Optimal uniform coefficient for closed Kähler surfaces

Determine the optimal uniform coefficient c_4^{\mathrm{Kah}} governing the ratio ||M||/\mathcal{A}_4(M) over all closed Kähler surfaces M, and decide whether the bidisk coefficient 3/(32π²) is valid for every closed Kähler surface of general type.

Background

The paper defines c_4{\mathrm{Kah}} as the supremum of ||M||/\mathcal{A}_4(M) over closed Kähler surfaces with positive scalar cost. The results establish the bounds 3/(32π²)≤c_4{\mathrm{Kah}}≤27/2. Bidisk quotients attain the lower bound, while the general argument yields the upper bound; the exact value and whether the lower bound is universal are left unresolved.

References

Determine c_4{\mathrm{Kah}}. In particular, decide whether the bidisk coefficient 3/(32\pi2) remains valid for every closed Kähler surface of general type.

— Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces  (2608.17335 - Min et al., 18 Aug 2026) in Section 5, Section 5 problem immediately following equation (5.2)

For each m\geq2, find the optimal constant C_m{\mathrm{can}} such that

\lVert X\rVert \leq C_m{\mathrm{can}} \int_X c_1(K_X)m

for every closed Kähler m-fold with nef canonical bundle.

— Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces  (2608.17335 - Min et al., 18 Aug 2026) in Section 5, Section 5 problem immediately following the discussion of the nef-canonical estimate