On slope unstable Fano varieties

Abstract: For Fano varieties, significant progress has been made recently in the study of $K$-stability, while the understanding of the weaker but more algebraic concept of $(-K)$-slope stability remains intricate. For instance, a conjecture attributed to Iskovskikh states that the tangent bundle of a Picard rank one Fano manifold is slope stable. Peternell-Wiśniewski and Hwang proved this conjecture up to dimension five in 1998, but Kanemitsu later disproved it in 2021. To address this gap in understanding, we present a method that aims to characterize the geometry associated with the maximal destabilizing sheaf of the tangent sheaf of a Fano variety. This approach utilizes modern advancements in the foliated minimal model program. In dimension two, our approach leads to a complete classification of $(-K)$-slope unstable weak del Pezzo surfaces with canonical singularities. As by-products, we provide the first conceptual proof that $\mathbb{P}¹ \times \mathbb{P}<sup>1$ and $\mathbb{F}_1$ are the only $(-K)$-slope unstable nonsingular del Pezzo surfaces, recovering a classical result of Fahlaoui in 1989. We also uncover a phenomenon that does not occur for Fano manifolds: there exists a del Pezzo surface with type A singularities admitting a weak Kähler-Einstein metric, yet whose tangent sheaf is slope unstable.

• The paper provides a complete classification of ($-K$)-slope unstable weak del Pezzo surfaces, utilizing the foliated minimal model program (fMMP) approach.
• The authors addressed a strengthened version of the Peternell–Wisniewski conjecture using a formal framework consisting of slopes and contractions.
• A pivotal observation: the $(-K)$-slope instability in singular surfaces does not mandate slope polystability.
• follow_up_questions
• Can we extend the foliated minimal model program to higher dimensions beyond surfrvaces, avoiding the limitations of the $F$-dlt singularities?
• In dimension three or higher, could more complex unstable weak del Pezzo surfaces admit a similar classification approach with advanced geometric intuition and classification methods?
• Given a Fano variety with unstable tangent bundle, is it possible to identify a precise range for the slope intervals observed in the classification?
• Why does the slope instability property mean for the overall development of algebraic and geometric theories, and does it help us cast more insight into higher dimensions?
• Find recent papers about slope unstable Fano varieties and birational classification.

Context and motivation

The paper studies the $(-K)$-slope stability of the tangent sheaf of Fano varieties over $\mathbb{C}$. For a Fano manifold $X$ of dimension $n$, slope stability requires $\mu(E) < \mu(T_X) = (-K_X)^n/n$ for every subsheaf $E \subseteq T_X$. Two conjectures frame the problem. The first, attributed to Iskovskikh, predicts that a Picard rank one Fano manifold has slope stable tangent bundle; this holds up to dimension five but was disproved by Kanemitsu via a 14-dimensional horospherical example whose maximal destabilizing sheaf has rank two and is induced by a non-trivial fibration. The second, due to Peternell–Wisniewski, predicts that instability of $T_X$ is always witnessed by the relative tangent bundle of a $K_X$-negative extremal contraction; this is known in dimensions two and three by classification-based arguments.

The authors sharpen the second conjecture into a birational version: for a Fano manifold with unstable tangent bundle, is the maximal destabilizing sheaf $F \subseteq T_X$ induced birationally by the relative tangent of an extremal contraction? Their main contribution is a framework to address this question using the foliated minimal model program (fMMP), together with a complete classification in dimension two that yields the first conceptual proof of Fahlaoui's 1989 result that $\mathbb{P}^1 \times \mathbb{P}^1$ and $\mathbb{C}$0 are the only $\mathbb{C}$1-slope unstable nonsingular del Pezzo surfaces.

Method: foliated MMP on the maximal destabilizing sheaf

The key structural input is that the maximal destabilizing sheaf $\mathbb{C}$2 in the Harder–Narasimhan filtration of $\mathbb{C}$3 is an algebraically integrable foliation with rationally connected leaves, and its canonical divisor $\mathbb{C}$4 — defined by $\mathbb{C}$5 — is not pseudo-effective. Indeed, if $\mathbb{C}$6 were pseudo-effective, then since $\mathbb{C}$7 is nef one would obtain $\mathbb{C}$8, contradicting $\mathbb{C}$9. Consequently, running a $X$0-MMP on the Mori dream space $X$1 terminates at a Mori fiber space $X$2.

