text:
\section{Introduction}
\subsection{Goals of this series}
In this paper and subsequent papers in this series we build a ``machine'' to transpose results of categorical and noncommutative Hodge theory into a logarithmic context. The main applications are as follows:
\begin{enumerate}
\item\label{it:hodge1} Interpret logarithmic structures on the Hodge theory of open varieties in terms of the Hochschild calculus of some category.
\item Re-interpret the ``integral Hodge theory'' structure \cite{bms} of Bhatt, Morrow and Scholze using Kaledin's noncommutative point of view \cite{kaledin1} on $p$-adic cyclic objects.
\item Introduce a logarithmic structure on the Hochschild homology of families of $A_\infty$ algebras over a base with suitably logarithmic behavior at the boundary: this should include Fukaya categories, viewed as families over the Novikov base. (This would in particular imply regularity of the noncommutative Gauss-Manin connection for such families of categories: see also \cite{pvv}, where this is proven for smooth, proper algebras.)
\item Write down a ``logarithmic Tate-Shafarevich'' motive $B\bar{\mathbb{E}}_{log},$ for $\bar{\mathbb{E}}$ the universal nodal elliptic curve, with the classifying space of the nodal fiber interpreted in a logarithmic way.
\item\label{it:microlocal} Write down a candidate for the mirror on the $B$ side for ``microlocal'' versions of SYZ mirror symmetry.
\end{enumerate}
In this paper, we address parts \ref{it:hodge1}.\ and \ref{it:microlocal}.\ above. Namely, given an open variety $U$ with a choice of ``nice'' partial compactification $V$ and boundary divisor $D : = V\setminus U$, we write down a symmetric monoidal Abelian category $\qcoh_{log}$ of logarithmic coherent (log-coherent) sheaves $\qcoh_{log}(U, V, D)$ which acts as the ``substrate'' for logarithmic theory, and prove a Hochschild-Kostant-Rosenberg result for its Hochschild homology. The author's point of view on this category comes from a mirror symmetry (in fact, a ``coherent-constructible correspondence'') result, \cite{vtor}, and the geometric interpretation afforded by the mirror comparison is our main computational tool throughout the paper. The category constructed turns out to be a very close cousin of the \emph{parabolic category of the infinite root stack} defined by Talpo and Vistoli, \cite{talpo-vistoli}, and as such should have a generalization in a wider logarithmic context. Talpo and Vistoli's category is a category of sheaves on a certain stack (not of finite type). The category of logarithmic coherent sheaves is the category of coherent sheaves on an \emph{almost stack}: a putative geometric object which combines properties of a stack and of Faltings' \emph{almost geometry}, \cite{faltings} and \cite{gabber-romero}.

\subsection{Logarithmic Hodge theory}
Say $U$ is a smooth algebraic variety of finite type. Its de Rham cohomology $H^*_{dR}(U)$ can be computed as the hypercohomology of the complex $(\Omega^*,d)$. If $U$ is closed, then homology groups of each sheaf of $k$-forms $H^*\Omega^k$ are finite-dimensional and moreover, the Grothendieck spectral sequence for the hypercohomology degenerates at the $E2$ term: this is the famous Hodge-to-de Rham degeneration theorem of Deligne-Illusie, \cite{deligne-illusie}. On the other hand if $U$ is open and smooth, then $H^*_{dR}(U)$ is finite-dimensional but the spectral sequence used to compute it is not. It is reasonable therefore to hope to ``reduce'' the a priori infinite-dimensional spaces $H^i\Omega^k(U)$ to a finite-dimensional piece, which is sufficient to recover $H^*_{dR},$ and perhaps also recover Hodge-to de Rham degeneration. Such a reduction is provided by \emph{log geometry}. Namely, fix a normal-crossings compactification $X$ of $U$ (something that can always be done in characteristic zero), with normal crossings boundary divisor $D : = X\setminus U$. Define the sheaf of \emph{log-one-forms} $\Omega^1_{log}(X, D)$ on $X$ to be one-forms $\xi$ which have at most first-order poles on curves transversal to $D$ (this condition can be imposed locally and multiplied by local functions on $X$, so $\Omega^1_{log}$ is a coherent sheaf). Define $\Omega^*_{log}(X, D)$ to be the exterior power algebra of $\Omega^1_{log}$ over the sheaf of rings $\oo$. Then $\Omega^*_{log}$ inherits a differential from $\Omega^*(U)$. Define the log Hodge homology, $H^{p, q}_{log}(X, D): = H^p\Omega^q_{log}(X, D)$ and $H^*_{\t{log-dR}}$ for the hypercohomology of $(\Omega^*_{log}, d)$ on $X$. Then we have the following results:
\begin{cthm}{Hablicsek, \cite{hablicsek}}
  \begin{enumerate}
  \item     $H^*_{\t{log-dR}}(X, D)\cong H^*_{dR}(X)$
  \item The log Hodge-to de Rham spectral sequence $H^{p, q}_{log}(X, D)\implies H^*_{\t{log-dR}}(X, D)$ degenerates.
    \end{enumerate}
\end{cthm}
This theorem has a generalization to the case where the compactification $X$ has mild singularities (but is still in some sense ``log-smooth''), for example when it is \emph{toroidal}: i.e., the pair $(X, D)$ looks locally like a pair of the form $(X_\Sigma, D_\Sigma)$ for $\Sigma$ a toric fan, $X_\Sigma$ the corresponding toric variety and $D_{\Sigma}: = X_{\Sigma}\setminus T$ the toric boundary. More generally, there is a theory of \emph{log structure} on $X$ which admits a Hodge and a de Rham theory which generalizes these two cases.

For $X$ a proper variety, both the Hodge cohomology $H^{p, q}(X)$ and the de Rham cohomology $H^*_{dR}(X)$ can be interpreted (up to regrading) as Hochschild invariants in a certain category: namely, there are \emph{Hochschild-Kostant-Rosenberg} (HKR) isomorphisms $$\bigoplus_{q-p = k}H^{p, q}(X)\cong HH_*\qcoh(X)$$ expressing Hodge homology as a Hochschild homology invariant and $$\bigoplus_{i \equiv \epsilon \t{ mod } 2} H^i_{dR}(X)\cong HP_\epsilon\qcoh(X)$$ expressing de Rham homology as a periodic cyclic homology invariant, both in terms of the category of quasicoherent sheaves on $X$. Multiplication on both then comes from the symmetric monoidal structure on $\qcoh(X)$.

Now it was observed by Talpo and Vistoli \cite{talpo-vistoli}, building on work of Borne and Vistoli \cite{borne-vistoli} that the log-Hodge homology groups can be naturally computed as Tor groups in a certain category $\qcoh_{par}(X, D)$ of \emph{parabolic sheaves}: sheaves on the ``infinite root stack'' (and this can be generalized for a certain larger class of log structures). However, it is not quite the case that $HH_*\qcoh_{par}(X, D)$ recovers log-Hodge homology.

In this paper we introduce a refinement of the Talpo-Vistoli category $\qcoh_{par}(X, D)$ for $X$ a toroidal compactification of $U = X\setminus D$. We define the {\bf log-coherent category}, $\qcoh_{log}(U, X, D)$, a symmetric monoidal category constructed in a similar way to $\qcoh_{par}(X, D)$ and in fact (in a suitable sense) contained in it, but with a subtle difference involving Faltings' \emph{almost mathematical formalism} \cite{faltings} (originally used in the context of $p$-adic Hodge theory). For this category, we prove log analogues of HKR isomorphisms in characteristic zero: namely
\begin{thm}\label{formstheorem}
  $$HH_*\qcoh_{log}(U, X, D) \cong \bigoplus H^{p, q}_{log}(U, X, D),$$ compatibly with differentials on both sides and
  $$HP_\epsilon\qcoh_{log}(U, X, D) \cong \bigoplus_{i = \epsilon\t{ mod }2} H_{dR}(U).$$
\end{thm}
Note that both $HH_*$ and $HP_*$ can be computed on the level of the derived category. We show that a class of transformations that is known to produce isomorphisms on log-forms also gives equivalences of derived symmetric monoidal categories. Namely, we say that a map of toroidal varieties with boundary $\tau:(U', X', D')\to (U, X, D)$ is a (strictly) toroidal modification if the map $U'\to U$ is an isomorphism and on the neighborhood of the fiber over any $x\in X$ it coincides with an equivariant map of toric varieties of equal dimension (viewed as varieties with boundary). We have the following invariance result.
\begin{thm}\label{invariance}
  For $\tau: (U', X', D')\to (U, X, D)$ a strictly toroidal modification, the derived pullback functor $$D^b\tau^*:D^b\qcoh_{log}(U', X', D')\to D^b\qcoh_{log}(U, X, D)$$ is a symmetric monoidal equivalence of symmetric monoidal dg categories.
\end{thm}
\noindent Note that this is not true in general on the level of Abelian categories, or for more general birational maps.

In future work, we will re-interpret log-polyvector fields as Hochschild cohomology of $\qcoh_{log}$, and interpret the Gauss-Manin connection in terms of this formalism, with regularity provided for free. Perhaps the most interesting application of this formalism is the mixed- and positive-characteristic local case. Here the key question is as follows: suppose that $K$ is a DVR with residue ring $k$ and ring of integers $\oo.$ We can view $\spec(\oo)$ as a partial compactification of $\spec(K),$ and for any variety $X$ over $K$, a model $X_\oo$ is a partial compactification (over $\spec(\oo)$) of $X$. The model is said to \emph{have semistable reduction} if the special fiber $X_k$ is a normal-crossings divisor in $X_\oo.$ It is a problem of immense interest to understand an analogue of log-Hodge theory for the pair $(X_\oo, X_k).$ The state of the art answer to this problem for $K$ a $p$-adic field comes from a paper by Bhatt, Morrow and Scholze, which uses perfectoid formalism to define ``integral $p$-adic Hodge theory'', a Dieudonn\'e module structure associated to such a pair which captures what should be its log Hodge theory (in particular, comparison results with \'etale cohomology). Now it is a result of Kaledin \cite{kaledin1} that the Hochschild homology theory of any category $\C$ over $\zz_p$ (more generally, any cyclic object over $\zz_p$) gives rise to a filtered Dieudonn\'e module. Thus, the cyclic module associated to $HH_*\qcoh_{log}(X_K, X_\oo, X_k)$ produces precisely a (filtered) Dieudonn\'e module, which we hope recovers the integral $p$-adic Hodge theory with significantly less work. We will treat the positive and mixed-characteristic cases in the next installment of this series. In the present paper, to make things easier we will restrict our attention to the characteristic-zero setting.
\subsection{Microlocal mirror symmetry}
The origin of this category, as well as our main computational tool, comes from the author's work on microlocal mirror symmetry, \cite{vtor}. Here, we get the following results:
  Let $(U, X, D)$ be the toroidal space with boundary, where $U = \gg_m^n$ is the complex torus and $X$ a toric variety. Let $S : = \frac{\rr^n}{\zz^n}$ be the real torus, which we view as a group, and let $\operatorname{Shv}(S)$ be the category of all topological sheaves on $S$.
  \begin{thm}
    There is an equivalence of categories $$D^b\qcoh_{log}(U, X, D)\cong D^b\operatorname{Shv}(S).$$ Moreover, this is an equivalence of symmetric monoidal dg categories with tensor product on $D^b\qcoh_{log}(U, X, D)$ is intertwined with the convolution product on $\operatorname{Shv}(S)$ with respect to the group structure on $S$.
  \end{thm}
Now $\operatorname{Shv}(S)$ is a ``microlocal Fukaya category'' for the symplectic cylinder $T^*S (= \rr^n\times (S^1)^n)$, and any projective toric variety can be viewed via the moment map as a Strominger-Yau-Zaslow degeneration of a toric fibration over the $n$-dimensional disk. Microlocal Fukaya categories at the moment are defined for cotangent bundles \cite{nadler-zaslow} and are a work in progress for more general (exact) symplectic varieties: see \cite{kontsevich-soibelman} and \cite{nadler-pants}. Nevertheless, we make the following conjecture:
  Suppose that $\S$ is a Lagrangian torus fibration over a real base $B$ which is part of an SYZ model. In particular, the dual torus fibration $\S^\vee$ over $B$ has almost complex structure which we assume to be integrable, and endowed with a degeneration $\bar{\S}^\vee$ over a compactification $\bar{B}$ of the base. Write $\S^\vee_\t{sing}$ for the boundary $\bar{\S}^\vee\setminus \S$.
\begin{conj}
  In contexts where the microlocal Fukaya category $\operatorname{Fuk}_\mu(\S)$ on $\S$ can be defined, there is an equivalence of derived categories,
  $$D^b\qcoh_{log}(\S^\vee, \bar{\S}^\vee, \S^\vee_\t{sing})\cong \operatorname{Fuk}_\mu(\S).$$
\end{conj}

\section{Preliminaries}
\subsection{Spaces with boundary}
Suppose $k$ is an algebraically closed field of characteristic zero, and work in the category of schemes (at first, of finite type) over $k$. In this section, the main geometric object of interest will be a ``space with boundary'', i.e. a triple $(U, V, D)$ with $V$ a separated normal variety of finite type, $D\subset V$ a Cartier (or $\qq$-Cartier) divisor and $U = V\setminus D$ a smooth Zariski open. A priori, $V$ is not assumed to be compact: if it is compact, the triple $(U, V, D)$ is called a ``compactification triple''. When convenient, we will abbreviate the data $(U, V, D)$ of a space with boundary and just use $V$ with the interior and boundary $U, D$ implicit.

