Abstract
We establish a new extension result for twisted canonical forms defined on a hypersurface with simple normal crossings of a projective manifold. Some of the examples presented in the appendix show that the bounds we obtain for the extension are sharp.
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1 Introduction
Since it was established in [28], the Ohsawa–Takegoshi extension theorem turned out to be a fundamental tool in complex geometry. As of today, there are uncountable many proofs and refinements of the original result and even more applications to both complex analysis and algebraic geometry. Very roughly, the set-up is as follows: u is a canonical form defined on a sub-variety \(Y\subset X\) with values in a Hermitian bundle\(F\rightarrow X\). We are interested in the following two main questions:
- \(Q_1.\):
-
Does the section u extend to X?
- \(Q_2.\):
-
If the answer to the previous question is “yes,” can one construct an extension whose \(L^2\) norm is bounded by the \(L^2\) norm of u, up to an universal constant?
If Y is non-singular, then the results in e.g., [23] give—practically optimal—curvature conditions for the bundle F such that the answer to both questions above is affirmative. We refer to the articles [2, 3, 5, 8, 11, 13, 14, 17, 24,25,26,27, 29,30,31,32,33,34, 36] for many interesting developments and applications.
The case of a singular sub-variety Y turns out to be significantly more difficult and the most complete results obtained so far only treat the qualitative aspect of the extension problem, that is to say the question \(Q_1\), cf. [7].
In this article we are concerned with the question \(Q_2\). We obtain a few quantitative results for extension of twisted forms defined on sub-varieties Y, which have simple normal crossings. Our main motivation is the Conjecture in [14]. To begin with we fix some notations/conventions.
Let X be a non-singular, projective manifold and let \(Y:= \sum _{i=1}^N Y_i\) be a divisor with simple normal crossings. Let \((L, h_L)\) be a Hermitian line bundle on X, endowed with a metric \(h_L\). The following assumptions will be in force throughout this article:
-
(a)
The usual curvature requirements are satisfied
$$\begin{aligned} \Theta _{h_L}(L)\geqslant 0, \quad \Theta _{h_L}(L)\geqslant \delta \Theta _{h_Y} (Y), \end{aligned}$$where \(\delta > 0\) is a positive real number and \(h_Y\) is a smooth metric on the bundle corresponding to \({\mathcal {O}}(Y)\). Let s be the canonical section of \({\mathcal {O}}(Y)\) with the normalization condition \(|s|_{h_Y} ^2 \leqslant e^{-\delta }\).
-
(b)
We write locally \(h_L =e^{-\varphi _L}\) and \(h_Y = e^{-\varphi _Y}\). The singularities of the metric \(h_L\) of L are of the following type:
$$\begin{aligned} \varphi _L= \sum _j r_j\log |f_j|^2+ \tau _L, \end{aligned}$$where \(f_j\) are local holomorphic functions such that they are not identically zero when restricted to any of the components of Y and \(r_j>0\) is positive number. Moreover, we assume that \(\tau _L\) is non-singular.
-
(c)
Let \(u\in H^0\big (X, (K_X+ Y+ L)\otimes {\mathcal {O}}_X/{\mathcal {O}}_X(-Y)\big )\) be a twisted canonical form defined over Y. There exists a covering \((\Omega _i)\) of X with coordinate sets such that the restriction of the section \(\displaystyle u|_{\Omega _i}\) of u admits an extension \(U_i \in H^0 (\Omega _i, K_X+ Y+ L)\), which belongs to the multiplier ideal of \(h_L\), i.e.,
$$\begin{aligned} \int _{\Omega _i}|U_i|^2e^{-\varphi _L- \varphi _Y}< + \infty . \end{aligned}$$(1.0.1)
We note that near a non-singular point of Y the existence of \(U_i\) follows from the usual \(L^2\) hypothesis of OT theorem provided that u belongs to the multiplier ideal sheaf of \(h_L|_Y\). But this may no longer be true in a neighborhood of singular point of Y.
In addition to the natural hypotheses (a), (b), and (c) above we collect next two other requirements we need to impose for some of our statements to hold.
-
(i)
We assume that there exists an open subset \(V_{\textrm{sing}}\) of X containing the singularities of Y such that the following hold.
- (i.\(\alpha \)):
-
There exists a snc divisor \(W= \sum W_j\) on X such that the singularities of the restriction of the metric \(h_L\) of L to \(V_{\textrm{sing}}\) are as follows:
$$\begin{aligned} \varphi _L= \sum _j \left( 1- \frac{1}{k_j}\right) \log |z_j|^2+ \tau _L, \end{aligned}$$where \(k_j\) are positive integers, and \(z_j\) are the local equations of W. The local weight \(\tau _L\) above is assumed to be bounded.
- (i.\(\beta \)):
-
The curvature of the restriction of \(h_L|_{V_{\textrm{sing}}}\) is greater than \(\displaystyle C_{\textrm{sing}}\omega _{{\mathcal {C}}}|_{V_{\textrm{sing}}}\), where \(C_\textrm{sing}\) is a positive constant, and \(\omega _{{\mathcal {C}}}\) is a fixed Kähler metric with conic singularities on \(\big (X, \sum (1-1/k_i)W_i\big )\), i.e., locally we have
$$\begin{aligned} \omega _{\mathcal {C}}\simeq \sum _{i=1}^r\frac{\sqrt{-1}dz_i\wedge \textrm{d}\overline{z}_i}{|z_i|^{2-\frac{2}{k_i}}} + \sum _{i\geqslant r+1}\sqrt{-1}dz_i\wedge \textrm{d}\overline{z}_i \end{aligned}$$(1.0.1)that is to say, \(\omega _{\mathcal {C}}\) is quasi-isometric with the RHS of (1.0.1), where \(z_1\dots z_r=0\) is the local equation of the divisor \(W=\sum _{i=1}^r W_i\).
-
(ii)
There exists an open subset \(V_{\textrm{sing}}\) of X containing the singularities of Y such that the curvature of the restriction of \(h_L|_{V_{\textrm{sing}}}\) is identically zero.
In this context our first result states as follows:
Theorem 1.1
We assume that the metric \(h_L= e^{-\varphi _L}\) of L and the section u verifies the requirements (a), (b), (c) as well as (i) above. Then u extends to X, and for each \(1\geqslant \alpha > 0\) there exists a section \(U \in H^0 (X, K_X +Y+L)\) such that \(U |_Y =u\) and we have the estimates
where \(\omega _{{\mathcal {C}}}\) is the reference metric on X and the constant C depends on \(\alpha \), the geometry of \((V_{\textrm{sing}}, \omega _{{\mathcal {C}}})\), the positivity constant and the upper bound for \(\textrm{Tr} \,_{\omega _{{\mathcal {C}}}}dd^c\tau _L\) in (i.\(\alpha \)).
Remark 1.2
The precise dependence of the constant C in (1.1.1) of the quantities mentioned in Theorem 1.1 can be easily extracted from the proof we present in Sect. 5.
Remark 1.3
It is very likely that our arguments work under more general circumstances, e.g., one can probably establish the same result in the absence of the hypothesis (b) (via the regularization procedure due to Demailly, cf. [12]). But so far it is unclear to us how to remove the local strict positivity hypothesis in (i.\(\beta \)), or the fact that the singularities of \(\displaystyle h_L|_{V_{\textrm{sing}}}\) are assumed to be of conic type.
A result pointing in the direction of the preceding Remark 1.3 is the following.
Theorem 1.4
Let X be a projective manifold and let \((L, h_L)\) be a line bundle such that the usual conditions (a)–(c) are satisfied, respectively. We assume moreover that \(\Theta _{h_L} (L) \leqslant C_1 \omega \) on \(V_{\textrm{sing}}\) for some constant \(C_1\). Here \(\omega \) is a non-singular Kähler metric on X. Then there exists a section \(U \in H^0 (X, K_X +Y+L)\) such that \(U |_Y =u\) with the estimates
where C depends only on the geometry of \((V_{\textrm{sing}}, \omega )\) and \(C_1\).
Remark 1.5
We stress on the fact that in the statement Theorem 1.4 the strict positivity of the curvature of \((L, h_L)\) on \(V_\textrm{sing}\) is not part of hypothesis.
In conclusion, Theorems 1.1 and 1.4 are providing an extension of u whose \(L^2\) norm is bounded by the usual quantity outside the singularities of Y, and by an ad hoc \(L^{p}\) norm near \(Y_{\textrm{sing}}\), for any \(p\in [1, 2[\). The example proving Claim 4 in Appendix shows that this type of estimates are sharp.
