Existence of stable bundles with prescribed Chern classes on simply-connected Calabi–Yau threefolds: the Douglas–Reinbacher–Yau conjecture and extremal bundles

On a simply-connected Calabi–Yau threefold $X$, which Chern data $(r, c_1, c_2, c_3)$ are realized by slope-stable reflexive sheaves? The Douglas–Reinbacher–Yau (DRY) conjecture predicts that any data satisfying two explicit numerical conditions — a strengthened Bogomolov-type equality fixing $c_2$ in terms of an ample class, and a cubic inequality bounding $c_3$ — are realized by a stable reflexive sheaf of rank $r$. The physically central extremal case ($c_1(V)=0$, $c_2(V)=c_2(T_X)$: heterotic compactifications with exact anomaly cancellation and no five-branes, where the net number of generations is $\tfrac{1}{2}\left|\int_X c_3(V)\right|$) forces the ample class to satisfy $\tilde{H}^2=\tfrac{24-r}{24r}\,c_2(T_X)$. Gao–He–Yau (2015) introduced the generalized Hartshorne–Serre construction, for which stability and the required bounds on the arithmetic genus of curves in a fixed curve class remain the two open ingredients. Despite partial results (elliptically fibered threefolds by Andreas–Curio; classes $n[D^2]$ on any Calabi–Yau threefold with a very ample divisor by Wu–Yau; monad constructions on favourable complete intersection Calabi–Yaus by Gao–He–Yau), the general conjecture, the extremal existence question, and the genus-bound program all remain open.

Background

A Calabi–Yau (CY) threefold $X$ is a compact Kähler three-dimensional complex manifold with trivial canonical bundle $K_X$ and $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$; it is called simply-connected when $\pi_1(X)=0$. All known examples, e.g. quintic hypersurfaces in projective space and complete intersection Calabi–Yaus (CICYs, complete intersections of products of projective spaces with sums of positive-degree hypersurfaces), are simply-connected. A torsion-free coherent sheaf $E$ of rank $r$ on a polarized $(X, H)$ is slope-(semi)stable if $\mu(F)=\frac{c_1(F)\cdot H^2}{\mathrm{rk}(F)}<(=)\,\mu(E)$ for every proper coherent subsheaf $0<F<E$; a reflexive sheaf is torsion-free and coincides with its double dual, and on a smooth threefold its non-locally-free locus has codimension at least 3. The Bogomolov inequality, a necessary condition for slope stability, states that $\left(2r\,c_2(E)-(r-1)\,c_1(E)^2\right)\cdot H^2\ge 0$ for every slope-stable $E$ and ample $H$. By the Donaldson–Uhlenbeck–Yau theorem, a polystable holomorphic bundle with $c_1=0$ (resp. $c_1$ orthogonal to all $(1,1)$-classes) admits a Hermitian–Yang–Mills connection — a unitary connection whose curvature is of type $(1,1)$ and primitive with respect to the Kähler form — which is the supersymmetry condition for heterotic string compactifications. Heterotic anomaly cancellation requires $c_2(V)+[W]=c_2(T_X)$ for the gauge bundle $V$ plus possible five-brane classes $[W]$ (effective curve classes carried by five-branes in the heterotic string); a stable vector bundle with $c_1(V)=0$ and $c_2(V)=c_2(T_X)$ (no five-branes) is called extremal, and then the net number of families is $\tfrac{1}{2}\left|\int_X c_3(V)\right|$.

Constructing stable bundles with prescribed Chern classes on a threefold is much harder than on a surface: for surfaces, Gieseker (1988) and Artamkin (1990), building on Taubes (1984), proved that for $c_2>2p_g+4$ every suitable class is realized by a rank-2 (higher-rank) stable bundle with $c_1=0$ ($p_g=h^0(K_S)$ is the geometric genus); no threefold analogue at that level of generality is known. Existing threefold tools are: (i) monads on CICYs — a monad is a two-step complex $L_1\to L_2\to L_3$ of direct sums of line bundles whose middle cohomology $V=\ker(d_2)/\mathrm{im}(d_1)$ is a vector bundle with computable Chern classes; such bundles are stable for special choices of the summands (Gao–He–Yau 2015 enumerate positive monads, i.e. monads built from positive line bundles, of rank $\le 5$ on favourable CICYs, those for which the ambient-space divisor classes restrict to independent classes in $H^2(X)$); (ii) the spectral cover construction of Friedman–Morgan–Witten on elliptically fibered CY threefolds (stable bundles built from a divisor of the elliptic fibration — the spectral cover — together with a line bundle on it); (iii) the (generalized) Hartshorne–Serre construction: from a Cohen–Macaulay curve $C\subset X$ (a curve with no embedded points), generically locally complete intersection, the sequence $0\to H^0(C,\omega_C)\otimes\mathcal{O}_X\to V\to I_C\to 0$, where $\omega_C$ is the dualizing sheaf of $C$ and $I_C$ its ideal sheaf, yields a sheaf with $c_2(V)=[C]$ and $c_3(V)=2g(C)-2$, where $g$ is the arithmetic genus; (iv) deformation methods in the spirit of Huybrechts (1995), who gave sufficient conditions for $\mathcal{O}_X\oplus T_X$ to deform to a stable bundle; Wu–Yau (2014) extended this approach, giving explicit conditions under which $\mathcal{O}_X\oplus I_C$ deforms to a stable reflexive sheaf for a Cohen–Macaulay curve $C\subset X$.

