\newcommand{\cat}[1]{\mathsf{#1}}

\newcommand{\Sets}[0]{{\mathsf{Set}}}
\newcommand{\Set}[0]{{\mathsf{Set}}}
\newcommand{\sets}[0]{{\mathsf{Set}}}
\newcommand{\set}{{\mathsf{Set} }}
\newcommand{\Poset}[0]{\mathsf{Poset}}
\newcommand{\GSets}[0]{{G\dash\mathsf{Set}}}
\newcommand{\Groups}[0]{{\mathsf{Group}}}
\newcommand{\Grp}[0]{{\mathsf{Grp}}}

% Modifiers
\newcommand{\pre}[0]{{\mathsf{pre}}}
\newcommand{\fn}[0]{{\mathsf{fn}}}
\newcommand{\smooth}[0]{{\mathsf{sm}}}
\newcommand{\Aff}[0]{{\mathsf{Aff}}}
\newcommand{\Ab}[0]{{\mathsf{Ab}}}
\newcommand{\Assoc}[0]{\mathsf{Assoc}}
\newcommand{\Ch}[0]{\mathsf{Ch}}
\newcommand{\Coh}[0]{{\mathsf{Coh}}}
\newcommand{\Comm}[0]{\mathsf{Comm}}
\newcommand{\Cor}[0]{{\mathsf{Cor}}}
\newcommand{\Fin}[0]{{\mathsf{Fin}}}
\newcommand{\Free}[0]{\mathsf{Free}}
\newcommand{\Perf}[0]{\mathsf{Perf}}
\newcommand{\Unital}[0]{\mathsf{Unital}}
\newcommand{\eff}[0]{\mathsf{eff}}
\newcommand{\derivedcat}[1]{\mathbf{D} {#1} }
\newcommand{\Cx}[0]{\mathsf{Ch}}
\newcommand{\Stable}[0]{\mathsf{Stab}}
\newcommand{\ChainCx}[1]{\mathsf{Ch}\qty{ #1 } }
\newcommand{\Vect}[0]{{ \mathsf{Vect} }}
\newcommand{\loc}[0]{{\mathsf{loc}}}
\newcommand{\Bun}{{\mathsf{Bun}}}
\newcommand{\bung}{{\mathsf{Bun}_G}}


% Rings
\newcommand{\Local}[0]{\mathsf{Local}}
\newcommand{\Fieldsover}[1]{{ \mathsf{Fields}_{#1} }}
\newcommand{\Field}[0]{\mathsf{Field}}
\newcommand{\Number}[0]{\mathsf{Number}}
\newcommand{\Global}[0]{\mathsf{Global}}
\newcommand{\Ring}[0]{\mathsf{Ring}}
\newcommand{\CRing}[0]{\mathsf{CRing}}
\newcommand{\DedekindDomain}[0]{\mathsf{DedekindDom}}
\newcommand{\DVR}[0]{\mathsf{DVR}}

% Modules
\newcommand{\modr}[0]{{\mathsf{Mod}\dash\mathsf{R}}}
\newcommand{\modsleft}[1]{\mathsf{#1}\dash\mathsf{Mod}}
\newcommand{\modsright}[1]{\mathsf{Mod}\dash\mathsf{#1}}
\newcommand{\mods}[1]{{\mathsf{#1}\dash\mathsf{Mod}}}
\newcommand{\torsors}[1]{{\mathsf{#1}\dash\mathsf{Torsors}}}
\newcommand{\torsorsright}[1]{\mathsf{Torsors}\dash\mathsf{#1}}
\newcommand{\torsorsleft}[1]{\mathsf{#1}\dash\mathsf{Torsors}}
\newcommand{\comods}[1]{{{#1}\dash\mathsf{coMod}}}
\newcommand{\bimod}[2]{({#1}, {#2})\dash\mathsf{biMod}}
\newcommand{\Mod}[0]{{\mathsf{Mod}}}

\newcommand{\zmod}[0]{{\mathbb{Z}\dash\mathsf{Mod}}}
\newcommand{\rmod}[0]{{\mathsf{R}\dash\mathsf{Mod}}}
\newcommand{\kmod}[0]{{\mathsf{k}\dash\mathsf{Mod}}}
\newcommand{\gmod}[0]{{\mathsf{G}\dash\mathsf{Mod}}}
\newcommand{\grMod}[0]{{\mathsf{grMod}}}
\newcommand{\gr}[0]{{\mathsf{gr}\,}}
\newcommand{\mmod}[0]{{\dash\mathsf{Mod}}}
\newcommand{\Rep}[0]{{\mathsf{Rep}}}