However, the constant \(C_\alpha (V_{\textrm{sing}}, \omega _{{\mathcal {C}}})\) in Theorem 1.1 involves the geometry of the local pair \((V_{\textrm{sing}}, \omega _{{\mathcal {C}}})\), or if one prefers, the restriction of \(h_L\) to \(V_{\textrm{sing}}\). Moreover, we only allow singularities of conic type for \(h_L|_{V_{\textrm{sing}}}\). In order to try to “guess” the type of estimates one could hope for in general, we make the following observation. Let \(\Omega \subset V_{\textrm{sing}}\) be a coordinate open subset. The restriction of the RHS of (1.1.1) to \(\Omega \) is given by the following expression:
where \(\prod f_j= 0\) is the local equation of \(Y\cap \Omega \) and \(\prod z_i= 0\) is the equation of W. Therefore the second term of RHS of (1.1.1) is equivalent to
where \(\omega \) is a smooth Kähler metric on X. So from this point of view the following important—and very challenging—problem is natural.
Conjecture 1.6
We assume that the metric \(h_L= e^{-\varphi _L}\) of L and the section u verifies the requirements (a), (b), (c). Then for each \(1\geqslant \alpha > 0\), there exists an section \(U \in H^0 (X, K_X +Y+L)\) such that \(U |_Y =u\) with the estimates
where \(\omega \) is a reference smooth Kähler metric on X and \(C_\alpha (V_{\textrm{sing}})\) only depends on \((V_{\textrm{sing}}, \omega )\) and the restriction of the metric \(h_L|_Y\) to Y.
It is our belief that the most subtle part of the previous conjecture would be to have an accurate estimate for the constant \(C_\alpha (V_{\textrm{sing}})\).
Our next two results are of experimental nature and therefore we have decided to formulate them for surfaces only, so that we have \(\dim (X)= 2\). The same type of statements hold in arbitrary dimension, as one can easily convince oneself. The method of proof is completely identical to the case we explain here in detail, so for simplicity’s sake we stick to the case of surfaces.
We fix next few more notations adapted to the pair (X, Y).
Let \((\Omega _i)_{i\in I}\) be covering of X with open coordinate subsets. By the simple normal crossing hypothesis we can choose coordinates \(z_i= (z_i^1, z_i^2)\) such as
for each \(i\in I\) and some p (depending on i). Let \((\theta _i)_{i\in I}\) be a partition of unity subordinate to \((\Omega _i)_{i\in I}\).
Since we assume that X is a complex surface, the components of Y are smooth curves. The singular set of Y (i.e., the mutual intersections of its components) consists of a finite number of points of X, denoted by \(p_1,\dots , p_s\).
We assume that \(\Omega _i\) is refined enough so that the section \(u|_{\Omega _i}\) is given by
for some holomorphic function \(f_i\). On overlapping open subsets, different expressions (1.6.3) are gluing only modulo a 2-form divisible by the equation of the divisor Y.
Let p be one of the singular points of Y, assumed to be the center of some \(\Omega _i\). We denote by \(t_i:= z_i^1\cdot z_i^2\); this is—by our previous conventions—the local equation of the cross \(Y\cap \Omega _i\). We can interpret the function (= \(n-2\)-form in general) \(f_i\) as a local section of the bundle \(L|_{\Omega _i}\), and as such we can consider its derivative
with respect to the Chern connection of L. The result is a 1-form on \(\Omega _i\).
Given the hypothesis in our following statement, it is possible to construct an extension of u by applying the result in [23]. However, here we obtain different types of estimates.
Theorem 1.7
Let X be a smooth projective surface, and let \((L, h_L)\) be a line bundle such that the usual curvature conditions (a) and (c) are satisfied. Assume moreover that \(h_L\) is non-singular and for each \(i\in I_{\textrm{sing}}\) we have \(f_i\in (z_i^1, z_i^2)\), in other words our section vanishes on the set singular points of Y.
Then there exists a section \(U \in H^0 (X, K_X +Y+L)\) enjoying the following properties:
-
(1)
\(U|_{Y} =u\).
-
(2)
There exists a constant \(C(X, V_{\textrm{sing}})> 0\) such that we have
$$\begin{aligned} \frac{1}{C(X, V_{\textrm{sing}})}\int _X|U|^2e^{-\varphi _Y-\varphi _L}&\leqslant \int _{Y\setminus V_{\textrm{sing}}}\log ^2(\max |s_{j}|^2)\left| \frac{u}{\textrm{d}s}\right| ^2e^{-\varphi _L}\nonumber \\&\quad \quad + \int _{Y\cap V_{\textrm{sing}}}\log ^2(\max |s_{j}|^2) |\partial _{\varphi _L} u|^2e^{-\varphi _L}. \end{aligned}$$(1.7.1)
We obtain the same type of result provided that the bundle \((L, h_L)\) is flat near \(Y_{\textrm{sing}}\), as follows.
Theorem 1.8
Let X be a smooth projective surface, and let \((L, h_L)\) be a line bundle such that the curvature and \(L^2\) conditions (a), (b), (c) as well as the additional property (ii) are satisfied.
Then there exists a section \(U \in H^0 (X, K_X +Y+L)\) enjoying the following properties:
-
(1)
\(U|_{Y} =u\).
-
(2)
We have
$$\begin{aligned} \frac{1}{C(X, V_{\textrm{sing}})}\int _{X\setminus V_{\textrm{sing}}}|U|^2e^{-\varphi _Y-\varphi _L}&\leqslant \int _{Y\setminus V_{\textrm{sing}}}\log ^2(\max |s_{j}|^2)\left| \frac{u}{\textrm{d}s}\right| ^2e^{-\varphi _L}\nonumber \\&\quad + \int _{Y\cap V_{\textrm{sing}}}\log ^2(\max |s_{j}|^2) |\partial _{\varphi _L} u|^2e^{-\varphi _L}. \end{aligned}$$(1.8.1)
Our next statement is confined to the two-dimensional case.
Theorem 1.9
Let X be a smooth projective surface, and let \((L, h_L)\) be a line bundle such that the usual curvature and \(L^2\) conditions (a), (b), and (c) are satisfied, respectively. We assume moreover that the following hold.
-
(1)
There exists a component \(Y_1\) of Y which intersects \(\cup _{j\ne 1} Y_j\) in a unique point \(p_1\), such that \(u(p_1)\ne 0\).
-
(2)
The restriction \((L, h_L)|_{Y_1}\) is Hermitian flat.
Then the section u admits an extension U satisfying the same estimates as in Theorem 1.7.
The raison d’être of the previous Theorems 1.7, 1.8, and 1.9 is that the inequality (1.8.1) is meaningful even in the absence of the additional hypothesis these statements contain. Because of the variety of contexts in which an extension of u verifying the estimates of type (2) of Theorem 1.8 can be obtained, it is very tempting to formulate the following.
Conjecture 1.10
Let (X, Y) be a smooth projective pair, where X is a surface and Y is an snc divisor. Let \((L, h_L)\) be a line bundle such that the properties (a), (b), and (c) are satisfied. Then there exists a section \(U \in H^0 (X, K_X +Y+L)\) enjoying the following properties:
-
(1)
The section U is an extension of u.
-
(2)
We have
$$\begin{aligned} \frac{1}{C(X, V_{\textrm{sing}})}\int _{X\setminus V_\textrm{sing}}|U|^2e^{-\varphi _Y-\varphi _L}&\leqslant \int _{Y\setminus V_\textrm{sing}}\log ^2(\max |s_{j}|^2)\left| \frac{u}{\textrm{d}s}\right| ^2e^{-\varphi _L}\nonumber \\&\quad + \lim _{\varepsilon \rightarrow 0}\int _{Y\cap V_{\textrm{sing}}}\log ^2(\max |s_{j}|^2) |\partial _{\varphi _{L, \varepsilon }} u|^2e^{-\varphi _{L, \varepsilon }}, \end{aligned}$$(1.10.1)where \(\varphi _{L, \varepsilon }:= \log \left( \varepsilon ^2e^{\phi }+ e^{\varphi _L}\right) \) is a non-singular approximation of \(h_L\).
In the sequel of this article we will formulate the precise higher dimensional version of this conjecture, and we will explain its impact on the extension of the pluricanonical forms.
In Appendix A by Bo Berndtsson some examples are given that indicate that the estimates (2) in Conjecture 1.10 are most likely the best one could hope for: without the \(\log \) factor, this conjecture is simply wrong. Moreover, the example given in order to prove Claim 4 shows that the factor \(e^{-\varphi _L}\) in (1.6.1) cannot be replaced by the slightly less singular weight \(e^{-(1-\varepsilon )\varphi _L}\), for any \(\varepsilon > 0\). Finally, Appendix contains a comparison with a one-dimensional problem (the fat point), intended to highlight the origin of the difficulties in a very simple setting.
1.1 Organization of the Paper
In the second section we explain the main ideas involved in the proof of our results. The next section is dedicated to the revision and slight improvement of the usual a-priori inequalities. Our principal contribution to the Ohsawa–Takegoshi artisan industry is in Sect. 4, where the necessary tools from geometric analysis are recalled/developed. The proof of the results stated above is presented in Sect. 5.
