Nilpotent orbits of a generalization of Hodge structures.

*(English)*Zbl 1136.32011This paper concerns a generalisation of Hodge structures and their variations which first appeared in work of S. Cecotti and C. Vafa [Nuclear Phys., B 367, No. 2, 359–461 (1991); Commun. Math. Phys. 158, No. 3, 569–644 (1993; Zbl 0787.58049)] on supersymmetric field theories. The abstract notion has been studied by the first author under the name TERP-structure [J. Reine Angew. Math. 555, 77–161 (2003; Zbl 1040.53095)].

A TERP-structure consists of a twistor (a holomorphic vector bundle on \(\mathbb P^1\)) together with additional data generalising the ingredients of a polarised Hodge structure. More precisely (Definition 3.1), a TERP-structure of weight \(w\in\mathbb Z\) is a quadruple \((H,H'_{\mathbb R},\nabla,P)\), where \(H\) is a holomorphic vector bundle on \(\mathbb C\) equipped with a flat meromorphic connection \(\nabla\) with a pole of order at most \(2\) at \(0\), a flat real subbundle \(H'_{\mathbb R}\) of the restriction \(H'\) of \(H\) to \(\mathbb C^*\) such that \(H'=H'_{\mathbb R}\otimes\mathbb C\) and a flat, bilinear, \((-1)^w\)-symmetric non-degenerate pairing \(P:H_z\times H_{-z}\to\mathbb C\) for \(z\in\mathbb C^*\) such that \(P\) maps \((H'_{\mathbb R})_z\times (H'_{\mathbb R})_{-z}\) to \(i^w{\mathbb R}\) and the induced pairing on sections has the property that \(z^{-w}P\) is defined and non-degenerate at \(0\). The bundle \(H\) can be naturally extended to a bundle on \(\mathbb P^1\) and \(\nabla\) then naturally extends with a pole of order \(2\) at \(\infty\).

The main purpose of the present paper is to generalise the concept of nilpotent orbits for Hodge structures and the correspondence relating these to mixed Hodge structures (due to W. Schmid [Invent. Math. 22, 211–319 (1973; Zbl 0278.14003)] and to E. Cattani, A. Kaplan and W. Schmid [Invent. Math. 67, No. 1, 101–115 (1982; Zbl 0516.14005); Ann. Math. (2) 123, 457–535 (1986; Zbl 0617.14005); Astérisque 179–180, 67–96 (1989; Zbl 0705.14006)]). The notion of a nilpotent object is simple to generalise (Definition 4.1), but that of a polarised mixed Hodge structure is more complicated due to the possibility of the singularity of \(\nabla\) at \(0\) being irregular; the details of this are carried out in Definition 9.1 and involve the requirement that the formal decomposition of \((H,\nabla)\) at \(0\) can be done without ramification (see Definition 8.1). The focus of the paper is the conjecture (Conjecture 9.2) that a TERP-structure which does not require a ramification is a mixed TERP-structure if and only if it induces a nilpotent orbit. The main result of the paper (Theorem 9.3) is to prove (1) that the conjecture is true if the TERP-structure is regular singular and (2) that the implication “only if” is true for any TERP-structure. The implication “only if” in the regular singular case has already been proved by the author [op. cit., Theorem 7.20], so the present paper is concerned primarily with proving the converse result in the regular singular case and (2).

One can also consider \(r\to\infty\) instead of \(r\to 0\); the counterpart of a nilpotent orbit is then a Sabbah orbit (Definition 4.1 again). This nomenclature derives from a definition of C. Sabbah [preprint, 1999, arXiv:math/9805077] of a filtration \(F^\bullet_{Sab}\) on \(H^\infty\). The author proves (Theorem 7.3) that a TERP-structure induces a Sabbah orbit if and only if a twisted version of the filtration \(F^\bullet_{Sab}\) gives rise to a polarised mixed Hodge structure.

In addition to the papers of Cecotti and Vafa, key references include C. Sabbah’s paper [Polarizable twistor \(\mathcal D\)-modules. Astérisque 300. Paris: Société Mathématique de France. (2005; Zbl 1085.32014)] in which he generalises C. Simpson’s notion of a variation of twistor structures and a central degeneration result contained in T. Mochizuki’s paper [Mem. Am. Math. Soc. 869 (2007; Zbl 1259.32005)].