% Vector Spaces and Bundles
\newcommand{\VectBundle}[0]{{ \Bun\qty{\GL_r} }}
\newcommand{\VectBundlerk}[1]{{ \Bun\qty{\GL_{#1}} }}
\newcommand{\VectSp}[0]{{ \VectSp }}
\newcommand{\VectBun}[0]{{ \VectBundle }}
\newcommand{\VectBunrk}[1]{{ \VectBundlerk{#1} }}
\newcommand{\Bung}[0]{{ \Bun\qty{G} }}


% Algebras
\newcommand{\Hopf}[0]{\mathsf{Hopf}}
\newcommand{\alg}[0]{\mathsf{Alg}}
\newcommand{\Alg}[0]{{\mathsf{Alg}}}
\newcommand{\scalg}[0]{\mathsf{sCAlg}}
\newcommand{\cAlg}[0]{{\mathsf{cAlg}}}
\newcommand{\calg}[0]{\mathsf{CAlg}}
\newcommand{\liegmod}[0]{{\mathfrak{g}\dash\mathsf{Mod}}}
\newcommand{\liealg}[0]{{\mathsf{Lie}\dash\mathsf{Alg}}}
\newcommand{\kalg}[0]{{\mathsf{Alg}_{/k} }}
\newcommand{\kAlg}[0]{{\mathsf{Alg}_{/k} }}
\newcommand{\kSch}[0]{{\mathsf{Sch}_{/k}}}
\newcommand{\rAlg}[0]{{\mathsf{Alg}_{/R}}}
\newcommand{\ralg}[0]{{\mathsf{Alg}_{/R}}}
\newcommand{\CCalg}[0]{{\mathsf{Alg}_{\mathbb{C}} }}
\newcommand{\cdga}[0]{{\mathsf{cdga} }}
\newcommand{\dga}[0]{{\mathsf{dga} }}
\newcommand{\Poly}[0]{{\mathsf{Poly} }}

% Schemes and Sheaves
\newcommand{\Loc}[0]{\mathsf{Loc}}
\newcommand{\Ringedspace}[0]{\mathsf{RingSp}}
\newcommand{\RingedSpace}[0]{\mathsf{RingSp}}
\newcommand{\LRS}[0]{\Loc\RingedSpace}
\newcommand{\DCoh}[0]{{\mathsf{DCoh}}}
\newcommand{\QCoh}[0]{{\mathsf{QCoh}}}
\newcommand{\Cov}[0]{{\mathsf{Cov}}}
\newcommand{\sch}[0]{{\mathsf{Sch}}}
\newcommand{\presh}[0]{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}
\newcommand{\prest}[0]{ {\underset{ \mathsf{pre} } {\mathsf{St} } } }
\newcommand{\Descent}[0]{{\mathsf{Descent}}}
\newcommand{\Desc}[0]{{\mathsf{Desc}}}
\newcommand{\FFlat}[0]{{\mathsf{FFlat}}}
\newcommand{\Perv}[0]{\mathsf{Perv}}
\newcommand{\smsch}[0]{{ \smooth\Sch }}
\newcommand{\Sch}[0]{{\mathsf{Sch}}}
\newcommand{\Schf}[0]{{\mathsf{Schf}}}
\newcommand{\Sh}[0]{{\mathsf{Sh}}}
\newcommand{\St}[0]{{\mathsf{Stack}}}
\newcommand{\Vark}[0]{{\mathsf{Var}_{/k} }}
\newcommand{\Var}[0]{{\mathsf{Var}}}
\newcommand{\Open}[0]{{\mathsf{Open}}}

% Homotopy
\newcommand{\CW}[0]{{\mathsf{CW}}}
\newcommand{\sset}[0]{{\mathsf{sSet}}}
\newcommand{\sSet}[0]{{\mathsf{sSet}}}
\newcommand{\ssets}[0]{\mathsf{sSet}}
\newcommand{\hoTop}[0]{{\mathsf{hoTop}}}
\newcommand{\hoType}[0]{{\mathsf{hoType}}}
\newcommand{\ho}[0]{{\mathsf{ho}}}
\newcommand{\SHC}[0]{{\mathsf{SHC}}}
\newcommand{\SH}[0]{{\mathsf{SH}}}
\newcommand{\Spaces}[0]{{\mathsf{Spaces}}}
\newcommand{\Spectra}[0]{{\mathsf{Sp}}}
\newcommand{\Sp}[0]{{\mathsf{Sp}}}
\newcommand{\Top}[0]{{\mathsf{Top}}}