1.2 In Memoriam Nessim Sibony
This article is dedicated to a mentor and colleague, Nessim Sibony, who has left us in 2021. His untimely death has affected us all deeply, leaving a void in the complex geometry community. He was a highly accomplished mathematician and a colleague who was inspirational to generations. He will be remembered not only as a revered collaborator with his profound understanding of the field of complex geometry but also as a dear friend with his sharp grasp of a wide range of subjects from literature, history, and to life in general.
2 An Overview of the Arguments
Our results are obtained by combining the method in [28] with the method in [2]. In order to highlight the main arguments as well as some of the difficulties, we only discuss here the case of a non-singular metric \(h_L\) on L. In general the whole scheme of the proof becomes more technical, since the regularization procedure we have to use for the metric is quite tricky to implement in the presence of the singular hypersurface Y.
We start with a quick review of the usual case.
2.1 The Case of a Non-singular Hypersurface Y
Let \(\xi \) be a L-valued form of type (n, 1). We denote by \(\gamma _\xi := \star \xi \) its Hodge dual (induced by an arbitrary Kähler form on X).
Consider the functional
associated to the current \({\overline{\partial }} \left( \frac{u}{s_Y}\right) \).
We decompose \(\xi = \xi _1+ \xi _2\) according to \(\textrm{Ker}({\overline{\partial }})\) and \(\textrm{Ker}({\overline{\partial }})^\perp \). It turns out that we have the equality
which is not completely obvious, given that the current defining \({\mathcal {F}}\) is not in \(L^2\).
We have \(\frac{u}{\textrm{d}s_Y}\in L^2(e^{-\varphi _L})\), hence it is enough to find an upper bound for
This is done by the next estimate, which is derived in [4] via the \(\partial {\overline{\partial }}\)—Bochner method due to Siu cf. [39]
In conclusion, we have
and the “estimable” extension will be obtained by using the solution of the equation \({\overline{\partial }} \left( \frac{u}{s_Y}\right) = {\overline{\partial }} v\). We define \(U:= s_Y v\) and then we have
2.1.1 Difficulties in the Case of an snc Hypersurface Y
In our setting we have \(Y= \bigcup Y_i\), and the difficulty steams from the fact that the functional
becomes a sum of expressions involving forms with log poles. We have
in general, so the previous arguments are breaking down.
Nevertheless we do have
near the singularities of Y for any reasonable metric \(\omega \). This means that we have to find an estimate of the \(L^\infty \) norm of \(\gamma _{\xi _1}|_{V_{\textrm{sing}}\cap Y}\) in terms of the RHS of (2.0.5).
To this end, we use a procedure due to Donaldson–Sun in [15]. This consists in the following simple observation. Assume that the support of \(\xi \) is contained in \(X\setminus V_{\textrm{sing}}\). Then we have
in other words, the restriction of \(\xi _1\) to \(V_{\textrm{sing}}\) is harmonic. As we learn from function theory, harmonic functions satisfy the mean value inequality: this is what we implement in our context, and it leads to the proof of Theorem 1.1.
The drawback of this method is that in the end, the constant measuring the \(L^2\) norm of the extension is far from being as universal as in the case \(Y_{\textrm{sing}}= \emptyset \). This is due to the fact that the quantity \(\Delta ''|\xi _1|^2\) has a term with wrong sign involving the trace of the curvature of \((L, h_L)\) with respect to the metric \(\omega \) on X. This trace is not bounded e.g., if \(h_L\) is singular and \(\omega \) is a fixed, non-singular Kähler metric. It is for this reason that the singularities of \(h_L\) and those of \(\omega _{{\mathcal {C}}}\) must be the same in Theorem 1.1.
3 A-Priori Inequalities Revisited
We first recall the following estimate, which is essentially due to [4].
Theorem 3.1
Let \((X, \omega )\) be a Kähler manifold, and Y be simple normal crossing divisor in X. Let L be a line bundle on X with a non-singular metric \(h_L\) such that
for some \(\delta >0\) small enough, where \(h_Y\) is a smooth metric on \({\mathcal {O}}_X (Y)\) such that \(|s_Y|^2_{h_Y}\leqslant e^{-\delta }\). Let \(\xi \) be a smooth (n, 1) form with compact support and with values in L. We denote by \(\gamma _\xi := \star \xi \) the image of \(\xi \) by the Hodge operator. Then we have
where C is a numerical constant and \(\tau \) is an arbitrary real number.
Before giving the proof of Theorem 3.1 we notice that it implies the following statement:
Theorem 3.2
Let \((X, \omega )\) be a Kähler manifold, and Y be simple normal crossing divisor in X. Let L be a line bundle on X with a non-singular metric \(h_L\) such that
for any \(\delta >0\) small enough, where \(h_Y\) is a smooth metric on \({\mathcal {O}}_X (Y)\). Let \(\xi \) be a smooth (n, 1) form with compact support and with values in L. We denote by \(\gamma _\xi := \star \xi \) the image of \(\xi \) by the Hodge operator. Then we have
where \(s_{Y}\) is the canonical section of \({\mathcal {O}}(Y)\), normalized in a way that works for the proof.
Proof of Theorem 3.1
We note that this improves slightly the estimate of Bo Berndtsson in [4], but the proof is virtually the same. Nevertheless, we will provide a complete treatment for the convenience of the reader.
To start with, we recall the following "\(\partial {\bar{\partial }}\)-Bochner formula.”
Lemma 3.3
([38]) Let \(\xi \) be a (n, 1)-form with values in \((L, h_L)\) and compact support. We denote by \(\gamma _\xi = \star \xi \) the Hodge \(\star \) of \(\xi \) with respect to a Kähler metric \(\omega \). Let
be the \((n-1, n-1)\)-form on X corresponding to \(\xi \), where \(c_{n-1}= \sqrt{-1}^{(n-1)^2}\) is the usual constant. Then we have the equality
We apply this in the following context. Consider the function \(w:= \log \frac{1}{|s_Y|^2+ \tau ^2}\). A quick computation gives
where \(\theta _Y:= \Theta _{h_Y}\left( {\mathcal {O}}(Y)\right) \) is the curvature of the bundle \({\mathcal {O}}(Y)\) with respect to the metric \(h_Y\).
We multiply the equality (3.3.2) with w and integrate the resulting top form over X. The left-hand side term is equal to the difference of two terms
and
and we see that (3.3.5) is the term we have to estimate.
We drop the positive terms on the right-hand side of (3.3.2) and we therefore get
A first observation is that the curvature term (3.3.6) is negative, by the hypothesis of Theorem 3.1. Moreover, by Stokes formula we have
so we see that modulo the second term on the RHS of (3.3.7), we are done.
In order to take care of it we use Cauchy–Schwarz inequality and we obtain
Thus the new term to bound is
and as observed in [4], the quantity (3.3.9) is less singular that the LHS of (3.3.6), which was our initial problem.
In order to obtain a bound for (3.3.9) we consider the function
We have
and we use the same procedure as before, but with \(w_1\) instead of w. The analog of (3.3.4) and (3.3.5) read as
and this is good, because the second term in (3.3.11) is the one we are now after. We skip some intermediate steps because they are absolutely the same as in the preceding consideration, except that \(w_1\) appears instead of w. After integration by parts, the new “bad term,” i.e., the analog of the RHS of (3.3.7) in our current setting is
for which we use Cauchy–Schwarz inequality and the observation is that \(\partial w_1\wedge \overline{\partial w_1}\) coincides with \(\frac{\sqrt{-1}\partial w\wedge {{\bar{\partial }}}w}{w^2}\).
As a result of this second part of the proof we infer that we have
Then Theorem 3.1 follows, by combining (3.3.13) with (3.3.8). \(\square \)
Remark 3.4
Actually we can use the second part of the proof of Theorem 3.2 in order to get the estimates
where \(\sigma \) is a holomorphic section of a line bundle \((F, h_F)\) endowed with a non-singular metric \(h_F\). The constant “C” in (3.4.1) depends on the norm of the curvature of \((F, h_F)\). Thus, we obtain an estimate of the norm of \(\gamma _\xi \) in the tangential directions of \(\sigma =0\) with respect to the Poincaré-type measure associated to \(\sigma \). If the curvature of \((L, h_L)\) is greater than the some (small) multiple of \(\Theta _{h_F}(F)\), then we can remove the term \(|\xi |^2\) in the formula (3.4.1).
4 Geometric Analysis Methods and Results
In this section, we follow the notations in Theorem 1.1. Moreover, we ask that the metric \(\varphi _L\) satisfies the requirements (a), (b), (c) and \((i.\alpha )\). In particular, we don’t assume \((i.\beta )\) for \(\varphi _L\).
Let \(\xi \) be a L-valued form of (n, 1) type such that \(\textrm{Supp}(\xi ) \subset X\setminus (V_{\textrm{sing}}\cup |H|)\). We recall that here \(V_{\textrm{sing}}\) is an open subset of X containing the singularities of Y, and H is a hyperplane section containing the singularities of the metric \(h_L\).
