关于不可约辛流形的Boucksom-Zariski分解与负数有界性
On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity
AI总结：
本文研究了不可约辛流形上Boucksom-Zariski分解的性质，并证明了在光滑情况下负数有界猜想的有效版本，从而得出关于辛流形上大线包的有效应性结论。
摘要：
Zariski分解在代数曲面理论中具有重要作用，因其广泛应用。对于不可约辛流形，Boucksom通过Beauville-Bogomolov-Fujiki二次型给出了其除子的Zariski分解特征。近年来，不同类型的奇性辛流形已被广泛研究。本文首先证明了在最大的可能框架下，即具有辛奇性的流形中，Boucksom-Zariski分解适用于有效除子。另一方面，在射影曲面的情况下，最近的研究表明，Zariski分解中伪有效整除子系数的有界性与负数有界猜想之间存在严格关系。在本文中，我们证明在射影不可约辛流形的情况下也能观察到类似现象。我们进一步在光滑情况下证明了负数有界猜想的有效版本。结合这些结果，我们获得了关于辛流形上Boucksom-Zariski分解分母的信息。从这样的界中，我们轻易地得出了一个关于射影辛流形上大线包的有效birational性结论，回答了F.Charles提出的问题。
英文摘要：
Zariski decomposition plays an important role in the theory of algebraic surfaces due to many applications. For irreducible symplectic manifolds Boucksom provided a characterization of his divisorial Zariski decomposition in terms of the Beauville-Bogomolov-Fujiki quadratic form. Different variants of singular holomorphic symplectic varieties have been extensively studied in recent years. In this note we first show that the ``Boucksom-Zariski'' decomposition holds for effective divisors in the largest possible framework of varieties with symplectic singularities. On the other hand in the case of projective surfaces, it was recently shown that there is a strict relation between the boundedness of coefficients of Zariski decompositions of pseudoeffective integral divisors and the bounded negativity conjecture. In the present note, we show that an analogous phenomenon can be observed in the case of projective irreducible symplectic varieties. We furthermore prove an effective analog of the bounded negativity conjecture in the smooth case. Combining these results we obtain information on the denominators of ``Boucksom-Zariski'' decompositions for holomorphic symplectic manifolds. From such a bound we easily deduce a result of effective birationality for big line bundles on projective holomorphic symplectic manifolds, answering to a question asked by F. Charles.
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