The Douglas–Reinbacher–Yau (DRY) conjecture ('Branes, Bundles and Attractors: Bogomolov and Beyond', arXiv:math/0604597, 2006), motivated by attractor-flow arguments in type II string compactification (moduli-space flows that stabilize charged black-hole solutions), is the proposed general existence criterion. In its primary form (Conjecture 1.1 of Douglas–Reinbacher–Yau): let $X$ be a simply-connected CY threefold, $r>1$ an integer, and $c_i\in H^{2i}(X,\mathbb{Z})$ ($i=1,2,3$); write $\mathrm{ch}_2=\frac{c_1^2-2c_2}{2}$ and $\mathrm{ch}_3=\frac{c_1^3-3c_1c_2+3c_3}{6}$ for the Chern-character components of a sheaf with those Chern classes. If there exists an ample class $\tilde{H}\in H^2(X,\mathbb{R})$ with $\frac{1}{2r^2}\left(2r\,c_2-(r-1)\,c_1^2-\frac{r^2}{12}\,c_2(X)\right)=\tilde{H}^2$ and $\frac{1}{6r^2}\left(c_1^3+3r\left(r\,\mathrm{ch}_3-\mathrm{ch}_2\,c_1\right)\right)<\frac{2^{5/2}}{3}\,r\,\tilde{H}^3$, then there exists a stable reflexive sheaf of rank $r$ on $X$ with exactly these Chern classes, stable with respect to some (the authors add: indeed any) ample class. In the case $c_1=0$ this reads: $c_2=r\,\tilde{H}^2+\frac{r}{24}\,c_2(X)$ and $c_3<\frac{2^{9/2}}{3}\,r\,\tilde{H}^3=\frac{16\sqrt{2}}{3}\,r\,\tilde{H}^3$; these are the forms restated by Gao–He–Yau (2015, with extremal specialization $\frac{24-n}{24n}\,c_2(T_X)=H^2$, i.e. their eq. (4.39)) and by Wu–Yau (2014). Data satisfying the DRY conditions also satisfy the Bogomolov inequality, so the conjecture is consistent with all known necessary conditions.

(Early versions of the DRY preprint additionally conjectured a 'strong Bogomolov inequality' — an improved universal inequality for stable bundles on simply-connected surfaces with trivial or ample canonical bundle; counterexamples of Costa–Miró-Roig and Nakashima on surfaces refuted that inequality side-conjecture, Jardim's example of a rank-3 stable bundle on a generic quintic threefold (presented by the authors themselves in Appendix B of v4) showed the improvement does not extend to Calabi–Yau threefolds, and the side-conjecture was withdrawn from the final preprint version v4. It is logically independent of the existence Conjecture 1.1, which no counterexample is known to touch.)

Gao–He–Yau verified the numerical conditions against their database of stable monad bundles (finding explicit DRY solutions, with interesting exceptions on self-mirror threefolds — those whose Hodge diamond is symmetric under $(i,j)\leftrightarrow(j,i)$, in particular $\chi(X)=0$ — including the Schoen manifold, the fiber product of two rational elliptic surfaces) and observed that spectral-cover bundles on elliptic fibrations violate the DRY bounds in certain polarization regions — while being provably unable to produce extremal bundles at all (their Proposition 3.1: no stable $SU(n)$ bundle — a holomorphic bundle with structure group $SU(n)$, i.e. $c_1=0$ — from the spectral-cover construction on an elliptically fibered CY threefold is extremal).

Problem Statement

Prove or disprove the Douglas–Reinbacher–Yau conjecture in full generality, and settle its extremal specialization, as follows.

