Stabilization of Intersection Betti Numbers

• Stabilization of intersection Betti numbers is the phenomenon where fixed degree intersection cohomology groups become constant once parameters exceed a specific threshold.
• Universal product formulas and recursive algorithms rigorously control the stabilization process in moduli spaces of sheaves and complete intersections.
• This stabilization uncovers deep structural regularities in geometric representation theory, moduli theory, and syzygies, offering sharper bounds and generating function insights.

Stabilization of intersection Betti numbers refers to the phenomenon wherein, for a natural family of parameterized algebraic varieties (such as moduli spaces or schemes associated to increasingly positive data), the intersection Betti numbers in fixed degree eventually become independent of the parameter, once the parameter exceeds an explicit threshold. This stabilization encodes deep structural regularity in geometric representation theory, moduli theory, and the paper of syzygies, often controlled by universal product formulas or recursive decompositions.

1. Foundational Concepts and Definitions

Intersection Betti numbers, denoted $b_i^{\text{IH}}(X)$ for a complex algebraic variety $X$, are the graded dimensions of the rational intersection cohomology groups $IH^i(X;\mathbb{Q})$. For smooth varieties, these coincide with the ordinary Betti numbers; for singular varieties they capture the "correct" topological invariants that behave well under natural degenerations.

Stabilization occurs for families $\{X_t\}$ indexed by an unbounded parameter $t$ (for example, Chern classes, divisor degrees, or generator degrees in a complete intersection). The central assertion is that, for every fixed $k$, $b^{{\rm IH}}_k(X_t)$ becomes independent of $t$ for $t$ large enough, often matching canonical, explicitly computable stable sequences.

2. Stabilization for Moduli Spaces of One-Dimensional Sheaves

Let $S$ be a smooth projective surface and let $M_\beta(\chi)$ denote the coarse moduli space of $H$-semistable sheaves of pure dimension one, first Chern class $c_1(\mathcal{F}) = \beta$ in $\text{Num}(S)$, and Euler characteristic $\chi(\mathcal{F}) = \chi$. Generally, $M_\beta(\chi)$ can be singular, and intersection cohomology provides the natural home for stabilized Betti numbers.

The main result is the existence of a universal "stable" sequence $\{b_k^\infty\}$ of intersection Betti numbers such that for every fixed $k$, $b_k^{\text{IH}}(M_\beta(\chi)) = b_k^\infty$ once $\beta$ is sufficiently positive and certain genericity assumptions are met (Si et al., 23 Nov 2025). The product formula determining $\{b_k^\infty\}$ is derived from Göttsche’s generating function for Hilbert schemes of points:

$$\sum_{k\ge0} b_k^\infty\,q^k = \prod_{m=1}^\infty \frac{(1+q^{2m-1})^{b_1(S)}(1+q^{2m+1})^{b_1(S)}}{(1-q^{2m})^{b_2(S)+1}(1-q^{2m+2})},$$

which stabilizes the intersection Betti numbers in all degrees $k$ up to a range determined by the very-ampleness of $\beta$ and the codimension of the non-integral locus in $|\beta|$. For Enriques surfaces and bielliptic surfaces, analogous stabilization results hold for refined structures such as perverse Hodge numbers (Si et al., 23 Nov 2025).

3. Betti Number Stabilization for Moduli of Higher Rank Sheaves

Consider the moduli space $M_{\mathbb{P}^2, H}(r, aH, c_2)$ of slope-stable sheaves of rank $r$ and first Chern class $aH$ over $\mathbb{P}^2$, with $r\geq 2$, $a$ coprime to $r$, and $c_2$ the second Chern class. For fixed $r,a$, the $2N$-th intersection Betti number stabilizes for all $c_2\ge N + C(r,a)$, where

$$C(r, a) = \left\lfloor \tfrac{r-1}{2r} a^2 + \tfrac{1}{2} (r^2 + 1) \right\rfloor.$$

Thus, $b_{2N}(c_2) = b_{2N}(c_2 + 1) = \cdots$ for all $c_2\ge N+C(r,a)$ (1908.09977). In this context, ordinary and intersection Betti numbers agree because the associated moduli spaces are smooth, a consequence of the coprimality hypothesis eliminating strictly semistable sheaves.

The limit of the stabilized Poincaré polynomials is

$$P_\infty(t) = \prod_{i=1}^\infty (1 - t^{-2i})^{-3},$$

corresponding to an infinite product determined by the class of the affine line—reflecting the structure of a Heisenberg algebra action on the cohomology.

