Now, let $\epsilon>0$ be the minimum of the corresponding values from Theorems
3.10 and 5.5 for $k\leq k_{0}-1$. We put
$\epsilon_{0}\mathrel{\mathop{\ordinarycolon}}=\frac{\epsilon}{2}$. Let
$y_{0}\in Y$ and $y_{1},y_{2}\in B_{y_{0}}^{Y}(\epsilon_{0})$. We decompose
the integral in (5.44) into two parts: the first one over
$B_{y_{0}}^{Y}(\epsilon)$, and the second one is over its complement, which we
denote by $Q$. Clearly, for $y_{3}\in Q$, we get

${\rm{dist}}(y_{1},y_{3})+{\rm{dist}}(y_{3},y_{2})\geq\epsilon.$ (5.46)

Hence, from Theorems 4.7, 5.1.b), 5.7, Proposition 2.8 and (1.5), we see that
the contribution from the integration over $Q$ is smaller than
$\exp(-c\sqrt{p}(1+{\rm{dist}}(y_{1},y_{2})))$ for some constant $c>0$.
Consequently, only the integration over $y_{1}\in B_{y_{0}}^{Y}(\epsilon)$ is
non-negligible. To evaluate it, we apply Theorem 4.7. We calculate the
integral over the pull-back with respect to the exponential map of our
differential forms. We use the notations introduced before Theorem 3.10. After
the change of variables $Z\mapsto\sqrt{p}Z$, an estimate, similar to the one
which bounded the integral over $Q$, Lemma 2.10 and the second part of Lemma
3.1, applied for $n\mathrel{\mathop{\ordinarycolon}}=m$, we see that (5.2)
holds for

$J_{k,r}^{E}(Z,Z^{\prime}_{Y})\mathrel{\mathop{\ordinarycolon}}=\sum_{a+b=r}\mathcal{K}_{n,m}^{EP}\big{[}J_{k,a}^{E,0},I_{A_{k,p}^{-1},b}^{Y}\big{]},$
(5.47)

where $I_{A_{k,p}^{-1},b}^{Y}$ are the polynomials associated to
$(\frac{1}{p^{n-m+k}}A_{k,p})^{-1}$ as in Theorem 3.5. From the corresponding
statements about the parity of $J_{k,r}^{E,0}$, $I_{A_{k,p}^{-1},r}^{Y}$ from
Theorem 3.5 and Lemmas 3.1, 5.10, we deduce that the parity of $J_{k,r}^{E}$
coincides with $k+r$. From Proposition 3.9 and (5.47), we deduce that

$\deg(J_{k,r}^{E})\leq\max\Big{\\{}\deg(J_{k,a}^{E,0})+\deg(I_{A_{k,p}^{-1},b}^{Y})\Big{\\}},$
(5.48)

where the maximum is taken over $a+b=r$. From Proposition 3.9, Lemma 5.10,
(5.35), (5.48) and the bound on the degree of $J_{k,a}^{R,0}$ from Lemma 5.10,
we deduce the needed bound on the degree of $J_{k,r}^{E}$.

Directly from (3.7), (3.11), (3.12), (5.41) and (5.47), we see that the
formula (5.3) holds for $J_{k,0}^{E}$. Moreover, under assumption (1.10), by
Theorem 5.7, for $i=0,1$, we have

$I_{A_{k,p}^{-1},i}^{Y}=\frac{1}{(2\pi)^{k}\cdot k!}\cdot J_{k,i}^{Y|Y},$
(5.49)

where $J_{k,i}^{Y|Y}$ are the polynomials from Theorem 3.10, associated to
$X,Y\mathrel{\mathop{\ordinarycolon}}=Y$ and
$F\mathrel{\mathop{\ordinarycolon}}={\rm{Sym}}^{k}(N^{1,0})^{*}\otimes\iota^{*}(F)$.
From this, (3.38), (3.39), we conclude that

$J_{k,1}^{E}(Z,Z^{\prime}_{Y})=\frac{1}{(2\pi)^{k}\cdot
k!}\mathcal{K}_{n,m}^{EP}\big{[}J_{k,1}^{E,0},1\big{]}.$ (5.50)

We deduce (5.4) from this, Lemma 5.10, (3.21) and (5.35).