A map of spaces with boundary $\tau:(U', V', D')\to (U, V, D)$ is, for the purposes of this paper, a generically finite map of schemes $\tau:V'\to V$ with $\tau^{-1}(D)_{red} = D'$ (equivalently, $\tau^{-1}(U) = U'$), and such that $\tau\mid U'$ is \'etale. A map of spaces with boundary is said to be \emph{\'etale} if it is \'etale on all of $V'$: in particular, the \'etale maps to a triple $(U, V, D)$ form a site which coincides with the \'etale site on $V$. A map $\tau:(U', V', D')\to (U, V, D)$ is called \emph{ramified \'etale} if it is finite (and \'etale on $U$). A map of spaces with boundary is called a \emph{covering} if it is proper. Our construction of the category $\qcoh_{log}(U, V, D)$ will be \'etale local on $V$, and based on the interplay between \'etale (on $V$) and ramified coverings. The crucial result is the following:
\begin{lm}\label{lm:finite-extension}
  Fix a space with boundary $(U, V, D).$ Then for any \'etale map of interiors $\tau_0:U'\to U,$ there is a unique space with boundary $(U', V', D')$ and ramified \'etale map $\tau: (U', V', D')\to (U, V, D)$ extending $\tau_0.$
\end{lm}
\begin{proof}
  Say $\tau_0:U'\to U$ is an \'etale map. It suffices to work locally, so WLOG we assume $V$ is affine with ring of functions $\oo_V$. Now since $D$ is $\qq$-Cartier, $U$ is also (possibly after passing to smaller affine neighborhood) affine. A finite map is locally affine, so $U'$ is \'etale with ring of functions $\oo_U'$ an \'etale extension of $U$. Let $\oo_V'$ be the normal closure of $\oo_V$ in $\oo_U'$. Since \'etale maps are normal, this is a normal ring, giving a finite \'etale extension $\tau_1:V'\to V$. Since the pullback of a $\qq$-Cartier divisor is $\qq$-Cartier, the triple $(U', V', D')$ with $D'$ the pullback of $D$ satisfies our requirements. The normalcy requirement on $V'$ guarantees that this is the unique such extension.
\end{proof}
We can consider any space as a space with trivial boudary. There is an evident notion of product: if $(U, V, D), (U', V', D')$ are spaces with boundary, their product, abbreviated as $V\times V'$, is the space $(U\times U', V\times V', V\times D'\cup D\times V'.)$.

We could also work with \emph{formal} spaces with boundary, consisting of triples $(U, V, D)$ with $V$ a formal variety, $D\subset V$ a closed divisor locally defined by a single function, and $U = V\setminus D$ \emph{formally} taken to be the punctured neighborhood. Since there are some difficulties with defining a category of formal punctured varieties, we can formulate all the definitions above just in terms of the pair $(V, D)$. In particular, in this context we make the following definition:
\begin{defi}
  A map $\tau:(U', V', D')\to (U, V, D)$ of formal spaces with boundary is a proper map $\tau:V'\to V$ with $\tau^{-1}(D) = D'$ and with ramification locus contained scheme-theoretically in a finite thickening of $D$. It is \emph{ramified \'etale} if $V'\to V$ is finite.
\end{defi}
In this case Lemma \ref{lm:finite-extension} becomes tautological, as we formally define an \'etale map $U'\to U$ in this context as a ramified \'etale map of triples $(U', V', D')\to (U, V, D).$ These naturally form a site, where $U'\times_U U''$ for $(U', V', D')$ and $(U'', V'', D'')$ two ramified \'etale triples defined as $(V''', D''')$ with $V'''$ the proper transform of $V'\times_V V''$. In terms of this definition, all our results for triples $(U, V, D)$ of finite type will directly extend to formal spaces with boundary.

\subsection{Normal-crossings and toroidicity}
We will be interested in spaces with boundary locally modeled on two basic examples.
\begin{enumerate}
\item The ``$k$-dimensional full normal-crossings triple'', $$NC_k : = (\aa^k_{f_k}, \aa^k, V_{f_k},)$$ with $f_k$ the defining polynomial of the cross of the hyperplane cross, $$f_k : = x_1\cdots x_k$$ (for $x_1, \dots, x_k, x_{k+1}, \dots, x_n\in \oo_{\aa^n}$ the canonical coordinates). A generalized normal crossings triple is a space with boundary of the form $NC_k \times \aa^{n-k}.$
\item The ``affine toric triple'' $T_\sigma : = (T, X, X\setminus T)$ for $X$ a $k$-dimensional affine toric variety with \emph{full-dimensional} toric fan $\sigma$ (a single open cone in the real vector space $X^*(T)^\vee_\rr$) and $T$ the torus acting on $X$ with open orbit. A \emph{toroidal triple} is a triple of the form $T_\sigma\times \aa^{n-k}$ (for $\sigma$ a $k$-dimensional toric cone). For a more general toric fan $\Sigma,$ a toric triple is the collection $T_\Sigma: = (T, X_\Sigma, X_\Sigma\setminus T)$ for $X_\Sigma$ the toric variety associated to $\Sigma.$
\end{enumerate}
Each irreducible component of the boundary divisor of a normal-crossings, respectively, toroidal scheme has naturally induced normal-crossings, respectively, toroidal structure, and this induces a natural stratification of $V$ in both the normal-crossings and the toroidal case by closed strata $V_j$ of codimension $j$ with $V_0 = V$ and $V_{j+1}$ inductively defined as the $n-j-1$-dimensional boundary of $V_j$. Define in this case $U_j : = V_j\setminus V_{j+1}$ the open strata: these are smooth. In both cases, there is a unique $0$-dimensional orbit, $V_k = : \{x_0\}$.

\begin{defi} A \emph{formal} full normal-crossings, resp., affine toric triple is the formal neighbrohood near $x_0$ of a normal-crossings, resp., affine toric triple.
\end{defi}
\begin{defi}
 We say that a space with boundary $(U, V, D)$ is normal-crossings, respectively toroidal, if it is \'etale locally isomorphic to $NC_k\times \aa^{n-k}$ or $T_\sigma\times \aa^{n-k}$ with $\sigma$ a $k$-dimensional cone. (And similarly for formal schemes, in the \'etale topology for formal schemes.)
\end{defi}

\begin{defi}
  A map of spaces with boundary $(U', V', D')\to (U, V, D)$ is called a (strict) \emph{toroidal modification} if the map $U'\to U$ is an isomorphism and there is an \'etale cover $V_i$ of $V$ with $V_i': = V_i\times_V V'$ such that the map $V_i'\to V_i$ is isomorphic to the map of toric triples $$\one\times \beta:\aa^k\times X_\Sigma\to \aa^k\times X_\sigma,$$ for $X_\Sigma\to X_\sigma$ a proper equivariant map of toric varieties.
A map of spaces with boundary $(U', V', D')\to (U, V, D)$ is called a (strict) \emph{normal-crossings modification} if it is a strict normal crossings modification and both $V, V'$ are smooth.
\end{defi}
The simplest strict normal-crossings modification is the blow-up of $V$ at a $k$-dimensional normal-crossings stratum.
\begin{defi}
  A map of spaces with boundary $(U', V', D')\to (U, V, D)$ is called a \emph{weak} toroidal, resp., normal-crossings, modification if it is an isomorphism on $U$ and \'etale locally the map $(U', V', D')\to (U, V, D)$ becomes a toroidal, resp., normal-crossings modification after extending $D$ to a larger (toroidal, resp., normal-crossings) divisor $D_+$.
\end{defi}
The basic example of a weak normal-crossings mofidication is the blow-up of $\aa^1\times NC^1$ (i.e., the triple $(\gg_m\times \gg_m, \aa^1\times \gg_m, \aa^1)$) at a point of the boundary. In some sense, if we are interested in ``asymptotic'' geometry of an open variety $U$, it is sufficient to study normal-crossings compactifications. Namely, it follows from Hironaka's resolution of singularities theorem \cite{hironaka} that any smooth open variety $U$ has a normal-crossings compactification $X$. The \emph{weak factorization} theorem, proved by Wlodarczyk, \cite{wlodarczyk}, implies that any two compactifications of $U$ are related by a sequence of weak toroidal modifications. In fact, it has a strengthening of the following form
\begin{thm}[Strong factorization \cite{akmw}]
  For any two normal-crossings compactifications $X, Y$ of $U$, there is an integer $N$ and a sequence of compactificatiosn $X_i, Y_i$ with $$X = X_0\from \dots \from X_N = Y_N\to Y_{N-1}\to Y_{N-2}\to \dots\to Y_0 = Y$$ such that each arrow is a weak normal-crossings modification.
\end{thm}
Moreover, we can take these arrows to be blow-ups. Unfortunately, strong normal-crossings (or indeed, strong toroidal) modifications are not sufficient for a factorization result: already the blow-up of $\aa^1\times NC^1$ above cannot be birationally factored into strong toric morphisms.
\subsection{Ramification}
Recall that the \'etale fundamental group of a $k$-dimensional torus $T^k$ is $\hat{\zz}^k$. A generating collection of covers is the set of covers $\tau_N:T^{(N)}\to T$ with $t_i\mapsto t_i^N,$ with $N$ running over $\nn^\times.$ The \'etale fundamental group of $\aa^{n-k}$ is trivial, so we once again have a fundamental collection of covers $\tau_N\times \one:(T^k\times \aa^{n-k})^{(N)}\to (T^k\times \aa^{n-k}).$ Note that these covers are independent of choice of basis, as they are characterized by being the maximal connected \'etale covers of exponent dividing $N$.

Given a standard toroidal triple $(T\times \aa^{n-k}, T_\sigma\times \aa^{n-k}, D)$, the covers $(T^k\times \aa^{n-k})^{(N)}$ extend uniquely to ramified triples $(T_\sigma\times \aa^{n-k})^{(N)}.$ In the normal-crossings case, the corresponding maps are $$\tau_N:(x_1,\dots, x_k, x_{k+1}, \dots, x_n)\mapsto (x_1^N, \dots, x_k^N, x_{k+1}, \dots, x_n).$$
In the formal setting, any ramified cover of $(T_\sigma)_{\hat{x_0}}$ also factors through one of the $\tau_N$ (this can be seen e.g. over $\cc$ by noting that any ramified formal map can be extended to a tubular neighborhood). Now suppose that $(U, V, D)$ is a toroidal space with boundary and $\tau:(U', V', D')\to (U, V, D)$ is a ramified Galois covering, with (disrete) Galois group $\Gamma$. Let $x\in V$ be a point. Choose a preimage $\tilde{x}\in V',$ and write $$\Gamma_{\tilde{x}}\subset \Gamma$$ for the stabilizer of $\tilde{x}.$ Choose a toroidal local neighborhood $X_\sigma\times \aa^k$ of $\tilde{x}$, together with choice of toroidal coordinates on $\gg_m^k\times \aa^{n-k}.$ Then we get $$\pi_1(V'_{\hat{\tilde{x}}})\cong \hat{\zz}^k.$$ The group $\Gamma_{\tilde{x}}$ is a quotient of $\hat{\zz}^k$ by some cofinite group isomorphic (non-canonically) to $\prod_{i = 1}^k \zz/N_i.$ Write $|\Gamma|_x$, the ``ramification level of $\Gamma$ at $x$'', for the l.c.m. of the $N_i$. This is the maximal $N$ such that the map $V'\to V$ factors in any toroidal neighborhood $T_\sigma\times \aa^{n-k}$ of $x$ through the $N$-power cover $(T_\sigma\times \aa^{n-k})^{(N)}$, and in particular does not depend on choices.