We consider the orthogonal decomposition
where \(\xi _1\in \textrm{Ker}({{\bar{\partial }}})\) and \(\xi _2\in \textrm{Ker}({{\bar{\partial }}})^\perp \) with respect to the fixed Kähler metric \(\omega _{{\mathcal {C}}}\) with conic singularities on X and the given metric \(h_L\) on L.
\(\bullet \) The convention during the current section is that we denote by “C” any constant which depends in an explicit way of the quantities we will indicate.
4.1 Orthogonal Decomposition, I: Approximation
In the following sections we will use an approximation statement, for which the context is as follows:
We can write
where each \(\Omega _m\) is a Stein domain with smooth boundary. Let \(\omega _m\) be a complete metric on \(\Omega _m\). Corresponding to each positive \(\delta \) we introduce
it is a complete metric on \(\Omega _m\) such that \(\omega _{m, \delta }> \omega _{{\mathcal {C}}}\) and \(\lim _{\delta \rightarrow 0}\omega _{m, \delta }= \omega _{{\mathcal {C}}}\) for each m.
We remark that the \(L^2\) norm of \(\xi \) with respect to \(\omega _{m, \delta }\) and \(h_L|_{\Omega _m}\) is finite, given the pointwise monotonicity of the norm of (n, 1)-forms. Then we can decompose the restriction of \(\xi \) to each \(\Omega _m\) as follows:
We establish next the following statement.
Lemma 4.1
We have
uniformly on compact sets of \(X\setminus H\).
The proof is based on the monotonicity of the \(L^2\) norms
for each \(m, \delta \) and for any form \(\rho \) of type (n, 1) with values in L. The details are as follows:
Proof
Let \(K\subset X\setminus (\varphi _L= -\infty )\) be a compact subset. In what follows we are using the notation “\(\varepsilon \)” to indicate the set of parameters \((m, \delta )\), and we assume that \(m\gg 0\) so that \(K\subset \Omega _m\).
We first notice that for each parameter \(\varepsilon \) the form \(\xi _1^{(\varepsilon )}\) is smooth, and that it verifies the equation
where \(\Delta _\varepsilon ''\) is the Laplace operator on (n, 1)-forms with values in \((L, h_L)\) and \((\Omega _m, \omega _{m, \delta })\). We also have
given the fact that (4.1.1) is orthogonal. It follows that the family
is uniformly bounded in \({\mathcal {C}}^\infty \) norm. We can therefore extract a limit \(\xi _1^{(0)}\) as \(\varepsilon \rightarrow 0\), uniform on compact subsets by the usual diagonal process. We remark that we have
given that each form \(\xi _1^{(\varepsilon )}\) is \({{\bar{\partial }}}\)-closed, combined with (4.1.4).
On the other hand, let \(\rho \) be a \({{\bar{\partial }}}\)-closed form of (n, 1)-type with values in L. We assume moreover that \(\rho \) is \(L^2\) with respect to \(\omega _{{\mathcal {C}}}\) and \(h_L\). Then we equally have
for each \(\varepsilon = (m, \delta )\), and open subset \(\Omega _m\). We infer that
for each value of m and \(\varepsilon \).
Let \((K_l)\) be an increasing exhaustion of \(X{\setminus } (\varphi _L= -\infty )\) by relatively compact sets. If \(m\gg 0\) (depending on l) then we have
by Cauchy inequality combined with (4.1.7). It follows that
By letting \(\varepsilon \rightarrow 0\) we infer that for each fixed l we have
Next, the inequality (4.1.6) shows that \(\xi _2^{(0)}\) is \(L^2\)-integrable with respect to \((L, h_L)\) and \((X, \omega _{{\mathcal {C}}})\). It follows that we have
as \(l\rightarrow \infty \) since both \(\rho \) and \(\xi _2^{(0)}\) are in \(L^2\).
In other words, the form \(\xi _2^{(0)}\) is orthogonal to \(\textrm{Ker} \,{{\bar{\partial }}}\) and since we have
our lemma is proved (thanks to the uniqueness of such decomposition). \(\square \)
4.2 Orthogonal Decomposition, II: Mean Value Inequality
We analyze here the behavior of \(\xi _1\) restricted to the set \(V_{\textrm{sing}}\). During the current subsection we make the following conventions.
-
(i)
We work with respect to the Kähler metric \(\omega _{{\mathcal {C}}}\) exclusively on \(V_{\textrm{sing}}\subset X\) (this will be understood even if we do not mention it explicitly) and with respect to the Hermitian metric \(h_{L}\) defined in the previous section on L.
-
(ii)
We denote by \(\xi \) a (n, 1) form with values in L such that we have
$$\begin{aligned} \textrm{Supp}(\xi ) \subset X\setminus (V_{\textrm{sing}}\cup |H|). \end{aligned}$$We use the notations in (4.0.2) for its orthogonal decomposition with respect to \((\omega _{{\mathcal {C}}}, h_L)\).
-
(iii)
We fix \(\frac{1}{2}V_{\textrm{sing}} \Subset V_{\textrm{sing}}\) an open set of compact support in \(V_{\textrm{sing}}\), and \(\frac{1}{2}V_{\textrm{sing}}\) contains the singular locus of Y.
In this subsection we establish the next result.
Theorem 4.2
We have the mean value type inequality
where \(C(V_{\textrm{sing}})\) here is a constant which only depends on the allowed quantities, i.e., the geometry of \((V_{\textrm{sing}}, \omega _{{\mathcal {C}}})\) as well as \(\tau _L\) in the assumption \(\mathrm (i.\alpha )\).
The norm of \(\xi _1\) in (4.2.1) is measured with the conic metric \(\omega _{{\mathcal {C}}}\).
The proof of Theorem 4.2 unfolds as follows (cf. [15, 22] for similar computations). In order to simplify the notations, we drop the \(e^{-\varphi _L}\) in (4.2.1), and write \(|\xi _1|^2\) to express the pointwise norm of \(\xi _1\) with respect to \(\omega _{{\mathcal {C}}}\) and \(h_L\). First we show that there exists a constant C such that
where we denote by |W| the support of the divisor W. This is the main reason why we have to assume that the singularities of \(h_L\) and \(\omega \) are “the same” in Theorem 1.1.
After this, we establish a differential inequality satisfied by the function \(|\xi _1|^2\) in the complement of the set
This is standard, and it combines nicely with (4.2.2) and Moser iteration process to give (4.2.1).
Proof of (4.2.2)
First we establish the crucial boundedness result (4.2.2). Let \(z_1,\dots , z_n\) be a set of local coordinates defined on a open subset \(\Omega \subset V_{\textrm{sing}}\). We assume that the \((z_i)_{i=1\dots n}\) are adapted to the pair (X, W), meaning that the local equation of \(\Omega \cap W\) is
for some \(p\leqslant n\). By hypothesis, the weight of the metric \(h_L\) can be written as
where \(k_i\) are positive integers and \(\tau \) is a smooth function defined on \(\Omega \).
The restriction of \(\xi _1\) to \(\Omega \) has the following properties:
where \(\phi \) is any smooth (n, 0) form with compact support in \(\Omega \) which is \(L^2\)-integrable, and such that \({{\bar{\partial }}}\phi \) is in \(L^2\) as well.
These properties have a very neat interpretation in terms of ramified covers, as follows. Let
be the usual local covering map corresponding to the divisor \(\sum _{i=1}^p\Big (1-\frac{1}{k_i}\Big ) W_i\). We define the (n, 1)-form \(\eta \) on \(\widehat{\Omega }\) as follows:
and a first remark is that we have
where g is the inverse image of the conic metric \(\displaystyle g:= \pi ^\star \omega _{{\mathcal {C}}}\). The relation (4.2.7) is an immediate consequence of the change of variables formula, combined with the expression of \(\varphi _L\) in (4.2.3). It follows that
on \(\widehat{\Omega }\) (this is true pointwise outside the support of W, and it extends across W by using [12, Chap VIII, Lemma 7.3]).
Let \(\alpha \) be a smooth (n, 0)-form on \(\widehat{\Omega }\) with compact support. We claim that we have
Indeed this is clear, given the equality
where \(\widehat{\alpha }:= {\prod _{j=1}^p w_j^{k_j-1}}\alpha \). On the right-hand side of (4.2.10) we can assume that \(\widehat{\alpha }\) is the inverse image of a (n, 0) form with compact support on \(\Omega \). Indeed, let f be an element of the group G acting on \(\widehat{\Omega }\). Then we have
since all the other objects involved in the integral in question are invariant by inverse image. It follows that
where \(\pi ^\star \phi := \frac{1}{|G|}\sum _{f\in G}f^\star \widehat{\alpha }\). Then our claim follows by the third property in (4.2.4).