% Infty Cats
\newcommand{\Finset}[0]{{\mathsf{FinSet}}}
\newcommand{\Cat}[0]{\mathsf{Cat}}
\newcommand{\Grpd}[0]{{\mathsf{Grpd}}}
\newcommand{\inftyGrpd}[0]{{ \underset{\infty}{ \mathsf{Grpd}} }}
\newcommand{\Fun}[0]{{\mathsf{Fun}}}
\newcommand{\Kan}[0]{{\mathsf{Kan}}}
\newcommand{\Monoid}[0]{\mathsf{Mon}}
\newcommand{\Arrow}[0]{\mathsf{Arrow}}
\newcommand{\quasiCat}[0]{{ \mathsf{quasiCat} } }
\newcommand{\inftycat}[0]{{ \underset{\infty}{ \Cat}  }}
\newcommand{\inftycatn}[1]{{ \underset{(\infty, {#1})}{ \Cat}  }}
\newcommand{\core}[0]{{ \mathsf{core} }}

% New?
\newcommand{\Prism}[0]{\mathsf{Prism}}
\newcommand{\Solid}[0]{\mathsf{Solid}}
\newcommand{\WCart}[0]{\mathsf{WCart}}

% Motivic
\newcommand{\Torsor}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}}
\newcommand{\Torsorleft}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}}
\newcommand{\Torsorright}[1]{{\mathsf{Torsor}\dash\mathsf{#1} }}
\newcommand{\Quadform}[0]{{\mathsf{QuadForm}}}
\newcommand{\HI}[0]{{\mathsf{HI}}}
\newcommand{\DM}[0]{{\mathsf{DM}}}
\newcommand{\hoA}[0]{{\mathsf{ho}_*^{\scriptstyle \AA^1}}}
\newcommand\Tw[0]{\mathsf{Tw}}
\newcommand\SB[0]{\mathsf{SB}}
\newcommand\CSA[0]{\mathsf{CSA}}
\newcommand{\CSS}[0]{{ \mathsf{CSS} } }

% Unsorted
\newcommand{\FGL}[0]{\mathsf{FGL}}
\newcommand{\FI}[0]{{\mathsf{FI}}}
\newcommand{\Fuk}[0]{{\mathsf{Fuk}}}
\newcommand{\Lag}[0]{{\mathsf{Lag}}}
\newcommand{\Mfd}[0]{{\mathsf{Mfd}}}
\newcommand{\Riem}[0]{\mathsf{Riem}}
\newcommand{\Wein}[0]{{\mathsf{Wein}}}
\newcommand{\gspaces}[1]{{#1}\dash{\mathsf{Spaces}}}
\newcommand{\deltaring}[0]{{\delta\dash\mathsf{Ring}}}
\newcommand{\dgens}[1]{\gens{\gens{ #1 }}}
\newcommand{\ctz}[1]{\, {\converges{{#1} \to\infty}\longrightarrow 0} \, }
\newcommand{\conj}[1]{{\overline{{#1}}}}
\newcommand{\complex}[1]{{#1}_{*}}

\newcommand{\floor}[1]{{\left\lfloor #1 \right\rfloor}}
\newcommand{\fourier}[1]{\widehat{#1}}
\newcommand{\embedsvia}[1]{\xhookrightarrow{#1}}
\newcommand{\openimmerse}[0]{\underset{\scriptscriptstyle O}{\hookrightarrow}}
\newcommand{\weakeq}[0]{\underset{\scriptscriptstyle W}{\rightarrow}}

\newcommand{\fromvia}[1]{\xleftarrow{#1}}
\newcommand{\generators}[1]{\left\langle{#1}\right\rangle}
\newcommand{\gens}[1]{\left\langle{#1}\right\rangle}
\newcommand{\globsec}[1]{{{\Gamma}\qty{#1} }}
\newcommand{\Globsec}[1]{{{\Gamma}\qty{#1} }}

\newcommand{\equalsbecause}[1]{\overset{#1}{=}}
\newcommand{\congbecause}[1]{\overset{#1}{\cong}}
\newcommand{\congas}[1]{\underset{#1}{\cong}}
\newcommand{\isoas}[1]{\underset{#1}{\cong}}
\newcommand{\addbase}[1]{{ {}_{\pt} }}
\newcommand{\ideal}[1]{\mathcal{#1}}
\newcommand{\adjoin}[1]{ { \left[ {#1} \right] } }
\newcommand{\polynomialring}[1]{ { \left[ {#1} \right] } }
\newcommand{\powerseries}[1]{ { \left[ {#1} \right] } }
\newcommand{\functionfield}[1]{ { \left( {#1} \right) } }
\newcommand{\htyclass}[1]{ { \left[ {#1} \right] } }
\newcommand{\formalpowerseries}[1]{ { \left[\left[ {#1} \right] \right] } }
\newcommand{\formalseries}[1]{ { \left[\left[ {#1} \right] \right] } }
\newcommand{\qtext}[1]{{\quad \operatorname{#1} \quad}}
\newcommand{\abs}[1]{{\left\lvert {#1} \right\rvert}}
\newcommand{\stack}[1]{\mathclap{\substack{ #1 }}} 