\subsection{Affine inverse limits}
We will use the index $N$ loosely as belonging to a countable filtered index poset $I$ over which we take inverse limits. In particular will implicitly allow replacing it by a filtered sub-index set when comparing different inverse limits if they are connected by maps on some generating sub-index sets. Later, we will use the specific index set of multiplicative positive integers, $I = \nn^\times,$ with $N\succeq N'$ when $N\mid N'.$ Say $X$ is an algebraic variety.

\begin{defi}
  A \emph{tower} $X^{(*)}$ over $X$ is a collection of spaces $$X\from X^{(1)}\from X^{(2)}\from \cdots$$ indexed by a countable filtered index poset $I$. We say a tower is \emph{affine} if $X$ is affine and each $X^{(N)}$ is affine. We say a tower is \emph{locally affine} if $X^{(*)}\mid U$ is affine for every affine open $U\subset X$.
\end{defi}
Given a locally affine tower $X^{(*)}$ over $X$, define its \emph{affine limit} to be the inverse limit of the diagram $X^{(*)}$ in the category of schemes (not necessarily of finite type) affine over $X$. This limit exists, and is presented by the \emph{direct} limit of sheaves of rings of quasicoherent sheaves, $\underset{\to}{\text{lim}}_I \oo^{(N)}$ over $X$. Similarly for spaces with boundary, define:
\begin{defi}
  A tower $(U^{(*)}, V^{(*)}, D^{(*)})$ over a space with boundary $(U, V, D)$ is a diagram of spaces with boundary indexed by $I$. It is \'etale, respectively, ramified \'etale if each $V^{(n)}\to V$ is \'etale, respectively, finite.
\end{defi}
A ramified \'etale map is finite, therefore affine. Given a tower of ramified \'etale spaces with boundary $(U^{(*)}, V^{(*)}. D^{(*)}),$ write $(U^{(\infty)}, V^{(\infty)}, D^{(\infty)})$ for the ``triple'' (in general, no longer of finite type) with $U^{(\infty)}, V^{(\infty)}$ defined as the corresponding affine inverse limits and $D^{(\infty)}$ the closed subvariety locally defined by the sheaf of ideals $\mathcal{I}_D^\infty: = \underset{\to}{\text{lim}}_I I^{(N)}.$

A type of tower of particular interest for us will be the tower $\Gamma^{(*)}$ where $\Gamma^{(*)}$ is a collection of finite groups: generally quotients of a profinite group of \'etale deck transforms. In this case, $\Gamma^{(\infty)}$ will no longer in general be discrete, but it will be a group (as the Hopf algebra structures on extend to a Hopf structure on the direct limit). The example that is most important for us is the collection of quotients of the (profinite completion of) the integers, $\Gamma^N = \zz/N.$ In this case we compute the (direct) system of rings of functions on the $\Gamma_N$ as $$\Gamma_\infty : = \underset{\to}{\text{lim}}_{N\in \nn^\times} k\left[\frac{1}{N}\zz\right]\cong k\left[\qq/\zz\right].$$ (Here we are using algebraic closedness: more generally, the two sides differ by a Tate twist.)

\subsection{Almost local sheaves}
Suppose $X$ is a scheme of finite type and $X^{(\infty)}$ is an affine scheme (possibly of infinite type) with sheaf of functions $\oo_\infty$ over $X$. In particular, we assume that $X^{(\infty)}$ has an affine cover $X_i^{(\infty)}$ with affine intersections (in fact, this picture makes sense more generally in the quasi-separated context, see \cite{gabber-romero}). Let $\mathcal{I}$ be a sheaf of ideals in $\oo_\infty.$ Suppose $\mathcal{I}$ is flat, and \emph{idempotent} $\mathcal{I}^2 = \mathcal{I}\subset \oo$ (a phenomenon that does not occur on irreducible varieties of finite type). Further, suppose that we have the following identity:
$$\mathcal{I}\otimes \hom(\mathcal{I}, \F) = \mathcal{I}\otimes \F$$ for any $\oo_\infty$-module $\F$. Then we define the category of \emph{almost local quasicoherent sheaves} $\qcoh(X^{(\infty)})$ on $X^{(\infty)}$ (with respect to $\mathcal{I}$) as the full subcategory of $\qcoh(X^{(\infty)})$ consisting of sheaves $\F$ such that $\mathcal{I} \F\to \F$ is an isomorphism. We say a sheaf $\F$ is \emph{almost zero} if $\F\otimes \mathcal{I} \cong 0$. Write $\qcoh^{a.l.}(\oo_\infty),$ resp., $\qcoh^{a.z.}(\oo_\infty)$ for the almost local, respectively, almost zero, subcategories. The category of almost zero sheaves on $V$ is equivalent to the category of sheaves of modules on $\oo_\infty/\mathcal{I}.$ Note that unlike the case for the boundary of the finite-type scheme, the property of being a pushforward from the boundary on $\oo/\mathcal{I}$ is stable under extension, i.e.\ a Serre subcategory. In particular, the category of almost-zero sheaves $\qcoh^{a.z.}(V^{(\infty)})$ is a Serre subcategory.  We have a pair of adjoint functors $$\t{forg}:\qcoh^{a.l.}(X^{(\infty)})\leftrightarrows \qcoh(X^{(\infty)}):-\otimes \mathcal{I},$$ and this adjunction is a piece of a Serre quotient sequence of functors $\qcoh^{a.z.}(\oo_\infty)\to \qcoh(\oo_\infty)\to \qcoh^{a.l.}(\oo_\infty).$

If we have an affine algebraic group $\Gamma_\infty$ acting fiberwise on $\spec \oo_\infty$ and preserving $\mathcal{I},$ the group $\Gamma_\infty$ acts on the (compactly generated) category $\qcoh(X^{(\infty)}).$ An equivariant object with respect to this action is a quasicoherent sheaf $\F$ on $X^{(\infty)}$ together with a fiberwise trivialization of the pullback $\mu^*\F$ for $\mu:\Gamma_\infty\times X^{(\infty)}\to X^{(\infty)}$ the action map, satisfying an associativity condition (see e.g. section 2 of \cite{vtor}). Define the category $\qcoh^{a.l.}(X^{\infty})^{\Gamma_\infty}$ to be the category of all $\Gamma_\infty$-equivariant sheaves $\F$ on $X^{(\infty)}$ which satisfy the almost locality condition.

\section{Log-coherent sheaves}
We are ready to define our new quasicoherent category. The definition in this paper is stacky by nature, but because of our use of Faltings' almost category formalism, we need to be careful about sheaf properties which are taken for granted in the theory of ordinary stacks. Absent a good theory of almost stacks (which the author hopes will soon exist), we build the category up from a suitable system of \'etale open covers ``by hand''.
Let $(U, V, D)$ be a space with boundary.
\begin{defi} Call a tower $(U^{(N)}, V^{(N)}, D^{(N)})$ of ramified Galois covers filtering the Galois group of $U$ an \emph{exhaustive} Galois ramified tower.
\end{defi}
As before, $N$ belongs to some filtered poset which we implicitly allow replacing by a filtered sub-index set when comparing different towers. Fix an exhaustive Galois ramified tower. Let $x_0$ be a point of the open stratum $U$ and choose (noncanonically) a compatible system of lifts $x^{(N)}\in V^{(N)}$. Let $\Gamma_N$ be the group of deck transformations of $V^{(N)}$ relative to $x^{(N)}$. The inverse limit of the $\Gamma_N$ is the geometric Galois group of $U$. The rings of functions $\oo(\Gamma_N)$ form a directed system of (Hopf) algebras, and write $\oo_\infty^\Gamma : = \bigsqcup \oo(\Gamma_N)$. The coproduct and counit on the $\Gamma_N$ extend uniquely to a coproduct and counit on $\oo_\infty$, and we define the group scheme $\Gamma_\infty := \spec\oo_\infty^\Gamma.$ Similarly, note that the $V^{(N)}$ are affine over $V$ and for any affine $V'\subset V$ the sheaves of rings $\Gamma(\oo V^{(N)}, V')$ form a directed system of quasicoherent sheaves of
rings over $V$; these glue to an affine scheme $V^{(\infty)}$ over $V$. Because of uniqueness of normal extension, $\Gamma^{(N)}$ acts on $V^{(N)}$, and one can check that $\Gamma^{(\infty)}$ acts on $V^{(\infty)}.$ Note that neither $V^{(\infty)}$ nor $\Gamma_\infty$ depends on the particular sequence of $U^{(N)}$, so long as it is filtering (since both are colimits in the same category, in different filtering sub-diagrams of the same diagram). In particular, we can define canonically the category of equivariant quasicoherent sheaves, $\qcoh(V^{(\infty)})^{\Gamma_\infty}$ (this should be thought of as sheaves on the stack $V_{\t{orbi-}\,\infty} : = V^{(\infty)}/\Gamma_\infty$). The data of equivariance can be defined locally over any affine cover of $V$ (with a glueing condition). Note further that the sheaves associated to the reduced preimages $D^{(N)}$ of $D$ extend to a sheaf of ideals $\mathcal{I}_{D^{(\infty)}}$ on $V^{(\infty)}$. We define the category $\t{Pre}\qcoh_{\t{log}}$ of \emph{pre log-coherent} sheaves associated to the pair $(V, D)$ as the category of $\Gamma_\infty$-equivariant sheaves $\F$ on $V^{(\infty)}$ such that the map $\F\otimes \mathcal{I}\to\F$ is an isomorphism.