In conclusion, the form \(\eta \) is harmonic on \(\widehat{\Omega }\) with respect to the metric g and the weight \(e^{-\tau \circ \pi }\). It is in particular bounded, and this is equivalent to (4.2.2). \(\square \)
We choose geodesic local coordinates \(\displaystyle (z_i)_{i=1,\dots , n}\) for the Kähler metric \(\omega _{{\mathcal {C}}}\) locally near a point \(x_0\in V_{\textrm{sing}}{\setminus } W\). Let \(e_L\) be a local holomorphic frame of L, such that the induced weight \(\phi \) of \(h_{L}\) verifies the relations
We express \(\xi _1\) locally with respect to these coordinates
where \(dz:= dz^1\wedge \dots \wedge dz^n\). We then have
The formula for the Laplace operator is \(\Delta ''= \textrm{Tr} \,_{\omega _{{\mathcal {C}}}}\sqrt{-1}\partial {\bar{\partial }}\) and so we have
where we denote by \(\big (g_{p\overline{q}}\big )\) the coefficients of the metric \(\omega _{{\mathcal {C}}}\) with respect to the geodesic coordinates \((z_i)\) and by g the corresponding matrix.
In order to obtain an intrinsic expression of the terms containing the second derivative in the RHS of (4.2.16), we recall that we have
hence the next equality holds at \(x_0\)
where \(\square := [{{\bar{\partial }}}, {{\bar{\partial }}}^\star ]\) is the Laplace operator acting on L-valued forms of (n, 1) type. The formula (4.2.20) is only valid for closed forms, which is the case for \(\xi _1\). Also, we have \(\xi _{\overline{\alpha }, p\overline{q}}= \xi _{\overline{q}, p\overline{\alpha }}\) and therefore (twice the real part of) the first term on the RHS of (4.2.20) coincides with the second one on the RHS of (4.2.16).
Next, since the metric \(\omega _{{\mathcal {C}}}\) is Kähler we have \(g^{\overline{\alpha }\delta }_{, \delta \overline{\gamma }}= -g_{\alpha \overline{\gamma }, \delta \overline{\delta }}\) hence we obtain
where \({\mathcal {R}} _{\alpha \overline{\beta }}\) are the coefficients of the Ricci tensor of \(\omega _{{\mathcal {C}}}\).
The last term in (4.2.20) is simply \(\displaystyle \theta _{\alpha \overline{\beta }}\xi _{\overline{\alpha }}\overline{\xi _{\overline{\beta }}}\) where \(\theta _{\alpha \overline{\beta }}\) are the coefficients of \(\Theta _{h_L}(L)\).
Again by the Kähler hypothesis the term (4.2.17) is equal to
and therefore we obtain
by collecting the previous equalities at \(x_0\).
The Ricci curvature of the metric \(\omega _{{\mathcal {C}}}\) is uniformly bounded, so the function
(where the norm is measured with respect to \(\omega _{{\mathcal {C}}}\) and \(h_L\)) verifies the following properties:
-
(1)
We have \(\sup _{V_{\textrm{sing}}\setminus W}f< \infty \), and moreover f is smooth on \(V_{\textrm{sing}}\setminus W\).
-
(2)
The following differential inequality holds true
$$\begin{aligned} \Delta ''f^2\geqslant |\nabla f|^2- Cf^2 \end{aligned}$$(4.2.27)where C is a constant depending on the Ricci curvature of the metric \(\omega _{{\mathcal {C}}}\) and the trace of \(dd^c\tau \) with respect to it.
Indeed the inequality at the point (2) follows from (4.2.23), since we have
Based on (1) and (2) we can conclude in two ways: either show that Schoen–Yau mean value inequality holds for functions f which verify these properties (the proof would be a simple adaptation of the arguments presented in [37]), or use the Moser iteration procedure. In what follows, we use Moser procedure.
We show next that the following statement holds true.
Lemma 4.3
Let f be the function defined in (4.2.26). Then there exists a constant \(C_1\) depending on C and \((V_{\textrm{sing}}, \omega _{{\mathcal {C}}})\) only such that we have
Remark that the main point here is that the constant \(C_1\) is independent of the \(\sup \) norm in (1). After establishing this statement we are basically done, i.e., this implies Theorem 4.2 announced at the beginning of the current section.
Proof of Lemma 4.3
Let \(\rho \) be a function which is equal to 1 on \(\displaystyle 1/2V_{\textrm{sing}}\) and whose support is in \(V_{\textrm{sing}}\). Then we have
where the norm of the differential in (4.3.2) is measured with respect to the reference metric \(\omega \) on X.
Following [3], there exists a family of functions \((\Xi _\varepsilon )_{\varepsilon > 0}\) associated to the analytic subset \(W= (h_L= \infty )\cap V_{\textrm{sing}}\) such that \(\textrm{Supp}(\Xi _\varepsilon )\subset V_{\textrm{sing}}{\setminus } W\) and for each compact subset \(K\subset X\setminus W\) we have \(\Xi _\varepsilon |_K= 1\) if \(\varepsilon < \varepsilon (K)\) is small enough. Moreover we have
as \(\varepsilon \rightarrow 0\). We recall very briefly the construction: let \(\rho _\varepsilon \) be a function which is equal to one on the interval \([0, \varepsilon ^{-1}]\) and which equals zero on \([1+ \varepsilon ^{-1}, \infty [\). Then we define
where \(s_W\) is the sections whose zero set is W. Then with respect to the conic metric \(\omega _{{\mathcal {C}}}\) we have
up to a constant, from which (4.3.3) follows (we get a similar inequality for the Laplacian of \(\Xi _\varepsilon \)).
The existence of \((\Xi _\varepsilon )_{\varepsilon > 0}\) combined with the second property in (4.3.2) allows us to deal with the fact that f is not necessarily smooth.
The proof which follows is rather standard, but we will nevertheless provide a complete treatment for convenience. We refer to [16] for a general discussion about Moser iteration method, and more specifically to [22] where this is implemented in a context which is very similar to ours here.
We multiply the differential inequality (4.2.27) with \(\Xi _\varepsilon \rho ^2\) and integrate the result over X; we infer that we have
On the LHS we integrate by parts. The terms containing derivatives of \(\Xi _\varepsilon \) are
and they tend to zero precisely because of the uniform boundedness property (1) of f, together with (4.3.3). These terms are vanishing as \(\varepsilon \rightarrow 0\), and the inequality (4.3.6) becomes
On the other hand we write
which combined with (4.3.8) gives
where the constant C in (4.3.10) only depends on the norm of the first and second derivatives of \(\rho \).
The following version of Sobolev inequality is a direct consequence of [21], page 153.
Theorem 4.4
There exists a constant \(C>0 \) such that the following holds
for any function f on X.
We therefore infer that we have
In order to obtain estimates for higher norms, we use (4.4.1) for \(f:= \Xi _\varepsilon \rho f^{\frac{p}{2}}\) and we obtain
We show now that the second term of the right-hand side of (4.4.3) verifies the inequality
This is done using integration by parts: we have
so we have to obtain a bound for the term
We have
and as before, the terms involving \(\nabla (\Xi _\varepsilon )\) tend to zero as \(\varepsilon \rightarrow 0\). We therefore get the inequality (4.4.4). Remark that we are using the inequality (4.2.27) in order to obtain (4.4.6).
We define \(V_i:= (1/2+ 1/2^i)V_{\textrm{sing}}\) and let \(\rho _i\) be a cutoff function such that \(\rho _i= 1\) on \(V_{i+1}\) and such that \(\textrm{Supp}(\rho _i)\subset V_i\). Then we have \(|\nabla \rho _i|\leqslant C2^i\), and by using (4.4.4) combined with the usual iteration process, Lemma 4.3 follows. \(\square \)
Theorem 4.2 is proved. \(\square \)
Remark 4.5
Actually a careful examination of the proof shows that one can obtain a constant “C” in Lemma 4.3 as follows:
If necessary, this can be obtained by adapting the arguments of Schoen–Yau in [37, p. 75].
5 Proof of the Main Results
5.1 Proof of Theorem 1.1
We consider the “usual” family of cutoff functions
where \(\rho \) is a function defined on the set of positive real numbers such that \(\rho = 1\) on [0, 1] and \(\rho = 0\) on \([2, \infty [\).
We will show here that the following a-priori inequality holds
where \(\xi \) is a (n, 1)-form with values in \(Y+ L\) whose support is contained in the complement of \(V_{\textrm{sing}}\cup H\), and \(U_0\) is an arbitrary holomorphic extension of u, cf. [7]. Also, we denote by \(\phi _L\) the metric
on the bundle \(L+ Y\). We will see that the constant \(C_\varepsilon (U_0)\) in (5.0.3) is explicit, and it converges to the RHS of (1.6.1) as \(\varepsilon \rightarrow 0\). Note that all the integrals above are at least well defined, given the condition we impose on the support of \(\xi \).