A technical caveat is acknowledged explicitly: since $X$3 may fail to be $X$4-dlt, the most general setting in which an fMMP can be run, the authors instead run the classical log MMP. On surfaces they prove that any $X$5-negative extremal contraction contracts only $X$6-invariant curves — via adjunction for non-invariant divisors, $X$7 for a non-invariant curve $X$8 — so the process is foliated in effect. They also note that the analogous statement in higher dimensions would follow from the foliated cone theorem if $X$9 had $n$0-dlt singularities, which is not available in general; this is the main obstacle to extending the method beyond surfaces.

Another essential tool is the Property $n$1 modification of an algebraically integrable foliation: a birational morphism $n$2 with $n$3 $n$4-factorial and klt, such that $n$5 is induced by an equidimensional morphism over a smooth base, the pair $n$6 is log canonical, and $n$7. This modification allows a delicate analysis of foliated singularities along leaves using precise adjunction formulas for canonical foliation surface singularities.

Classification of unstable weak del Pezzo surfaces

For a nonsingular weak del Pezzo surface $n$8 with unstable $n$9, the $\mu(E) < \mu(T_X) = (-K_X)^n/n$0-MMP yields either $\mu(E) < \mu(T_X) = (-K_X)^n/n$1 or $\mu(E) < \mu(T_X) = (-K_X)^n/n$2. In the former case, $\mu(E) < \mu(T_X) = (-K_X)^n/n$3, and tracing the sequence of blowups down to the relative minimal model $\mu(E) < \mu(T_X) = (-K_X)^n/n$4 over $\mu(E) < \mu(T_X) = (-K_X)^n/n$5 (with $\mu(E) < \mu(T_X) = (-K_X)^n/n$6 forced by weak del Pezzo conditions), the authors track the invariant $\mu(E) < \mu(T_X) = (-K_X)^n/n$7, which changes by $\mu(E) < \mu(T_X) = (-K_X)^n/n$8 under blowups depending on whether the center is a foliation singularity. Since $\mu(E) < \mu(T_X) = (-K_X)^n/n$9 on $E \subseteq T_X$0 and $E \subseteq T_X$1 on $E \subseteq T_X$2, each singular fiber must arise from a 2-blowup — a pair of blowups, first at a point $E \subseteq T_X$3 and then at the intersection of the exceptional divisor with the strict transform of the fiber through $E \subseteq T_X$4 — and there are at most three such fibers because $E \subseteq T_X$5 forces $E \subseteq T_X$6.

The resulting classification states:

Surface	Description	Destabilizing slope
$E \subseteq T_X$7, $E \subseteq T_X$8	Hirzebruch surface	$E \subseteq T_X$9
$T_X$0	one 2-blowup on $T_X$1	$T_X$2
$T_X$3	two 2-blowups on distinct fibers	$T_X$4
$T_X$5	three 2-blowups on distinct fibers	$T_X$6

In all cases $T_X$7 is induced by the canonical fibration $T_X$8, confirming the birational form of the Peternell–Wisniewski conjecture in dimension two. Note that $T_X$9 need not equal the relative tangent sheaf itself when a fiber becomes non-reduced after a 2-blowup; it is the pullback foliation of $K_X$0.

Passing to minimal resolutions gives the classification of weak del Pezzo surfaces with canonical singularities: such a surface has unstable tangent sheaf only if it is induced from one of the nonsingular surfaces above, and consequently the tangent sheaf of any weak del Pezzo surface with canonical singularities is always $K_X$1-slope semistable. A further analysis of possible $K_X$2-curve configurations restricts the singularities of an unstable canonical del Pezzo surface to the types $K_X$3 ($K_X$4), $K_X$5, $K_X$6, and $K_X$7.