The proof of the fact that $J_{k,r}^{E}$ vanishes at least up to order $k$
along ${}^{2m}\times{}^{2m}\subset{}^{2n}\times{}^{2m}$ proceeds along the
same lines as the corresponding result from Lemma 5.10. In total, this
finishes the proof of Theorem 5.3 for
$k\mathrel{\mathop{\ordinarycolon}}=k_{0}$.

From the expansion (5.2), we see that we can improve (5.45) to (5.1). This
finishes completely the proof of Theorem 5.1.a) for
$k\mathrel{\mathop{\ordinarycolon}}=k_{0}$. ∎

###### Proof of Lemma 5.9.c).

First of all, an easy verification shows the following identity

$\frac{1}{p^{n}}B_{p}^{\perp,k}=\frac{1}{p^{m-\frac{k}{2}}}{\rm{E}}_{k,p}\circ\frac{1}{p^{n-m+\frac{k}{2}}}({\rm{Res}}_{Y}\circ\nabla^{k}B_{p}^{X,kY}).$
(5.51)

On the level of Schwartz kernels, the identity (5.51) basically means that

$\frac{1}{p^{n}}B_{p}^{\perp,k}(x_{1},x_{2})=\int_{y\in
Y}\frac{1}{p^{m-\frac{k}{2}}}{\rm{E}}_{k,p}(x_{1},y)\cdot\frac{1}{p^{n-m+\frac{k}{2}}}({\rm{Res}}_{Y}\circ\nabla^{k}B_{p}^{X,kY})(y,x_{2})dv_{Y}(y).$
(5.52)

From Theorems 3.10 and 5.1, for $k\leq k_{0}-1$, Proposition 2.8 and (5.52),
we see that Theorem 5.1.b) holds for $k=k_{0}$.

From (5.52), the expansion (5.5) follows from the same reasoning as we used in
(5.47), except that we need to rely on the third part of Lemma 3.1 and
$J_{k,r}^{\perp}$ are defined as follows

$J_{k,r}^{\perp}(Z,Z^{\prime})\mathrel{\mathop{\ordinarycolon}}=\sum_{a+b=r}\mathcal{K}_{n,m}^{ER}\big{[}J_{k,a}^{E},J_{k,b}^{R,0}\big{]}.$
(5.53)

From (3.24), (3.26), (5.3), (5.4), (5.28), (5.29) and (5.53), we deduce (5.7)
and (5.8). The parity statement about the polynomials $J_{k,r}^{\perp}$
follows from Lemma 3.1 and the corresponding statements for $J_{k,a}^{E}$ and
$J_{k,b}^{R,0}$, proved in Theorem 5.3 and Lemma 5.10. The vanishing up to
order $k$ of the polynomials $J_{k,r}^{\perp}$ follows from (3.24), (5.53) and
the corresponding statements for $J_{k,a}^{E}$ and $J_{k,b}^{R,0}$, proved in
Theorem 5.3 and Lemma 5.10. From Proposition 3.9 and (5.53), we deduce that

$\deg(J_{k,r}^{\perp})\leq\max\Big{\\{}\deg(J_{k,a}^{E})+\deg(J_{k,b}^{R,0})\Big{\\}},$
(5.54)

where the maximum is taken over $a+b=r$. From Lemma 5.10, (5.54) and the bound
on the degree of $J_{k,a}^{R,0}$ (resp. $J_{k,a}^{E}$) from Lemma 5.10 (resp.
Theorem 5.3), we deduce the needed bound on the degree of $J_{k,r}^{\perp}$. ∎

###### Proof of Theorems 5.1, 5.3, 5.5, 5.7.

It follows directly from Lemma 5.9 by induction and the fact that for $k<0$,
the statements of Theorems 5.1.b) and 5.5 are void. ∎

###### Proof of Theorem 1.3 .

First of all, let us establish that under the assumption (1.10), there is
$C>0$, such that as $p\to\infty$, we have