The proof of (5.0.3) will be presented along the following line of arguments.
\(\bullet \) Consider a (n, 1)-form \(\xi \) as above together with the orthogonal decomposition \(\frac{\xi }{s_Y}= \xi _1+ \xi _2\) we have already discussed in detail in the previous section. Then we show that we have
up to a numerical constant. This will be done by an approximation argument, using Lemma 4.1 as well as Theorem 3.1.
\(\bullet \) The norm of the functional on the LHS of (5.0.3) is evaluated in two steps on the set \(V_{\textrm{sing}}\) we use Theorem 4.2, combined with a few simple calculations. In the complement \(X\setminus V_{\textrm{sing}}\) the arguments are rather standard: we will use (5.0.5).
The remaining part of the current section is organized as follows. We first show that (5.0.3) implies the existence of an “estimable extension” of u. Then we prove that the estimate (5.0.3) holds true.
5.1.1 Functional Analysis
Our method relies on the next statement.
Theorem 5.1
Let \(u\in H^0 (Y, (K_X +Y+ L) |_Y)\) be a holomorphic section. We assume that there is a constant \(C_\varepsilon (U)\) such that for any L-valued smooth test form \(\xi \) of type (n, 1) with compact support in \(X\setminus (V_{\textrm{sing}}\cup |W|)\) the a-priori inequality (5.0.3) holds. Then u admits an extension \(U\in H^0 (X, K_X+Y+ L)\) such that
Proof
This is done as in the classical case, by considering the vector subspace
of the \(L^2_{n,0}(X, Y+L)\) forms, endowed with the scalar product induced by
The functional
is well defined and bounded on E by (5.0.3), hence it extends by Hahn–Banach. The representation theorem of Riesz implies that there exists some form
such that we have
for all test forms \(\xi \in E\) and such that
Equation (5.1.5) shows that we have
on \(X\setminus V_{\textrm{sing}}\). On the other hand, the form
is in \(L^2(X\setminus V_{\textrm{sing}})\): this is clear for the first term, as for the second one it is a consequence of (5.1.6).
We infer that the form
extends holomorphically on \(X\setminus V_{\textrm{sing}}\). This implies that \(v|_{X\setminus V_{\textrm{sing}}}\) is non-singular, in particular v equal zero when restricted to \(Y{\setminus } V_\textrm{sing}\)—given the estimates in (5.1.6).
Therefore we infer the equality
We remark that \(U_\varepsilon \) extends to X by theorem of Hartog’s. This is clear if X is a surface cf. e.g., [19, Theorem 2.3.2]. The general case follows as a consequence of this, by a simple argument of slicing which we will not detail here.
Finally, the estimate for the \(L^2\) norm of U is deduced from (5.1.6): we have
modulo a quantity which tends to zero as \(\varepsilon \rightarrow 0\). The conclusion follows. \(\square \)
5.1.2 End of the Proof
We prove now the inequality (5.0.3). As we have already mentioned, one of the main part of the proof is based on the a-priori estimate (5.0.5) which we derive here from Theorem 3.2 combined with the results established in the first part of Sect. 3.
We start with the following technical result, which plays a key role in the arguments to come. In order to simplify the notations, we write \(\xi \) instead of the quotient \(\displaystyle \frac{1}{s_Y}\xi \).
Proposition 5.2
Consider the orthogonal decomposition \(\xi = \xi _1+ \xi _2\). Then the following hold: for each positive \(\varepsilon \) we have
where N is the number of components of Y.
Remark that we have the equality \(\displaystyle |{{\bar{\partial }}}^\star \mu |^2e^{-\varphi _L}= |{{\bar{\partial }}}^\star \xi |^2e^{-\phi _L}\) if \(\mu = \frac{1}{s_Y}\xi \), so the estimate (5.2.1) is precisely what we have to prove.
Proof
We recall the context in Sect. 3: we have considered an exhaustion
where each \(\Omega _m\) was a Stein domain with smooth boundary, endowed with the family of complete metrics \(\omega _{m, \delta }\) cf. (4.0.4). The restriction of \(\xi \) to each \(\Omega _m\) decomposes as follows:
according to \((\Omega _m, \omega _{m, \delta })\) and \((L, h_L)\).
We apply the inequality in Theorem 3.1 for \(\xi _1^{(m,\delta )}\) and we get
Indeed we can use Theorem 3.1 in this context even if the form does not have compact support because the metric \(\omega _{m, \delta }\) is complete. This has another consequence: we have the equality \({{\bar{\partial }}}^\star (\xi _1^{(m,\delta )})= {{\bar{\partial }}}^\star (\xi )\). For the inequality (5.2.4) we have used the inequality
where N is the number of components of Y.
Let \(K\subset X\) be any open set with compact closure in \(X{\setminus } H\); for any \(m\geqslant m_0(K)\) we have \(\overline{K}\subset \Omega _m\) so the inequality (5.2.4) implies
Now the support of \(\xi \) is a compact contained in \(X\setminus H\), so if m is large enough the boundary of \(\Omega _m\) is disjoint from \(\textrm{Supp}(\xi )\). A limit process (i.e., \(\delta \rightarrow 0, m\rightarrow \infty \)), together with Lemma 4.1 implies that we have
The compact subset K in (5.2.6) is arbitrary, so Proposition 5.2 is proved. \(\square \)
We are now ready to finish the proof of Theorem 1.1. Consider the integral
which up to a sign equals
We decompose as usual \(\frac{\xi }{s_Y}= \xi _1+ \xi _2\) and then (5.2.8) becomes
We split its evaluation into two parts. The first one is
and by Cauchy–Schwarz inequality the square of its absolute value is smaller than
where \(K_\varepsilon \) is the support of the function \(\rho '\left( \frac{|s_Y|^2}{\varepsilon ^2}\right) \). We remark that we have
on the set \(K_\varepsilon \). Therefore, the second factor of the product (5.2.11) is smaller than
by Proposition 5.2.
The rest of the integral (5.2.9) is analyzed as follows. For simplicity we assume that \(V_{\textrm{sing}}= \Omega \) is a coordinate subset and the expression we have to evaluate is
This is bounded by the quantity
and Hölder inequality shows that (5.2.15) is smaller than the product of
with
The limit of the quantity (5.2.17) as \(\varepsilon \rightarrow 0\) is equal to
As for the (5.2.16), we use Theorem 4.2 together with our previous considerations and it follows that it is smaller than
It is at this point that we are using the positivity assumption (i): we have
where \(C_{\textrm{sing}}\) is the (positive) lower bound for the positivity of \((L, h_L)|_{V_{\textrm{sing}}}\). By Bochner formula we get
Thus we obtain the expected estimate for the functional (5.2.9) and Theorem 1.1 is proved. We remark that the contribution of the singularities of Y to the estimate in this result is
where C is a constant depending on \((X, V_{\textrm{sing}}, {\omega _{{\mathcal {C}}}})\) and \(\alpha \in [0, 1]\) is an arbitrary positive real which is smaller than 1.
Remark 5.3
The quantity (5.2.22) is part of the term estimating
A slight modification of the proof shows that we can get a similar estimate for the integral
for any strictly positive real \(\tau \).
Remark 5.4
Actually one can replace the curvature condition (i) with the following: there exists a constant \(C_{\textrm{sing}}> 0\) such that we have
pointwise on \(V_{\textrm{sing}}\). The estimate for the extension we obtain in the end is the same, but we are using a twisted Bochner formula instead of (5.2.21).
5.2 Proof of Theorem 1.4
We follow the notations in Theorem 1.8. Since \({{\bar{\partial }}}\xi _1 =0\), the Hodge relation \(\partial ^\star _{\varphi _L} = [{{\bar{\partial }}}, \Lambda _{\omega }]\) implies that
Let \(\cup \Omega _i\) be a small Stein cover of \(V_{\textrm{sing}}\). The standard \(L^2\)-estimate as well as the Bochner inequality (5.5.7) imply that we can find \(G_i\) satisfying
where C is a uniform constant independent of \(h_L\).
We would like to control the \(C^0\)-norm of \(G_i\). Note first that, as \(i\Theta _{h_L}(L) \) is bounded on \(V_{\textrm{sing}}\), we can found a bounded function \(\varphi _L '\) on \(\Omega _i\) such that
Therefore \(\varphi _L -\varphi _L'\) is the real part of some holomorphic function \(f_i\) on \(\Omega _i\). Replacing \(\xi _1\) by \(\xi _1 \cdot e^{-\frac{f_L}{2}}\), we can suppose in the beginning that the \(C^2\)-norm of \(\varphi _L\) is bounded by the constant \(C_1\).
