{
  "id": "1906.03432",
  "version": "v1",
  "published": "2019-06-08T10:30:14.000Z",
  "updated": "2019-06-08T10:30:14.000Z",
  "title": "The LLV decomposition of hyper-Kaehler cohomology",
  "authors": [
    "Mark Green",
    "Yoon-Joo Kim",
    "Radu Laza",
    "Colleen Robles"
  ],
  "comment": "46 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Looijenga-Lunts and Verbitsky showed that the cohomology of a compact hyper-Kaehler manifold $X$ admits a natural action by the Lie algebra $\\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact Kaehler manifolds. In this paper, we determine the Looijenga-Lunts-Verbitsky (LLV) decomposition for all known examples of compact hyper-Kaehler manifolds. As an application, we compute the Hodge numbers of the exceptional OG10 example starting only from the knowledge of the Euler number $e(X)$, and the vanishing of the odd cohomology of $X$. In a different direction, we establish the so-called Nagai's conjecture for all known examples of hyper-Kaehler manifolds. More importantly, we prove that, in general, Nagai's conjecture is equivalent to a representation theoretic condition on the LLV decomposition of the cohomology $H^*(X)$. We then notice that all known examples of hyper-Kaehler manifolds satisfy a stronger, more natural condition on the LLV decomposition of $H^*(X)$: the Verbitsky component is the dominant representation in the LLV decomposition of $H^*(X)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-08T10:30:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "llv decomposition",
      "hyper-kaehler cohomology",
      "compact hyper-kaehler manifold",
      "nagais conjecture",
      "hyper-kaehler manifolds satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}