\newcommand{\localize}[1]{ \left[ { \scriptstyle { {#1}\inv}  } \right]}
\newcommand{\primelocalize}[1]{ \left[ { \scriptstyle { { \qty{ {#1}^c } }\inv}  } \right]}
\newcommand{\plocalize}[1]{\primelocalize{#1}}
\newcommand{\sheafify}[1]{ \left( #1 \right)^{\scriptscriptstyle \mathrm{sh}} }
\newcommand{\complete}[1]{{ {}^{ \hat{#1} } }}
\newcommand{\takecompletion}[1]{{ \overbrace{#1}^{\widehat{\hspace{4em}}}  }}
\newcommand{\pcomplete}[0]{{ {}^{ \wedge }_{p} }}
\newcommand{\kv}[0]{{ k_{\hat{v}} }}
\newcommand{\Lv}[0]{{ L_{\hat{v}} }}

\newcommand{\twistleft}[2]{{ {}^{#1} #2 }}
\newcommand{\twistright}[2]{{ #2 {}^{#1} }}
\newcommand{\liesover}[1]{{ {}_{/ {#1}} }}
\newcommand{\liesabove}[1]{{ {}_{/ {#1}} }}
\newcommand{\slice}[1]{_{/ {#1}} }
\newcommand{\quotright}[2]{ {}^{#1}\mkern-2mu/\mkern-2mu_{#2} }
\newcommand{\quotleft}[2]{ {}_{#2}\mkern-.5mu\backslash\mkern-2mu^{#1} }
\newcommand{\invert}[1]{{ \left[ { \scriptstyle \frac{1}{#1} } \right] }}
\newcommand{\symb}[2]{{ \qty{ #1 \over #2 } }}


\newcommand\cartpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} }  }}
\newcommand\sumpower[1]{{ {}^{ \scriptscriptstyle\oplus^{#1} }  }}
\newcommand\prodpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} }  }}
\newcommand\tensorpower[2]{{ {}^{ \scriptstyle\otimes_{#1}^{#2} }  }}
\newcommand\tensorpowerk[1]{{ {}^{ \scriptscriptstyle\otimes_{k}^{#1} }  }}
\newcommand\derivedtensorpower[3]{{ {}^{ \scriptstyle {}_{#1} {\otimes_{#2}^{#3}} }  }}
\newcommand\smashpower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} }  }}
\newcommand\fiberpower[2]{{ {}^{ \scriptscriptstyle\fiberprod{#1}^{#2} }  }}
\newcommand\powers[1]{{ {}^{\cdot #1} }}
\newcommand\skel[1]{{ {}^{ (#1) } }}
\newcommand\ltranspose[1]{{ {}^t{#1}} }
\newcommand\tmf{ \mathrm{tmf} }
\newcommand\taf{ \mathrm{taf} }
\newcommand\TAF{ \mathrm{TAF} }
\newcommand\TMF{ \mathrm{TMF} }

\newcommand{\BO}[0]{{\B \Orth}}
\newcommand{\EO}[0]{{\mathsf{E} \Orth}}
\newcommand{\BG}[0]{{\B G}}
\newcommand{\EG}[0]{{\mathsf{E} G}}
\newcommand{\BP}[0]{{\operatorname{BP}}}
\newcommand{\BU}[0]{{\operatorname{BU}}}

\newcommand{\MO}[0]{{\operatorname{MO}}}
\newcommand{\MSO}[0]{{\operatorname{MSO}}}
\newcommand{\MSpin}[0]{{\operatorname{MSpin}}}
\newcommand{\MSp}[0]{{\operatorname{MSpin}}}
\newcommand{\MString}[0]{{\operatorname{MString}}}
\newcommand{\MStr}[0]{{\operatorname{MString}}}
\newcommand{\MU}[0]{{\operatorname{MU}}}


\newcommand{\KO}[0]{{\operatorname{KO}}}
\newcommand{\KU}[0]{{\operatorname{KU}}}