Thanks to (5.4.2), \(F_i:= \gamma _{\xi _1} - G_i\) is a holomorphic \((n-1, 0)\)-form on \(\Omega _i\). Recall that \(\xi _1\) is \(\Delta ^{''}\)-harmonic on \(V_{sing}\). Then
Using (5.4.2), the \(H^{-2}\)-norm of \( \Delta ^{''} (\omega \wedge F_i) \) on \(\frac{3}{4}\Omega _i\) is bounded by \(C\int _X |{{\bar{\partial }}}^\star \xi _1|^2_{\omega , h_L}\textrm{d}V_{\omega }\). Since \(F_i\) is holomorphic and the \(C^2\)-norm of \(\varphi _L\) is bounded, \( \Delta ^{''} (\omega \wedge F_i) \) is continuous. Therefore the \(C^0\)-norm of \( \Delta ^{''} (\omega \wedge F_i) \) on \(\frac{3}{4}\Omega _i\) is bounded by \(C\int _X |{{\bar{\partial }}}^\star \xi _1|^2_{\omega , h_L}\textrm{d}V_{\omega }\). As a consequence, the elliptic regularity implies that the
where \(C'\) is a uniform constant.
Now we can prove the theorem. Theorem 3.2 and (5.4.4) imply that
Then the mean value inequality for holomorphic functions implies that
Together with (5.4.4), we have thus
As a consequence, we have
Like the arguments as in the above theorems, this implies the existence of an holomorphic extension U with the estimates
where \(C'\) depends only on the geometry of \((V_\textrm{sing}, \omega )\) and \(C_1\).
5.3 Proof of Theorem 1.7
We are using the notations from the previous section, so \(\omega \) is a fixed reference Kähler metric on X.
By hypothesis, the metric \(h_L\) is non-singular and in this case the equality
is immediate.
We remark that the restriction \(\displaystyle \frac{u}{\textrm{d}s_Y}\Big |_{Y_j}\) is holomorphic, for each component \(Y_j\) of Y. This is where the vanishing of u on the singularities of Y is used. We decompose the restriction of \(\displaystyle \gamma _{\xi _1}\) to \(Y_j\) as follows:
where \(\alpha _j\) is holomorphic and \(\beta _j\) is orthogonal to the space of L-valued holomorphic top forms on \(Y_j\). Then we have
Let \(x_0\) be a singular point of Y. We then have coordinates \((z_1, z_2)\) defined on a open subset \(x_0\in V\), centered at \(x_0\) and such that \((z_k= 0)= Y_k\cap V\) for each \(k=1, 2\). We equally fix a trivialization of \(L|_V\) and let \(\varphi _L\) be the corresponding weight of the metric \(h_L\). We write
and let \(\theta \) be a function which is equal to 1 near \(x_0\) and such that \(\textrm{Supp}(\theta )\subset V\).
Thus we have
together with a similar equality on \(V\cap Y_2\). We can write
and then we observe that the left-hand side term of (5.4.12) is \(\partial _{\varphi _L}\)—exact on \(Y_1\). Therefore we have
since \(\alpha _1\) is holomorphic. We infer that we have
and all that we still have to do is to apply the Cauchy–Schwarz inequality to each of the two terms of the RHS of the inequality above.
A last remark is that we have
by the definition of \(\alpha _1\) and \(\beta _1\). We use the a-priori inequality and we conclude as in Theorem 1.1. \(\square \)
Remark 5.5
In the absence of hypothesis \(u|_{Y_{\textrm{sing}}}\not \equiv 0\) the evaluation of the term (5.4.7) near the singularities of Y is problematic. In the decomposition (5.4.8), we write \(\beta _j= {{\bar{\partial }}}^\star (\tau _j)\), and then the question is to estimate the quotient
at the points of \(Y_{\textrm{sing}}\). This does not seem to be possible, since we only have the norm \(W^{1,2}\) of \(f_j\) at our disposal. Indeed, the quantity \({{\bar{\partial }}}\beta _j\) is equal to the restriction of the form \({{\bar{\partial }}}\gamma _{\xi _1}\) to \(Y_j\). In Question 5.7 we provide a few more precisions about this matter.
5.4 Proof of Theorem 1.8
This is another set-up in which the considerations above work, as follows. We recall that the metric of \(h_L\) of L satisfies the hypotheses (a) and (b) at the beginning and moreover \((L, h_L)\) is flat near the singularities of Y, i.e.,
Then we get an estimable extension as follows. Let \(\omega \) be a fixed Kähler metric on X. As in the proof of the preceding result Theorem 1.7, we will use the method of Berndtsson [2], so the quantity to be bounded is
Integration by parts shows that it is enough to obtain a mean value inequality for the function
This is done according to the same principle as before. In the first place the differential inequality satisfied by \(\displaystyle |\partial ^\star _{\varphi _L}\xi _1|^2\) is as follows:
for some constant \(C> 0\) which only depends on (the curvature of) \(\displaystyle \omega |_{V_{\textrm{sing}}}\). We will not detail the calculation here because this is very similar with the one in the proof of Theorem 1.1. However, we highlight next the main differences:
-
(1)
It is not necessary to introduce any regularization of the metric, since by hypothesis (5.5.1) the restriction \(h_L|_{V_{\textrm{sing}}}\) is non-singular.
-
(2)
Without any additional information about \((L, h_L)\), the term \(\left\langle \partial ^\star _{\varphi _L}\xi _1, \square \partial ^\star _{\varphi _L}\xi _1\right\rangle \) is problematic. Actually (5.5.1) is needed precisely in order to deal with it: the curvature of \((L, h_L)|_{V_{\textrm{sing}}}\) equals zero, then we have \(\square \partial ^\star _{\varphi _L}\xi _1= 0\) pointwise on \(V_{\textrm{sing}}\). In general we have the term
$$\begin{aligned} \left\langle [{{\bar{\partial }}}, \Lambda _{\Theta _{h_L}(L)}]\xi _1, \partial ^\star _{\varphi _L}\xi _1\right\rangle \end{aligned}$$which appears in the computation and seems impossible to manage.
-
(3)
In the evaluation of the Laplacian of the norm of a harmonic tensor we have two terms: the gradient of the tensor, and several curvature terms corresponding to the metric on the ambient manifold and to the twisting, respectively. Here we do not have any contribution from L, and the term involving the curvature of \(\omega \) is taken care by the constant \(-C\) in (5.5.4).
Anyway, the inequality (5.5.4) can be re-written as
and this combined with Moser iteration procedure shows that we have
Finally, the term that one (almost) never uses in Bochner formula shows that we have
and we thus obtain the inequality
Then we conclude as in Theorem 1.7. \(\square \)
Remark 5.6
Actually in the proof of Theorem 1.8 only needs to evaluate the \(L^2\) norm
of \({{\bar{\partial }}}\gamma _{\xi _1}|_{Y}\). One might try to use a similar method as the one in Sect. 2, but there are serious difficulties to overcome.
5.5 Proof of Theorem 1.10
By hypothesis we know that Y has one component \(Y_1\) which only intersects \(\cup _{i\ne 1} Y_i\) in a unique point \(p_0\) such that \(u(p_0)\ne 0\). We also assume that \(\displaystyle L|_{Y_1}\) is flat, in the sense that there exists a section \(\tau \) such that \(\tau (p_0)\ne 0\) and \(\partial _{\varphi _L} \tau =0\). Then we argue as follows:
Let \(\omega \) be a fixed, reference metric on X. On each component \(Y_j\) of Y we solve the equation
where \(\alpha _j\) is holomorphic (1, 0) form and \(\beta _j\) is of type (1, 1) on \(Y_j\). We note that by elliptic regularity the form \(\beta _j\) is smooth.
We have \(\beta _j= f_j\omega |_{Y_j}\) and then the equality
follows by the residues formula. Here we denote by \(\displaystyle \sigma _j:= \prod _{i\ne j} s_{Y_i}\).
In case \(j=1\), the sum above only has one term, by hypothesis. Since we have \({{\bar{\partial }}}^\star (\tau \omega )= 0\), we can modify the solution \(f_1\) so that the global sum of residues is zero.
Question 5.7
Let p be one of the intersection points of two curves \(Y_1\cap Y_2\) in X. The analog of the a-priori inequality in Sect. 2 gives
provided that the bundle \((L, h_L)\) has the right curvature hypothesis, let us assume this holds for the moment. In (5.7.1) we denote by C a constant which we can compute explicitly. This a-priori inequality is obtained by considering the (2, 2)-form with values in L
whose \(\star \) coincides with the section \(f_j\), and use the procedure Theorem 2.1 for the function \(w:= \frac{1}{|s_1|^2+ |s_2|^2}\). The curvature requirements this induces will most likely be
We have
which explains (5.7.1).
Anyway, by equality (5.6.2) we control the norm
Then the question is: can we find a smooth section \(\widetilde{f}_j\) of L such that it equals \(f_j\) on Y and such that
where C in (5.7.6) is universal?