Ruling out the Picard rank one outcome

The technically central part of the paper excludes $K_X$8, i.e., the possibility that the $K_X$9-MMP ends at a del Pezzo surface $F \subseteq T_X$0 of Picard rank one with canonical singularities. Assuming this case, the Property $F \subseteq T_X$1 modification and adjunction arguments show that $F \subseteq T_X$2 has exactly one dicritical singularity, located at a nonsingular point, while all other points are terminal foliation singularities. The dicritical point must be nonsingular because any contracting curve through a dicritical point would move in a covering family (by the contraction theorem for terminal foliations), contradicting birationality of the MMP.

Using Shokurov's complexity criterion, the authors then prove that $F \subseteq T_X$3 is a toric foliation with at most two singular points: the log Calabi–Yau pair constructed from the boundary components has complexity strictly less than one, forcing a toric structure. A case-by-case analysis of the sixteen Gorenstein toric del Pezzo surfaces reduces to three candidates (nos. 6d, 8c, 9). For each, the arithmetic of the invariant $F \subseteq T_X$4 combined with the constraint that no blowup center may lie over the dicritical point or on any $F \subseteq T_X$5-curve bounds the number of admissible blowups strictly below what is required ($F \subseteq T_X$6 for no. 8c, $F \subseteq T_X$7 or $F \subseteq T_X$8 for no. 9), yielding contradictions in all cases. Hence $F \subseteq T_X$9 always, completing the classification.

A new phenomenon for singular Fano varieties

The singularity classification produces a result with no analogue among smooth Fano manifolds. By the Kobayashi–Hitchin correspondence, $\mathbb{P}^1 \times \mathbb{P}^1$0-polystability of $\mathbb{P}^1 \times \mathbb{P}^1$1 implies $\mathbb{P}^1 \times \mathbb{P}^1$2-slope polystability of $\mathbb{P}^1 \times \mathbb{P}^1$3 for Fano manifolds. Odaka–Spotti–Sun showed that del Pezzo surfaces with canonical singularities of type $\mathbb{P}^1 \times \mathbb{P}^1$4 or $\mathbb{P}^1 \times \mathbb{P}^1$5 admit weak Kähler–Einstein metrics. However, the classification here shows that such surfaces can have slope unstable tangent sheaf: contracting all $\mathbb{P}^1 \times \mathbb{P}^1$6-curves of the surface $\mathbb{P}^1 \times \mathbb{P}^1$7 from the two-blowup family yields a del Pezzo surface with $\mathbb{P}^1 \times \mathbb{P}^1$8 admitting a weak KE metric yet with unstable $\mathbb{P}^1 \times \mathbb{P}^1$9. Thus the implication from $\mathbb{C}$00-polystability to slope polystability fails for singular Fano varieties equipped with the weak KE notion — a concrete disanalogy between the smooth and singular theories.

Limitations and open questions

The paper is candid about the scope of its methods. The restriction to surfaces is essential: the argument that extremal contractions are foliated relies on surface adjunction, and the general higher-dimensional analogue would require $\mathbb{C}$01-dlt singularities, which cannot currently be guaranteed. Extending the classification to dimension three is described as already intricate due to singularities and geometric complexity. The authors also formulate a weaker, seemingly more approachable problem: whether the maximal destabilizing sheaf of an unstable Fano manifold always has rank at least two. This holds when $\mathbb{C}$02 and is compatible with all known examples, but remains open in general. Finally, the birational characterization of the maximal destabilizing sheaf (the strengthened Peternell–Wisniewski problem) is verified only in dimension two; whether the fMMP framework suffices in higher dimensions is left open.

Conclusion

This paper establishes a complete classification of $\mathbb{C}$03-slope unstable weak del Pezzo surfaces with canonical singularities, proving that instability is always induced by the canonical fibration to $\mathbb{C}$04 and providing the first conceptual proof of Fahlaoui's classification of unstable nonsingular del Pezzo surfaces. Methodologically, it demonstrates that the foliated MMP, combined with Property $\mathbb{C}$05 modifications and fine control of foliated surface singularities, offers a viable strategy for the birational study of slope instability. The discovery that weak Kähler–Einstein metrics do not force slope polystability on singular del Pezzo surfaces marks a substantive divergence between the stability theories of smooth and singular Fano varieties.