K(π,1)K(\pi, 1)-neighborhoods and comparison theorems

A technical ingredient in Faltings' original approach to p-adic comparison theorems involves the construction of $K(\pi, 1)$-neighborhoods for a smooth scheme X over a mixed characteristic dvr with a perfect residue field: every point of X has an open neighborhood whose general fiber is a $K(\pi, 1)$ scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in p-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether Normalization Lemma.Comment: 24 page

Spherical multiple flags

For a reductive group G, the products of projective rational varieties homogeneous under G that are spherical for G have been classified by Stembridge. We consider the B-orbit closures in these spherical varieties and prove that under some mild restrictions they are normal, Cohen-Macaulay and have a rational resolution.Comment: 16 page

F-Split and F-Regular Varieties with a Diagonalizable Group Action

Let $H$ be a diagonalizable group over an algebraically closed field $k$ of positive characteristic, and $X$ a normal $k$-variety with an $H$-action. Under a mild hypothesis, e.g. $H$ a torus or $X$ quasiprojective, we construct a certain quotient log pair $(Y,\Delta)$ and show that $X$ is F-split (F-regular) if and only if the pair $(Y,\Delta)$ if F-split (F-regular). We relate splittings of $X$ compatible with $H$-invariant subvarieties to compatible splittings of $(Y,\Delta)$, as well as discussing diagonal splittings of $X$. We apply this machinery to analyze the F-splitting and F-regularity of complexity-one $T$-varieties and toric vector bundles, among other examples.Comment: 40 page

Allegories of destruction: “woman” and “the Jew” in Otto Weininger's 'Sex and character'

This article investigates the constructions of masculinity, femininity, and Jewishness and their interrelation in Otto Weininger's widely discussed book Geschlecht und Charakter (Sex and Character, 1903). Departing from previous scholarship, I argue that not only the commonalities between Weininger's images of “Woman” and “the Jew,” but also their hitherto largely ignored differences, are crucial for an understanding of Weininger's views and their relation to his historical context. Reading Weininger through the lens of Critical Theory suggests viewing “the Woman” and “the Jew” as outward projections of different, but related contradictions within the constitution of the modern subject itself. More specifically, “Woman” comes to embody the threat to the (masculine) bourgeois individual emanating from its own embodied existence, from “nature” and libidinal impulses. “The Jew,” on the other hand, comes to stand for historical developments of modern society that make themselves more keenly felt towards the end of the nineteenth century and threaten to undermine the very forms of individuality and independence that had previously been produced by this society. Such a reading of Geschlecht und Charakter not only can help illuminate the crisis of the bourgeois individual at the turn of the twentieth century, but also could contribute to ongoing discussions on why modern society, although based on seemingly universalist conceptions of subjectivity, continues to produce difference and exclusion along the lines of gender and race

A refinement of the Deligne-Illusie theorem

We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$ which is liftable to $W_2(k)$, the truncations $\tau_{[a,b]}(\Omega^\bullet_{X/k})$ are decomposable for $a\leq b < a + p-1$. Consequently, the first nonzero differential in the conjugate spectral sequence $$E^{ij}_2 = H^i(X', \Omega^j_{X'/k}) \Rightarrow H^{i+j}_{\rm dR}(X/k)$$ appears no earlier than on page $p$. We deduce new criteria for the degeneration of the Hodge spectral sequence in favorable cases. The main technical result belongs purely to homological algebra. It concerns certain commutative differential graded algebras whose cohomology algebra is the exterior algebra, dubbed by us abstract Koszul complexes, of which the de Rham complex in characteristic $p$ is an example.Comment: 7 page