\newcommand{\smashprod}[0]{\wedge}

\newcommand{\ku}[0]{{\operatorname{ku}}}
\newcommand{\hofib}[0]{{\operatorname{hofib}}}
\newcommand{\hocofib}[0]{{\operatorname{hocofib}}}
\DeclareMathOperator{\Suspendpinf}{{\Sigma_+^\infty}}


\newcommand{\Loop}[0]{{\Omega}}
\newcommand{\Loopinf}[0]{{\Omega}^\infty}
\newcommand{\Suspend}[0]{{\Sigma}}
\newcommand*\dif{\mathop{}\!\operatorname{d}}
\newcommand*{\horzbar}{\rule[.5ex]{2.5ex}{0.5pt}}
\newcommand*{\vertbar}{\rule[-1ex]{0.5pt}{2.5ex}}
\newcommand\Fix{ \mathrm{Fix} }
\newcommand\places[1]{ \mathrm{Pl}\qty{#1} }
\newcommand\Ell{ \mathrm{Ell} }
\newcommand\homog{ { \mathrm{homog} } }
\newcommand\Kahler[0]{\operatorname{Kähler}}
\newcommand\Prinbun{\mathrm{Bun}^{\mathrm{prin}}}
\newcommand\aug{\fboxsep=-\fboxrule\!\!\!\fbox{\strut}\!\!\!}
\newcommand\compact[0]{\operatorname{cpt}}
\newcommand\hyp[0]{{\operatorname{hyp}}}
\newcommand\jan{\operatorname{Jan}}
\newcommand\curl{\operatorname{curl}}
\newcommand\kbar{ { \bar{k} } }
\newcommand\ksep{ { k\sep } }
\newcommand\mypound{\scalebox{0.8}{\raisebox{0.4ex}{\#}}}
\newcommand\rref{\operatorname{RREF}}
\newcommand\RREF{\operatorname{RREF}}
\newcommand{\Tatesymbol}{\operatorname{TateSymb}}
\newcommand\tilt[0]{ {}^{ \flat } }
\newcommand\vecc[2]{\textcolor{#1}{\textbf{#2}}}
\newcommand{\Af}[0]{{\mathbb{A}}}
\newcommand{\Ag}[0]{{\mathcal{A}_g}}
\newcommand{\Ahat}[0]{\hat{ \operatorname{A}}_g }
\newcommand{\Ann}[0]{\operatorname{Ann}}
\newcommand{\Arg}[0]{\operatorname{Arg}}
\newcommand{\Art}[0]{\operatorname{Art}}
\newcommand{\BB}[0]{{\mathbb{B}}}
\newcommand{\Betti}[0]{{\operatorname{Betti}}}
\newcommand{\CC}[0]{{\mathbb{C}}}
\newcommand{\CF}[0]{\operatorname{CF}}
\newcommand{\CH}[0]{{\operatorname{CH}}}
\newcommand{\CP}[0]{{\mathbb{CP}}}
\newcommand{\CY}{{ \text{CY} }}
\newcommand{\Cl}[0]{{ \operatorname{Cl}} }
\newcommand{\Crit}[0]{\operatorname{Crit}}
\newcommand{\DD}[0]{{\mathbb{D}}}
\newcommand{\DSt}[0]{{ \operatorname{DSt}}}
\newcommand{\Def}{\operatorname{Def} }
\newcommand{\Diffeo}[0]{{\operatorname{Diffeo}}}
\newcommand{\Diff}[0]{\operatorname{Diff}}
\newcommand{\Disjoint}[0]{\displaystyle\coprod}
\newcommand{\Disk}[0]{{\operatorname{Disk}}}
\newcommand{\Dist}[0]{\operatorname{Dist}}
\newcommand{\Div}[0]{\operatorname{Div}}
\newcommand{\EE}[0]{{\mathbb{E}}}
\newcommand{\EKL}[0]{{\mathrm{EKL}}}
\newcommand{\unram}[0]{{\scriptscriptstyle\mathrm{un}}}
\newcommand{\Emb}[0]{{\operatorname{Emb}}}
\newcommand{\minor}[0]{{\operatorname{minor}}}
\newcommand{\Et}{\text{Ét}}
\newcommand{\trace}{\operatorname{tr}}
\newcommand{\Norm}{\operatorname{Nm}}
\newcommand{\Extpower}[0]{\bigwedge\nolimits}
\newcommand{\Extalgebra}[0]{\bigwedge\nolimits}
\newcommand{\Extalg}[0]{\Extalgebra}