$\displaystyle\Big{|}\big{\|}{\rm{E}}_{k,p}\big{\|}-\frac{1}{p^{\frac{n-m+k}{2}}}\cdot\frac{1}{\sqrt{k!\cdot(2\pi)^{k}}}\Big{|}\leq\frac{C}{p^{\frac{n-m+k+2}{2}}},$
(5.55)
$\displaystyle\Big{|}\big{\|}{\rm{Res}}_{k,p}\big{\|}-p^{\frac{n-m+k}{2}}\cdot\sqrt{k!\cdot(2\pi)^{k}}\Big{|}\leq
Cp^{\frac{n-m+k-2}{2}}.$

Indeed, remark that from Theorem 5.7 and Proposition 2.9 that for any
$k\in\mathbb{N}$, there are $p_{1}\in\mathbb{N}$, $C>0$, such that for any
$p\geq p_{1}$, we have

$\Big{\|}\frac{1}{p^{n-m+k}}A_{k,p}-k!\cdot(2\pi)^{k}\cdot
B_{k,p}^{Y}\Big{\|}\leq\frac{C}{p}.$ (5.56)

In particular, we conclude that

$\Big{|}\big{\|}A_{k,p}\big{\|}-p^{n-m+k}\cdot k!\cdot(2\pi)^{k}\Big{|}\leq
Cp^{n-m+k-1}.$ (5.57)

Using Lemma 5.11 and similar arguments, we see that

$\Big{|}\big{\|}A_{k,p}^{-1}\big{\|}-\frac{1}{p^{n-m+k}}\cdot\frac{1}{k!\cdot(2\pi)^{k}}\Big{|}\leq\frac{C}{p^{n-m+k+1}}.$
(5.58)

From (4.2) and the fact that
${\rm{Res}}_{k,p}\circ{\rm{E}}_{k,p}=B_{k,p}^{Y}$, we have the following
identities

$({\rm{E}}_{k,p})^{*}\circ{\rm{E}}_{k,p}=\big{(}(A_{k,p})^{*}\big{)}^{-1},\qquad{\rm{Res}}_{k,p}\circ({\rm{Res}}_{k,p})^{*}=A_{k,p}.$
(5.59)

Clearly, we have
$\|({\rm{E}}_{k,p})^{*}\circ{\rm{E}}_{k,p}\|=\|{\rm{E}}_{k,p}\|^{2}$ and
$\|{\rm{Res}}_{k,p}\circ({\rm{Res}}_{k,p})^{*}\|=\|{\rm{Res}}_{k,p}\|^{2}$.
The bounds (5.55) now follows from this observation, (5.57), (5.58) and
(5.59).

Now, from (5.55) and the fact that
${\rm{Res}}_{k,p}\circ{\rm{E}}_{k,p}=B_{k,p}^{Y}$, we conclude that there are
$p_{1}\in\mathbb{N}$, $C>0$, such that for any $k\in\mathbb{N}$, $p\geq
p_{1}$, $g\in
H^{0}_{(2)}(Y,{\rm{Sym}}^{k}(N^{1,0})^{*}\otimes\iota^{*}(L^{p}\otimes F))$,
$g\neq 0$, we have

$\bigg{|}\frac{\|{\rm{E}}_{k,p}(g)\|_{L^{2}(X,L^{p}\otimes
F)}}{\|g\|_{k,L^{2}(Y,L^{p}\otimes
F)}}-\frac{1}{p^{\frac{n-m+k}{2}}}\cdot\frac{1}{\sqrt{k!\cdot(2\pi)^{k}}}\bigg{|}\leq\frac{C}{p^{\frac{n-m+k+2}{2}}}.$
(5.60)

Now, let us fix $f\in H^{0}_{(2)}(X,L^{p}\otimes F)$, and denote by $[f]$ the
element it represents in the quotient space $H^{0}_{(2)}(X,L^{p}\otimes
F)/H^{0}_{(2)}(X,L^{p}\otimes F\otimes\mathcal{J}_{Y}^{k+1})$. Directly from
the definition, we have

$\big{\|}[f]\big{\|}_{L^{2}(X,L^{p}\otimes
F)}^{2}=\sum_{l=0}^{k}\|B_{l,p}^{\perp}f\|_{L^{2}(X,L^{p}\otimes F)}^{2}.$
(5.61)

