We show that the Tate module of the $\ell$-adic unramified cohomology $H^3_{\nr}(X, {\Q_\ell}/{\Z_\ell})$ of a smooth projective surface $X$ over a local or strictly local field $k$ of residue characteristic prime to $\ell$ vanishes under suitable conditions. We apply this to answer some questions concerning the left and right kernels of the Brauer--Manin pairing for open subvarieties of $X$. To prove the vanishing theorem, we establish Bloch's formula for the Chow group of codimension-two cycles on a semistable model of $X$, and derive applications to the restriction map for such cycles to the closed fiber of the model.
Let $\mathscr{O}_K$ be a mixed characteristic complete DVR with perfect residue field $k$, and let $\mathfrak{X}$ be a proper formal scheme over $\mathscr{O}_K$ with semistable reduction. Answering a question of Fontaine and Jannsen, a classical theorem of Hyodo and Kato gives a rational identification between the log...
Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold of Picard number one. We prove that, if $X$ has terminal singularities, then $(-K_X)^4\le648$, with equality precisely for $\mathbb{P}(1,1,1,1,2)$. If the singularities are canonical, we obtain $(-K_X)^4\le7332$. Under the additional assumption that a...
Let $K/\mathbb{Q}_p$ be a finite extension, and let $X/K$ be an algebraic variety. We present a new proof that the unipotent completion of the \'etale fundamental group of $X_{\bar{K}}$ is a pro-de Rham representation, and show that it is pro-semi-stable whenever $X$ has semi-stable \'etale cohomology. We also reprove,...
The (naive) de Rham cohomology of a zero-dimensional scheme is the homology of the K\"ahler differential algebra of its coordinate ring, viewed as a complex. Since it is well-known to vanish in higher degrees if the ring is quasi-homogeneous, we concentrate on the affine case. For a general affine $K$-algebra $R=P/I$,...
Given a smooth projective curve $X$ over $\mathbb{C}$, we show that the twisted character variety of $X$ is equivalent to a component of the character variety of a stacky curve $\mathcal{X}$, constructed by adding an orbifold point to $X$. This allows to interpret the non abelian Hodge correspondence in degree $d$ over...
We study the category of constructible $p$-Kummer-\'{e}tale sheaves on the log adic generic fiber $(\mathrm{X},\mathcal{M}_{\mathrm{D}})_\eta$ for the stratification induced by a horizontal normal crossings divisor $\mathrm{D}$, showing that it is antiequivalent to a category of finitely generated unit Laurent crystals...
Dylan Pentland· 0 citations
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