\newcommand{\Extprod}[0]{\bigwedge\nolimits}
\newcommand{\Ext}{\operatorname{Ext} }
\newcommand{\FFbar}[0]{{ \bar{ \mathbb{F}} }}
\newcommand{\FFpn}[0]{{\mathbb{F}_{p^n}}}
\newcommand{\FFp}[0]{{\mathbb{F}_p}}
\newcommand{\FF}[0]{{\mathbb{F}}}
\newcommand{\FS}{{ \text{FS} }}
\newcommand{\Fil}[0]{{\operatorname{Fil}}}
\newcommand{\Flat}[0]{{\operatorname{Flat}}}
\newcommand{\Fpbar}[0]{\bar{\mathbb{F}_p}}
\newcommand{\Fpn}[0]{{\mathbb{F}_{p^n} }}
\newcommand{\Fppf}[0]{\mathrm{\operatorname{Fppf}}}
\newcommand{\Fp}[0]{{\mathbb{F}_p}}
\newcommand{\Frac}[0]{\operatorname{Frac}}
\newcommand{\GF}[0]{{\mathbb{GF}}}
\newcommand{\GG}[0]{{\mathbb{G}}}
\newcommand{\GL}[0]{\operatorname{GL}}
\newcommand{\GW}[0]{{\operatorname{GW}}}
\newcommand{\Gal}[0]{{ \mathsf{Gal}} }
\newcommand{\Gl}[0]{\operatorname{GL}}
\newcommand{\Gr}[0]{{\operatorname{Gr}}}
\newcommand{\HC}[0]{{\operatorname{HC}}}
\newcommand{\HFK}[0]{\operatorname{HFK}}
\newcommand{\HF}[0]{\operatorname{HF}}
\newcommand{\HHom}{\mathscr{H}\kern-2pt\operatorname{om}}
\newcommand{\HH}[0]{{\mathbb{H}}}
\newcommand{\HP}[0]{{\operatorname{HP}}}
\newcommand{\HT}[0]{{\operatorname{HT}}}
\newcommand{\HZ}[0]{{H\mathbb{Z}}}
\newcommand{\Hilb}[0]{\operatorname{Hilb}}
\newcommand{\Homeo}[0]{{\operatorname{Homeo}}}
\newcommand{\Honda}[0]{\mathrm{\operatorname{Honda}}}
\newcommand{\Hsh}{{ \mathcal{H} }}
\newcommand{\Id}[0]{\operatorname{Id}}
\newcommand{\Intersect}[0]{\displaystyle\bigcap}
\newcommand{\JCF}[0]{\operatorname{JCF}}
\newcommand{\RCF}[0]{\operatorname{RCF}}
\newcommand{\Jac}[0]{\operatorname{Jac}}
\newcommand{\II}[0]{{\mathbb{I}}}
\newcommand{\KK}[0]{{\mathbb{K}}}
\newcommand{\KH}[0]{ \K^{\scriptscriptstyle \mathrm{H}} }
\newcommand{\KMW}[0]{ \K^{\scriptscriptstyle \mathrm{MW}} }
\newcommand{\KMimp}[0]{ \hat{\K}^{\scriptscriptstyle \mathrm{M}} }
\newcommand{\KM}[0]{ \K^{\scriptstyle\mathrm{M}} }
\newcommand{\Kah}[0]{{ \operatorname{Kähler} } }
\newcommand{\LC}[0]{{\mathrm{LC}}}
\newcommand{\LL}[0]{{\mathbb{L}}}
\newcommand{\Lie}[0]{\operatorname{Lie}}
\newcommand{\Log}[0]{\operatorname{Log}}
\newcommand{\MCG}[0]{{\operatorname{MCG}}}
\newcommand{\MM}[0]{{\mathcal{M}}}
\newcommand{\MW}[0]{\operatorname{MW}}
\newcommand{\Mat}[0]{\operatorname{Mat}}
\newcommand{\NN}[0]{{\mathbb{N}}}
\newcommand{\NS}[0]{{\operatorname{NS}}}
\newcommand{\OO}[0]{{\mathcal{O}}}
\newcommand{\OP}[0]{{\mathbb{OP}}}
\newcommand{\OX}[0]{{\mathcal{O}_X}}
\newcommand{\Obs}{\operatorname{Obs} }
\newcommand{\obs}{\operatorname{obs} }
\newcommand{\Ob}[0]{{\operatorname{Ob}}}
\newcommand{\Op}[0]{{\operatorname{Op}}}
\newcommand{\Orb}[0]{{\mathrm{Orb}}}
\newcommand{\Conj}[0]{{\mathrm{Conj}}}
\newcommand{\Orth}[0]{{\operatorname{O}}}
\newcommand{\PD}[0]{\mathrm{PD}}
