Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces

Abstract: Let $M$ be a closed Kähler surface. We prove that every Riemannian metric $g$ on $M$ with $\operatorname{Sc}_g\geq-λ<sup>2$, where $λ\geq 0$, satisfies $$\lVert M\rVert\leq \frac{27}{2}\,λ^{4\operatorname{vol}_g(M).}$$ This proves Gromov's quantitative scalar-curvature--simplicial-volume conjecture for closed Kähler surfaces. We also construct infinitely many non-Kähler symplectic 4-manifolds of general type with positive simplicial volume for which the same estimate holds.

• The paper proves that every closed Kähler surface satisfies ||M|| ≤ (27/2)∫(Sc_g^-)² dvol_g, and therefore ||M|| ≤ (27/2)λ⁴vol_g(M) when Sc_g ≥ −λ².
• The paper combines a Yamabe-invariant reformulation, twisted Kähler–Einstein metrics, canonical-volume estimates, and surface classification to show that positive simplicial volume forces a surface to be of general type, while nonnegative Yamabe invariant implies vanishing simplicial volume.
• The paper identifies the optimal fixed-manifold coefficient 3/(32π²) for bidisk quotients and their blow-ups, extends the 27/2 estimate to infinitely many non-Kähler symplectic 4-manifolds, and leaves the optimal universal Kähler constant open.

Overview and main result

The paper proves Gromov's quantitative scalar-curvature–simplicial-volume conjecture for closed Kähler surfaces. For a closed oriented manifold $M$, the simplicial volume $\lVert M\rVert$ is the $\ell^1$-seminorm of the real fundamental class, a homotopy invariant that is multiplicative under finite covers and invariant under connected sum with simply connected manifolds in dimension four. Gromov conjectured that a lower scalar-curvature bound controls this invariant: for each $n$ there should be $c_n\geq 0$ such that every closed Riemannian $n$-manifold with $Sc_g\geq -\lambda^2$ satisfies $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$ (2608.17335).

The main theorem establishes this for all closed Kähler surfaces with the uniform constant $27/2$. More precisely, every Riemannian metric $g$ on a closed Kähler surface satisfies the integrated estimate

$\lVert M\rVert$0

where $\lVert M\rVert$1, and hence $\lVert M\rVert$2 whenever $\lVert M\rVert$3. The constant is uniform over all Kähler surfaces, including blow-ups, but the authors state plainly that it is not optimal on every fixed manifold: bidisk quotients admit the optimal coefficient $\lVert M\rVert$4. The paper also extends the estimate to infinitely many non-Kähler symplectic 4-manifolds of general type.

Yamabe reformulation of the conjecture

The first substantive reduction replaces the pointwise curvature hypothesis by metric-independent invariants. Defining the scalar cost $\lVert M\rVert$5, a key lemma proves the three-fold identity

$\lVert M\rVert$6

where $\lVert M\rVert$7 is the smooth Yamabe invariant. The proof uses Hölder's inequality to bound the Yamabe functional below by the negative part of scalar curvature, and constructs near-minimizing metrics by rescaling unit-volume Yamabe minimizers in conformal classes approaching $\lVert M\rVert$8. The lemma also proves that four formulations of Gromov's conjecture are equivalent: the pointwise bound, the scalar-cost bound, the Yamabe bound, and the integrated bound. This reduction means that for Kähler surfaces it suffices to prove $\lVert M\rVert$9.

Twisted Kähler–Einstein metrics and the nef-canonical estimate

The central geometric input works in all complex dimensions. If $\ell^1$0 is nef, the classes $\ell^1$1 are Kähler for every $\ell^1$2. Solving the negative-sign complex Monge–Ampère equation (the Aubin–Yau theorem) in these classes yields Kähler forms $\ell^1$3 satisfying the twisted Ricci equation $\ell^1$4, with $\ell^1$5 and $\ell^1$6 semipositive, so the associated metrics obey $\ell^1$7. Gromov's Main Inequality then gives $\ell^1$8, and since the volume is a cohomological polynomial in $\ell^1$9, one may let $n$0. The result is

$n$1

with vanishing simplicial volume whenever the canonical volume vanishes. Notably, the argument takes only a cohomological limit; it requires no convergence of the metrics themselves.

Proof of the surface theorem

The classification of Kähler surfaces supplies the remaining ingredients. If $n$2 is not of general type, Paternain–Petean's collapse theorem with sectional curvature bounded below, combined with Gromov's Ricci estimate, gives $n$3. If $n$4 is of general type with minimal model $n$5, then $n$6, so blow-up invariance gives $n$7, and LeBrun's formula gives $n$8, hence $n$9. Since $c_n\geq 0$0 is nef and big, the nef-canonical estimate in dimension two yields $c_n\geq 0$1, and the ratio $c_n\geq 0$2 produces the theorem. Two structural consequences follow: $c_n\geq 0$3 forces $c_n\geq 0$4 to be of general type, and $c_n\geq 0$5 forces $c_n\geq 0$6.

For closed four-manifolds with $c_n\geq 0$7, the paper defines the fixed-manifold coefficient $c_n\geq 0$8, which Lemma 2.1 identifies as the smallest valid Gromov constant on that smooth manifold. An estimate $c_n\geq 0$9 is thus equivalent to a scalar-curvature inequality with coefficient $n$0.