Now given an \'etale open $V'$ in $V$, consider the (not necessarily exhaustive) Galois ramified tower $V'\times_V V^{(N)}$, with action by the groups $\Gamma_N$. We can refine these to an exhaustive tower $(V')^{(N)}$ of $V'$, with Galois groups $\Gamma'_N$ mapping surjectively to $\Gamma_N.$ Taking limits, this glues to a map $j_\infty:(V')^{(\infty)}\to V^{(\infty)}$ compatible with actions with respect to the map of group schemes $\Gamma'_\infty\to \Gamma_\infty.$ In particular, pullback along $j_\infty$ gives a restriction functor $$j_\infty^* : \qcoh((V')^{(\infty)})^{\Gamma_\infty}\to \qcoh(V^{(\infty)})^{\Gamma_\infty}.$$ We use the notation $\F\mid V'$ for $j_\infty^*(\F)$ when there is no ambiguity. Since $j^*\mathcal{I} = \mathcal{I}',$ this functor preserves the property of almost locality, i.e.\ if $\mathcal{I}\otimes \F \to \F$ is an isomorphism then $\mathcal{I}'\otimes j^*\F\to j^*\F$ is as well. Thus we get a restriction functor $$j_\infty^*:\t{Pre}\qcoh_{\t{log}}(V,D)\to \t{Pre}\qcoh_{\t{log}}(V', D).$$ We define the category $\qcoh(V, D)$ as the \'etale sheafification of this presheaf of categories. I.e., an object of $\qcoh_{\t{log}}(V)$ is defined as a collection of objects $\F_i$ of $\qcoh_{\t{log}}(V_i)$ on some cover $V_i$ of $V$, together with compatible isomorphisms $\F_i\mid (V_i\cap V_j)\cong \F_j\mid (\V_i\cap V_j)$ with the usual descent condition for sheaves of (Abelian) categories: that the natural automorphism of $\F\mid (V_i\cap V_j\cap V_k)$ is the identity map for any triple of indices $(i, j, k)$. We similarly define a category of \emph{pre-parabolic} sheaves $$\t{Pre}\qcoh_{par}(U, V, D): = \qcoh(X^{(\infty)})^{\Gamma_\infty},$$ without the almost locality condition. We want to consider the limit of categories corresponding to smaller and smaller refinements $\{V_i\}:$ but in fact, it turns out that it always suffices to pass to a finite (and, indeed, a Zariski) cover, consisting of opens which \emph{admit maximal ramification} along the divisor $D$, i.e., such that the ramification level becomes infinitely divisible as we go up in the tower.
\begin{defi}
We say that $V$ admits maximal ramification if there exists a tower of ramified covering $V^{(N)}$ indexed by $N\in \nn^\times$, and such that $V^{(N)}$ has level a multiple of $N$ at each point $x\in V$. We say such a tower is \emph{maximally ramified}.
\end{defi}

\begin{prop}
  Each point $x\in V$ has a Zariski neighborhood which admits maximal ramification.
\end{prop}
\begin{proof}
It suffices to take a toroidal neighobhood.
\end{proof}
Now suppose the pair $(V,D)$ admits maximal ramification. Then (by basechange), so does the pair $(V', D')$ for any Zariski (or \'etale) neighborhood. We prove the following Lemma.
\begin{lm}
  Let $V^{(N,\t{ram})}$ be a Galois tower of ramified coverings of $V$ which attains maximal ramification. Let $\Gamma_N^{\t{ram}}$ be the corresponding tower of Galois groups. As before, let $V^{(\infty, \t{ram})}, \Gamma_\infty^{\t{ram}}$ be the inverse limit schemes in the locally affine sense, and let $\qcoh^{a.l.}(V^{(\infty, \t{ram})})$ be the category of sheaves $\F$ such that the map $\mathcal{I}^{(\infty, \t{ram})}\otimes\F \to \F$ is an isomorphism. Then the category $\qcoh^{a.l.}(V^{(\infty,\t{ram})})^{\Gamma_\infty\t{ram}}$ is canonically equivalent (as a symmetric monoidal category) to $\qcoh^{a.l.}(V^{(\infty)})^{\Gamma_\infty}$.
\end{lm}
\begin{proof}
$V^{(\infty)}$ is a principal bundle over $V^{(\infty,\t{ram})}$ with respect to the group $$\Gamma_{\infty}^{\t{unram}}: = \ker(\Gamma_\infty\to \Gamma_\infty^{\t{ram}}).$$ It follows that we have an equivalence of the $\Gamma_\infty^{\t{unram}}$-invariants of the pre-log-coherent category with the category coming from the ramified tower: $$\qcoh(V^{(\infty)})^{\Gamma_\infty^{\t{unram}}}\cong \qcoh(V^{(\infty, \t{ram})}).$$ Now the short exact sequence of groups $\Gamma_{\infty}^{\t{unram}}\to \Gamma_\infty\to \Gamma_\infty^{\t{ram}}$ gives an identification $$\qcoh(V^{(\infty)})^{\Gamma_\infty}\cong \qcoh(V^{(\infty, \t{ram})})^{\Gamma_\infty^{\t{ram}}}.$$
(see e.g. \cite{vtor}, section 2). Further, it is clear that this equivalence is symmetric monoidal and takes the equivariant boundary ideal $\mathcal{I}_\infty\in \qcoh(V^{(\infty)})^{\Gamma_\infty}$ to its ramified analogue $\mathcal{I}_\infty^{\t{ram}}\in \qcoh(V^{(\infty, \t{ram})})^{\Gamma_{\infty}^{\t{ram}}}$. Thus, it induces an equivalence on the subcategories of almost local objects.
\end{proof}
Now suppose that $V ( = (U, V, D))$ is a space with boundary which admits maximal ramification and $V_i$ is a Zariski (or an \'etale) cover of $V$. Then a maximally ramified tower $V^{(N)}$ over $V$ induces maximally ramified systems over the $V_i$, and it is clear that logarithmic categories defined in terms of restrictions of the same maximally ramified tower glue in either an \'etale or Zariski sense on $V$. Thus, the pre-log category of the space with boundary $V$ coincides with its log category (computed in terms of any cover, and hence also in the limit). Since the intersection of maximally ramifiable \'etale (or Zariski) subsets is maximally ramifiable, we see that the log-coherent category of an arbitrary space with boundary $V$ can be glued out of pre-log-coherent categories on any \'etale cover by maximally ramifiable opens. An analogous result holds for parabolic sheaves: when a space with boundary $V$ is maximally ramifiable, its category of pre-parabolic sheaves coincides with its category of parabolic sheaves, and these patch to give a definition of the \emph{parabolic coherent sheaf category} $$\qcoh_{par}(U, V, D)$$ for any space with boundary. This definition coincides with the notion in \cite{talpo-vistoli} of parabolic coherent sheaves on the infinite root stack (corresponding to the canonical log structure on the toroidal boundary).

\section{Toric mirror symmetry results}
Here we describe in detail the log-coherent category for a toric variety. Let $(T, X, D)$ be a toric variety, viewed as a torus with toric boundary. Let $T=\spec(k[\M])$ be a torus with $\M$ a lattice, $\mathcal{N} : = \M^\vee$, and $\Sigma$ a fan in $\mathcal{N}_\rr^\vee$ (a set of cones $\sigma\in \M_\rr^\vee$). The $N$-power maps $\tau_N:T\to T$ can be extended to ``geomoetric Frobenius'' maps $\tau_N:X\to X$, which attain maximal ramification at every point of $X$. Write $X^{(N)}$ for the tower of toric varieties, compactifying \'etale tower of tori $T^{(N)}$. These correspond to lattices $\mathcal{N}^{(N)} : = N\mathcal{N}$ and $\M^{(M)} : = \frac{1}{N}\M.$ The space $X$ has an atlas by affines $X_\sigma$ for $\sigma\in \Sigma,$ with rings of functions $\oo_\sigma : = k[\M\cap \sigma^\vee].$ The preimages $X_\sigma^{(N)} = X^{(N)}\times_X X_\sigma$ in the tower have rings of functions $k[\frac{1}{M}\M],$ so that the affine limit $X_\sigma^{(\infty)} = X^{(\infty)}\times_X X_\sigma$ is affine with ring of functions $k[\M_\qq\cap \sigma^\vee]$ (here we are using that $\underset{\to}{\text{lim}} \frac{1}{N}\M\cap \sigma^\vee = \M_\qq\cap \sigma^\vee$). Now the Galois groups $\Gamma_N = (\zz/N)^n$ are, as affine schemes, given by $\spec k[\M/\frac{1}{N}\M]$, and the limit is $$\Gamma_\infty : = \spec \frac{\M_\qq}{\M}.$$ (Note that there is also an action of the larger group $T^{(\infty)} = \spec k[M_\qq]$, representing the ``pro-toric equivariance,'' and the pro-Galois $\Gamma_\infty$ action is then induced from the mapping $\Gamma_\infty\to T^{(\infty)}$ induced from the quotient map on groups in the other direction.)

Now the category of $\Gamma_\infty$-equivariant spaces is (symmetric monoidally) equivalent (see \cite{vtor} section 2) to the category of $\frac{\M_\qq}{\M}$-\emph{graded} spaces. In particular, the action of $\Gamma_\infty$ on $X^{\infty}$ comes from $\frac{\M_\qq}{\M}$-graded (ring) structure on the $\sigma\cap \M_\qq,$ with grading via the composition $$(\sigma\cap \M_\qq)\to \M_\qq\to \frac{\M_\qq}{\M}.$$ The sheaf of ideals $\mathcal{I}$ is patched locally out of the (graded) $\oo_\sigma^{(\infty)}$-ideals $$\mathcal{I}_\sigma : = k[M_\qq\cap \mathring{\sigma}^\vee],$$ spanned by the interior of the corresponding cones.

\subsection{Microlocal mirror symmetry}
The main result of \cite{vtor} (for rational fans) can be written as follows: let $S(\M)$ be the topological torus associated to $\M = \mathcal{N}^\vee.$ Let $\operatorname{Shv}\left(S(M)\right)$ be the category of all topological sheaves on the torus with (bounded) derived category $$D^b\operatorname{Shv}(S(\M)).$$ Then if $\Lambda\subset T^*\left(S(\M)\right)$ is a polyhedral coisotropic subset of the tangent space, there is a notion from microlocal analysis (see \cite{kashapira}) of ``derived topological sheaves with a singular support condition'' $$D^b\operatorname{Shv}_\Lambda\left(S(\M)\right)$$ (usually, one considers constructible sheaves, but if $\Lambda$ is polyhedral, topological sheaves work as well). For $\Sigma$ a fan, define its ``support'' $|\Sigma|\subset \mathcal{N}$ to be the union of all its cones, $$|\Sigma| : = \sqcup_{\sigma\in \Sigma}\sigma\subset \mathcal{N}.$$

Using the fact that $S(\M)$ is a group and therefore has canonically trivial cotangent bundle, we have $T^*S(\M) \cong N_\rr\times S(\M).$ In terms of this condition, we define the polyhedral coisotropic subset $$\Lambda : = S(\M)\times |\Sigma| \subset T^*S(\M)$$, with $|\Lambda|$ over every point of $S(\M)$. Note also that if $X'\to X$ is a toric map of toric varieties, this corresponds to a map of cones $\Sigma'\to \Sigma$ (i.e., an inclusion of each cone $\sigma'$ in some cone $\sigma\in \Sigma$), and in particular an inclusion of support subsets, $|\Sigma'|\subset |\Sigma|\subset \mathcal{N}_\rr$. This inclusion is an equality if and only if $\Sigma'$ is a \emph{refinement} of $\Sigma,$ which is the case if and only if the map of toric varieties $X'\to X$ is \emph{proper}.
In \cite{vtor}, the author proves the following result.
\begin{thm}[Main theorem of \cite{vtor}]\label{mainfromvtor}
There is an equivalence of derived categories $$D^b\qcoh_{log}(T, X, D)\cong D^b\operatorname{Shv}_{\Delta}\left(S(\M)\right).$$ Further, if $\Sigma\to \Sigma'$ is a map of fans (corresponding to a toric map of toric varieties $X'\to X$) then we have an inequality on supports: $\Lambda\subset \Lambda'$, so $D^b\operatorname{Shv}_{\Delta}$ is naturally a (fully faithful, dg) subcategory of $D^b\operatorname{Shv}_{\Delta'}$. The inclusion functor then is compatible with the pullback functor on coherent categories $$D^b\qcoh_{log}(T, X', D')\to D^b\qcoh_{log}(T, X, D).$$
\end{thm}
The key consequence of this result is the following:
\begin{cor}
  Suppose $\beta:X'\to X$ is a proper toric map of toric varieties. Then derived pullback induces an equivalence of derived categories, $D^b\qcoh_{log}(X)\cong D^b\qcoh_{log}(X').$
\end{cor}
We deduce the following theorem.
\begin{thm}\label{toroidal-inv}
  Given any strictly toroidal map $\beta:(U', X', D')\to (U, X, D)$, the pullback map on derived categories $D^b\beta^*:D^b\qcoh_{log}(U, X, D)\to D^b\qcoh_{log}(U', X', D')$ is an equivalence of derived categories.
\end{thm}
\begin{proof}
First, suppose that $V'\to V$ is a map of toric varieties with $V$ affine and $U\subset V$ is an open subset. Let $U'$ be the preimage of $U$. Then we claim that the functor $D^b\qcoh_{log}(U)\to D^b\qcoh_{log}(U')$ is an equivalence of categories. This follows from the fact that the map $D^b\qcoh_{log}(V)\to D^b\qcoh_{log}(V')$ is fibered (locally) over $V$, and so the equivalence of derived categories is compatible with tensor product by any object of $\qcoh_{log}(V)$. In particular, $D^b\beta^*(\F)\otimes \oo_U\cong D^b\beta^*(\F\otimes \oo_U),$ which implies the lemma.