\newcommand{\PGL}[0]{\operatorname{PGL}}
\newcommand{\PP}[0]{{\mathbb{P}}}
\newcommand{\PSL}[0]{{\operatorname{PSL}}}
\newcommand{\Pic}[0]{{\operatorname{Pic}}}
\newcommand{\Pin}[0]{{\operatorname{Pin}}}
\newcommand{\Places}[0]{{\operatorname{Places}}}
\newcommand{\Presh}[0]{\presh}
\newcommand{\QHB}[0]{\operatorname{QHB}}
\newcommand{\QHS}[0]{\operatorname{QHS}}
\newcommand{\QQpadic}[0]{{ \QQ_p }}
\newcommand{\QQ}[0]{{\mathbb{Q}}}
\newcommand{\QQbar}[0]{{ \bar{ \mathbb{Q} } }}
\newcommand{\Quot}[0]{\operatorname{Quot}}
\newcommand{\RP}[0]{{\mathbb{RP}}}
\newcommand{\RR}[0]{{\mathbb{R}}}
\newcommand{\Rat}[0]{\operatorname{Rat}}
\newcommand{\Rees}[0]{{\operatorname{Rees}}}
\newcommand{\Reg}[0]{\operatorname{Reg}}
\newcommand{\Ric}[0]{\operatorname{Ric}}
\newcommand{\SF}[0]{\operatorname{SF}}
\newcommand{\SL}[0]{{\operatorname{SL}}}
\newcommand{\SNF}[0]{\mathrm{SNF}}
\newcommand{\SO}[0]{{\operatorname{SO}}}
\newcommand{\SP}[0]{{\operatorname{SP}}}
\newcommand{\SU}[0]{{\operatorname{SU}}}
\newcommand{\Sgn}[0]{{ \Sigma_{g, n} }}
\newcommand{\Sm}[0]{{\operatorname{Sm}}}
\newcommand{\SpSp}[0]{{\mathbb{S}}}
\newcommand{\Spec}[0]{\operatorname{Spec}}
\newcommand{\Spf}[0]{\operatorname{Spf}}
\newcommand{\Spinc}[0]{\mathrm{Spin}^{{c} }}
\newcommand{\Spin}[0]{{\operatorname{Spin}}}
\newcommand{\Sq}[0]{\operatorname{Sq}}
\newcommand{\Stab}[0]{{\operatorname{Stab}}}
\newcommand{\Sum}[0]{ \displaystyle\sum }
\newcommand{\Syl}[0]{{\operatorname{Syl}}}
\newcommand{\Sym}[0]{\operatorname{Sym}}
\newcommand{\Tor}[0]{\operatorname{Tor}}
\newcommand{\Homcx}[0]{\operatorname{Hom}^{\bullet}}
\newcommand{\Tr}[0]{\operatorname{Tr}}
\newcommand{\Ug}[0]{{\mathcal{U}(\mathfrak{g}) }}
\newcommand{\Uh}[0]{{\mathcal{U}(\mathfrak{h}) }}
\newcommand{\Union}[0]{\displaystyle\bigcup}
\newcommand{\U}[0]{{\operatorname{U}}}
\newcommand{\Wedge}[0]{\bigwedge}
\newcommand{\Wittvectors}[0]{{\mathbb{W}}}
\newcommand{\ZHB}[0]{\operatorname{ZHB}}
\newcommand{\ZHS}[0]{\mathbb{Z}\operatorname{HS}}
\newcommand{\ZZG}[0]{{\mathbb{Z}G}}
\newcommand{\ZZH}[0]{{\mathbb{Z}H}}
\newcommand{\ZZlocal}[1]{{ \ZZ_{\hat{#1}} }}
\newcommand{\plocal}[0]{{ \scriptsize {}_{ \localize{p}  } }}
\newcommand{\ZZpadic}[0]{{ \ZZ_p }}
\newcommand{\ZZ}[0]{{\mathbb{Z}}}
\newcommand{\ZZbar}[0]{{ \bar{ \mathbb{Z} } }}
\newcommand{\Zar}[0]{{\mathrm{Zar}}}
\newcommand{\ZpZ}[0]{\mathbb{Z}/p}
\newcommand{\abuts}[0]{\Rightarrow}
\newcommand{\ab}[0]{{\operatorname{ab}}}
\newcommand{\actsonl}[0]{\curvearrowleft}
\newcommand{\actson}[0]{\curvearrowright}
\newcommand{\adjoint}[0]{\dagger}
\newcommand{\adj}[0]{\operatorname{adj}}
\newcommand{\ad}[0]{\operatorname{ad}}
\newcommand{\afp}[0]{A_{/\FF_p}}
\newcommand{\annd}[0]{{\operatorname{ and }}}
