超Kähler上同调的LLV分解
The LLV decomposition of hyper-Kaehler cohomology
AI总结：
本文确定了所有已知紧超Kähler流形的LLV分解，提出了关于分解中权重的一般猜想及对第二Betti数的强界，并推广了Göttsche公式，同时建立并推广了Nagai猜想。
摘要：
Looijenga--Lunts和Verbitsky证明了紧超Kähler流形X的上同调允许Lie代数so(4, b_2(X)-2)的自然作用，推广了紧Kähler流形的Hard Lefschetz分解。本文确定了所有已知紧超Kähler流形例子的Looijenga--Lunts--Verbitsky (LLV)分解，并提出了关于LLV分解中出现的权重的猜想，特别地，它确定了超Kähler流形的第二Betti数b_2(X)的强界。具体地，在K3^[n]和Kum_n情形，我们给出了相关LLV表示的形式特征的生成级数，推广了这些超Kähler流形系列的Euler数、Betti数和Hodge数的著名Göttsche公式。对于O'Grady的两个例外情形，我们细化了其上同调的已知结果。特别地，我们指出LLV分解为O'Grady 10型超Kähler流形的Hodge数提供了简单证明。另一方面，对于所有已知超Kähler流形例子，我们建立了关于超Kähler流形退化的单值性的所谓Nagai猜想。更重要的是，我们指出Nagai猜想是迈向这里陈述的更一般、更自然猜想的第一步。最后，我们证明此新猜想被已知类型的超Kähler流形满足。
英文摘要：
Looijenga--Lunts and Verbitsky showed that the cohomology of a compact hyper-Kähler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact Kähler manifolds. In this paper, we determine the Looijenga--Lunts--Verbitsky (LLV) decomposition for all known examples of compact hyper-Kähler manifolds, and propose a general conjecture on the weights occurring in the LLV decomposition, which in particular determines strong bounds on the second Betti number $b_2(X)$ of hyper-Kähler manifolds. Specifically, in the $K3^{[n]}$ and $\mathrm{Kum}_n$ cases, we give generating series for the formal characters of the associated LLV representations, which generalize the well-known Göttsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-Kähler manifolds. For the two exceptional cases of O'Grady we refine the known results on their cohomology. In particular, we note that the LLV decomposition leads to a simple proof for the Hodge numbers of hyper-Kähler manifolds of O'Grady 10 type. In a different direction, for all known examples of hyper-Kähler manifolds, we establish the so-called Nagai's conjecture on the monodromy of degenerations of hyper-Kähler manifolds. More consequentially, we note that Nagai's conjecture is a first step towards a more general and more natural conjecture, that we state here. Finally, we prove that this new conjecture is satisfied by the known types of hyper-Kähler manifolds.
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