Generalized Bogomolov Inequalities

Technical Summary: Generalized Bogomolov Inequalities

Problem Statement
The paper addresses the generalization of the classical Bogomolov inequality, which asserts that for a semistable torsion-free sheaf $E$ on a complex projective manifold $X$ with respect to an ample class $h$, the discriminant $\Delta(E) = 2r c_2(E) - (r-1)c_1(E)^2$ satisfies $\int_X \Delta(E) \cdot h^{d-2} \geq 0$. The authors seek to extend this inequality to broader contexts where the polarization is not necessarily a power of an ample divisor, but rather a general cohomology class $\eta_{d-1} \in H^{d-1,d-1}(X)$ (or the nef cone in the analytic setting). Specifically, they investigate under what conditions a pair of cohomology classes $(\eta_{d-1}, \eta_{d-2})$ ensures that $\int_X \Delta(E) \cdot \eta_{d-2} \geq 0$ for all $\eta_{d-1}$-semistable sheaves.

Methodology
The authors introduce two new concepts to frame this problem:

1. Hodge-Riemann Pairs: A pair $(\eta_{d-1}, \eta_{d-2})$ is defined as a Hodge-Riemann pair if $\eta_{d-2}$ satisfies the Hodge-Riemann bilinear relations (having signature $(+, -, \dots, -)$ on the kernel of the map induced by $\eta_{d-1}$) and specific positivity conditions hold. This generalizes the classical case where $\eta_i = h^i$.
2. Bogomolov Pairs: A pair $(\eta_{d-1}, \eta_{d-2})$ is a Bogomolov pair if the Bogomolov inequality holds for all $\eta_{d-1}$-semistable sheaves.

The central conjecture posits that every Hodge-Riemann pair is a Bogomolov pair. The methodology employs a mix of algebraic and analytic techniques:

• Algebraic Geometry: Utilizing Schur polynomials, derived polynomials, and the geometry of ample vector bundles. The authors use the "R-twisted" bundle construction $A\langle th \rangle$ to relate Schur classes of vector bundles to classical ample classes.
• Complex Differential Geometry: Employing the Hitchin-Kobayashi correspondence (Hitchin-Kobayashi correspondence) and Hermitian-Einstein metrics. For stable bundles, they compute the pointwise discriminant with respect to a Gauduchon metric derived from the cohomology classes.
• Boundedness Theory: Using the boundedness of families of semistable sheaves to extend results from stable to semistable sheaves via induction on rank and deformation arguments.

Key Contributions and Results

1. Definition and Properties of Hodge-Riemann Pairs:
   The paper establishes that pairs formed by Schur polynomials of ample classes (or Kähler classes) constitute Hodge-Riemann pairs. Specifically, if $A$ is an ample vector bundle of rank $e \geq d-1$ and $\lambda$ is a partition of $d-1$, then $(s_\lambda(A), s'_\lambda(A))$ is a Hodge-Riemann pair. This extends to Kähler classes $\alpha_i$ in the analytic setting.

2. Proof of the Conjecture in Specific Cases:
   The authors prove that the conjecture (Hodge-Riemann pair $\implies$ Bogomolov pair) holds in several significant cases:

   • Schur Classes of Kähler Classes: For any Kähler classes $\alpha_1, \dots, \alpha_e$ with $e \geq d-1$ and partition $\lambda$, the pair $(s_\lambda(\alpha), s'_\lambda(\alpha))$ is a Bogomolov pair. The proof for stable bundles relies on the Hitchin-Kobayashi correspondence and a pointwise computation of the discriminant using the Hermitian-Einstein metric. This is extended to semistable sheaves via resolution and induction.
   • Threefolds and Complex Tori: The conjecture is proven for all Hodge-Riemann pairs on complex threefolds and complex tori.
   • Segre and Chern Classes of Ample Bundles: Under specific slope conditions (e.g., $s_d(A) > \mu_{\max, s_{d-1}(A)}(A)$), the pairs $(s_{d-1}(A), s_{d-2}(A))$ and $(c_{d-1}(A), c_{d-2}(A))$ are shown to be Bogomolov pairs. This involves lifting the problem to the projectivization $P(A)$ and applying the classical Bogomolov inequality there.

3. Boundedness of Semistable Sheaves:
   A major application of these results is a new boundedness theorem. The authors prove that if a set of classes $K' \subset \text{Amp}^{d-1}(X)$ is path-connected and compact, and can be connected to a known "boundedness class" via a path of Hodge-Riemann and Bogomolov pairs, then the set of $\eta_{d-1}$-semistable sheaves of fixed topological type is bounded. This implies the existence of finite-type moduli spaces for these generalized stability conditions.

4. Extension to Higgs Sheaves:
   The paper generalizes the Bogomolov inequality to Higgs sheaves. It is shown that if $(\Omega_{d-1}, \Omega_{d-2})$ is a Hodge-Riemann pair of differential forms, then for any $\Omega_{d-1}$-semistable Higgs sheaf admitting a Hermitian-Yang-Mills metric, the discriminant inequality holds pointwise and in cohomology.

Significance and Claims
The paper claims to provide the "right setting" for generalizing the Bogomolov inequality by introducing the notion of Hodge-Riemann pairs. The significance lies in:

• Unification: It unifies various known inequalities (classical Bogomolov, results for Schur classes) under a single conjectural framework.
• Moduli Theory: It establishes the boundedness of semistable sheaves with respect to a wide class of non-standard polarizations (specifically those derived from Schur polynomials of Kähler classes), which is a prerequisite for constructing moduli spaces.
• Analytic Techniques: It demonstrates the utility of the Hitchin-Kobayashi correspondence and Gauduchon metrics in proving algebraic inequalities for sheaves on Kähler manifolds.

The authors remain modest regarding the full conjecture, noting that while they prove it for specific classes (Schur polynomials, threefolds, tori, and bundles satisfying specific slope inequalities), the general case for arbitrary Hodge-Riemann pairs remains open. They explicitly state that the converse of certain implications (e.g., whether a limit of Hodge-Riemann pairs is always Hodge-Riemann) does not hold, highlighting the subtlety of the problem.