Further, we note that $D^b\qcoh_{log}$ is a sheaf of dg categories in the \'etale topology: this follows from the corresponding statement for the derived category of (ordinary) quasicoherent sheaves, which is known. The theorem now follows from Theorem \ref{mainfromvtor} and the definition of strict toroidal maps.
\end{proof}
\begin{cor}
  Suppose $(U, X, D)$ is a normal-crossings space with boundary. Given any blow-up map $\beta:X'\to X$ along a smooth center normally embedded in one of the normal-crossings components, the functor $D^bj^*:D^b\qcoh_{log}(X)\to D^b\qcoh_{log}(X')$ is fully faithful.
\end{cor}
\begin{proof}
  For any such blow-up map we can, \'etale locally near any point $x\in X$, enlarge the divisor $D$ so that $\beta$ is a blow-up along a full normal-crossings component. We then use that derived full faithfulness can be checked locally, and that the pullback map $\qcoh_{log}(U, X, D)\to \qcoh_{log}(U', X, D')$ is fully faithful if $D\subset D'$ is a normal crossings sub-divisor.
\end{proof}
\begin{rmk}
  In our definition of strict toroidal map, we used \'etale neighborhoods. We could instead use formal neighborhoods: indeed, the two definitions agree when the compactification $X$ is compact. When $X$ is not compact, we have a diagram of Serre categories

  \begin{diagram}D^b\qcoh_{log}\big(X'\setminus X'_x\big)^{nilp}&\rTo &D^b\qcoh_{log}(X')&\rTo&D^b\qcoh_{log}(X')_{\hat{x}}\\
    \uTo&&\uTo&&\uTo\\
    D^b\qcoh_{log}\big(X\setminus X_x\big)^{nilp}&\rTo &D^b\qcoh_{log}(X)&\rTo&D^b\qcoh_{log}(X)_{\hat{x}}
\end{diagram}
which by quasicompact induction reduces the problem to checking equivalence of the map $D^b\qcoh_{log}(U, X, D)_{\hat{x}}^{nilp}\to D^b\qcoh_{log}(U', X', D')_{\hat{x}}^{nilp}$ of support subcategories, which follows from the result on toric varieties by observing that a support condition can be expressed as vanishing of tensor product by a certain sheaf on $X$.
\end{rmk}
\subsection{The universal nodal elliptic curve}
A particularly interesting example of this equivalence is as follows. Let $(\bar{\mathbb{E}}, \mathbb{E}, \mathbb{E}_\infty)$ be the orbifold with boundary corresponding to the universal elliptic curve over the modular orbifold with boundary $(Y(1), X(1), \infty) = (\bar{\M_{1, 1}}, \M_{1, 1}, \infty)$. The modular orbifold has (ramified) Galois covers $X(N)$ classifying curves with level structure (a basis for the $N$-torsion lattice) and there is a universal family $\bar{\mathbb{E}}(N)$ over $X(N)$, which can also be viewed as an orbifold with boundary; indeed, if $N\ge 3,$ these orbifolds will be honest spaces (so that the formalism of this paper applies without modification). Let $PGL_2(\qq_p)$ be the $p$-adic topological group and let $K_N\subset PGL_2(\qq_p)$ be the level-$N$ subgroup (which only depends on the $p$-adic valuation of $N$). It is well-known that the Hecke algebra $K_N\backslash PGL_2(\qq_p)/K_N$ acts on $X_N$ by correspondences. This action extends to an action of the algebra $K_N'\backslash GL_2(\qq_p)/K_N'$ on the open universal family $\mathbb{E}(N)$. (Here $K_N'\subset GL_2(\qq_p)$ is a level group). However, this action does not extend to the compactification $\bar{\mathbb{E}}_N$. The reason for this is that the fiberwise $\ell$-power map $[\ell]: \mathbb{E}\to \mathbb{E}$ does not extend to $\bar{\mathbb{E}},$ but rather extends to a map $\bar{\mathbb{E}}_{+\ell}\to \bar{\mathbb{E}}$, where $\bar{\mathbb{E}}_{+\ell}$ is the $\ell-1$-fold blow-up of the univeral elliptic curve at the node of $\mathbb{E}_\infty$ (this is the nodal family classified by a map $X(1)\to \M_{1, \ell^2}$, with the $\ell^2$ points at the node arranged as sets of $\ell$th roots of unity in a necklace of $\ell$ copies of $\pp^1$). Theorem \ref{toroidal-inv} implies that the derived log-coherent category of $(\mathbb{E}, \bar{\mathbb{E}}_{+\ell}, \mathbb{E}_{\infty, +\ell})$ is equivalent to that of $(\mathbb{E}, \bar{\mathbb{E}}, \mathbb{E}_\infty),$ and in fact the action of the full Hecke algebra $K_N'\backslash GL_2(\qq_p)/K_N'$ can be reconstructed on a dg categorical level.

\section{Forms and Hochschild homology}
From this section onwards, we assume that the space with boundary $V$ has normal-crossings (rather than toroidal) compactification. Up to passing to derived categories, we can reduce to this case using Theorem \ref{invariance}. Recall that we defined the category $\qcoh_{par}$ of parabolic coherent sheaves as sheaves on the orbifold locally (on maximally ramifiable opens) given by $\qcoh(V^{(\infty)})^{\Gamma_\infty}.$ Now the category $\qcoh_{log}(V)$ is a subcategory of $\qcoh_{par}(V)$, hence is enriched in $\qcoh_{par}(V)$ (note that also, a fortiriori, it is enriched over the category of coherent sheaves on $V$). For a couple of sheaves $\F, \G\in \qcoh_{log}(V)$, write $\homu(\F, \G)\in \qcoh_{par}(V)$ and $\rhomu(\F, \G)\in D^b\qcoh_{par}(V)$, for these enriched versions of $\hom.$ Now while $\qcoh_{log}$ is not Morita equivalent to the category of modules over a ring, it nevertheless has a good notion of Hochschild homology and cohomology: these are defined more generally for any dg (or indeed spectrally enriched) category by Blumberg and Mandell in \cite{blumberg-mandell}. The homology is defined as a very large complex associated to a simplicial space with simplices parametrized by (multilinear combinations of) cycles of objects with composable morphisms
\[
\begin{tikzcd}[row sep=3.6em,column sep=1em]
& X_0\arrow[rr,"f_1"] && X_1 \arrow[dr,"f_2"] \\
X_r\arrow[ur,"f_0"]  &&  && X_2\arrow[dl, "f_3"] \\
& X_{r-1} \arrow[ul,"f_{r-1}"] && X_3 \arrow[ll, dotted, no head]
\end{tikzcd}
\]
(not necessarily commutative), and face and degeneracy morphisms come from composing edges or including a unit edge. Note that the diagram is cyclically asymmetric with the arrow $f_0$ playing a special role (corresponding to the basepoint in the standard cell decomposition of the circle). Here to make the definition precise we need to take a small cofibrantly generating piece of the category (for example, the derived category of countably generated modules should do the trick), and any two such choices produce canonically isomorphic homology, see \cite{blumberg-mandell}, so long as they both are contained in a third small subcategory. In particular, the $\qcoh_{log}$-enrichment of the arrow $f_0$ extends to an enrichment of the entire diagram, and we have a \emph{parabolic} Hochschild homology object $$CH_*^{par}(\qcoh_{log}V)$$ defined by the same diagram with the equivariant $V^{(\infty)}$ action taken into account. It is easy to see (analogously to the case of ordinary sheaves) that the Hochschild homology is \'etale local on the space with boundary $V$ (indeed, even formally local in an appropriate sense). Suppose WLOG $V$ is affine and maximally ramifiable. Define $T_\infty^\vee$ for the tangent sheaf of $V_\infty,$ given as a module by the quotient of $V_\infty^{\otimes 2}$ with image of $f\otimes g$ written $f dg$ with $f, g\in V_\infty$ and quotiented out by the standard relation $$x d(y_1 y_2) = xy_1 d y_2 + xy_2 dy_1.$$ This is a $\Gamma_\infty$-equivariant sheaf (it can be interpreted as $\tor_*^{V\times V}(\oo_\Delta, \oo_\Delta)$), and as it naturally lives in $V\times V$. Define $$(T_\infty^\vee)^{a.l.} : = \hom(\mathcal{I}, \mathcal{I}\otimes_{\oo_\infty}T_\infty^\vee).$$ This sheaf is naturally $\Gamma_\infty$-equivariant: write $T^\vee_{par}$ for the resulting object of $\qcoh_{par}$.

The main result of this section is the following theorem.
\begin{thm}
  \begin{enumerate}
  \item There is a natural map of complexes of parabolic sheaves $T_{log}^\vee[1]\to CH_*(\qcoh_{log})$.
  \item The map induced by graded commutative multiplication, $\Lambda^*T_{log}^\vee[1]\to CH_*^{par}(\qcoh_{log})$ is an isomorphism.
  \item Locally (for maximally ramifiable, affine $V$), we have $$R\Gamma\Big(V, CH_*^{par}\left(\qcoh_{log}(V)\right)\Big)\cong \Gamma(V, \Omega^*_{log}),$$ for $\Omega^*_{log}$ the sheaf of logarithmic differentials.
  \end{enumerate}
\end{thm}
To prove this, we compare to the internal Hochschild homology object $CH_*^{par}(\qcoh_{par}(V))$ of $\qcoh_{par}$ itself. On the one hand, we have a map $$T^\vee_{par}\to CH_*^{par}(\qcoh_{par}(V)),$$ induced by observing that $T^\vee(\Gamma^\infty) = 0$ in characteristic zero (indeed, a differential form on $\Gamma^\infty$ must factor through a differential form on some discrete $\Gamma^N$, hence is zero). On the other hand, we have a natural map $CH_*^{par}(\qcoh_{par})\to CH_*^{par}(\qcoh_{log}),$ given by taking a diagram of the form
\[
\begin{tikzcd}[row sep=3.6em,column sep=1em]
& X_0\arrow[rr,"f_1"] && X_1 \arrow[dr,"f_2"] \\
X_r\arrow[ur,"f_0"]  &&  && X_2\arrow[dl, "f_3"] \\
& X_{r-1} \arrow[ul,"f_{r-1}"] && X_3 \arrow[ll, dotted, no head]
\end{tikzcd}
\]
with $X_i\in \qcoh_{par}$ and replacing it with the diagram of $\mathcal{I}\otimes X_i.$ In particular, observe that for each such diagram, there is a map $\hom_{V^{(\infty)}}(\mathcal{I}, \hom(X_r, \mathcal{I}\otimes X_0))\to \hom(\mathcal{I}\otimes X_r, \mathcal{I}\otimes X_0)$ which is compatible with simplicial structure. This gives us a canonical map $\hom(\mathcal{I}, \mathcal{I}\otimes CH_*^{par}(\qcoh_{par}))\to CH_*^{par}(\qcoh_{log}V).$ Already on the level of $T^\vee_{par},$ this gives us logarithmic forms. To see this, up to application of an \'etale local change of coordinates we can assume that $V$ is a standard genearlized normal-crossings triple, $\aa^{n-k}\times NC_k.$ In fact, it will be enough for us to consider the simplest case $$V = NC_1 : = (\gg_m, \aa^1, \{0\}).$$ Here for any rational $q> 0$, we have a homogeneous form $\frac{1}{q}d x^{q} = x^{q-1}dx\in T^\vee_{par}.$ Now we have an obvious map $T^\vee_{par}\to \hom(\mathcal{I}, \mathcal{I}\otimes T^\vee_{par}).$ Now note that we have an additional map, which we call $d\log x:\mathcal{I} \to \mathcal{I}\otimes T^\vee_{par},$ which is the map sending $x^q\mapsto \frac{1}{q} d x^q$ for $q> 0$ (i.e. $x^q\in \mathcal{I}$). Thus we get a copy of $k[x^{\qq_+}]d\log x$ sitting inside $CH_*^{par}(\qcoh_{log})$.

