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1. Cohomological Invariants in Positive Characteristic

   https://doi.org/10.1093/imrn/rnaa321  

   Totaro, Burt (January 2021, International Mathematics Research Notices)

   Abstract We determine the mod $$p$$ cohomological invariants for several affine group schemes $$G$$ in characteristic $$p$$. These are invariants of $$G$$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $$p$$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $$p$$ Milnor K-theory of fields. 

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2. A horizontal Chern–Gauss–Bonnet formula on totally geodesic foliations

   https://doi.org/10.1007/s10455-022-09827-3  

   Baudoin, Fabrice; Grong, Erlend; Vega-Molino, Gianmarco (February 2022, Annals of Global Analysis and Geometry)

   Abstract Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors in Baudoin and Grong (Ann Glob Anal Geom 56(2):403–428, 2019). 

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3. Erratic birational behavior of mappings in positive characteristic

   https://doi.org/10.1002/mana.202200003  

   Cutkosky, Steven Dale (November 2023, Mathematische Nachrichten)

   Abstract Birational properties of generically finite morphisms of algebraic varieties can be understood locally by a valuation of the function field ofX. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation, there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields, it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing up. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in defect Artin–Schreier extensions which can have any prescribed distance. 

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4. Lower bounds on Hilbert–Kunz multiplicities and maximal F -signatures

   https://doi.org/10.1017/S0305004122000238  

   JEFFRIES, JACK; NAKAJIMA, YUSUKE; SMIRNOV, ILYA; WATANABE, KEI–ICHI; YOSHIDA, KEN–ICHI (March 2023, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract Hilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunz multiplicity. 

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5. Bernstein-Sato theory for singular rings in positive characteristic

   https://doi.org/10.1090/tran/8917  

   Jeffries, Jack; Núñez-Betancourt, Luis; Quinlan-Gallego, Eamon (July 2023, Transactions of the American Mathematical Society)

   The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case. 

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