An optimal extension theorem for 1-forms and the Lipman-Zariski conjecture.

*(English)*Zbl 1310.14008The authors study the problem of extending certain differential forms defined on a complex normal algebraic variety \(X\) to forms on a suitable desingularization \(\pi:\tilde X \to X\). They use their results to prove the famous Lipman-Zariski (L-Z) conjecture, which says: “if the tangent sheaf of \(X\) is locally free then \(X\) is smooth”. They do it for varieties \(X\) such that, for a suitable divisor \(D\) of \(X\), \((X,D)\) is a log canonical pair, using the terminology of the “minimal model program”, see [J. Kollár and S. Mori, Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. Cambridge: Cambridge University Press (1998; Zbl 0926.14003)].

More precisely, they prove the following theorem, which implies the result on the L-Z conjecture just cited. Let \(X\) be a normal complex variety, \(\pi:\tilde X \to X\) a log resolution, \(\tilde D\) a simple normal crossings divisor of \(\tilde X\) such that \(\pi _{\star }\Omega ^1_{\tilde X}(\log {\tilde D})\) is reflexive. Then, the L-Z conjecture holds for \(X\).

The mentioned reflexivity is equivalent to saying that logarithmic 1-forms defined on the dense open set of points of \(X\) where \(\pi_{\star}(\tilde D)\) has simple normal crossings extend to logarithmic 1-forms on \((\tilde X, \tilde D)\).

They also give a criterion in order that, given a log canonical pair \((X,D)\), having a log resolution \(\pi:{\tilde X} \to X\), the direct image \(\pi _{\star}\Omega ^1_{\tilde X}(\log {\tilde D})\) be reflexive. It requires that \(\mathrm{Supp}({\tilde D})\) be between two closed sets of \(\tilde X\), related to the exceptional set of \(\pi\) and a certain preimage of of the divisor \(D\).

They prove that such a statement is optimal, both with respect to the conditions imposed on the pole divisor \({\tilde D}\) and the degree (equal to one) of the forms used. More precisely, concerning the degrees, and citing from the abstract, they give an example of “a 2-form defined on the smooth locus of 2-form defined on the smooth locus of a three-dimensional log canonical pair \((X,\emptyset)\) which acquires a logarithmic pole along an exceptional divisor of discrepancy zero”.

They also present some (negative) results about the possible reflexivity of certain interesting sheaves involving differential forms when dealing with Kawamata log terminal singularities.

More precisely, they prove the following theorem, which implies the result on the L-Z conjecture just cited. Let \(X\) be a normal complex variety, \(\pi:\tilde X \to X\) a log resolution, \(\tilde D\) a simple normal crossings divisor of \(\tilde X\) such that \(\pi _{\star }\Omega ^1_{\tilde X}(\log {\tilde D})\) is reflexive. Then, the L-Z conjecture holds for \(X\).

The mentioned reflexivity is equivalent to saying that logarithmic 1-forms defined on the dense open set of points of \(X\) where \(\pi_{\star}(\tilde D)\) has simple normal crossings extend to logarithmic 1-forms on \((\tilde X, \tilde D)\).

They also give a criterion in order that, given a log canonical pair \((X,D)\), having a log resolution \(\pi:{\tilde X} \to X\), the direct image \(\pi _{\star}\Omega ^1_{\tilde X}(\log {\tilde D})\) be reflexive. It requires that \(\mathrm{Supp}({\tilde D})\) be between two closed sets of \(\tilde X\), related to the exceptional set of \(\pi\) and a certain preimage of of the divisor \(D\).

They prove that such a statement is optimal, both with respect to the conditions imposed on the pole divisor \({\tilde D}\) and the degree (equal to one) of the forms used. More precisely, concerning the degrees, and citing from the abstract, they give an example of “a 2-form defined on the smooth locus of 2-form defined on the smooth locus of a three-dimensional log canonical pair \((X,\emptyset)\) which acquires a logarithmic pole along an exceptional divisor of discrepancy zero”.

They also present some (negative) results about the possible reflexivity of certain interesting sheaves involving differential forms when dealing with Kawamata log terminal singularities.

Reviewer: Augusto Nobile (Baton Rouge)

##### MSC:

14B05 | Singularities in algebraic geometry |

14E30 | Minimal model program (Mori theory, extremal rays) |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

13H05 | Regular local rings |

32S05 | Local complex singularities |

32S45 | Modifications; resolution of singularities (complex-analytic aspects) |