We let ${\rm{Jet}}_{k,p}(f)=(g_{0},\ldots,g_{k})$, where $g_{i}\in
H^{0}_{(2)}(Y,{\rm{Sym}}^{i}(N^{1,0})^{*}\otimes\iota^{*}(L^{p}\otimes F))$.
From the definition of the map ${\rm{Jet}}_{k,p}$ from (1.16) and the
characterization of $g_{i}$ from (1.17), we see that

$g_{i}={\rm{Res}}_{i,p}(B_{i,p}^{\perp}f).$ (5.62)

We conclude from (5.55) and (5.62) that

$\big{\|}g_{i}\big{\|}_{k,L^{2}(Y,L^{p}\otimes
F)}=\Big{(}p^{\frac{n-m+k}{2}}\cdot\sqrt{k!\cdot(2\pi)^{k}}+O(p^{\frac{n-m+k-2}{2}})\Big{)}\cdot\big{\|}B_{i,p}^{\perp}f\big{\|}_{L^{2}(X,L^{p}\otimes
F)}.$ (5.63)

The result now follows from (5.60), (5.61), (5.63) and the definition of the
scalar product $\langle\cdot,\cdot\rangle_{{\rm{Jet}}_{k,p}}$ from (1.19). ∎

###  5.3 Higher order peak sections as a special case of extension theorem

The main goal of this section is to illustrate Theorems 5.1, 5.3 on a simple
example when the submanifold corresponds to a fixed point.

Let us denote the extension operator in this case by
${\rm{E}}_{k,p}^{\\{x\\}}$. It corresponds to

${\rm{E}}_{k,p}^{\\{x\\}}\mathrel{\mathop{\ordinarycolon}}{\rm{Sym}}^{k}T^{1,0}X_{x}^{*}\otimes(L^{p}\otimes
F)_{x}\to H^{0}_{(2)}(X,L^{p}\otimes F\otimes\mathcal{J}_{x}^{k}).$ (5.64)

Clearly, for any $v\in{\rm{Sym}}^{k}(N_{x}^{1,0})^{*}\otimes(L^{p}\otimes
F)_{x}$, the section
$s_{k,p}^{x,v}\mathrel{\mathop{\ordinarycolon}}={\rm{E}}_{k,p}^{\\{x\\}}(v)$
minimizes the $L^{2}$-norm among all sections from $H^{0}_{(2)}(X,L^{p}\otimes
F\otimes\mathcal{J}_{x}^{k})$, having $k$-th jet equal to $v$.

The sections $s_{k,p}^{x,v}$ were defined in complex geometry by Tian [38].
For $k=0$, they bear the name “peak sections”. Due to this reason, for $k\geq
1$, we call $s_{k,p}^{x,v}$ the higher order peak sections.

######  Theorem 5.12.

There are $c,C>0$, such that for any $x,y\in X$, $p\in\mathbb{N}$, $p\geq
p_{1}$, $v\in{\rm{Sym}}^{k}(N_{x}^{1,0})^{*}\otimes(L^{p}\otimes F)_{x}$, we
have

$\big{|}s_{k,p}^{x,v}(y)\big{|}_{\mathscr{C}^{r}}\leq
Cp^{\frac{r-k}{2}}\exp(-c\sqrt{p}{\rm{dist}}(x,y)).$ (5.65)

Moreover, there are $\epsilon>0$, $c,C>0$, such that for any $x\in X$, for any
$Z\in{}^{2n}$, $|Z|<\epsilon$, $p\in\mathbb{N}$, $p\geq p_{1}$,
$v\in{\rm{Sym}}^{k}(N_{x}^{1,0})^{*}\otimes(L^{p}\otimes F)_{x}$, we have

$\Big{|}s_{k,p}^{x,v}(\phi_{x}^{X}(Z))-(v\cdot Z^{\otimes
k})\cdot\exp\big{(}-\frac{\pi}{2}p|Z|^{2}\big{)}\Big{|}_{\mathscr{C}^{r}}\leq
C\cdot|v|\cdot p^{\frac{r}{2}-1}\cdot|Z|^{k}\cdot\exp(-c\sqrt{p}|Z|),$ (5.66)

where the $\mathscr{C}^{r}$-norm is taken with respect to $x,Z$.

