My Works on Relative Bott-Chern
1990 - 1991
\[\]
\[\huge{\color{Brown}{\mathbb{I.\ Relative\ Bott-Chern\ Secondary\ Characteristic\ Classes}}}\]
\[\]
\(\bigodot\ \Large{f:X\rightarrow\ Y:\ \mathrm{smooth\
morphism\ of\ Kaehler\ manifolds}}\)
\[\]
\(\bigodot\ \Large{\tau_f:\ \mathrm{hermitian\ metric\ on\ the\ relative\
tangent\ sheaf}\ \mathcal{T}_f}\)
\[\]
\(\bigodot\ \Large{(\mathcal{E},\rho):\ \mathrm{f-q.\,acyclic\ hermitian\
vector\ sheaf\ on}\ X}\)
\[\]
\(\bigodot\ \Large{\phi_{\mathrm{BC}}(\mathcal E.,\rho.):\
\mathrm{classical\ Bott-Chern\ secondary\ characteristic\ classes}}\)
\[\]
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Axiom\ 1\ (dd^c\mathrm{-Equation})}:}}\)
\[\]
\(\qquad \Large{\exists\
\mathrm{ch_{BC}}(\mathcal{E}, \rho, f, \tau_f)\in\widetilde A(Y),\ such\ that}\)
\[\LARGE{\boxed{d_Yd_Y^c \mathrm{ch_{BC}}(\mathcal{E}, \rho; f, \tau_f)=
f_*(\mathrm{ch}(\mathcal{E},\rho)
\,\mathrm{td}(\mathcal{T}_f,\tau_f))-\mathrm{ch}\,(f_*\mathcal{E},L^2(\tau_f,\rho))}}\]
\[\]
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Axiom\ 2\ (\mathrm{Base\ Change}):}}}\)
\[\]
\(\qquad\Large{For\qquad\qquad \begin{matrix} X\times_YY'&\buildrel {g_f} \over \longrightarrow &X\\
{f_g\downarrow}&&{\downarrow f}\\
Y'&\buildrel g\over\longrightarrow &Y,\end{matrix}}\)
\[\LARGE{\boxed{g^*\mathrm{ch_{BC}}(\mathcal{E}, \rho; f, \tau_f)=
\mathrm{ch_{BC}}(g_f^*\mathcal{E}, g_f^*\rho; f_g, \tau_{f_g})}}\]
\[\]
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Axiom\ 3\ (\mathrm{Uniqueness\ w.r.t.\ Metrized\ Sheaves}):}}}\)
\[\]
\(\qquad\Large{For\
exact\ sequence\ of\ f\ q.\,acyclic\ vector\ sheaves\ \mathcal{E}.:\ \
0\rightarrow \mathcal{E}_1\rightarrow \mathcal{E}_2\rightarrow \mathcal{E}_3\rightarrow 0\
with\ hermitian\ metrics\ \rho_j,}\)
\(\qquad\Large{f_*\mathcal{E}.:\ \
0\rightarrow f_*\mathcal{E}_1\rightarrow f_*\mathcal{E}_2\rightarrow f_*\mathcal{E}_3
\rightarrow 0: \ the\ direct\ image\ of\ \mathcal{E}_.\ with\ associated\
L^2\ metrics\ L^2(\tau_f,\rho)_j, }\)
\[\LARGE{\boxed{\begin{align*} \mathrm{ch_{BC}}&(\mathcal{E}_2, \rho_2; f, \tau_f)-
\mathrm{ch_{BC}}(\mathcal{E}_1, \rho_1; f, \tau_f)-
\mathrm{ch_{BC}}(\mathcal{E}_3, \rho_3; f, \tau_f)\\
&=f_*(\mathrm{ch_{BC}}(\mathcal{E}_., \rho_.)\,\mathrm{td}(\mathcal{T}_f,\tau_f))-
\mathrm{ch_{BC}}(f_*\mathcal{E}_., L^2(\tau_f,\rho)_.).\end{align*}}}\]
\[\]
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Axiom\ 4\ (\mathrm{Uniqueness\ w.r.t.\ Metrized\ Morphisms}):}}}\)
\[\]