Now we use the following results that follow from arguments in \cite{vtor}.
\begin{thm}
  \begin{enumerate}
  \item   $HH_*\qcoh_{log}(\pp^1, \gg_m, \{0,\infty\}) \cong k\oplus k[1].$
  \item   $HH_*\qcoh_{log}(\aa^1, \gg_m, \{0\})\cong k[x]\oplus k[x][1].$
  \end{enumerate}

\end{thm}
The first result follows from a computation of the self-tor of the skyscraper sheaf of the (in fact, anti-\footnote{because we are conflating $\qcoh_{log}$ and $\qcoh_{log}^{op},$ which necessitates a sign.}) diagonal circle $\Delta \subset S^1\times S^1$, which represents the identity functor. For the second argument, let $\operatorname{Shv}_+(S^1)$ be the category of sheaves with singular support the positive (co)tangent ray. Let $\Delta\in S^1\times S^1$ be the anti-diagonal. Then the category of dg functors $D^b\operatorname{Shv}_+(S^1)\to D^b\operatorname{Shv}_+(S^1)$ is classified by complexes of sheaves on $S^1\times S^1$ which have singular support contained in the quadrant $\rr_+\times \rr_+.$ The initial such object living over the (anti-)diagonal sheaf $\Delta$ is the pushforward of the constant sheaf on the semi-cone $$\frac{\{(x, y)\mid x-y> 0\}}{(x, y)\sim (x+1, y-1)}.$$ Now choosing tilted coordinates on the torus, this semi-cylinder becomes just a copy of $\rr\times S^1$, mapping to $S^1\times S^1.$ Using standard equivariance and push-pull arguments, the self-tor then becomes $\bigoplus_{k\in \zz}H_*(\rr_+\times S^1 \delta_{\rr_+ + k});$ the resulting space will be isomorphic to $k[x] H_*(S^1)$, where by tracing through the arguments, one can see that multiplication by $x$ on the two sides is intertwined. By using the same argument after quotienting out a finite subgroup of $S^1,$ we see that indeed, we have abstractly $CH_*^{par}(\qcoh_{log})\cong k[x^{\qq_+}].$ Thus we have a map $$d\log k[x^{\qq^+}]\to T^\vee_{log}\cong k[x^{\qq_+}],$$ which we know is injective and compatible with multiplication by $x^n$ by a restriction to $\gg_m$ argument. Further, we know that the two sides have a common finite-dimensional equivariant quotient (corresponding to the $\pp^1$ case). It follows that the map from the LHS to the RHS above is an isomorphism. We deduce the general case $D_1^k\times \aa^1$ from the case of $D_1$ and the space without boundary $\aa^1$ (where the statements are well-known) by applying a simple hybrid of the K\"unneth formula in topology and in algerbaic geometry. Now both sides embed injectively in $HH_*(\gg_m)$, and from this we deduce compatibility with differential. This completes our proof of the HKR isomorphisms, and implies Theorem \ref{formstheorem}.

\subsection{Computation of $HH_*(\qcoh_{par}(NC_1)).$}
  Using the above result and exactness of $CH_*$ for Serre short exact sequences of categories (see e.g. Blumberg, Gepner and Tabuada, \cite{blumberg-gepner-tabuada}), we can compute the Hochschild homology of the \emph{parabolic} category $HH_*(\qcoh_{par}(NC_1)).$ Namely, we have a Serre short exact sequence $\vect_{\qq/\zz}\to \qcoh_{par}(NC_1)\to \qcoh_{log}(NC_1),$ where $\vect_{\qq/\zz}$ is the category of $\qq/\zz$-graded vector spaces, with Hochschild homology $k^{\oplus \qq/\zz}.$ In particular, we see that we have an isomorphism $HH_{\ge 2}(\qcoh_{par}(NC_1))\to HH_{\ge 2}(\qcoh_{log}(NC_1)).$  On the level of $HH_1,$ we use that $\Gamma^\infty= \spec k[\qq/\zz]$ has trivial tangent bundle to see that $HH_1(\qcoh_{par}(NC_1))\cong T^\vee_{par}\cong k[x^{\ge -1}]dx,$ so that the map $HH_1(\qcoh_{par}(NC_1))\to HH_1(\qcoh_{log}(NC_1))$ is one-dimensional. By exactness, this leaves the remaining $k^{\oplus \qq/\zz\setminus 0}$ elements of $HH_0\vect_{\qq/\zz}$ to get added, so that $HH_0(\qcoh_{par}(NC_1))\cong HH_0(\qcoh_{log}(NC_1))\oplus k^{\oplus \qq/\zz\setminus 0}\cong x \cdot k[x]\oplus k^{\oplus \qq/\zz}.$ It should be possible to deduce from this the sheaf version of $HH_*(\qcoh_{par}(V))$ for any normal crossings space with boundary $V$ (by using a K\"unneth formula and \'etale locality), but the answer is much messier than the homology of $HH_*(\qcoh_{log}(V)).$

\nocite{*}

redpajama_set_name: RedPajamaArXiv


text:
Council 'just left me' in abuser's care
By Dan Rowley on 4th April 2019 Comments Off on Council 'just left me' in abuser's care
SOLENT NEWS & PHOTO AGENCY
Rex Case was jailed for abusing five girls, including three foster children
A woman abused by her foster carer as a child has waived her right to anonymity to criticise the council involved in her care.
Abbey Richards was five years old when she was sent to live with Rex Case in Southampton. He was eventually jailed for abusing her and four other girls.
A serious case review revealed he was allowed to continue fostering despite several allegations made against him.
The city council said it was now better at listening to children in care.
Case and his wife, who lived in Bassett, became registered foster carers for Hampshire County Council in the early 1970s and for Southampton City Council from 1997.
The review by Southampton Safeguarding Children Board, published in September last year, revealed there had been concerns about him before he was approved but there was a culture of giving carers the "benefit of the doubt".
It said, during his time as a foster carer, concerns were raised about sexual abuse and he was also sacked from his job at a supported hostel for inappropriate behaviour with a 17-year-old girl.
While Ms Richards was a child his care in 1998, his house was attacked by a mob who wrote the word "paedophile" on his wall but when her mother, Tracey Collins, contacted social services expressing concern about her daughter, she said they were prevented from speaking to each other.
Case continued to be a registered foster carer for five years and it was not until further complaints were made in 2010 that the police uncovered abuse going back decades.
Ms Richards testified against Case, and in 2012 he was jailed for 21 years. In 2016 he was jailed for a further four years after admitting further offences.
Southampton City Council apologised at the time but Ms Richards said the conviction did not give her as much of a sense of justice as she hoped.
'No counselling'
She told the BBC: "The big thing for me was that there was nobody. You were just left there [in foster care] and there was no one to talk to.
"All these questions go around in your head. How it was allowed to happen?
"I was never offered any counselling or anything like that – you're just left and you've got to try and fit in to society."
A spokesman for the two councils involved said the review identified "gaps" in the way carers were recruited in the 70s and 80s.
He said: "Since the historic cases referenced in the report, significant changes to regulation have been implemented to ensure that children in care are safeguarded, listened to and their wellbeing is prioritised."
Council 'just left me' in abuser's care added by Dan Rowley on 4th April 2019

redpajama_set_name: RedPajamaCommonCrawl


The Duplex Evo reimagines the DNA of its 90s predecessor. modernizing a vintage runner with a fresh. lightweight design. A fully redesigned EVA outsole optimizes the new Duplex for the street with increased flexibility. cushioning. and durability. This sleeker. more techy update gives it a look that calls back to its classic roots. This version features breathable knit mesh for comfort and style.


Recently there have been some quotes taken from Jeremy Renner (here) on his involvement in the third Captain America film, saying there is a good chance that Hawkeye will make an appearance in Steve Roger's next solo installment.
Then, the other day the Captain America: The Winter Soldier co-director Joe Russo hinted at some of the developments that are happening in the third installment over at Marvel Studios. Coincidently, Russo talked about the potential involvement of Hawkeye and other members of The Avengers, hinting at who could potentially show up in the third Captain America outing.
Anthony also confirmed that a new draft of the script has just come through and it confirms the Winter Soldier (Sebastian Stan) will have a substantial role to play: "The relationship between Cap and the Winter Soldier was never resolved at the end of the last movie. Thinking about where that relationship can go is a piece of the puzzle for us for sure. That character is a wonderfully, beautifully tragic figure in the sense of is he the world's most feared assassin or is he the world's longest serving POW.
Full steam ahead for the Russo boys by the sounds of things.


text:
Sonoma Raceway 'not at immediate risk' from North Bay fires
By Tony DiZinnoOct 9, 2017, 12:41 PM EDT
Photo: Sonoma Raceway
Unconfirmed reports circulated this morning on social media that Sonoma Raceway might be directly affected by significant fires overnight in Napa and Sonoma counties appear to be wide of the mark.
Sonoma Raceway released an update Monday morning confirming it is "not at immediate risk."
A statement from from Sonoma Raceway President and General Manager Steve Page as of 9:30 a.m. PT on Monday morning is below, with the track confirming additional updates will be shared as needed.
"All of us at Sonoma Raceway extend our heartfelt thoughts and prayers to those who have been touched by the devastating North Bay fires. Our facilities team and a number of local fire companies have been battling grassland fires on Cougar Mountain and elsewhere around our property, and at this point it does not appear any of the raceway's structures or other facilities are at immediate risk. We appreciate all of the good wishes that have flooded in from friends around the state and country."
Napa, Santa Rosa and Petaluma appear to be some of the areas significantly affected. Highway 121 just outside the track located on Arnold Dr. also was mentioned. More information is here via SFGate.com.
Sonoma Raceway hosts a round of the Monster Energy NASCAR Cup Series and Verizon IndyCar Series each year and in 2018 will also host the SCCA National Championship Runoffs in October.
Further south of Sonoma this weekend at Mazda Raceway Laguna Seca in Monterey, Calif., SRO hosts the California 8 Hours, a round of the Intercontinental GT Challenge, on a weekend that also features Mazda Motorsports' Global MX-5 Challenge. Pirelli World Challenge's Touring Car classes season finale, and the Pirelli GT3 Cup Trophy USA.
Sonoma sent through a few photos, below.


text:
Der Seltzbach () ist ein gut 33 km langer linker Zufluss der Sauer im Elsass (Region Grand Est).

Geographie

Verlauf
Der Seltzbach entspringt auf einer Höhe von etwa   in der Forêt de Gœrsdorf nordöstlich von Mitschdorf  in den Nordvogesen. Er fließt zunächst etwa 700 m durch ein Waldgebiet in südlicher Richtung. Danach führt sein Lauf an der Grenze zwischen den Gemeinden Mitschdorf und Lampertsloch entlang, vorbei an den Feldern und Wiesen des Gewanns In der Heimelsstraengen  auf seiner rechten und  dem Liebwald auf der linken Seite. Seine Laufrichtung ist nun nach Süd-Südwest ausgerichtet. Bei der Brehmmühle südöstlich von Mitschdorf und nördlich der Rue de Voyageurs (D677) knickt er scharf, fast rechtwinklig, nach Ost-Südost ab, seiner zukünftigen Hauptfließrichtung, welche er im Wesentlichen bis zu seiner Mündung beibehält.

Er erreicht nun den Westrand von Preuschdorf und wird dort auf seiner linken Seite von einem, aus dem Liebwald von Norden kommenden, Quellbach gestärkt. Etwas später wird er auf der rechten Seite von einem vom Kalmersberg kommenden Bach gestärkt, welcher auf manchen Karten  auch als der namensgebende Hauptquellbach angesehen wird. Östlich der Ortschaft wird er auf seiner rechten Seite vom Kinderlochgraben gespeist, welcher aus dem Südwesten kommt. Sein Lauf wird nun  für längere Zeit von der D28 begleitet. Er fließt nun durch eine Felder- und Wiesenlandschaft. Kurz bevor er Merkwiller-Péchelbronn erreicht, fließen ihm auf der linken Seite der Augraben und etwas später auf der rechten Seite der Willenbachgraben zu. Nachdem der Seltzbach den Ort verlassen hat, nimmt er auf seiner rechten Seite südlich von Wingertsfeld den Bruchgraben auf. Der Seltzbach passiert nun die Gemeinde Kutzenhausen. An der Stelle, wo ihm auf der rechten Seite der Sumpfgraben  zufließt, schwenkt der Seltzbach nach Nordosten ab und erreicht kurze Zeit später Soultz-sous-Forêts im Ortsteil Wolfsgarten.

