Related papers: Lichtenbaum-van Hamel duality for singular varieti…

I make some remarks on Hodge symmetry, and prove for instance that if $k$ is a perfect field of characteristic $p>0$ and $X/k$ smooth, proper and Hodge-Witt scheme, and Hodge de Rham sequence of $X$ degenerates at $E_1$ and $X$ has…

Let $\Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $\lambda$ of the field of algebraic numbers which is prime to p, consider the $\lambda$-adic pro-semisimple completion of $\Pi$ as an object…

Andreotti-Vesentini, Ohsawa, Gromov, Koll\'ar, among others, have observed that Hodge theory could be extended to (non compact) K\"ahler complete manifolds, within the L^2 framework. Also, many vanishing theorems on projective or K\"ahler…

We study the algebraic $K$-theory of smooth schemes over $W_n(\Bbbk)$, where $\Bbbk$ is a perfect field of characteristic $p>0$. For a $p$-adic smooth scheme $X_{\centerdot}$ over $W_{\centerdot}(k)$, we introduce complexes…

Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts…

Given a stratified variety X with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of "parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in…

We prove a topological version of abelian duality where the gauge groups are finite abelian. The theories are finite homotopy TFTs, topological analogues of the $p$-form $U(1)$ gauge theories. Using Brown-Comenetz duality, we extend the…

Sen's theorem on the ramification of a $p$-adic analytic Galois extension of $p$-adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmetic Sen operator. By developing $p$-adic Hodge theory for…

We study Torelli-type theorems in the Zariski topology for varieties of dimension at least 2, over arbitrary fields. In place of the Hodge structure, we use the linear equivalence relation on Weil divisors. Using this setup, we prove a…

In this article we study the cohomological and homological (due to Jannsen) Hodge conjecture for singular varieties. The motivation for studying singular varieties comes from the fact that any smooth projective variety X is birational to a…

In this document we consider an exact sequence of group varieties $e\to N\to G\to Q\to e$ over an algebraically closed field. We show that for $l\neq \mathrm{char}(k)$ a prime there exists an isomorphism of graded $\mathbb{Q}_l$-algebras…

We give a simple characterization of all perfectoid profinite \'{e}tale covers of abelian varieties in terms of the Hodge-Tate filtration on the $p$-adic Tate module. We also compute the geometric Sen morphism for all profinite $p$-adic Lie…

We show that the continuous \'etale cohomology groups $H^n_{\mathrm{cont}}(X,\mathbf{Z}_l(n))$ of smooth varieties $X$ over a finite field $k$ are spanned as $\mathbf{Z}_l$-modules by the $n$-th Milnor $K$-sheaf locally for the Zariski…

We apply the techniques developed in our previous work with Leung to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi-Yau variety $X$. In…

We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-etale site, which makes all constructions…

Let $(A, I)$ be a bounded prism, and $X$ be a smooth $p$-adic formal scheme over $\Spf(A/I)$. We consider the notion of crystals on Bhatt--Scholze's prismatic site $(X/A)_{\prism}$ of $X$ relative to $A$. We prove that if $X$ is proper over…

In this article, we develop an $L^{2}$-Hodge theory on complete $2n$-dimensional almost K\"{a}hler manifolds $(X,\omega)$. In the first part, we establish several identities for various Laplacians, generalized Hodge and Serre dualities, a…

We reprove Kuznetsov's "fundamental theorem of homological projective duality" using LG models and variation of GIT stability. This extends the validity of the theorem from smooth varieties to nice subcategories of smooth quotient stacks,…

Let $C$ be a complete, algebraically closed non-archimedean extension of $\mathbb{Q}_p$, and $X$ be a proper rigid-analytic variety over $C$. We show that the category of pro-\'etale vector bundles on $X$ is equivalent to the category of…

We define cohomological complexes of locally compact abelian groups associated with varieties over $p$-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact…