Demailly (2021) proposed an elliptic system of Hermitian–Yang–Mills type for the curvature tensor — a matrix Monge–Ampère equation coupled with a trace-free Hermite–Einstein condition with logarithmic 'friction' terms — designed so that a solution on the whole continuity interval t∈[0,1]t\in[0,1] endows an arbitrary ample vector bundle with a dual Nakano, hence Griffiths, positive Hermitian metric, which would settle the 1969 Griffiths conjecture that ampleness and Griffiths curvature positivity are equivalent. He proved ellipticity and invertible linearization near t=0t=0, but the thresholds governing invertibility depend on the unknown solution itself, and no a priori estimates ruling out blow-up before t=1t=1 are known. The problem is to resolve at least one of the following, in every case for an ample vector bundle over a projective manifold of dimension at least 2: (A) long-time existence on [0,1][0,1] for the original system; (B) long-time existence on [0,1][0,1] for a suitable modification of it; (C) the existence theory for the related extremal-metric program for the Monge–Ampère volume; or (D) a rigorous refutation of (A). Items (A)–(C) are the affirmative completions and (D) is the only accepted negative one: a refutation of (B) or (C) does not by itself resolve the problem. The four branches are defined precisely in the Problem Statement. The problem is open: the literature of 2020–2026 solves only special cases (Riemann surfaces, direct sums of ample line bundles on curves, vortex bundles, conformal classes, and partial low-rank results), and the Griffiths conjecture itself remains open in general.

Background

Let XX be a compact projective manifold of complex dimension nn and let E→XE\to X be a holomorphic vector bundle of rank rr. Algebro-geometrically, EE is called ample (in the sense of Hartshorne) when the tautological line bundle OP(E∗)(1)\mathcal O_{\mathbb P(E^*)}(1) on the projectivization P(E∗)\mathbb P(E^*) is ample. Differential-geometrically, a smooth Hermitian metric hh on EE has a Chern curvature tensor ΘE,h=i∇E,h2\Theta_{E,h}=\mathrm{i}\nabla^2_{E,h}, a (1,1)(1,1)-form with values in the Hermitian endomorphisms of EE; hh is Griffiths positive when ΘE,h(ξ,ξˉ)(v,vˉ)>0\Theta_{E,h}(\xi,\bar\xi)(v,\bar v)>0 for every nonzero tangent vector ξ∈T1,0X\xi\in T^{1,0}X and every nonzero v∈Exv\in E_x, and Nakano (respectively dual Nakano) positive when ΘE,h\Theta_{E,h} is positive as a Hermitian form on T1,0X⊗ET^{1,0}X\otimes E (respectively T1,0X⊗E∗T^{1,0}X\otimes E^*); both are stronger than Griffiths positivity. Griffiths positivity implies ampleness via the Kodaira embedding theorem, and for line bundles the notions coincide. A conjecture of Griffiths (1969) asserts the converse: every ample vector bundle carries a Griffiths positive Hermitian metric. This remains one of the central open problems of complex differential geometry; it is settled on compact Riemann surfaces but open for r≥2r\ge 2 in dimension n≥2n\ge 2.

The classical PDE route to such statements is the Hermitian–Yang–Mills (HYM) equation ωn−1∧ΘE,h=λ ωn⊗IdE\omega^{n-1}\wedge\Theta_{E,h}=\lambda\,\omega^{n}\otimes\mathrm{Id}_E for a Kähler form ω\omega, solvable by the Donaldson–Uhlenbeck–Yau theorem exactly for slope-polystable bundles. Ample bundles need not be stable, but Uhlenbeck and Yau proved that a 'cushioned' Hermite–Einstein equation, in which a logarithmic friction term −εlog⁡q~ε-\varepsilon\log\tilde q_\varepsilon is inserted in the right-hand side, is smoothly solvable for every EE; as ε→0\varepsilon\to 0 these metrics degenerate along the Harder–Narasimhan filtration of EE (the canonical filtration by subsheaves of maximal slope).

