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Register a new userA remark on the Hochschild-Kostant-Rosenberg theorem in characteristic p
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We prove a Hochschild-Kostant-Rosenberg decomposition theorem for smooth proper schemes $X$ in characteristic $p$ when $dim Xleq p$. The best known previous result of this kind, due to Yekutieli, required $dim X<p$. Yekutielis result follows from the observation that the denominators appearing in the classical proof of HKR do not divide $p$ when $dim X<p$. Our extension to $dim X=p$ requires a homological fact: the Hochschild homology of a smooth proper scheme is self-dual.
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We give counterexamples to the degeneration of the HKR spectral sequence in characteristic $p$, both in the untwisted and twisted settings. We also prove that the de Rham--$mathrm{HP}$ and crystalline--$mathrm{TP}$ spectral sequences need not degenerate.
We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevichs formality theorem, for the cup product, does not hold for Deligne--Mumford stacks in general. We discuss relationships with orbifold cohomology, extending Ruans cohomological conjectures. This employs a trivialization of the determinants in the case of a symplectic group action on a symplectic variety X, which requires (for the cup product) a nontrivial normalization missing in previous literature.
Beliakova-Putyra-Wehrli studied various kinds of traces, in relation to annular Khovanov homology. In particular, to a graded algebra and a graded bimodule over it, they associate a quantum Hochschild homology of the algebra with coefficients in the bimodule, and use this to obtain a deformation of the annular Khovanov homology of a link. A spectral refinement of the resulting invariant was recently given by Akhmechet-Krushkal-Willis. In this short note we observe that quantum Hochschild homology is a composition of two familiar operations, and give a short proof that it gives an invariant of annular links, in some generality. Much of this is implicit in Beliakova-Putyra-Wehrlis work.
Let $C$ be an irreducible, reduced, non-degenerate curve, of arithmetic genus $g$ and degree $d$, in the projective space $mathbf P^4$ over the complex field. Assume that $C$ satisfies the following {it flag condition of type $(s,t)$}: {$C$ does not lie on any surface of degree $<s$, and on any hypersurface of degree $<t$}. Improving previous results, in the present paper we exhibit a Castelnuovo-Halphen type bound for $g$, under the assumption $sleq t^2-t$ and $dgg t$. In the range $t^2-2t+3leq sleq t^2-t$, $dgg t$, we are able to give some information on the extremal curves. They are arithmetically Cohen-Macaulay curves, and lie on a flag like $Ssubset F$, where $S$ is a surface of degree $s$, $F$ a hypersurface of degree $t$, $S$ is unique, and its general hyperplane section is a space extremal curve, not contained in any surface of degree $<t$. In the case $dequiv 0$ (modulo $s$), they are exactly the complete intersections of a surface $S$ as above, with a hypersurface. As a consequence of previous results, we get a bound for the speciality index of a curve satisfying a flag condition.
We generalize the basic results of Vinbergs theta-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or good characteristic), confirming a conjecture of Popov in this case.
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