circ(X)$);

(3) $c_3(X)\le0$ (the three are not independent: (1) implies (2), and (2) implies (3) — the corrigendum derives (c) from (b) — so (1) is the strongest of the three);

and re-establish in the same generality the accompanying structure statements proven in the corrigendum under the extra hypotheses: the general fiber of $f$ is a torus, a primary Hopf surface, or an Inoue surface (Kodaira and hyperelliptic general fibers being excluded), together with the direct-image consequences $R^1f_*\mathcal O_X\simeq\mathcal O_C$ and $R^2f_*\mathcal O_X=0$ in the non-torus cases. Alternatively, refute the statement by an explicit counterexample: a compact complex threefold $X$ with $b_2(X)=0$, $a(X)=1$, and holomorphic algebraic reduction violating one of the requested conclusions (1)–(3) or one of the accompanying structure statements. In addition, and independently of the above, determine the exact scope of validity of [CDP98, Lemma 1.5] — for which fibrations $f:X\to C$ of smooth compact threefolds onto curves and locally free sheaves $F$ the higher direct images $R^if_*(F)$ are locally free — in particular whether it holds for the algebraic-reduction fibrations arising in the 1998 paper's context, or whether counterexamples exist already within that class. The common objective of the questions above is to complete the Campana–Demailly–Peternell classification program for the full class of $b_2=0$ threefolds, of which the currently treated $H^1=H^2=0$ case is the special case that suffices for the six-sphere application.

References

1. Frédéric Campana, Jean-Pierre Demailly, Thomas Peternell, The algebraic dimension of compact complex threefolds with vanishing second Betti number, Compositio Mathematica 112 (1998), 77–91, doi:10.1023/A:1000313214795, https://doi.org/10.1023/A:1000313214795
2. Frédéric Campana, Jean-Pierre Demailly, Thomas Peternell, The algebraic dimension of compact complex threefolds with vanishing second Betti number (corrigendum to the 1998 paper), Compositio Mathematica 156 (2020), 679–696, doi:10.1112/S0010437X19007802, arXiv:1904.11179, https://doi.org/10.1112/S0010437X19007802
3. Christian Lehn, Sönke Rollenske, Caren Schinko, The complex geometry of a hypothetical complex structure on $S^6$, Differential Geometry and its Applications 57 (2018), 121–137, doi:10.1016/j.difgeo.2017.10.015, arXiv:1912.09719, https://doi.org/10.1016/j.difgeo.2017.10.015
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