Er wechselt nun seine Richtung nach Westen, fließt durch eine Grünanlage und unterquert die Rue du Docteur Michel Deutsch (D264). Kurz darauf werden auf seiner linke Seite seine Wassermassen durch den Froeschwillerbach, auch Froeschwillerbaechel genannt, verstärkt. Der Seltzbach unterquert nun die Gleisanlagen der SNCF und fließt nun durch Felder und Wiesen, unterquert nordöstlich von Hohwiller die D263 und kommt dann in Hoffen an, wo er bei der Bostaetter Mühle die D52 kreuzt. Bei der Saegmuehle, nordöstlich von Leiterswiller, empfängt er auf seiner linken Seite den vom Norden kommenden Haussauerbach. Dort spaltet sich auf seiner Südseite der Eichelgraben ab und dann südlich von Oberroedern auf seiner Nordseite der Eilgraben. Der Seltzbach unterquert nun die D245 und vereinigt sich kurz darauf wieder mit den Eichelgraben. Gleich darauf spaltet sich ein weiterer südlicher Graben vom Seltzbach ab. Südwestlich von Buhl fließt auch der nördliche Eilgraben wieder in den Seltzbach zurück. Bald danach mündet der Seebach, auch Seebaechel genannt, von der linken Seite in den Seltzbach ein. Bei der Gerittsmühle erhält der  Seltzbach auf seiner linken Seite durch den, aus dem Norden kommenden Warsbach eine weitere Stärkung. Er erreicht nun Niederrœdern und wird dort auf seiner rechten Seite vom Reissergraben gespeist.

Er verlässt diesen Ort und schlängelt sich am Nordrand des Waldes Hesselbusch entlang, unterquert dann die N363 und erreicht den Nordrand von Seltz, wo er auf seiner linken Seite den von Norden kommenden Eberbach aufnimmt. Er fließt nun zwischen Seltz und Niedersand und mündet schließlich auf einer Höhe von   in die Sauer.

Zuflüsse
 Kinderlochgraben (rechts), 3,3 km
 Fussel (Augraben) (links), 3,7 km
 Willenbachgraben (rechts)
 Bruchgraben (rechts)
 Sumpfgraben (rechts), 5,8 km
 Froeschwillerbach (links), 6,9 km
 Schindelbach (rechts)
 Haussauerbach (links), 14,5 km
 Seebach (Seebaechel) (links), 10,8 km
 Warsbach (links), 9,7 km
 Reissergraben (rechts), 5,6 km
 Eberbach (links), 5,6 km

Hydrologie
An der Mündung des Seltzbaches in die Sauer beträgt die mittlere Abflussmenge (MQ) 1,8 m³/s; das Einzugsgebiet umfasst hier 219,8 km².

Am Pegel Niederroedern wurde über einen Zeitraum von 47 Jahren (1965–2011) die durchschnittliche jährliche Abflussmenge des Seltzbaches berechnet. Das Einzugsgebiet  entspricht an dieser Stelle mit 202 km² etwa 92 % des vollständigen Einzugsgebietes des Flusses.

Die höchsten Wasserstände werden in den Monaten Dezember – März gemessen. Ihren Höchststand erreicht die Abflussmenge mit 3,32 m³/s im Februar. Von April an geht die Schüttung Monat für Monat merklich zurück und erreicht ihren niedrigsten Stand im August mit 0,45 m³/s, um danach wieder von Monat zu Monat anzusteigen. Der jahresdurchschnittliche Wert ist 1,67 m³/s.

Der  monatliche mittlere Abfluss (MQ) des Seltzbaches in m³/s an der hydrologischen Station NiederroedernDaten aus den Werten der Jahre 1965–2011 berechnet

Einzelnachweise

Weblinks
 Débits caractéristiques le Seltzbach (französisch; PDF-Datei; 15 kB)
  (PDF; 994 kB)

Gewässer in den Vogesen

redpajama_set_name: RedPajamaWikipedia


text:
Navigating the Boundaries of Identities
"If every path you take comes back to you, then you will never move ahead" (Kung Fu Panda)
During a meeting at the Asia Institute for Science and Technology in Bangkok, Thailand, I approached a student to ask about Canada's foreign assistance in South Asia. However, the topic of conversation quickly changed when I couldn't take my eyes off the black plastic dot that decorated her forehead. Known as the bindi, the dot is a common accessory for young Indian women. In this case, its large size almost overwhelmed the symmetry of her eyes.
"That sure is a big bindi," I said. She laughed and lifted her eyebrows to animate the bindi into a playful wiggle. Everyone has a distinguishing trademark, and this was hers.
"I'm known for my big bindis," she explained.
I asked her opinion about the origin of the bindi. "For most modern Indian women, it is just a decoration like a nose ring or bangles," she said when I asked for her opinion on the origin of the bindi. "I suspect men invented it to distract other men's attention away from their wives' beautiful eyes. For me, a bindi is a decoy. It's the first thing that strangers see before our eyes meet. It gives me a chance to give them a quick look-over."
The foreign aid debate was put aside. The origin of the bindi had to be settled. A small group gathered around to listen to my theory. I had heard that the bindi originated in religious rituals. After prayers, Hindu priests always bless male and female devotees with red powder called the tika. Some scholars say that the tika is a religious symbol representing a drop of blood of demons killed by the goddess Durga, a marker of victory of good over evil. Now that's the kind of cosmic power everyone should be proud to carry around on their heads.
What about the bindi from a feminist perspective? According to custom in rural India, only married women have the right to wear it. Young girls are not allowed to do so, and widows must give up their bindis. The discussion was rapidly turning into a feminist critique of the bindi. We questioned why widows didn't have the right to adorn their heads any way they wish. One observer pointed out that maybe widows were happy to be free from the mark.
"I think the bindi was originally used to mark slaves," said a student, "and married women in India are slaves to their husbands." That remark promoted a lot of muttering in the affirmative, and it looked for a moment as if these students were going to take to the streets in a mass revolt.
Cooler heads prevailed at the end of the day, and there were no demonstrations. Some young women noted that old taboos are changing these days. It's unclear whether the bindi symbolizes women's spiritual power, marital bondage, or maybe some kind of natural fashion sense. The bindi is obviously in the eye of the beholder. Whatever its origins, this simple plastic dot on a person's forehead is clearly full of cultural meaning and deeply embedded in issues of identity. More research needs to be done.
For now, my conclusion is simple: bindi is as bindi does–and big bindis do it better.


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text:
washingtonpost.com > Politics
Wall Street shifting political contributions to Republicans
By Dan Eggen and Tomoeh Murakami Tse
Commercial banks and high-flying investment firms have shifted their political contributions toward Republicans in recent months amid harsh rhetoric from Democrats about fat bank profits, generous bonuses and stingy lending policies on Wall Street.
The wealthy securities and investment industry, for example, went from giving 2 to 1 to Democrats at the start of 2009 to providing almost half of its donations to Republicans by the end of the year, according to new data compiled for The Washington Post by the Center for Responsive Politics.
Commercial banks and their employees also returned to their traditional tilt in favor of the GOP after a brief dalliance with Democrats, giving nearly twice as much to Republicans during the last three months of 2009, the data show. At the same time, total political donations by the major banks and investment houses alike dropped in the waning months of that year.
The nascent shift came even before the White House announced proposals for a new tax on banks and a curb on some of their riskiest trading activities.
The proposals, offered last month, particularly alarmed Wall Street and have triggered renewed industry efforts to work with Democrats as well as Republicans on regulatory reform legislation that the bankers can live with, according to industry and government officials. Wall Street executives would prefer to engage with Democratic leaders now rather than face prolonged uncertainty about the rules to govern the industry, the sources said.
The new campaign contributions data underscore the political quandary facing Democrats, who want Wall Street donations to help fend off a GOP resurgence in congressional elections this fall but hope to distance themselves from an industry vilified by the public as greedy and ungrateful. President Obama has sought to strike a balance, calling outsize Wall Street bonuses "shameful" and "obscene" while also assuring business executives that he does not "begrudge people success or wealth."
Republicans, meanwhile, are soliciting Wall Street for donations with the argument that Democratic proposals would hurt the bottom lines of major financial institutions. House Minority Leader John A. Boehner (R-Ohio) told reporters this month that he was urging Wall Street executives to "help our team" oppose the "bizarre policies" coming out of the Obama administration.
One senior Republican staff member on Capitol Hill, who discussed contributions on the condition of anonymity, said: "Democrats in Washington are clearly trying to move legislation that would be very damaging to that industry. It was almost like there was a free ride time. But now they're starting to see the real negative impact of Democratic proposals."
Obama had unusually strong backing from Wall Street for a Democratic presidential candidate. He raised more than $18 million from bank and brokerage employees, for example, compared with rival John McCain's $10 million. (Obama did not accept money from PACs.) Prominent among Obama's bundlers -- individuals who raised at least $50,000 -- were private equity executives and hedge fund titans, including billionaire Kenneth C. Griffin of Citadel Investment Group, who had previously backed Republicans.
But Obama soon encountered stiff opposition from the financial industry -- and some fellow Democrats -- over proposals to curb executive pay, tighten rules on financial derivatives and create an agency to protect consumers of mortgages, credit cards and other financial products. Financial executives have also bristled at the president's increasingly populist tone over the past year, including his quip in December that he did not run for office to help "fat-cat bankers on Wall Street."
The industry has responded with its own change in attitude, according to contribution data and interviews. For some prominent executives, the final straw came in January, when Obama proposed a fee on big banks to recoup losses from the government's $700 billion program to bail out financial firms. When the president followed up a few days later with another plan to restrict the growth of large banks, some on Wall Street said they regretted their earlier support. "I'm not voting for him again," one said.
Still, others said they would not switch alliances just yet. "I understand people are not happy about this. Wall Street did pour a lot of money into the campaign, some of which I solicited," said one Wall Street executive and Democratic bundler, who spoke on the condition of anonymity because of the sensitivity of the topic. "Having said that, we're kind of responsible for a lot of what went down."
CONTINUED 1
© 2010 The Washington Post Company


Months after their first date, the couple discovered they had been classmates in preschool, and one year into their relationship Justin arranged to have the young students from their former school hold up signs that asked, "Will you marry me? How to boost the odds with a better profile: Use recent pictures (taken within the past year) and at least one good close-up headshot.Show that you're humble through a joke, a self-effacing story or a humorous anecdote. To make a strong first impression, use anecdotes instead of a string of adjectives describing yourself.
Derfor vil sider tilegnet dating for gifte være et godt alternative, for at peppe din hverdag op.We created an online profile for each of the top dating websites and spent over 30 hours comparing each site's matching capabilities, search features and messaging tools.We took personality tests, answered over 300 dating-related questions and downloaded apps for all the sites that offered them.Træt af at være single - Tilmeld dig og hvem ved, det kan være, du finder den store kærlighed i Odense.var det første dating site som tilbudt en i Phone app – og den er stadigvæk helt gratis.
Most gratifying were the responses from some of you saying, "Omigod, this is happening to me right now! I'm in a similar situation; however, I believe he's the first man I've ever truly loved.


