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The bounded-potential conjecture for nef classes on Calabi–Yau manifolds

Let XnX^n be a Calabi–Yau manifold and let α\alpha be a closed real (1,1)(1,1)-form such that [α]∈∂CX[\alpha]\in\partial\mathcal{C}_X is nef. Bounded-potential conjecture. There is a bounded function φ∈L∞(X)\varphi\in L^\infty(X) such that

α+−1 ∂∂‾φ⩾0\alpha+\sqrt{-1}\,\partial\overline{\partial}\varphi\geqslant 0

in the weak sense of currents on XX. This is equivalent to bounded potentials for a closed positive current with minimal singularities in [α][\alpha]. The conjecture remains open even for nef classes of square zero on K3K3 surfaces; for nef and big classes it follows from the analytic contraction conjecture above.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The bounded-potential conjecture for nef classes on Calabi–Yau manifolds

    Let XX be a Calabi–Yau manifold and let [α][\alpha] be a nef (1,1)(1,1)-class. Bounded-potential conjecture. Then [α][\alpha] contains a closed positive current with bounded potentials. The conjecture is open even for surfaces; the weaker assertion involving vanishing Lelong numbers is also open when [α][\alpha] is not big.

    source: Valentino Tosatti, “Semipositive line bundles and (1,1)-classes”, arXiv:2309.00580 (2023).

Sources & referencesView supporting material

Primary source

Valentino Tosatti, “Collapsing Calabi-Yau manifolds”, arXiv:2003.00673 (2020).

calabi-yau-manifoldskahler-geometrypluripotential-theorypositive-currents

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