4. Boij–Söderberg Decomposition and Complete Intersections

For graded modules over polynomial rings given by complete intersections, the stabilization phenomenon appears in the decomposition of their Betti diagrams. Specifically, let $I=(f_1,\dots,f_{c+1})$ be a homogeneous complete intersection with $\deg f_i = a_i$, and set $a = a_1 + \cdots + a_c$. The Betti diagram $B(a_1,\dots,a_{c+1})$ has a Boij–Söderberg decomposition into pure diagrams with rational coefficients. Gibbons–Huben–Stone establish that, once the last degree $$a_{c+1} > \max\{ a, T_1/z_1,\dots, T_\varepsilon/z_\varepsilon \}$$, this decomposition stabilizes: the number of pure summands, their ordered degree sequences, and the coefficients become fixed and polynomial (in $a_{c+1}$) (Gibbons et al., 2017). This “locking in” is realized via a three-phase recursive decomposition algorithm (Algorithm 2.2), with explicit closed-form expressions for the coefficients in the first and last positions and polynomial formulas conjecturally for the intermediate coefficients.

In codimension four complete intersections, the explicit stabilization threshold can be computed and, for all larger $a_4$, the decomposition has the same 12 pure diagrams, with the first and last five coefficients affine-linear in $a_4$ and the two middle coefficients quadratic in $a_4$. In all codimensions, the shape and degree-sequence chain of the Boij–Söderberg decomposition become independent of the last generator once it is sufficiently large.

5. Universal Product Formulas and Generating Functions

A unifying structural aspect is the appearance of universal product formulas and generating functions—both for intersection Betti numbers of Hilbert schemes (Göttsche’s formula) and for Betti numbers of moduli spaces parameterized by degree or Chern class. These generating functions package the cohomological data across the family's parameter:

• For Hilbert schemes of points on a surface $S$:

$$G_S(q, t) = \prod_{m=1}^\infty \frac{1}{(1 - q^m t^0)^{b_0(S)}(1 - q^m t^2)^{b_1(S)}(1 - q^m t^4)^{b_2(S)}}$$

• For moduli spaces as $c_2 \to \infty$, the criterion for stabilization is that certain weighted generating functions converge appropriately after a shift by top degree; the stable Betti numbers are then read off as coefficients in the resulting product (1908.09977).
• In the complete intersection case, as the last generator degree grows, the recursive algorithm ensures that the Betti diagram coefficients become polynomials, and their stabilized values can often be interpreted as coefficients of a generating function determined by the degrees of the generators (Gibbons et al., 2017).

6. Criteria and Thresholds for Stabilization

The precise range in which stabilization occurs is governed by geometric and combinatorial criteria:

• For one-dimensional sheaf moduli, stabilization of $b_k^{\text{IH}}$ holds provided $\beta$ is sufficiently positive to be $2k$–very ample and the non-integral locus has codimension $>k$, with explicit combinatorial bounds given in terms of the dimensions and ample divisors (Si et al., 23 Nov 2025).
• For sheaf moduli on $\mathbb{P}^2$, the threshold for $c_2$ is $N+C(r,a)$, with $C(r,a)$ explicitly determined by $r$ and $a$ (1908.09977).
• For complete intersections, the threshold depends on the earlier degrees and explicit formulas involving the combinatorial “remainders” in the decomposition process (Gibbons et al., 2017).

A plausible implication is that these stabilization thresholds are sharp in the generic case, but in some low-rank or low-degree settings, an even earlier stabilization is observed (e.g., for $r=2$ in the moduli of sheaves on $\mathbb{P}^2$).

7. Extensions: Perverse Hodge Numbers and Further Directions

Refinements of stabilization address not just intersection Betti numbers but finer invariants, such as perverse Hodge numbers, particularly in contexts where a perverse Leray filtration arises naturally (e.g., from a Hilbert–Chow or Hitchin-type fibration). For generic Enriques surfaces, perverse Hodge numbers of the moduli of one-dimensional sheaves stabilize to a universal sequence given by a similar product formula with multivariate generating function (Si et al., 23 Nov 2025).

Potential directions include:

• Generalizing to higher-dimensional varieties, moduli of objects in derived categories, or varieties with additional structure.
• Analyzing wall-crossing effects and their impact on stabilization in spaces with nontrivial strictly semistable loci.
• Sharpening or lowering stabilization thresholds via refined intersection-theoretic or combinatorial arguments.

The overarching structure is that stabilization reflects, and is often controlled by, universal algebraic or generating function identities, often tied to Heisenberg algebra actions or symplectic geometry of the moduli frameworks.