{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\varpi \\colon \\mathcal{X} \\to B$ be a complex analytic [[family|VKQSM3BCE4]] of compact [[Kähler manifolds|X60JG9AEQJ]] over a simply connected base $B$.\nFix $k$, let $H = H^k(X_0, \\C)$, and let $\\mathcal{P}^p \\colon B \\to \\operatorname{Gr}(f^p, H)$, $t \\mapsto F^p H^k(X_t)$, be the components of its [[period map|YGNXQVTWP3]].\nThen:\n\n1. Each $\\mathcal{P}^p$ is holomorphic.\n2. For each $t \\in B$ and each tangent vector $v \\in T_t B$, the differential $d\\mathcal{P}^p_t(v)$, viewed in $T_{F^p}\\operatorname{Gr}(f^p, H) = \\Hom(F^p, H/F^p)$, has image in $F^{p-1}/F^p$, where $F^j = F^j H^k(X_t)$.\n\n## Remarks\n\nEquivalently, if $s(t)$ is a holomorphic family of classes with $s(t) \\in F^p H^k(X_t)$, then the derivative of $s$ in any direction, taken in the fixed space $H$, lies in $F^{p-1} H^k(X_t)$.\nThe idea of the proof: represent $s(t)$ by closed forms of type $(r, k-r)$ with $r \\ge p$ on $X_t$.\nTo first order, a change of complex structure replaces $dz_j$ by $dz_j + \\sum_k \\varphi_{jk} d\\bar z_k$, so it lowers the holomorphic degree of a form by at most one.\nHence the derivative has components only in types $(r, k-r)$ with $r \\ge p - 1$.\n\nExample: for a family of curves and $k = 1$, $F^0 = H$, so condition 2 holds automatically.\nFor Calabi–Yau threefolds and $k = 3$, condition 2 says that the derivative of the line $F^3 = H^{3,0}$ lies in $F^2$, which is a genuine restriction.\n\n## Explanations\n\n### L1 human\n\nThe Hodge pieces of the cohomology move holomorphically as the complex structure varies.\nMoreover, each step of the Hodge filtration can only move \"one step down\" to first order.\nThis restricts which curves in the period domain can come from families of manifolds.\n\n### L2 human\n\nTransversality is the key constraint of variations of Hodge structure; it is built into their definition.\nPeriod domains are usually not Hermitian symmetric, and the transversality condition defines a horizontal distribution on them.\nCurvature estimates along horizontal directions give the Griffiths–Schmid results on period maps, for instance that horizontal holomorphic maps from a disc are distance decreasing for a suitable metric.\nThe derivative of $\\mathcal{P}^p$ is computed by the [[Kodaira–Spencer map|EWZ1G10MAY]] followed by cup product and contraction.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "YGNXQVTWP3"
    ],
    "kind": "statement",
    "msc": [
      "32G20",
      "14D07"
    ],
    "name": "Griffiths transversality",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Period_mapping"
      }
    ],
    "tag": "589GJMH5EF",
    "uses": [
      "YGNXQVTWP3",
      "VKQSM3BCE4",
      "X60JG9AEQJ",
      "CYPA882RB9",
      "EWZ1G10MAY"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/589GJMH5EF/",
  "links": [
    {
      "field": "about",
      "target": "YGNXQVTWP3"
    },
    {
      "field": "uses",
      "target": "YGNXQVTWP3"
    },
    {
      "field": "uses",
      "target": "VKQSM3BCE4"
    },
    {
      "field": "uses",
      "target": "X60JG9AEQJ"
    },
    {
      "field": "uses",
      "target": "CYPA882RB9"
    },
    {
      "field": "uses",
      "target": "EWZ1G10MAY"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/589GJMH5EF.md",
  "msc": [
    "32G20",
    "14D07"
  ],
  "name": "Griffiths transversality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/589/589GJMH5EF.md",
  "status": null,
  "tag": "589GJMH5EF",
  "uri": "m2://589GJMH5EF",
  "url": "https://tamnd.github.io/math2/wiki/griffiths-transversality/",
  "writing": {
    "tag": "589GJMH5EF",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}