Demailly (2021) built on this a continuity method designed to force positivity of the curvature itself. With ω0=Θdet⁡E>0\omega_0=\Theta_{\det E}>0 a Kähler metric in c1(E)c_1(E) and α>0\alpha>0 large, he studies the system

det⁡TX⊗E∗(TΘE,ht+(1−t) α ω0⊗IdE∗)1/r=ft ω0n,ω0n−1∧ΘE,ht∘=gt,\det\nolimits_{TX\otimes E^*}\bigl({}^{T}\Theta_{E,h_t}+(1-t)\,\alpha\,\omega_0\otimes\mathrm{Id}_{E^*}\bigr)^{1/r}=f_t\,\omega_0^{n},\qquad\omega_0^{n-1}\wedge\Theta^{\circ}_{E,h_t}=g_t,

where ΘE,h∘=ΘE,h−1rΘdet⁡E⊗IdE\Theta^{\circ}_{E,h}=\Theta_{E,h}-\frac{1}{r}\Theta_{\det E}\otimes\mathrm{Id}_E is the trace-free part of the curvature and det⁡TX⊗E∗\det_{TX\otimes E^*} is the determinant of the curvature viewed as a Hermitian form on TX⊗E∗TX\otimes E^*. The first equation is a matrix analogue of the complex Monge–Ampère equation solved by Yau (1978) for line bundles (the Calabi conjecture); the second restores determinacy, since for r>1r>1 the determinantal equation alone is underdetermined (one scalar equation for r2r^2 unknowns). Right-hand sides of the friction form ft=(det⁡H0/det⁡ht)λa0f_t=(\det H_0/\det h_t)^{\lambda}a_0 and gt=−ε(det⁡H0/det⁡ht)μ ω0n⊗log⁡h~t∘g_t=-\varepsilon(\det H_0/\det h_t)^{\mu}\,\omega_0^{n}\otimes\log\tilde h_t^{\circ}, with parameters λ≥0\lambda\ge 0, ε>0\varepsilon>0, μ∈R\mu\in\mathbb R, are compatible at t=0t=0 with the Uhlenbeck–Yau cushioned solution: here H0H_0 is a fixed reference Hermitian metric on EE with Θdet⁡E,det⁡H0=ω0\Theta_{\det E,\det H_0}=\omega_0, a0=ω0−ndet⁡TX⊗E∗(TΘE,h0+α ω0⊗IdE∗)1/r>0a_0=\omega_0^{-n}\det_{TX\otimes E^*}({}^{T}\Theta_{E,h_0}+\alpha\,\omega_0\otimes\mathrm{Id}_{E^*})^{1/r}>0 is chosen so that the cushioned metric h0h_0 (with det⁡h0=det⁡H0\det h_0=\det H_0) satisfies the determinantal equation at t=0t=0, and h~t∘\tilde h_t^{\circ} is the H0H_0-Hermitian endomorphism of EE associated with the normalized metric ht∘=ht(det⁡ht)−1/rh_t^{\circ}=h_t(\det h_t)^{-1/r}. Demailly proved (his Theorems 2.11 and 2.17) that the system is elliptic with invertible linearization — under a Hilbert–Schmidt norm bound on the symbol of the second equation's right-hand side — so a solution exists on some interval [0,t0)[0,t_0); and if a solution exists on all of [0,1][0,1] without explosion, then h1h_1 is dual Nakano positive, hence Griffiths positive, and the Griffiths conjecture follows. The obstacle, stated as open in his Remark 2.18, is that the thresholds ε≥ε0(ht)\varepsilon\ge\varepsilon_0(h_t) and λ≥λ0(ht)(1+μ2)\lambda\ge\lambda_0(h_t)(1+\mu^2) above which invertibility is proved depend on the unknown solution hth_t, and no a priori C0C^0 or higher-order estimates are known that would rule out blow-up at some t0<1t_0<1; he proposes investigating blow-up via 'destabilizing subsheaves' contradicting ampleness of a quotient, in analogy with Uhlenbeck–Yau. Demailly records two further directions toward an affirmative resolution of the problem, both still open in dimension ≥2\ge 2: uniform solution-independent bounds on the thresholds ε0,λ0\varepsilon_0,\lambda_0, which would let the continuity method close at t=1t=1; and a redesign of the right-hand sides in the spirit of Variants 2.19, including Variant 2.19(a)'s variable Kähler metric ωt=βt/(rα+1)\omega_t=\beta_t/(r\alpha+1) — whose linearized invertibility is proven only for small tt — and Variant 2.19(b)'s added trace-free friction terms. A companion extremal problem concerns the Monge–Ampère volume MAVol(E)=sup⁡h∫Xdet⁡TX⊗E∗((2π)−1TΘE,h)1/r\mathrm{MAVol}(E)=\sup_h\int_X\det_{TX\otimes E^*}((2\pi)^{-1}{}^{T}\Theta_{E,h})^{1/r} taken over metrics hh with ΘE,h\Theta_{E,h} dual Nakano positive; it satisfies the optimal bound MAVol(E)≤r−nc1(E)n\mathrm{MAVol}(E)\le r^{-n}c_1(E)^n, attainment and the structure of 'extremal metrics' (governed by a fourth-order Euler–Lagrange system) are unknown, and Demailly suspected this extremal route could be an alternative, possibly better, path to the conjecture.