\newcommand{\ann}[0]{\operatorname{Ann}}
\newcommand{\arccot}[0]{\operatorname{arccot}}
\newcommand{\arccsc}[0]{\operatorname{arccsc}}
\newcommand{\arcsec}[0]{\operatorname{arcsec}}
\newcommand{\bP}[0]{\operatorname{bP}}
\newcommand{\barz}{\bar{z} }
\newcommand{\bbm}[0]{{\mathbb{M}}}
\newcommand{\bd}[0]{{\del}}
\newcommand{\bigast}[0]{{\mathop{\text{\Large $\ast$}}}}
\newcommand{\bmgn}[0]{{ \bar{\mathcal{M}}_{g, n} }}
\newcommand{\bundle}[1]{\mathcal{#1}}
\newcommand{\by}[0]{\times}
\newcommand{\candim}[0]{\operatorname{candim}}
\newcommand{\chp}[0]{\operatorname{ch. p}}
\newcommand{\ch}[0]{\operatorname{ch}}
\newcommand{\character}[0]{\operatorname{ch}}
\newcommand{\cl}[0]{{ \operatorname{cl}} }
\newcommand{\codim}[0]{\operatorname{codim}}
\newcommand{\cohdim}[0]{\operatorname{cohdim}}
\newcommand{\coim}[0]{\operatorname{coim}}
\newcommand{\coker}[0]{\operatorname{coker}}
\newcommand{\cok}[0]{\operatorname{coker}}
\newcommand{\cone}[0]{\operatorname{cone}}
\newcommand{\conjugate}[1]{{\overline{{#1}}}}
\newcommand{\connectsum}[0]{\mathop{ \Large\mypound }}
\newcommand{\const}[0]{{\operatorname{const.}}}
\newcommand{\converges}[1]{\overset{#1}}
\newcommand{\convolve}[0]{\ast}
\newcommand{\correspond}[1]{\theset{\substack{#1}}}
\newcommand{\covers}[0]{\rightrightarrows}
\newcommand{\projresolve}[0]{\rightrightarrows}
\newcommand{\injresolve}[0]{\leftleftarrows}
\newcommand{\covol}[0]{\operatorname{covol}}
\newcommand{\cpt}[0]{{ \operatorname{compact} } }
\newcommand{\crit}[0]{\operatorname{crit}}
\newcommand{\cross}[0]{\times}
\newcommand{\dR}[0]{\mathrm{dR}}
\newcommand{\dV}{\,dV}
\newcommand{\dash}[0]{{\hbox{-}}}
\newcommand{\da}[0]{\coloneqq}
\newcommand{\ddd}[2]{{\frac{d #1}{d #2}\,}}
\newcommand{\ddim}[0]{\operatorname{ddim}}
\newcommand{\ddt}{\tfrac{\dif}{\dif t}}
\newcommand{\ddx}{\tfrac{\dif}{\dif x}}
\newcommand{\dd}[2]{{\frac{\partial #1}{\partial #2}\,}}
\newcommand{\definedas}[0]{\coloneqq}
\newcommand{\del}[0]{{\partial}}
\newcommand{\diagonal}[1]{\Delta}
\newcommand{\Diagonal}[1]{\Delta}
\newcommand{\diag}[0]{\operatorname{diag}}
\newcommand{\diam}[0]{{\operatorname{diam}}}
\newcommand{\diff}[0]{\operatorname{Diff}}
\newcommand{\discriminant}[0]{{\Delta}}
\newcommand{\disc}[0]{{\operatorname{disc}}}
\newcommand{\disjoint}[0]{{\textstyle\coprod}}
\newcommand{\dist}[0]{\operatorname{dist}}
\newcommand{\dlog}[0]{\operatorname{dLog}}
\newcommand{\dom}[0]{\operatorname{dom}}
\newcommand{\dual}[0]{ {}^{ \vee }}
\newcommand{\dmu}{\,d\mu}
\newcommand{\dr}{\,dr}
\newcommand{\ds}{\,ds}
\newcommand{\dtheta}{\,d\theta}
\newcommand{\dt}{\,dt}
\newcommand{\du}{\,du}
\newcommand{\dw}{\,dw}
\newcommand{\dxi}{\,d\xi}
\newcommand{\dx}{\,dx}
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