\(\qquad\Large{For\ g: Y\rightarrow W\ smooth\ with\ g-q.acyclic\ f_*\mathcal{E},\ \mathrm{td_{BC}}(f,g):\ classical\ Bott-Chern\ of\ 0\rightarrow \mathcal{T}_f\rightarrow \mathcal{T}_{g\circ f}
\rightarrow f^*\mathcal{T}_g\rightarrow 0,}\)
\[\LARGE{\boxed{\begin{align*} \mathrm{ch_{BC}}&(\mathcal{E}, \rho; g\circ f, \tau_{g\circ f})-
\mathrm{ch_{BC}}(f_*\mathcal{E}, L^2(\tau_f,\rho); g, \tau_g)-
g_*(\mathrm{ch_{BC}}(\mathcal{E}, \rho; f, \tau_f)\,\mathrm{td}(\mathcal{T}_g,\tau_g))\\
&=(g\circ f)_*(\mathrm{ch}(\mathcal{E}, \rho)\,\mathrm{td_{BC}}(f,g))-
\mathrm{ch_{BC}}((g\circ f)_*\mathcal{E}_., L^2(\tau_{g\circ f},\rho_.)\end{align*}}}\]
\[\]
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Axiom\ 5\ (\mathrm{Deformation\ to\ the\ Normal\ Cone}):}}}\)
\[\]
\(\qquad\Large{For\ the\ deformation\ to\ the\ normal\ cone\ associated\
to\quad \Big(B_{X\times\{0\}} Z\times\mathbb P^1\Big)/(Y\times\mathbb P^1),\qquad
in\ \widetilde A(Y),}\)
\[\LARGE{\boxed{\begin{align*}
d_Yd_Y^c&\int_{\mathbb P^1}\mathrm{ch}_{\mathrm{BC}}(E_.,\rho_.;G,\tau_{G})[\log|z|^2]\\
=&\mathrm{ch}_{\mathrm{BC}}(E_.|_{W_0},\rho_.|_{W_0};g_0,\tau_{g_0})
-\mathrm{ch}_{\mathrm{BC}}(E_.|_{W_\infty^1},\rho_.|_{W_\infty^1};g_\infty,\tau_{g_\infty})\\
&+\int_{\mathbb P^1}G_*\big(\mathrm{ch}(E_.,\rho_.)\mathrm{td}(\mathcal T_G(-\log\infty),\tau_G)\big)
[\log|z|^2]-
\int_{\mathbb P^1}(\mathrm{ch}(G_*E_.,L^2(\tau_G,\rho_.)))
[\log|z|^2]\end{align*}}}\]
\[\]
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Axiom\ 5\ (\mathrm{Deformation\ to\ the\ Normal\ Cone})}':}}\)
\[\]
\(\qquad\Large{For\ the\ deformation\ to\ the\ normal\ cone\ associated\
to\quad \Big(B_{X\times\{0\}} Z\times\mathbb P^1\Big)/(Y\times\mathbb P^1),\qquad
in\ \widetilde A(Y),}\)
\[\LARGE{\boxed{\begin{align*}~&\lim_{t\to
\infty}\Big(\big(\mathrm{ch}_{\mathrm{BC}}(E_t,\rho_t;g_t,\tau_{g_t})
-\mathrm{ch}_{\mathrm{BC}}(E_t(-X),\rho_t';g_t,\tau_{g_t})\big)\\
&+\big(\mathrm{ch}_{\mathrm{BC}}((g_t)_*(E_t);
L^2(\rho_t,\tau_t),\gamma_t)-\mathrm{ch}_{\mathrm{BC}}((g_t)_*(E_t(-X));
L^2(\rho_t',\tau_t),\gamma_t')\big)\Big)\\
=&\big(\mathrm{ch}_{\mathrm{BC}}(E_\infty,\rho_\infty;g_\infty,\tau_{g_\infty})-
\mathrm{ch}_{\mathrm{BC}}(E_\infty(-X),\rho_\infty';g_\infty,\tau_{g_\infty})\big)+\\
&\big(\mathrm{ch}_{\mathrm{BC}}((g_\infty)_*(E_\infty);
L^2(\rho_\infty,\tau_\infty),\gamma_\infty)
-\mathrm{ch}_{\mathrm{BC}}((g_\infty)_*(E_\infty(-X));
L^2(\rho_\infty',\tau_\infty),\gamma_\infty')\big)\end{align*}}}\]
\[\]
\[\]