text:
G20 pledge climate action but make few commitments
Leaders posed for a picture by the famed Trevi Fountain on Sunday morning
The leaders of the world's richest economies have agreed to pursue efforts to limit global warming with "meaningful and effective actions".
But the agreement from the G20 summit in Rome made few concrete commitments, disappointing activists.
Host nation Italy had hoped that firm targets would be set before the COP26 summit in Glasgow, which has now begun.
UK PM Boris Johnson, who is hosting COP26 said leaders' promises without action were "starting to sound hollow".
"These commitments... are drops in a rapidly warming ocean," Mr Johnson said.
Watch US President Biden say Russia and China "basically didn't show up" on climate matters
US President Joe Biden said there were a "series of very productive meetings" at the G20, adding that he was "looking forward to continuing to make progress" on climate issues in Glasgow.
He also said China and Russia - who are among the world's biggest polluters - "basically didn't show up" on matters of the climate, which was "disappointing".
The G20 group, made up of 19 countries and the European Union, accounts for 80% of the world's emissions.
The communiqué, or official statement released by the leaders, also makes no reference to achieving net zero by 2050.
Net zero means reducing greenhouse gas emissions as much as possible, until a country is absorbing the same amount of emissions from the atmosphere that it is putting out.
Italian Prime Minister Mario Draghi did however say in his closing statement that all of the G20 countries are committed to reaching the target by the mid-century. Scientists say this must be achieved by 2050 to avoid a climate catastrophe, and most countries have agreed to this.
However China - the world's biggest polluter - and Russia have pushed that target out to 2060.
Neither China's President Xi Jinping, nor Russia's Vladimir Putin were in Rome for the conference, instead joining via video link. Their absence raised concerns that a deal would not be easy to reach.
Mr Putin said his plan was to "not just to reach carbon neutrality, but to make sure that within the next three decades, the amount of greenhouse gas emissions in Russia is lower than, for instance, that of our neighbours and colleagues in the EU."
The communiqué does pledge to stop financing new, unabated coal plants internationally by the end of this year - a strong message to coal-reliant nations such as China, Russia and India. But the G20 leaders stopped short of agreeing to end coal power in their own nations.
Mr Draghi said that they were closer to reaching the goal of providing $100bn a year in climate financing to developing countries - a key pledge which has been pushed out to 2023.
António Guterres, the UN's secretary-general, tweeted that he was leaving Rome with his "hopes unfulfilled" but not buried.
While I welcome the #G20's recommitment to global solutions, I leave Rome with my hopes unfulfilled — but at least they are not buried. Onwards to #COP26 in Glasgow to keep the goal of 1.5 degrees alive and to implement promises on finance and adaptation for people & planet. pic.twitter.com/c1nhIDbA8m
— António Guterres (@antonioguterres) October 31, 2021
Mr Johnson said it had been a "reasonable" G20, with some progress made, but there was still so much work to do that "right now, the Paris agreement and the hope that came with it is just a piece of paper".
Leaders are due to leave Rome and arrive in Glasgow on Sunday night ahead of their high-level climate talks on Monday at COP26.
Mr Johnson had a gloomy message ahead of the conference: "If Glasgow fails, then the whole thing fails."
Simple guide to climate change
Why the COP26 climate summit is important
Activists had expressed their disappointment over the communiqué, feeling it was not tough enough.
Oscar Soria, of the activist network Avaaz, told the agency there was "little sense of urgency" coming from the group, adding: "There is no more time for vague wish-lists, we need concrete commitments and action."
However, others have been keen to paint the G20 as a stepping stone to COP26, which will see delegates from nearly 200 countries gather to discuss cutting carbon emissions.
A US official told reporters the G20 was about "helping build momentum" before the leaders head to Glasgow, while France's President Emmanuel Macron told newspaper Journal du Dimanche that "nothing is ever written before a COP".
Watch Boris Johnson warn "if Glasgow fails, then the whole thing fails"
On Saturday, the first day of the two-day summit, focus was on Covid and the economy, with an agreement being reached on a global tax rate which will see the profits of large businesses taxed at least 15%.
It follows concern that multinational companies are re-routing their profits through low tax jurisdictions.
After years of talking, the planet is now at least 1.1C hotter than the pre-industrial level. Climate scientist want warming restricted to 1.5C to prevent climate catastrophe.
COP26 climate summit - The basics
Climate change is one of the world's most pressing problems. Governments must promise more ambitious cuts in warming gases if we are to prevent greater global temperature rises.
The summit in Glasgow is where change could happen. You need to watch for the promises made by the world's biggest polluters, like the US and China, and whether poorer countries are getting the support they need.
All our lives will change. Decisions made here could impact our jobs, how we heat our homes, what we eat and how we travel.
Read more about the COP26 summit here.
Why China's climate policy matters to us all
What action are the biggest carbon polluters taking?
'Moment of truth' as world meets at climate summit
9US police hunt monkeys missing after motorway crash
10'A Met Police officer secretly filmed me naked'


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Spitz's Manga Minute: Mahou Shoujo Madoka Magica: The Different Story vol. 1
Tomoe Mami is a rookie Magical Girl, whose philosophy differs from her allies when it comes to fighting the Witches who prey upon Humanity.
While she does what she must to ensure peoples lives are saved, her life has felt lonesome since she became a Magical Girl, and what she desires most is a friend.
Then one day she meets a girl named Sakura Kyouko; a spunky young girl inexperienced at Witch hunting.
Is this fellow magical girl a reliable ally, and even more, could she be the friend Tomoe Mami has longed for?
What I thought: "A terrific piece
of backstory, if a little short."
(Note: This is a review of the first volume
of The Different Story, not the series as a whole.)
Coming out of the tv anime and films, I thought that the story told was airtight for what it set out to accomplish, and did an excellent job at giving you the feeling that this story takes place in a tangible but mysterious world, and that there was a potential for there to be a lot of story out there that hasn't been told.
That being said, while the motivations of these characters made sense given what backstory you were given in the anime, Kyouko and Mami were still given the least attention of the quartet, and I was stoked to find out that more was explained in this series of manga volumes.
As far as I've read or heard, this story is canon, and it doesn't do anything that would have me believing otherwise, which is a huge plus, and I'm happy with what details this manga gives us as to who these two girls were and what they were like before what took place in the anime, but with that being said, I would have liked to have seen more.
The first volume is about 150 pages, and while it informs you of the types of people Kyouko and Mami were in their early days fighting Witches together, which was quite interesting and enjoyable to read and leads you up to the present day (of around the spot where the anime begins), it doesn't give you as much backstory as I would have hoped for.
Kyouko is maybe my favorite character in the series, and just seeing how she behaved and saw the world before her contract with Kyuubey made it well worth the read, not a whole lot is told here that you didn't already know coming out of the anime, and this is even more-so the case with Mami, whose background is briefly explained here but not much else.
The art is clean and detailed, and the new designs for Kyouko and some of the supporting characters are all terrific. I thought some of the action scenes were a little difficult to follow, but the story told here is perfectly coherent (if you've already seen the anime), wraps up nicely, and has me curious to see where the second volume goes.
I wouldn't recommend this series to anyone who is unfamiliar with the story told in the anime just because things related to the nature of the world aren't explained in detail here, but if you're a fan of the series, and especially if you have a soft spot for Kyouko or Mami, I highly recommend you check it out.
From → Manga
Spitz's Year End Wrap Up 2014 | Spitz's Soapbox
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So About Them "Videogames", July 2014 »


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Athletic Medical Services & Physical Forms
High School Sports on Pittsburgh Post-Gazette
Charles Lickert Memorial Award
Senior Citizens: Gold Card Free Admission
Meet our Athletic Director!
Senior Locker Decorating Policy
Employment Clearances and Requirements
MASD Venues
Schedulestar Schedules and Alerts Instructions
Stadium Project 2020
2021-2022 Athletic Event Season Passes
STUDENT & ADULT SEASON PASSES!
MASD will now offer STUDENTS and ADULTS the opportunity to buy season passes for ALL home athletic events for the entire school year!
Prices: $30 for adults and $15 for students
Take advantage of significant discounts off of entry prices at the gates.
ONLINE PURCHASES ONLY by clicking here ---> Purchase Season Passes!
Once purchase is completed, season pass cards are available for pickup in the athletic office Monday-Friday from 9AM-11AM on school days.
Passes are not applicable for away games. reserved side seating at varsity football games or any WPIAL or PIAA playoff event. Passes will NOT be replaced if lost or stolen!
WINTER SPORTS PHYSICAL EXAM DATE!
Winter sports physicals will be given on Saturday November 6th at the High School in Room 161. Physical examinations are free! Each student wishing to participate in a winter sport must have a physical examination on file in the athletic ofice prior to participation.
Students who have previously received a physical examination dated after June 1, 2021 can participate in athletics with the completion and submission of the Re-Certification form!
Athletic Event Tickets!
Facebook Athletics Page
Order past football games on DVD!
Mr. Charley Kiss
Athletic Director/Business Affairs
CKiss@mckasd.net
Welcome to the McKeesport Area School District Tiger Athletic's Page! We hope you find this website a useful tool for learning about our teams! Be sure to check out our District calendar to see practice and game times. In partnership with athletes, coaches, and parents, it is the mission of McKeesport Area School District to create an environment that will complement and enrich the educational experience for all students.
The McKeesport Area School District's athletic program is both voluntary and competitive. In order to participate, students must be physically healthy, academically qualified, and willing to make a significant personal commitment to maximize the development of their knowledge, skills and attitudes relative to the particular sport.
For many decades, McKeesport Area School District has been home to outstanding student-athletes and has a strong history of competitive teams. We are proud of the athletic accomplishments and the tradition of academic success of our students.
The Athletic Department of the McKeesport Area School District is committed to excellence both on and off the playing field.
Once a Tiger........Always a Tiger!


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Rita Ora in a Cropped Black Jumper and a Dark Blue Denim Jacket NYC 01/30/2023
Rita Ora in an All-black Sweatsuit and Carrying Miu Miu Bag in New York 01/31/2023
Hot Sales for 'Operation Napoleon' with 'GOT' Star Iain Glen
Amal Clooney Steps Down as Britain's Special Envoy as She Criticizes U.K. Government
Amal Clooney has resigned as Britain's special envoy on media freedom to protest the U.K. government's plans to break international law.
Clooney, an internationally-respected human rights lawyer, has joined several officials stepping down over the government's internal market bill, which ministers have admitted breaks international law by breaching Britain's withdrawal agreement with the European Union.
The wife of Hollywood superstar George Clooney denounced the move in a letter to U.K. Foreign Secretary Dominic Raab and announced she could no longer represent Britain and its government, calling Prime Minister Boris Johnson's actions "lamentable."
In her letter, released to media outlets via the International Bar Association in London, she writes, "I have been dismayed to learn that the government intends to pass legislation – the internal market bill – which, if enacted, would, by the government's own admission, 'break international law…' "
"Although the government has suggested that the intended violation of international law is 'specific and limited', it is lamentable for the U.K. to be speaking of its intention to violate an international treaty signed by the prime minister less than a year ago."
Clooney claimed she had held back from stepping down as a special envoy but she "received no assurance that any change of position is imminent", and that left her with "no alternative but to resign from my position."
She continued, "I am disappointed to have to do so because I have always been proud of the U.K.'s reputation as a champion of the international legal order, and of the culture of fair play for which it is known. However, very sadly, it has now become untenable for me, as special envoy, to urge other states to respect and enforce international obligations while the U.K. declares that it does not intend to do so itself."
The 42-year-old took up the envoy post last year (19).
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Sounds fun! I understand how you feel about both of those products lol. I started using the Low Porosity Leave In Conditioner in place of the Generic PM Leave In and I like it. As for the Hello Hydration, still havent found a natural product like it. Looking forward to hearing about your results!


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De herfsterebia (Erebia neoridas) is een vlinder uit de onderfamilie Satyrinae van de familie Nymphalidae, de vossen, parelmoervlinders en weerschijnvlinders.

De soort komt voor in het oosten van de Pyreneeën, bergachtige gebieden van zuidoost Frankrijk en het aansluitende deel van Italië. De vlinder heeft een voorvleugellengte van 19 tot 23 millimeter. De soort vliegt van juli tot september. De vlinder vliegt op hoogtes van 500 tot 1600 meter boven zeeniveau. De habitat bestaat uit grasland met struiken en open plekken in bossen.

De waardplanten van de herfsterebia komen uit de grassenfamilie. De soort overwintert als rups.

Externe link

Satyrinae
Dier uit het Palearctisch gebied
IUCN-status niet bedreigd