MathDB

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

About 6 years old · traced to

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 1Other wordings

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).

References

Primary source

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

calabi-yau-manifoldskahler-geometrypluripotential-theorypositive-currents
Posted

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.