[Paper Review] An obstruction to the existence of constant scalar curvature Kähler metrics

Julius Ross, Richard Thomas|ArXiv.org|Dec 29, 2004

Geometry and complex manifolds28 references14 citations

TL;DR

This paper introduces a new obstruction to the existence of constant scalar curvature Kähler (cscK) metrics on polarized complex manifolds by defining a slope semistability condition based on subschemes. It proves that if a manifold admits a cscK metric, then its slope must be at least as large as that of any subscheme; this obstruction applies to del Pezzo surfaces, projective bundles, and blow-ups, providing explicit examples of Kähler classes without cscK metrics even with trivial automorphism groups.

ABSTRACT

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope $μ$ for a projective manifold and for each of its subschemes, and show that if $X$ is cscK then $μ(Z)\leμ(X)$ for all subschemes $Z$. This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as del Pezzo surfaces and projective bundles. If $\PP(E) o B$ is a projective bundle which admits a cscK metric in a rational Kähler class with sufficiently small fibres, then $E$ is a slope semistable bundle (and $B$ is a slope semistable polarised manifold). The same is true for \emph{all} rational Kähler classes if the base $B$ is a curve. We also show that the slope inequality holds automatically for smooth curves, canonically polarised and Calabi Yau manifolds, and manifolds with $c_1(X)<0$ and $L$ close to the canonical polarisation.

Motivation & Objective

• To establish a new algebraic obstruction to the existence of constant scalar curvature Kähler (cscK) metrics on polarized complex manifolds.
• To define a slope invariant for subschemes of a manifold and show that cscK metrics imply slope semistability.
• To provide explicit examples of Kähler classes—especially on del Pezzo surfaces and projective bundles—that do not admit cscK metrics despite trivial automorphism groups.
• To extend the understanding of K-stability and its relation to cscK metrics through geometric and algebraic methods.

Proposed method

• Define the slope μ(Z) for a subscheme Z of a polarized manifold (X,L) using intersection theory on the blow-up of X along Z.
• Introduce a modified slope μ_c(O_Z, L) depending on a parameter c related to the Seshadri constant ε(Z,L), which measures the positivity of L near Z.
• Use asymptotic Riemann-Roch expansions and intersection formulas on the blow-up to derive a criterion for slope destabilization.
• Apply the slope inequality μ_c(O_Z, L) < μ(X,L) to detect obstructions to K-semistability and thus to cscK metrics.
• Analyze specific cases such as projective bundles, blow-ups of P², and del Pezzo surfaces to demonstrate the obstruction in practice.
• Leverage known results on Kähler-Einstein metrics and stability for canonical and anticanonical polarizations to verify the slope condition in special cases.

Experimental results

Research questions

• RQ1 Does every polarized manifold admitting a cscK metric satisfy a slope semistability condition defined via subschemes?
• RQ2 Can slope semistability be used to obstruct the existence of cscK metrics in Kähler classes where the Calabi-Futaki invariant vanishes?
• RQ3 For projective bundles P(E) → B, under what conditions on E and B does slope semistability imply the existence of a cscK metric?
• RQ4 Are there explicit examples of Kähler classes on surfaces with trivial automorphism group that do not admit cscK metrics, and can slope semistability detect them?
• RQ5 Does slope semistability recover or extend known obstructions such as K-stability or the existence of Kähler-Einstein metrics?

Key findings

• The paper proves that if a polarized manifold (X,L) admits a cscK metric, then it must be slope semistable, meaning μ(Z) ≤ μ(X) for all subschemes Z.
• For projective bundles P(E) → B, if a cscK metric exists in a rational Kähler class with sufficiently small fibers, then E must be slope semistable; this holds for all rational classes if B is a curve.
• The slope inequality holds automatically for smooth curves, canonically polarized manifolds, Calabi-Yau manifolds, and manifolds with c₁(X) < 0 and L close to the canonical class.
• Explicit examples are constructed where (X,L) has trivial automorphism group but no cscK metric exists, such as P² blown up at ≥4 points in certain polarizations.
• The blow-up of Pⁿ at a point is slope unstable with respect to polarizations where the exceptional divisor is large, showing obstruction in such cases.
• A -2 curve on a del Pezzo surface can destabilize the manifold with respect to certain polarizations, providing a concrete obstruction example.