A TERP-structure consists of a twistor (a holomorphic vector bundle on \(\mathbb P^1\)) together with additional data generalising the ingredients of a polarised Hodge structure. More precisely (Definition 3.1), a TERP-structure of weight \(w\in\mathbb Z\) is a quadruple \((H,H'_{\mathbb R},\nabla,P)\), where \(H\) is a holomorphic vector bundle on \(\mathbb C\) equipped with a flat meromorphic connection \(\nabla\) with a pole of order at most \(2\) at \(0\), a flat real subbundle \(H'_{\mathbb R}\) of the restriction \(H'\) of \(H\) to \(\mathbb C^*\) such that \(H'=H'_{\mathbb R}\otimes\mathbb C\) and a flat, bilinear, \((-1)^w\)-symmetric non-degenerate pairing \(P:H_z\times H_{-z}\to\mathbb C\) for \(z\in\mathbb C^*\) such that \(P\) maps \((H'_{\mathbb R})_z\times (H'_{\mathbb R})_{-z}\) to \(i^w{\mathbb R}\) and the induced pairing on sections has the property that \(z^{-w}P\) is defined and non-degenerate at \(0\). The bundle \(H\) can be naturally extended to a bundle on \(\mathbb P^1\) and \(\nabla\) then naturally extends with a pole of order \(2\) at \(\infty\).

The main purpose of the present paper is to generalise the concept of nilpotent orbits for Hodge structures and the correspondence relating these to mixed Hodge structures (due to W. Schmid [Invent. Math. 22, 211–319 (1973; Zbl 0278.14003)] and to E. Cattani, A. Kaplan and W. Schmid [Invent. Math. 67, No. 1, 101–115 (1982; Zbl 0516.14005); Ann. Math. (2) 123, 457–535 (1986; Zbl 0617.14005); Astérisque 179–180, 67–96 (1989; Zbl 0705.14006)]). The notion of a nilpotent object is simple to generalise (Definition 4.1), but that of a polarised mixed Hodge structure is more complicated due to the possibility of the singularity of \(\nabla\) at \(0\) being irregular; the details of this are carried out in Definition 9.1 and involve the requirement that the formal decomposition of \((H,\nabla)\) at \(0\) can be done without ramification (see Definition 8.1). The focus of the paper is the conjecture (Conjecture 9.2) that a TERP-structure which does not require a ramification is a mixed TERP-structure if and only if it induces a nilpotent orbit. The main result of the paper (Theorem 9.3) is to prove (1) that the conjecture is true if the TERP-structure is regular singular and (2) that the implication “only if” is true for any TERP-structure. The implication “only if” in the regular singular case has already been proved by the author [op. cit., Theorem 7.20], so the present paper is concerned primarily with proving the converse result in the regular singular case and (2).

One can also consider \(r\to\infty\) instead of \(r\to 0\); the counterpart of a nilpotent orbit is then a Sabbah orbit (Definition 4.1 again). This nomenclature derives from a definition of C. Sabbah [preprint, 1999, arXiv:math/9805077] of a filtration \(F^\bullet_{Sab}\) on \(H^\infty\). The author proves (Theorem 7.3) that a TERP-structure induces a Sabbah orbit if and only if a twisted version of the filtration \(F^\bullet_{Sab}\) gives rise to a polarised mixed Hodge structure.

In addition to the papers of Cecotti and Vafa, key references include C. Sabbah’s paper [Polarizable twistor \(\mathcal D\)-modules. Astérisque 300. Paris: Société Mathématique de France. (2005; Zbl 1085.32014)] in which he generalises C. Simpson’s notion of a variation of twistor structures and a central degeneration result contained in T. Mochizuki’s paper [Mem. Am. Math. Soc. 869 (2007; Zbl 1259.32005)].

Reviewer: P. E. Newstead (Liverpool)

##### MSC:

32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

32G20 | Period matrices, variation of Hodge structure; degenerations |

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

53C28 | Twistor methods in differential geometry |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

81T60 | Supersymmetric field theories in quantum mechanics |

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