Siu’s invariance of plurigenera: a one-tower proof.

*(English)*Zbl 1122.32014Let \(X \rightarrow \Delta\) be a smooth projective family of complex manifolds over the unit disc. For any \(t \in \Delta\), the \(m\)-th plurigenus of the fibre \(X_t\) is defined as the dimension of the space of global sections of the \(m\)-th power of the canonical bundle \(K_{X_t}\). It has been a long-standing conjecture that the plurigenera are invariant in families, i.e., do not depend on \(t\). This conjecture has been proven by Y.-T. Siu in two seminal papers [Invent. Math. 134, No. 3, 661–673 (1998; Zbl 0955.32017) and Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. Berlin: Springer (2002; Zbl 1007.32010)].

The paper under review gives a simplified proof of Siu’s theorem. The main theorem is as follows: Let \(X \rightarrow \Delta\) be a smooth projective family of complex manifolds over the unit disc. Let \(L\) be a line bundle over \(X\), endowed with a (possibly singular) Hermitian metric such that the curvature current is semi-positive and such that the restriction of the metric to the central fibre \(X_0\) is well-defined. Denote by \(I(h| _{X_0})\) the multiplier ideal associated to the restriction of the metric. Then any global section of \((m K_{X_0} + L) \otimes I(h| _{X_0})\) on \(X_0\) extends to \(X\). In words, any global section on the central fibre that satisfies a \(L^2\)-integrability condition with respect to the restricted metric extends to the total space.

This statement is slightly more general than the corresponding result in Siu’s article who made the hypothesis that the section is bounded with respect to the metric. Note furthermore that the case where \(L\) is trivial implies the invariance of plurigenera. The main technical tool of the proof is the extension theorem due to T. Ohsawa and K. Takegoshi [Math. Z. 195, 197–204 (1987; Zbl 0625.32011)]: given a section on the central fibre, the section extends to the total space if we find a metric on \(m K_X + L\) with semi-positive curvature current and such that the section satisfies the \(L^2\)-integrability condition. Vaguely speaking, this metric is obtained as follows: for \(A\) a sufficiently ample line bundle on \(X\), it is clear that every section of \(m K_{X_0} + L + A\) on the central fibre extends to the total space. The task is then to construct inductively global sections on \(k (m K_X + L) + A\) using the Ohsawa-Takegoshi theorem at each step. When \(k\) goes to infinity, this tower of sections of ever higher multiples will give the metric we are looking for (provided we assure boundedness by some universal constant).

The technical details of this shorter proof are still quite intricate, but are exposed very clearly in the article. Note that the technique of the proof depends heavily on the existence of the ample line bundle \(A\), i.e., the fact that we are dealing with a projective family of manifolds. Nevertheless Y.-T. Siu conjectures that the invariance of plurigenera still holds for a family of compact Kähler manifolds. Note furthermore that so far there is no algebraic proof of the invariance of plurigenera, cf. [R. Lazarsfeld, Positivity in Algebraic Geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 49. Berlin: Springer (2004; Zbl 1093.14500)] for an algebraic proof in the case of varieties of general type.

The paper under review gives a simplified proof of Siu’s theorem. The main theorem is as follows: Let \(X \rightarrow \Delta\) be a smooth projective family of complex manifolds over the unit disc. Let \(L\) be a line bundle over \(X\), endowed with a (possibly singular) Hermitian metric such that the curvature current is semi-positive and such that the restriction of the metric to the central fibre \(X_0\) is well-defined. Denote by \(I(h| _{X_0})\) the multiplier ideal associated to the restriction of the metric. Then any global section of \((m K_{X_0} + L) \otimes I(h| _{X_0})\) on \(X_0\) extends to \(X\). In words, any global section on the central fibre that satisfies a \(L^2\)-integrability condition with respect to the restricted metric extends to the total space.

This statement is slightly more general than the corresponding result in Siu’s article who made the hypothesis that the section is bounded with respect to the metric. Note furthermore that the case where \(L\) is trivial implies the invariance of plurigenera. The main technical tool of the proof is the extension theorem due to T. Ohsawa and K. Takegoshi [Math. Z. 195, 197–204 (1987; Zbl 0625.32011)]: given a section on the central fibre, the section extends to the total space if we find a metric on \(m K_X + L\) with semi-positive curvature current and such that the section satisfies the \(L^2\)-integrability condition. Vaguely speaking, this metric is obtained as follows: for \(A\) a sufficiently ample line bundle on \(X\), it is clear that every section of \(m K_{X_0} + L + A\) on the central fibre extends to the total space. The task is then to construct inductively global sections on \(k (m K_X + L) + A\) using the Ohsawa-Takegoshi theorem at each step. When \(k\) goes to infinity, this tower of sections of ever higher multiples will give the metric we are looking for (provided we assure boundedness by some universal constant).

The technical details of this shorter proof are still quite intricate, but are exposed very clearly in the article. Note that the technique of the proof depends heavily on the existence of the ample line bundle \(A\), i.e., the fact that we are dealing with a projective family of manifolds. Nevertheless Y.-T. Siu conjectures that the invariance of plurigenera still holds for a family of compact Kähler manifolds. Note furthermore that so far there is no algebraic proof of the invariance of plurigenera, cf. [R. Lazarsfeld, Positivity in Algebraic Geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 49. Berlin: Springer (2004; Zbl 1093.14500)] for an algebraic proof in the case of varieties of general type.

Reviewer: Andreas Höring (Paris)

##### MSC:

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

32L10 | Sheaves and cohomology of sections of holomorphic vector bundles, general results |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |

14D99 | Families, fibrations in algebraic geometry |

14J40 | \(n\)-folds (\(n>4\)) |

32J27 | Compact Kähler manifolds: generalizations, classification |