\(\huge{\heartsuit\ \color{Red}{\mathrm{Axiom\ 6\ (\mathrm{Uniqueness\ for}\ \mathbb P^1-\mathrm{Bundles}):}}}\)
\[\]
\[\qquad\LARGE{\boxed{\mathrm{ch_{BC}}(\mathcal{E},\rho;f,\tau_f)'s\ are\ uniquely\ determined\ by\ their\ values\ at\
\mathcal O_{{\mathbb P}^1_Y}\ and\ \mathcal O_{{\mathbb P}^1_Y}(1)}}\]
\[\]
\[\]
\(\huge{\spadesuit\ \color{Blue}{\mathrm{Theorem\ (Relative\ Bott-Chern\ Classes\
w.r.t.\ Smooth\ Morphisms):}}}\)
\[\]
\[\qquad\LARGE{\boxed{\,\exists !\ \mathrm{ch_{BC}}(\mathcal{E},\rho;f,\tau_f)\,\in\,\widetilde A(Y)\
which\ satisfies\ the\ axioms\ above\,}}\]
\[\]\[*********************************************************************************************\]
\[\]
\[\]
\[\huge{\color{Brown}{\mathbb{II.\ Application:\ Arithmetic\ Grothendieck-Riemann-Roch\ Theorem}}}\]
\[\]
\[\]
\(\bigodot\ \Large{f:X\rightarrow Y:\ \mathrm{smooth\ morphism\ of\
regular\ arithmetic\ varieties}}\)
\[\]
\(\bigodot\ \Large{K^\mathrm{Ar}_0(X):\ \mathrm{arithmetic} K\mathrm{-group}}\)
\[\]
\(\bigodot\ \Large{\mathrm{CH}_\mathrm{Ar}(X)_{\bf Q}:\ \mathrm{arithmetic\ Chow\ ring\ with}\
\mathbb Q\mathrm{-coefficients}}\)
\[\]
\(\bigodot\ \Large{f_{\mathrm{CH}}^\mathrm{Ar}:\ \mathrm{push-out\ morphism\ of\ arithmetic\ Chow\ ring}}\)
\[\]
\(\bigodot\ \Large{\phi_\mathrm{Ar}:\ \phi-\mathrm{arithmetic\ characteristic\ class}}\)
\[\]
\(\huge{\heartsuit\ \color{red}{\mathrm{Push-Out\
Morphism\ for\ Arithmetic}\ K-\mathrm{Groups:}}}\)
\[\]
\(\qquad\Large{For\ an\ f-q.acyclic\ hermitian\ vector\
(\mathcal{E},\rho),\ and\ any\ power\ series\ R(x)\in\mathbb R[[X]],\ define}\)
\[\LARGE{\boxed{
f_{K}^{\mathrm{Ar},R}(\mathcal E,\rho):=
(f_*\mathcal{E},L^2(\tau_f,\rho))+\mathrm{ch_{BC}}(\mathcal{E},\rho;f,\tau_f)}}+a(f_*(\mathrm{ch}(E)\mathrm{td}(T_f)R(T_f)))\]
\[\]
\(\huge{\spadesuit\ \color{Blue}{\mathrm{Arithmetic\ Grothendieck-Riemann-Roch\ Theorem\ for\ Smooth\ Morphisms:}}}\)
\[\]
\(\qquad\LARGE{{\exists\ !\ power\ series\ R(x)\in\mathbb R[[X]],\ s.t.\ for\ smooth}\ f:X\rightarrow Y\ {of\
regular\ arithmetic\ varieties,}}\)
\[\]\(\qquad\LARGE{{
the\ following\ diagram\ is\ commutative:}}\)
\[\LARGE{\boxed{\begin{matrix} K^\mathrm{Ar}_0(X)&\buildrel \mathrm{ch_\mathrm{Ar}()td_\mathrm{Ar}(\mathcal{T}_f^\mathrm{Ar})}\over \longrightarrow&
\mathrm{CH}_\mathrm{Ar}(X)_{\bf Q}.\\
f^{\mathrm{Ar},R}_K \downarrow &&\downarrow f^\mathrm{Ar}_\mathrm{CH}\\
K^\mathrm{Ar}_0(Y)&\buildrel \mathrm{ch_{Ar}}\over\longrightarrow&\mathrm{CH}_\mathrm{Ar}(Y)_{\bf Q}\end{matrix}}}\]
\[\]
\[\]
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