##
**Invariance for multiples of the twisted canonical bundle.**
*(English)*
Zbl 1122.32013

This paper deals with fiber-to-family extensions of holomorphic sections of multiples of twisted canonical bundles over smooth projective families defined over the disc. Its main result is the following Theorem 1.1.

Theorem 1.1. Let \(X\to\Delta\) be a smooth projective family over a disc \(0\in\Delta\), \(L\to X\) a line bundle endowed with a possibly singular Hermitian metric \(\tilde h\), and suppose that (i) the curvature form \(\Theta_{\tilde h}(L)\geq0\) is nonnegative as a current (i.e., \((L,\tilde h)\) is pseudo-effective), (ii) the restriction \(\tilde h_{X_0}\) of \(\tilde h\) to the central fiber \(X_0\) of \(X\) is well-defined (i.e., the local weights of \(\tilde h\) are locally summable on \(X_0\)), and (iii) the multiplier ideal sheaf \(I(X_0,\tilde h_{X_0})\) is trivial (i.e., any weight function \(e^{-\tilde\varphi}\) of \(\tilde h\) is square summable on the central fiber \(X_0\)). Then the restriction map \(H^0(X,m(K_X+L))\to H^0(X_0,m(K_{X_0}+L))\) of holomorphic sections is surjective, where \(m\geq1\) is any integer, and \(K\) denotes the canonical bundles.

The proof of Theorem 1.1 follows the line of Y.-T. Siu’s work [Complex geometry. Springer, 223–277 (2002; Zbl 1007.32010)] as simplified by M. Păun [J. Differ. Geom. 76, No. 3, 485–493 (2007; Zbl 1122.32014)], and relies on Siu’s formulation of the Ohsawa-Takegoshi extension theorem [T. Ohsawa and K. Takegoshi, Math. Z. 195, 197–204 (1987; Zbl 0625.32011)] for square summable holomorphic sections that states in this context (Theorem 2.1 here) that there is a universal bound of the \(L^2\)-extension from the central fiber to the family for holomorphic sections of a somewhat positive line bundle \((L,h)\) twisted by the canonical bundle \(K\). The goal is to manufacture so many sections of various line bundles and their attendant singular Hermitian metrics with somewhat positive curvature in an inductive process playing out the Ohsawa-Takegoshi extension with its estimate that a final metric \(g\) on \((m-1)K_X+mL\) can be produced that is not too singular so that it still makes square summable the section \(s\) of \(m(K_{X_0}+L)\) that has to be extended, and yet its curvature is positive enough to make the Ohsawa–Takegoshi extension applicable to find an extension \(\tilde s\) of \(s\). In doing so one borrows a camel in the form of an ample line bundle \(A\to X\), puts its positivity to work to engender sections, then plays down its contributions, and returns the camel in the limit.

The paper discusses further extension results as well. Its presentation could have been better, but it reads well enough.

Theorem 1.1. Let \(X\to\Delta\) be a smooth projective family over a disc \(0\in\Delta\), \(L\to X\) a line bundle endowed with a possibly singular Hermitian metric \(\tilde h\), and suppose that (i) the curvature form \(\Theta_{\tilde h}(L)\geq0\) is nonnegative as a current (i.e., \((L,\tilde h)\) is pseudo-effective), (ii) the restriction \(\tilde h_{X_0}\) of \(\tilde h\) to the central fiber \(X_0\) of \(X\) is well-defined (i.e., the local weights of \(\tilde h\) are locally summable on \(X_0\)), and (iii) the multiplier ideal sheaf \(I(X_0,\tilde h_{X_0})\) is trivial (i.e., any weight function \(e^{-\tilde\varphi}\) of \(\tilde h\) is square summable on the central fiber \(X_0\)). Then the restriction map \(H^0(X,m(K_X+L))\to H^0(X_0,m(K_{X_0}+L))\) of holomorphic sections is surjective, where \(m\geq1\) is any integer, and \(K\) denotes the canonical bundles.

The proof of Theorem 1.1 follows the line of Y.-T. Siu’s work [Complex geometry. Springer, 223–277 (2002; Zbl 1007.32010)] as simplified by M. Păun [J. Differ. Geom. 76, No. 3, 485–493 (2007; Zbl 1122.32014)], and relies on Siu’s formulation of the Ohsawa-Takegoshi extension theorem [T. Ohsawa and K. Takegoshi, Math. Z. 195, 197–204 (1987; Zbl 0625.32011)] for square summable holomorphic sections that states in this context (Theorem 2.1 here) that there is a universal bound of the \(L^2\)-extension from the central fiber to the family for holomorphic sections of a somewhat positive line bundle \((L,h)\) twisted by the canonical bundle \(K\). The goal is to manufacture so many sections of various line bundles and their attendant singular Hermitian metrics with somewhat positive curvature in an inductive process playing out the Ohsawa-Takegoshi extension with its estimate that a final metric \(g\) on \((m-1)K_X+mL\) can be produced that is not too singular so that it still makes square summable the section \(s\) of \(m(K_{X_0}+L)\) that has to be extended, and yet its curvature is positive enough to make the Ohsawa–Takegoshi extension applicable to find an extension \(\tilde s\) of \(s\). In doing so one borrows a camel in the form of an ample line bundle \(A\to X\), puts its positivity to work to engender sections, then plays down its contributions, and returns the camel in the limit.

The paper discusses further extension results as well. Its presentation could have been better, but it reads well enough.

Reviewer: Imre Patyi (Atlanta)

### MSC:

32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |

32G10 | Deformations of submanifolds and subspaces |

32D15 | Continuation of analytic objects in several complex variables |

14D06 | Fibrations, degenerations in algebraic geometry |

### Keywords:

extensions of pluricanonical sections; invariance of plurigenera; pseudo-effective line bundle; singular metric; multiplier ideal sheaf
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\textit{B. Claudon}, Ann. Inst. Fourier 57, No. 1, 289--300 (2007; Zbl 1122.32013)

### References:

[1] | Demailly, J.-P., private communication |

[2] | Demailly, J.-P.; Peternell, T.; Schneider, M., Compact complex manifolds with numerically effective tangent bndles, J. Alg. Geom., 3, 295-345 (1994) · Zbl 0827.14027 |

[3] | Oshawa, T.; Takegoshi, K., On the extension of \(L^2\) holomorphic functions, Math. Z., 195, 197-204 (1987) · Zbl 0625.32011 · doi:10.1007/BF01166457 |

[4] | Paun, M., Siu’s Invariance of Plurigenera : a One-Tower Proof, preprint (2005) · Zbl 1122.32014 |

[5] | Siu, Y.-T., Complex Geometry, 223-277 (2002) · Zbl 1007.32010 |

[6] | Takayama, S., Pluricanonical systems on algebraic varieties of general type, preprint (2005) |

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