##
**An ultimate proof of Hoffmann-Totaro’s conjecture.**
*(English)*
Zbl 1506.11046

Given an anisotropic quadratic form \(\varphi\) over a field \(F\), the restriction \(\varphi_K\) is isotropic where \(K\) is the function field of its underlying quadric. The maximal dimension of a totally isotropic subform, denoted by \(i_1(\varphi)\), was conjectured by Hoffmann (originally in characteristic 2) and Totaro (arbitrary characteristic, see [B. Totaro, J. Algebr. Geom. 17, No. 3, 577–597 (2008; Zbl 1144.11031), p. 596]) to be at most \(2^m\) where \(m\) is the the maximal integer for which \(2^m | \dim \varphi-i_1(\varphi)\). The conjecture implies, in particular, that \(i_1(\varphi)\leq \frac{1}{2} \dim \varphi\), something that is well known in the nonsingular case (where \(i_1(\varphi)\) coincides with the first Witt index), and the bound is sharp when \(\varphi\) is a Pfister form.

The conjecture was proven by the author of this current paper to hold true when \(\operatorname{char}(F)\neq 2\) in [N. A. Karpenko, Invent. Math. 153, No. 2, 455–462 (2003; Zbl 1032.11016)], using Steenrod operations on modulo 2 Chow groups. In the works of Primozic, Haution and Scully, much of the machinery was adapted to the characteristic 2 case, and the conjecture remained open only in the case of singular, but not totally singular, quadratic forms over fields of characteristic 2, a case which the current paper resolves in the positive (as expected).

The conjecture was proven by the author of this current paper to hold true when \(\operatorname{char}(F)\neq 2\) in [N. A. Karpenko, Invent. Math. 153, No. 2, 455–462 (2003; Zbl 1032.11016)], using Steenrod operations on modulo 2 Chow groups. In the works of Primozic, Haution and Scully, much of the machinery was adapted to the characteristic 2 case, and the conjecture remained open only in the case of singular, but not totally singular, quadratic forms over fields of characteristic 2, a case which the current paper resolves in the positive (as expected).

Reviewer: Adam Chapman (Tel Hai)

### MSC:

11E04 | Quadratic forms over general fields |

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

14C25 | Algebraic cycles |

PDFBibTeX
XMLCite

\textit{N. A. Karpenko}, J. Eur. Math. Soc. (JEMS) 24, No. 12, 4385--4398 (2022; Zbl 1506.11046)

Full Text:
DOI

### References:

[1] | Elman, R., Karpenko, N., Merkurjev, A.: The Algebraic and Geometric Theory of Quadratic Forms. Amer. Math. Soc. Colloq. Publ. 56, American Mathematical Society, Providence (2008) Zbl 1165.11042 MR 2427530 · Zbl 1165.11042 |

[2] | Haution, O.: Duality and the topological filtration. Math. Ann. 357, 1425-1454 (2013) Zbl 1284.14013 MR 3124937 · Zbl 1284.14013 |

[3] | Haution, O.: On the first Steenrod square for Chow groups. Amer. J. Math. 135, 53-63 (2013) Zbl 1267.14012 MR 3022956 · Zbl 1267.14012 |

[4] | Haution, O.: Detection by regular schemes in degree two. Algebr. Geom. 2, 44-61 (2015) Zbl 1322.14036 MR 3322197 · Zbl 1322.14036 |

[5] | Hoffmann, D. W.: Diagonal forms of degree p in characteristic p. In: Algebraic and Arith-metic Theory of Quadratic Forms, Contemp. Math. 344, American Mathematical Society, Providence, 135-183 (2004) Zbl 1074.11023 MR 2058673 |

[6] | Hoffmann, D. W., Laghribi, A.: Isotropy of quadratic forms over the function field of a quadric in characteristic 2. J. Algebra 295, 362-386 (2006) Zbl 1138.11012 MR 2194958 · Zbl 1138.11012 |

[7] | Karpenko, N., Merkurjev, A.: Essential dimension of quadrics. Invent. Math. 153, 361-372 (2003) Zbl 1032.11015 MR 1992016 · Zbl 1032.11015 |

[8] | Karpenko, N. A.: On the first Witt index of quadratic forms. Invent. Math. 153, 455-462 (2003) Zbl 1032.11016 MR 1992018 · Zbl 1032.11016 |

[9] | Karpenko, N. A.: Motives and Chow groups of quadrics with application to the u-invariant (after Oleg Izhboldin). In: Geometric Methods in the Algebraic Theory of Quadratic Forms, Lecture Notes in Math. 1835, Springer, Berlin, 103-129 (2004) Zbl 1185.11026 MR 2066516 · Zbl 1185.11026 |

[10] | Primozic, E.: Motivic Steenrod operations in characteristic p. Forum Math. Sigma 8, Paper No. e52, 25 (2020) Zbl 1460.14054 MR 4176756 · Zbl 1460.14054 |

[11] | Scully, S.: Hoffmann’s conjecture for totally singular forms of prime degree. Algebra Number Theory 10, 1091-1132 (2016) Zbl 1350.11049 MR 3531363 · Zbl 1350.11049 |

[12] | Totaro, B.: Birational geometry of quadrics in characteristic 2. J. Algebraic Geom. 17, 577-597 (2008) Zbl 1144.11031 MR 2395138 · Zbl 1144.11031 |

[13] | Vishik, A.: Motives of quadrics with applications to the theory of quadratic forms. In: Geo-metric Methods in the Algebraic Theory of Quadratic Forms, Lecture Notes in Math. 1835, Springer, Berlin, 25-101 (2004) Zbl 1047.11033 MR 2066515 · Zbl 1047.11033 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.