######  Remark 5.13.

For $r\leq 2$, similar results were obtained by Tian [38, Lemma 1.2]. For
$k=0$, this statement follows directly from Dai-Liu-Ma [10].

###### Proof.

The first statement follows directly by applying Theorem 5.1.a) for
$Y\mathrel{\mathop{\ordinarycolon}}=\\{y_{0}\\}$. Remark that for $Y$ as
above, the second fundamental form vanishes, so the second term of the
asymptotic expansion from Theorem 5.3 vanishes according to (5.4). The second
statement of Theorem 5.12 then follows from this remark, the application of
Theorem 5.3 for $Y\mathrel{\mathop{\ordinarycolon}}=\\{y_{0}\\}$ and $r=1$ and
the use of Remark 5.6.d). ∎

## References

  * [1] J.-M. Bismut.  Demailly’s asymptotic morse inequalities: A heat equation proof.  J. Funct. Anal., 72(2):263 – 278, 1987.
  * [2] J.-M. Bismut and G. Lebeau.  Complex immersions and Quillen metrics.  Publ. Math. IHES, 74(1):1–291, 1991.
  * [3] J.-M. Bismut and E. Vasserot.  The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle.  Comm. Math. Phys., 125(2):355–367, 1989.
  * [4] M. Bordemann, E. Meinrenken, and M. Schlichenmaier.  Toeplitz quantization of Kähler manifolds and $gl(N)$, $N\to\infty$ limits.  Commun. Math. Phys., 165(2):281–296, 1994.
  * [5] T. Bouche.  Sur les inégalités de Morse holomorphes lorsque la courbure du fibré en droites est dégénérée.  Ann. Sc. Norm. Super. Pisa, 18(4):501–523, 1991.
  * [6] J. Cao, J.-P. Demailly, and S.-I. Matsumura.  A general extension theorem for cohomology classes on non reduced analytic subspaces.  Sci. China, Math., 60(6):949–962, 2017.
  * [7] J. Cao and M. Păun.  On extension of pluricanonical forms defined on the central fiber of a Kähler family, ArXiv: 2012.05063.  2020\.
  * [8] D. Catlin.  The Bergman kernel and a theorem of Tian.  In Proceedings of the 40th Taniguchi symposium, Katata, Japan, June 23–28, 1997, pages 1–23. Boston, MA: Birkhäuser, 1999.
  * [9] D. Coman and G. Marinescu.  On the first order asymptotics of partial Bergman kernels.  Ann. Fac. Sci. Toulouse, Math. (6), 26(5):1193–1210, 2017.
  * [10] X. Dai, K. Liu, and X. Ma.  On the asymptotic expansion of bergman kernel.  J. Diff. Geom., 72(1):1–41, 2006.
  * [11] J.-P. Demailly.  Estimations $L^{2}$ pour l’opérateur $\overline{\partial}$ d’un fibre vectoriel holomorphe semi-positif au-dessus d’une variété Kaehlerienne complete.  Ann. Sci. Éc. Norm. Supér. (4), 15:457–511, 1982.
  * [12] J.-P. Demailly.  Analytic methods in algebraic geometry, volume 1.  Somerville, MA: International Press; Beijing: Higher Education Press, 2012.
  * [13] J.-P. Demailly.  Complex Analytic and Differential Geometry.  2012\.
  * [14] J.-P. Demailly.  Extension of holomorphic functions defined on non reduced analytic subvarieties.  In The legacy of Bernhard Riemann after one hundred and fifty years. Volume I, pages 191–222. Somerville, MA: International Press; Beijing: Higher Education Press, 2016.
  * [15] J. Eichhorn.  The boundedness of connection coefficients and their derivatives.  Math. Nachr., 152:145–158, 1991.
  * [16] S. Finski.  Semiclassical Ohsawa-Takegoshi extension theorem and asymptotic of the orthogonal Bergman kernel, arXiv: 2109.06851.  2021\.
  * [17] S. Finski.  Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions, arXiv:2201.04102.  2022\.
  * [18] N. Große and C. Schneider.  Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces.  Math. Nachr., 286(16):1586–1613, 2013.
  * [19] E. Heintze and H. Karcher.  A general comparison theorem with applications to volume estimates for submanifolds.  Ann. Sci. Éc. Norm. Supér. (4), 11(4):451–470, 1978.
  * [20] G. Hosono.  The optimal jet $L^{2}$ extension of Ohsawa-Takegoshi type.  Nagoya Math. J., 239:153–172, 2020.
  * [21] Z. Lu.  On the Lower Order Terms of the Asymptotic Expansion of Tian-Yau-Zelditch.  Amer. J. Math., 122(2):235–273, 2000.
  * [22] X. Ma and G. Marinescu.  Holomorphic Morse inequalities and Bergman kernels, volume 254 of Progr. Math. Birkhäuser Verlag Basel, 2007.
  * [23] X. Ma and G. Marinescu.  Generalized Bergman kernels on symplectic manifolds.  Adv. in Math., 217(4):1756–1815, 2008.
  * [24] X. Ma and G. Marinescu.  Toeplitz operators on symplectic manifolds.  J. Geom. Anal., 18(2):565–611, 2008.
  * [25] X. Ma and G. Marinescu.  Exponential estimate for the asymptotics of Bergman kernels.  Math. Ann., 362(3-4):1327–1347, 2015.
  * [26] X. Ma and W. Zhang.  Bergman Kernels and symplectic reduction.  Astérisque 318, 2008.
  * [27] L. Manivel.  A theorem of $L^{2}$ extension of holomorphic sections of a Hermitian bundle.  Math. Z., 212(1):107–122, 1993.
  * [28] J. McNeal and D. Varolin.  Extension of Jets With $L^{2}$ Estimates, and an Application, ArXiv: 1707.04483.  2017\.
  * [29] T. Ohsawa.  On the extension of $L^{2}$ holomorphic functions. II.  Publ. Res. Inst. Math. Sci., 24(2):265–275, 1988.
  * [30] T. Ohsawa and K. Takegoshi.  On the extension of $L^{2}$ holomorphic functions.  Math. Z., 195:197–204, 1987.
  * [31] F. T. Pokorny and M. Singer.  Toric partial density functions and stability of toric varieties.  Math. Ann., 358(3-4):879–923, 2014.
  * [32] D. Popovici.  $L^{2}$ extension for jets of holomorphic sections of a Hermitian line bundle.  Nagoya Math. J., 180:1–34, 2005.
  * [33] H. Randriambololona.  Hauteurs pour les sous-schémas et exemples d’utilisation de méthodes arakeloviennes en théorie de l’approximation diophantienne (thèse).  2002\.
  * [34] S. Rao and R. Zhang.  $L^{2}$ extension theorem for jets with variable denominators.  C. R., Math., Acad. Sci. Paris, 359(2):181–193, 2021.
  * [35] J. Ross and M. Singer.  Asymptotics of partial density functions for divisors.  J. Geom. Anal., 27(3):1803–1854, 2017.
  * [36] J. Ross and R. Thomas.  An obstruction to the existence of constant scalar curvature Kähler metrics.  J. Differ. Geom., 72(3):429–466, 2006.
  * [37] T. Schick.  Manifolds with boundary and of bounded geometry.  Math. Nachr., 223:103–120, 2001.
  * [38] G. Tian.  On a set of polarized Kähler metrics on algebraic manifolds.  J. Diff. Geom., 32(1):99–130, 1990.
  * [39] X. Wang.  Canonical metrics on stable vector bundles.  Commun. Anal. Geom., 13(2):253–285, 2005.
  * [40] S. Zelditch.  Szegö kernels and a theorem of Tian.  Internat. Math. Res. Notices, 1998(6):317–331, 1998.
  * [41] S. Zelditch and P. Zhou.  Interface asymptotics of partial Bergman kernels on $S^{1}$-symmetric Kähler manifolds.  J. Symplectic Geom., 17(3):793–856, 2019.
  * [42] K. Zhu.  Analysis on Fock spaces, volume 263 of Grad. Texts Math. New York, NY: Springer, 2012.

Siarhei Finski, CNRS-CMLS, École Polytechnique F-91128 Palaiseau Cedex,
France.

E-mail<EMAIL_ADDRESS>