Setup. Let $X$ be a simply-connected Calabi–Yau threefold (compact Kähler, $K_X$ trivial, $h^1(\mathcal{O}_X)=0$). The original conjecture is stated for simply-connected complex threefolds with trivial canonical bundle; since condition (1) below supplies an ample (hence Kähler) class, and on a compact Kähler threefold with trivial canonical bundle simple connectivity forces $h^1(\mathcal{O}_X)=h^2(\mathcal{O}_X)=0$, the two formulations cover the same threefolds. Let $r>1$ be an integer and let $c_i\in H^{2i}(X,\mathbb{Z})$, $i=1,2,3$. Define $\mathrm{ch}_2=\frac{c_1^2-2c_2}{2}\in H^4(X,\mathbb{Q})$ and $\mathrm{ch}_3=\frac{c_1^3-3c_1c_2+3c_3}{6}\in H^6(X,\mathbb{Q})$, the Chern-character components of a rank-$r$ sheaf with total Chern class $1+c_1+c_2+c_3$.

Question A (DRY conjecture, general). Suppose there exists an ample class $\tilde{H}\in H^2(X,\mathbb{R})$ such that

$$\frac{1}{2r^{2}}\left(2r\,c_{2}-(r-1)\,c_{1}^{2}-\frac{r^{2}}{12}\,c_{2}(X)\right)=\tilde{H}^{2}$$

and

$$\frac{1}{6r^{2}}\left(c_{1}^{3}+3r\left(r\,\mathrm{ch}_{3}-\mathrm{ch}_{2}\,c_{1}\right)\right)<\frac{2^{5/2}}{3}\,r\,\tilde{H}^{3}.$$

(Throughout, these two displayed relations are referred to as conditions (1) and (2), in this order.) Must there exist a stable reflexive sheaf $E$ on $X$ of rank $r$ with $c_1(E)=c_1$, $c_2(E)=c_2$, $c_3(E)=c_3$, stable with respect to some ample class of $X$? (The primary source additionally conjectures stability with respect to every ample class; proving the 'some' version already answers the existence question.) A counterexample is an explicit simply-connected Calabi–Yau threefold $X$, a rank $r>1$, and Chern data $c_1,c_2,c_3$ satisfying (1)–(2) for some ample $\tilde{H}$, with no stable reflexive sheaf of rank $r$ on $X$ carrying those Chern classes. In the case $c_1=0$, conditions (1)–(2) are equivalent to

$$c_{2}=r\,\tilde{H}^{2}+\frac{r}{24}\,c_{2}(X),\qquad c_{3}<\frac{16\sqrt{2}}{3}\,r\,\tilde{H}^{3}.$$

Question B (extremal bundles). For the extremal constraint $c_1=0$ and $c_2=c_2(T_X)$, condition (1) forces

$$\tilde{H}^{2}=\frac{24-r}{24r}\,c_{2}(T_X),$$

which can hold only for $r<24$ (the source infers this from ampleness of $\tilde{H}$ whenever $c_2(T_X)$ is positive, more precisely nef) and requires the class $\frac{24-r}{24r}\,c_2(T_X)$ to be the square of an ample class $\tilde{H}\in H^2(X,\mathbb{R})$. Determine for which simply-connected Calabi–Yau threefolds $X$ and ranks $r$ this ample-class condition holds, and whether then a stable holomorphic vector bundle $V$ with $c_1(V)=0$, $c_2(V)=c_2(T_X)$ exists for $c_3(V)$ in the range (2) — the DRY conjecture predicts at least a stable reflexive sheaf, and the source's extremal bundles are locally free — in particular, whether stable extremal bundles exist on every such $(X, r)$, and how the realizable values of $c_3(V)$ (equivalently, the achievable net generation numbers $\tfrac{1}{2}\left|\int_X c_3(V)\right|$) are distributed.

Question C (genus bounds). For which simply-connected Calabi–Yau threefolds $X$ and fixed classes $[C]\in H_2(X,\mathbb{Z})$ (viewed via Poincaré duality as classes in $H^4(X,\mathbb{Z})$), represented by (Cohen–Macaulay, generally locally complete intersection) curves $C\subset X$, do effective bounds $g(C)\le B([C])$ on the arithmetic genus hold, and for which $(X,[C])$ does no such bound exist? Do genus bounds at the class $[C]=c_2(T_X)$ of the extremal case decide the extremal-existence question of Question B, namely whether stable extremal bundles with $c_3(V)=2g(C)-2$ in the range (2) exist for every rank $r$ for which $\tilde{H}^2=\frac{24-r}{24r}\,c_2(T_X)$ is solvable by an ample class $\tilde{H}$?

The source paper additionally lists as open the asymptotics of Donaldson–Thomas invariants (enumerative invariants counting stable sheaves) for heterotic bundle moduli, the statistical distribution of Chern numbers under extremal transitions, and the polarization-dependence of spectral-cover stability; these adjacent directions are context rather than part of the questions posed above, since they remain meaningful whichever way Questions A–C are resolved.

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