Minimal singular metrics on nef adjoint bundles

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Minimal singular metrics on nef adjoint bundles

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Recursos

Take a nef adjoint bundle K_H + Θ on a projective klt pair. You can now explain why no minimal semipositive metric on its pullback has a logarithmic pole, and which hypothesis rules each pole out.

The problem

Limits of almost-semipositive metrics can acquire poles. For a bare nef bundle the pole is real: on a ruled surface over an elliptic curve, the divisor metric can be minimal and still have a positive Lelong number along the divisor [0:08]. Gongyo–Matsumura asked whether adjoint bundles avoid this. For klt pairs the answer is yes, on every projective log resolution and without bigness.

The extremal argument, move by move

Normalize [0:52]. Write N = K_V + B. The klt coefficients satisfy β_j < 1, so e^{−b} is integrable. The gauge Q_m(s) = ∫|s|^{2/m}e^{−a/m−b} is then a canonical density. klt vanishing gives a fixed twist A with mN + A globally generated. Assume ν(φ,p) > 0, and pick s_m with the largest value at p among sections of gauge one.
Small mass [1:52]. τ_m = (1/m)log|s_m|² has a semipositive limit, and minimality bounds that limit by φ + C. So the mass ε_m of the sublevel set {g_m < −R} tends to zero.
Localized ∂̄ [2:25]. Cut s_m down to that set and correct with u. The bound ∫|u|²e^{−S−f(g)} ≤ Cε_m holds with C independent of m and of the open set. The slope c makes e^{−cφ−b} non-integrable at p, so u(p) = 0 and w_m(p) = s_m(p).
Hölder and contradiction [3:09]. Q_m(w_m)^m ≤ C′ε_m(1+I)^d → 0. Rescaling w_m gives a section of gauge one that is taller at p than s_m, which contradicts the choice of s_m. So ν ≡ 0.

Why adjoint matters, and where it leads

A bare nef bundle offers neither the canonical density nor the fixed twist [3:51]. With zero Lelong numbers, Skoda's theorem gives 𝓘(tφ) = O_V for every t > 0. The same kind of metrics power the interior H¹-injectivity theorem [4:20]: for C₁ strictly between C₀ and C₂, the map H¹(K_V+L₁) → H¹(K_V+L₀) is injective, so sections restrict surjectively onto E. They also give semiampleness for klt fourfolds with nef adjoint and χ(O_X) ≠ 0 [5:12].

The move to carry into your own proofs: to rule out a pole at p, maximize at p among normalized sections, then show that the cut-off correction is a cheaper section with the same value there.

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