Problem Statement

The problem is Demailly's explicitly open analytic program toward the Griffiths conjecture, restricted to the dimensions where it is still unresolved. Let XX be a compact projective manifold of dimension n≥2n\ge 2 and let EE be an ample holomorphic vector bundle over XX. Establish long-time existence on the whole continuity interval t∈[0,1]t\in[0,1] for at least one of: (A) the original Demailly system — the determinantal Monge–Ampère equation coupled with the trace-free Hermite–Einstein condition displayed in the Background, with right-hand sides of the friction form (2.16)–(2.16°); (B) an explicitly modified design built from Demailly's Variants 2.19 (that is, any modification of the trace-free equation, via Variant 2.19(a)'s variable Kähler metric ωt=βt/(rα+1)\omega_t=\beta_t/(r\alpha+1) and/or Variant 2.19(b)'s added trace-free friction terms, that keeps the determinantal equation, preserves ellipticity and invertible linearization, and yields a dual Nakano positive metric on [0,1][0,1]); or (C) the related fourth-order extremal-metric system for the Monge–Ampère volume. Alternatively, rigorously refute (A), i.e. long-time existence for the original Demailly system; that refutation is the only accepted negative resolution — a refutation of (B) or (C) does not by itself resolve the problem. All requested ingredients serve this single objective: a smooth solution on [0,1][0,1] yields a dual Nakano positive (hence Griffiths positive) Hermitian metric on the arbitrary ample bundle EE, and thereby proves the Griffiths conjecture in full generality. A valid negative answer is a rigorous obstruction to the original Demailly system — for instance an explicit ample bundle EE with ample determinant for which every smooth solution family hth_t of that system, with the friction right-hand sides (2.16)–(2.16°), explodes before t=1t=1 — which would refute long-time existence for that system while leaving the modified designs, the extremal-metric program, and the Griffiths conjecture itself undecided. A proof merely that the invertibility thresholds ε0,λ0\varepsilon_0,\lambda_0 of Theorem 2.17 cannot be made solution-independent would block only the continuity-method route based on that theorem with these right-hand sides, and would not by itself constitute a resolution, since Demailly's Remark 2.18(a) treats such uniformity as a sufficient route whose failure motivates redesigning the right-hand sides. The case n=1n=1 (compact Riemann surfaces), resolved affirmatively by Murakami building on Pingali's reduction, and the line-bundle case r=1r=1, which is Yau's solution of the Calabi conjecture, are settled and form no part of the problem.

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