On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity

1. Problem Statement

The paper addresses two interconnected problems in the theory of algebraic geometry, specifically concerning Irreducible Symplectic (IS) varieties (also known as Hyperkähler manifolds in the smooth case) and their singular generalizations:

1. Generalization of Zariski Decomposition: The classical Zariski decomposition, a fundamental tool for linear series on algebraic surfaces, was generalized by Boucksom to smooth irreducible symplectic manifolds using the Beauville-Bogomolov-Fujiki (BBF) quadratic form ($q_X$). The authors investigate whether this decomposition holds for singular irreducible symplectic varieties (specifically those with symplectic singularities) and whether it can be defined for Weil divisors (which may not be $\mathbb{Q}$-Cartier) in the most general framework.
2. Bounded Negativity and Effective Birationality: On surfaces, the boundedness of coefficients in Zariski decompositions is linked to the Bounded Negativity Conjecture (boundedness of self-intersections of curves). The authors seek to establish an analogous "effective" bound for the denominators of Zariski decomposition coefficients in higher-dimensional IS varieties. This bound is crucial for proving effective birationality results (determining an explicit integer $m$ such that $|mL|$ defines a birational map) for big line bundles on these varieties, answering a question posed by F. Charles.

2. Methodology

The authors employ a combination of Hodge theory, intersection theory on singular spaces, and lattice theory.

• Extension of the BBF Form to Weil Divisors:
  • In the singular case, varieties are not necessarily $\mathbb{Q}$-factorial, meaning Weil divisors may not be $\mathbb{Q}$-Cartier. Standard intersection theory fails for non-Cartier divisors.
  • The authors define a quadratic form $q_{X,\sigma}$ on Weil divisors by restricting the variety to a smooth open subset $U$ (removing the "dissident locus" $\Sigma_0$ where singularities are worse than ADE type).
  • They utilize a resolution of singularities $\pi: Y \to X$ and define the form via the Poincaré dual of the closure of the restriction of the divisor to $Y$. They prove this definition is independent of the resolution and compatible with linear equivalence (Proposition 2.11).
• Intersection Product Property:
  • Following Bauer's approach for surfaces, they prove that the extended quadratic form $q_{X,\sigma}$ acts as an intersection product on the group of Weil divisors. Specifically, they show that for distinct prime divisors $D, D'$, $q_{X,\sigma}(D, D') \geq 0$.
  • This property allows the application of general theorems (from [7]) to guarantee the existence of a unique Boucksom-Zariski decomposition for effective Weil $\mathbb{Q}$-divisors.
• Bounding Denominators via Lattice Theory:
  • To bound the denominators of the decomposition coefficients, the authors relate the problem to the bounded negativity of prime exceptional divisors.
  • They utilize results by Druel and Markman regarding the geometry of prime exceptional divisors and their contraction.
  • They analyze the discriminant group $A_X = H^2(X, \mathbb{Z})^\vee / H^2(X, \mathbb{Z})$ of the cohomology lattice. By studying the divisibility of primitive classes and the action of the BBF form, they derive a lower bound for the self-intersection $q(E)$ of any prime exceptional divisor $E$.

3. Key Contributions and Results

A. Existence of Boucksom-Zariski Decomposition in the Singular Case

Theorem 1.1: The authors prove that for any compact Kähler variety $X$ with symplectic singularities and a reflexive symplectic form $\sigma$, every effective Weil $\mathbb{Q}$-divisor $D$ admits a unique decomposition:
$$D = P(D) + N(D)$$
where:

• $P(D)$ is $q_{X,\sigma}$-nef (non-negative intersection with all effective Cartier divisors).
• $N(D)$ is $q_{X,\sigma}$-exceptional (the Gram matrix of its components is negative definite).
• $q_{X,\sigma}(P(D), N(D)) = 0$.
• The natural map on global sections $H^0(X, \mathcal{O}_X(kP(D))) \to H^0(X, \mathcal{O}_X(kD))$ is surjective for sufficiently divisible $k$.

This result extends Boucksom's work from smooth manifolds to the largest possible framework (singular varieties with symplectic singularities) and handles non-$\mathbb{Q}$-Cartier divisors.

B. Effective Bound on Denominators

Theorem 1.2 & Corollary 4.10: The authors establish an explicit upper bound for the denominators of the coefficients in the Boucksom-Zariski decomposition of pseudoeffective Cartier divisors on a smooth projective IS variety $X$ of Picard number $\rho(X)$:
$$\text{Denominator Bound} \leq (4 \cdot \text{Card}(A_X))^{\rho(X)-1}!$$
where $\text{Card}(A_X)$ is the cardinality of the discriminant group of the intersection lattice.

• Significance: This proves an effective version of the Bounded Negativity Conjecture for IS varieties, showing that the self-intersection of any irreducible divisor is bounded from below by $4 \cdot \text{Card}(A_X)$.

C. Effective Birationality

Corollary 1.3: Using the denominator bound, the authors derive an explicit integer $m$ such that the linear system $|mL|$ is birational onto its image for any big line bundle $L$ on a smooth projective IS variety of dimension $2n$:
$$m \geq \frac{1}{2}(2n+2)(2n+3)(4 \cdot \text{Card}(A_X))^{\rho(X)-1}!$$
This provides a positive, effective answer to a question by F. Charles regarding the birationality of big line bundles on these varieties.

4. Significance and Impact

1. Unification of Smooth and Singular Cases: The paper successfully bridges the gap between smooth irreducible symplectic manifolds and their singular counterparts (moduli spaces of sheaves, minimal models). It demonstrates that the powerful tool of Zariski decomposition, driven by the BBF form, remains valid even when the variety is not $\mathbb{Q}$-factorial.
2. Explicit Bounds in Higher Dimensions: While the Bounded Negativity Conjecture is open for general surfaces and fails in higher dimensions for general varieties, this paper proves a strong, explicit boundedness result for the specific and important class of Irreducible Symplectic varieties.
3. Answering F. Charles's Question: The result provides the first explicit bound for the birationality of big line bundles on IS varieties. While C. Birkar later proved the existence of such a bound for general Calabi-Yau varieties with klt singularities, his result was non-effective (the integer $m$ was not given). This paper fills that gap for the IS case.
4. Uniformity: The bounds depend on topological invariants (Picard number, discriminant group size) and are uniform across deformation families, making them applicable to the entire moduli space of such varieties.

In summary, the paper establishes a robust theoretical framework for divisor theory on singular symplectic varieties and translates this into concrete, computable bounds for geometric properties like birationality, significantly advancing the understanding of linear series on these high-dimensional spaces.