Le Potier Function in Stability Theory

• The Le Potier function is a key invariant that defines sharp upper bounds on Chern class ratios for slope-semistable sheaves on projective surfaces.
• It generalizes classical Bogomolov–Gieseker inequalities to twisted settings and higher Picard rank, aiding in the understanding of Bridgeland stability conditions.
• Recent research extends its definition and highlights its central role in controlling wall-crossing phenomena and the structure of geometric stability chambers.

The Le Potier function is a fundamental tool in the study of slope-semistable sheaves on complex projective surfaces, encoding sharp upper bounds on Chern class ratios and playing an essential role in the geometric description of Bridgeland stability conditions. It characterizes the relationship between the slope and normalized second Chern class for $H$-semistable sheaves, and generalizes classical Bogomolov–Gieseker-type inequalities to twisted settings and higher Picard rank. Recent work extends its definition, validates its sharp behavior on free abelian quotients, and establishes its centrality in the structure of geometric stability chambers (Dell, 2023).

1. Classical Definition and Properties

Let $X$ be a smooth projective surface over $\mathbb{C}$, $$H \in \mathrm{Amp}(X) \subset \mathrm{NS}_\mathbb{R}(X)$$ an ample divisor, and $B \in \mathrm{NS}_\mathbb{R}(X)$ a real divisor class. For any nonzero torsion-free sheaf $F$ on $X$, define the $H$-slope

$\mu_H(F) = \frac{H \cdot \ch_1(F)}{H^2\,\ch_0(F)}$

and the twisted Chern character

$\ch^B(F) = \ch(F)\,e^{-B},$

so that

$\mu_H^B(F) = \mu_H(F) - \frac{H\cdot B}{H^2}.$

The (twisted) Le Potier function

$$\Phi_{X,H,B} \colon \mathbb{R} \to \mathbb{R}\cup\{-\infty\}$$

is defined by

$$\Phi_{X,H,B}(x) = \limsup_{\substack{\mu_H(F)\to x \ F\, H\text{-semistable}}} \frac{H^{0}\bigl(\ch_2(F)-B\cdot\ch_1(F)\bigr)}{H^2\,\ch_0(F)}.$$

Introducing the normalized expression

$$\nu_{H,B}(F) = \frac{\ch_2(F) - B\cdot\ch_1(F)}{H^2 \ch_0(F)},$$

one can equivalently write

$$\Phi_{X,H,B}(x) = \limsup_{\mu_H(F)\to x} \nu_{H,B}(F).$$

In the untwisted case ($B = 0$), this recovers the classical Le Potier function

$\Phi_{X,H}(x) = \Phi_{X,H,0}(x).$

The Bogomolov–Gieseker inequality provides a universal upper bound: $$\Phi_{X,H,B}(x)\leq \frac{1}{2}\left(x - \frac{H\cdot B}{H^2}\right)^2 - \frac{1}{2}\frac{B^2}{H^2}.$$

2. Extension to Arbitrary Picard Rank

Early studies by Fu–Li–Zhao focused primarily on the Picard rank one case. The generalization by Dell extends the full definition and the property of sharp upper bounds to arbitrary Picard rank, yielding the following: $\Phi_{X,H,B}$ is the minimal upper-semicontinuous function such that for every $H$–semistable sheaf $F$,

$\nu_{H,B}(F) \leq \Phi_{X,H,B}\big(\mu_H(F)\big).$

Further, for varieties that are quotients by finite groups, if $\pi:X\to Y = X/G$ is a free quotient and $H_Y, B_Y$ are pulled back from $Y$ to $X$, then

$\Phi_{Y,H_Y,B_Y} = \Phi_{X,\pi^* H_Y, \pi^* B_Y},$

ensuring that the function is compatible and can be computed on any finite abelian cover.

When $X \to \operatorname{Alb}(X)$ is the Albanese morphism and $H,B$ are induced from the abelian variety, the explicit formula is

$$\Phi_{X,H,B}(x) = \frac{1}{2}\left( x - \frac{H\cdot B}{H^2} \right)^2 - \frac{1}{2} \frac{B^2}{H^2}.$$

3. Fu–Li–Zhao Conjecture and Counterexamples

Fu–Li–Zhao conjectured (Conjecture 4.4 in their work, restated as Conjecture 3.4 in (Dell, 2023)) that for a polarized surface $(S,H)$ with irregularity $q(S) = 0$, the Le Potier function $\Phi_{S,H}$ must be discontinuous at $0$. Dell provides direct counterexamples using Beauville-type surfaces

$S = (C_1 \times C_2)/G,$

where each curve $C_i$ has genus $\geq2$ and $G$ acts freely, resulting in $q(S) = p_g(S) = 0$. For a suitable ample divisor $H_S$, the function attains

$$\Phi_{S,H_S}(x) = \frac{1}{2} x^2 \quad \text{for all } x \in \mathbb{R},$$

which is continuous at $0$. Thus, Beauville-type and analogous bielliptic surfaces provide direct counterexamples, disproving the conjecture in general.

4. Role in Bridgeland Stability and Geometric Chamber Parametrization

For any surface $X$, geometric Bridgeland stability conditions are those where all skyscraper sheaves $\mathcal{O}_x$ are stable of the same phase. When normalized via the $\widetilde{\mathrm{GL}}^+(2)$-action such that $Z(\mathcal{O}_x) = -1$ and $\phi(\mathcal{O}_x) = 1$, there is a homeomorphism

$$\operatorname{Gstab}(X) \cong \left\{ (H,B,\alpha,\beta) \mid H\in\mathrm{Amp}(X),\; B\in\mathrm{NS}_\mathbb{R}(X),\; (\alpha,\beta)\in\mathbb{R}^2,\; \alpha > \Phi_{X,H,B}(\beta) \right\} \times \widetilde{\mathrm{GL}}_2^+(\mathbb{R}).$$

The central charge and tilted heart are given by

$$Z_{H,B,\alpha,\beta}(E) = (\alpha - i\beta) H^2 \ch_0(E) + (B + iH)\cdot \ch_1(E) - \ch_2(E),$$

and the heart is produced by tilting $\operatorname{Coh}(X)$ at the slope-$\beta$ torsion pair.

The sharp inequality $\alpha > \Phi_{X,H,B}(\beta)$ is necessary and sufficient for all skyscraper sheaves to be stable and for the support property to hold. The boundary of the geometric chamber aligns with the graph $\alpha = \Phi_{X,H,B}(\beta)$, and further walls resulting from contractions of rational curves manifest when $H$ is no longer ample. If $\Phi_{X,H,B}$ were discontinuous, such as in rational or K3 cases, "Le Potier walls" could arise, but Dell's counterexamples show this is not invariably the case—only nef boundaries can induce geometric walls.

5. Applications and Centrality in Stability Theory

The Le Potier function now serves as the key invariant for describing the precise boundary and structure of the geometric chamber in the stability manifold for any projective surface—extending beyond rank one to arbitrary Picard rank. It plays a determining role in wall-crossing phenomena, structure of stability conditions, and the control of semistability in moduli theory. Its continuity or lack thereof directly influences the existence and character of "Le Potier walls," with the explicit counterexamples fundamentally clarifying the range of its possible behaviors. The direct correspondence between geometric chambers and configurations where $\alpha > \Phi_{X,H,B}(\beta)$ places the function at the center of ongoing and future investigations into stability conditions and wall structures in the derived category context (Dell, 2023).