Bidisk quotients and optimal coefficients

For closed bidisk quotients $n$1 and their blow-ups $n$2, the paper computes the exact data: Bucher–Karlsson's value $n$3, the Chern identity $n$4 (valid even for irreducible lattices, since the product splitting descends to parallel line bundles whose Chern forms have vanishing squares), and LeBrun's formula $n$5. Consequently $n$6 and $n$7, so every metric on $n$8 with $n$9 satisfies $Sc_g\geq -\lambda^2$0, and this coefficient is optimal for the fixed manifold. For the unblown-up quotient, equality is attained by the product metric of sectional curvature $Sc_g\geq -\lambda^2$1, which has scalar curvature $Sc_g\geq -\lambda^2$2, volume $Sc_g\geq -\lambda^2$3, and $Sc_g\geq -\lambda^2$4. Two caveats are stated explicitly: for $Sc_g\geq -\lambda^2$5 optimality refers only to the infimum defining $Sc_g\geq -\lambda^2$6, and the bidisk family shows every uniform coefficient must be at least $Sc_g\geq -\lambda^2$7 but does not decide whether $Sc_g\geq -\lambda^2$8 is sharp.

Two further remarks extend the framework: the coefficient $Sc_g\geq -\lambda^2$9 is constant along finite étale towers of minimal surfaces of general type, and holomorphic surface bundles over curves of genus at least two satisfy $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$0 using Kotschick's signature estimate.

Symplectic extension and non-Kähler examples

For symplectic 4-manifolds, LeBrun's exact Yamabe formula is unavailable, but his Seiberg–Witten curvature estimate in terms of monopole classes still gives $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$1, where $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$2 is the squared norm of the largest vector in the convex hull of the monopole classes. The paper proves a criterion: if $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$3 is a symplectic 4-manifold of general type with $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$4 whose minimal model has the same $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$5 as a minimal Kähler surface $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$6 with $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$7, then the same $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$8 estimate holds. The proof uses the Seiberg–Witten blow-up formula to place $$\lVert M\rVert\leq c_n\lambda^n\operatorname{vol}_g(M)$$9 in the monopole hull.

To realize the criterion, the authors construct, for each $27/2$0, an infinite family $27/2$1 of minimal symplectic 4-manifolds by symplectically summing a surface bundle $27/2$2 with fiber $27/2$3 (monodromy a single Dehn twist, so $27/2$4 and $27/2$5) with iterated symplectic sums $27/2$6 built from a telescoping triple. The resulting $27/2$7 have odd $27/2$8, hence admit no Kähler structure; they have nonamenable fundamental group; and a degree-one map $27/2$9 together with Gromov's additivity theorem gives $g$0 with a bound independent of $g$1. Comparing with $g$2, which has the same $g$3 and larger simplicial volume, yields $g$4 for all sufficiently large $g$5.

Failure of the canonical-volume bridge in higher dimensions

The nef-canonical estimate holds in every complex dimension, but the scalar-curvature conclusion requires controlling canonical volume by scalar cost. Proposition 5.1 shows this fails for every $g$6: a smooth hypersurface $g$7 of degree $g$8 has ample canonical bundle with $g$9, yet by the Gromov–Lawson classification it is simply connected, non-spin, and admits positive scalar curvature, so $\lVert M\rVert$00, $\lVert M\rVert$01, and $\lVert M\rVert$02. Thus no constant $\lVert M\rVert$03 with $\lVert M\rVert$04 exists for nef and big canonical bundles when $\lVert M\rVert$05. The paper notes this does not affect Gromov's conjecture itself, since both quantities vanish; it only rules out estimates factoring through canonical volume, leaving open whether other invariants could serve.

Open questions on sharp constants

Three problems are posed. First, determine the optimal uniform coefficient $\lVert M\rVert$06, currently bracketed by

$\lVert M\rVert$07

and in particular decide whether the bidisk coefficient holds for every Kähler surface of general type. Second, for each $\lVert M\rVert$08, find the optimal constant $\lVert M\rVert$09 with $\lVert M\rVert$10 for nef canonical bundles; in dimension one the optimum is $\lVert M\rVert$11. Third, determine the spectrum $\lVert M\rVert$12 of simplicial volumes of closed Kähler surfaces — Heuer–Löch showed every nonnegative rational occurs among four-manifolds, but not within the Kähler category — and ask whether for every closed oriented manifold $\lVert M\rVert$13 there is a Kähler surface with equal simplicial volume.

Conclusion

The paper proves Gromov's quantitative simplicial-volume conjecture for closed Kähler surfaces with the uniform constant $\lVert M\rVert$14, via a Yamabe reduction, twisted negative Kähler–Einstein metrics on nef canonical classes, and LeBrun's Seiberg–Witten computation of the Yamabe invariant. It identifies the optimal fixed-manifold coefficient for bidisk quotients, extends the estimate to an infinite family of non-Kähler symplectic 4-manifolds of general type, and shows that the canonical-volume-to-scalar-cost bridge used in the proof exists only in complex dimension two. The sharpness of the uniform constant and the Kähler simplicial-volume spectrum remain open.