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Appendix A: Further Results and Examples (by Bo Berndtsson)
Appendix A: Further Results and Examples (by Bo Berndtsson)
In this appendix we will study two very simple model examples of \(L^2\)-extension from a non-reduced or singular variety. It is the second example (in Sect. 3) that is most relevant to the subject of the main paper. The main point is to show that it is not possible to obtain an estimate that is substantially better than Theorems 1.1 and 1.7. The role of the first example is to show that similar difficulties appear already in an even simpler situation, that can be analyzed in a more complete way.
In the first example we consider the space
of holomorphic functions in the disk \(\Delta \) that are square integrable against a weight \(e^{-\phi }\). Given numbers \(a_k\), \(k=0, 1,...N-1\), we will compute the minimal norm of a function \(h\in A^2\) that satisfies \(h^{(k)}(0)=a_k\). The formula we give is exact but not very explicit; it contains the Bergman kernel for the space and various metrics derived from it. Therefore we will also discuss to what extent it is possible to estimate it in more concrete terms, and show that the most optimistic estimates fail.
In the second example we consider the unit polydisk U in \({\mathcal {C}}^2\) and the singular variety \(V=\{z\in U; z_1z_2=0\}\) in U. We again denote by \(A^2_\phi (U)\) the Bergman space of holomorphic functions in U that are square integrable against the weight \(e^{-\phi }\). The extension problem is now to find a function \(h\in A^2\) that restricts to a given function on V, i.e., satisfies \(h=f_1\) when \(z_1=0\) and \(h=f_2\) when \(z_2=0\), where \(f_1\) and \(f_2\) are holomorphic functions of one variable satisfying \(f_1(0)=f_2(0)\). In this case we get only an estimate for the minimal extension, and we conclude by an example (basically due to Ohsawa, [35]) which (perhaps) can serve as a motivation for the statement in Theorem 1.4.
1.1 Extension from a (Fat) Point in the Unit Disk
Let \(A^2=A^2_\phi (\Delta )\) be defined as above and let for \(k=0, 1, 2...\)
It follows from elementary Hilbert space theory that there is a unique function h of minimal norm in \(A^2\) satisfying \(h^{(k)}(0)=a_k\) for \(k=0,...N-1\). Write
where \(r_1\in E_1\) and \(h_0\perp E_1\). Then we write
with \(r_2\in E_2\) and \(h_1\perp E_2\). Continuing this way we get
with \(h_k\in E_k\ominus E_{k+1}\) and \(r_N\) in \(E_N\). That h has minimal norm means that h is orthogonal to \(E_N\), so \(r_N=0\). By orthogonality we have
so the problem amounts to estimating the norms of \(h_k\).
The spaces \(E_k\ominus E_{k+1}\) are one dimensional. Let \( e_k\) be an element of unit length. We start with a simple lemma from [18] whose proof follows almost directly from the definitions.
Lemma A.1
For \(k=0\), \(|e_0(0)|^2=B_0(0)\), the (diagonal) Bergman kernel at the origin. For \(k\geqslant 1\) \(|e_k^{(k)}(0)|^2=:B_k(0)\) can be viewed as a ‘higher order Bergman kernel’ and we refer to [18] for interesting applications of this idea. By a classical formula of Bergman, [1], we have
so the first-order Bergman kernel is strongly related to the Bergman metric. (We are abusing notation by identifying the metric with its density; more properly we should write
Since \(h^{(k)}(0)=a_k\) and \(h_j^{(k)}(0)=0\) for \(j>k\), we get
Then, since \(h_k\) is a multiple of \(e_k\) and \(h_k^{(k)}(0)=b_k\), we have
Recalling that \(e_k\) has norm 1 and that \(|e_k^{(k)}(0)|^2=B_k(0)\) we find that the norm of the minimal extension is given by
When \(N=1\) this is just the standard formula
which shows that estimates from above of the norm of the minimal extension are equivalent to estimates from below of the (usual) Bergman kernel. The next case is \(N=2\). Then we use (A.1.1) and find (since \(h_0=(a_0/e_0(0)) e_0\))
Here we think of \(a_1-a_0 e_0'(0)/e_0(0)\) as a 1-form and the second term in the right-hand side is its norm for the Bergman metric. Since the off-diagonal Bergman kernel \(B_0(z,w)\) is holomorphic in z and antiholomorphic in w, and \(e_0(z)=B_0(z,0)\), we have
Hence we can also write
By the standard Ohsawa–Takegoshi theorem, \(B_0(0)\geqslant C^{-1} e^{\phi (0)}\), with a universal constant C, so we get the slightly more explicit estimate
Because of the following lemma we can replace the norm with respect to the Bergman metric by the Euclidean norm.
Lemma A.2
For any (1-form) a, we have at the origin of \(\Delta \)
where \(|a|^2\) denotes the norm with respect to the Euclidean metric.
Proof
By (A.1.1) and Lemma A.1, at the origin
Now choose \(f=ze_0\). Since \(|z|<1\) and \(e_0\) has norm 1, f has norm less that 1. Moreover, \(f'(0)= e_0(0)\). Hence \(\omega _B\geqslant 1\), which proves the lemma. \(\square \)
Thinking of \(\log B_0(z)\) as an approximation of \(\phi \), (A.1.3) suggests that one might also have the inequality
but we shall see later that this does not hold.
We next discuss briefly estimates for larger values of N. We first note that there is a version of Lemma A.2 for all k, which is proved in much the same way.
Lemma A.3
Proof
Recall that
Take \(f=z^k e_0\). Then f has norm less than 1 and \(f^{(k)}(0)= k! e_0(0)\). Thus
\(\square \)
From the lemma and (A.1.2) we get the estimate for the norm of the \(L^2\)-minimal extension
where
As we have seen, this is difficult to estimate even when \(N=2\) and it is clear that the complexity grows with higher values of k and N.
1.2 Examples
We focus on the estimates for \(N=2\), i.e., extension of a first-order jet. The most naive conjecture for an explicit estimate would be
Claim 1: There is no constant C independent of \(\phi \) such that for all subharmonic \(\phi \), (A.3.1) holds.
For this, take \(\phi (z)=-2m\Re (z)\) and put \(g=e^{mz}h\). Take \(a_0=1, a_1=0\). If (A.3.1) held we would get
Hence
and \( g'(0)=m\). This contradicts Cauchy’s estimates for the derivative.
The estimate
might seem more plausible since we estimate \(\Vert h\Vert ^2\) by the connection in \(A^2_\phi \), \(h'-h\partial \phi \) instead of just \(h'(0)\).
Claim 2: There is no constant C independent of \(\phi \) such that for all subharmonic \(\phi \) the minimal extension satisfies (A.3.2).
Assume that \(\phi (0)=0\) and that \(\phi \) is smooth. Then there is a small annulus \(U=\{ r_1<|z|<r_2\}\) around the origin, where \(\phi >-1\). Choose \(\varepsilon >0\) so small that \(\phi +\varepsilon \log |z| >-1\) in the annulus too. Define a function \(\psi \) in the disk by \(\psi =\max (\phi +\varepsilon \log |z|, -1)\), in the disk where \(|z|<r_2\), and \(\psi =\phi +\varepsilon \log |z|\) where \(|z|>r_1\). Since the two definitions agree on overlaps, \(\psi \) is a well-defined subharmonic function, and \(\psi =-1\) in a small neighborhood of the origin.
If (A.3.2) holds for \(\psi \), then (A.3.1) also holds for \(\psi \) since \(\partial \psi \) vanishes near the origin. Letting \(\varepsilon \rightarrow 0\) (or just taking \(\varepsilon \) sufficiently small) we get an extension that satisfies (A.3.1) for \(\phi \). By the first claim, this is impossible.
1.3 A Singular Variety in the Bidisk
In this section we study \(L^2\)-extension from the variety
in the unit bidisk U, and we use the notation from the introduction. Following the scheme in the previous section we let
and
Let f be a holomorphic function on V, and let h be the holomorphic extension of f to U of minimal norm. Again we write
where \(h_0\perp E_1\) and \(r_1\in E_1\). Continuing as before we write
where \(h_1\in E_1\ominus E_2\) and \(r_2\in E_2\). Then \(h=h_0+h_1+ r_2=h_0+h_1\) since, by minimality, h is orthogonal to \(E_2\). Moreover, \(h_0\perp h_1\), so
The holomorphic function f on V is given by a pair \((f_1,f_2)\), where \(f_1\) is holomorphic on \(\{z_1=0\}\) and \(f_2\) is holomorphic on \(\{z_2=0\}\), and \(f_1(0)=f_2(0)=: a_0\).
Since \(h_0\) is orthogonal to \(E_1\) and \(h_0(0)=a_0\) we have that
whereas in the previous section \(e_0\) is a function of unit norm orthogonal to \(E_1\). Then \(|e_0(0)|^2=B_0(0)\), the (diagonal) Bergman kernel at the origin, and we get
just as before.
We next turn to \(h_1\), which is the \(L^2\)-minimal extension of \({\tilde{f}}:=f-h_0\). We can not give an exact formula for the norm of \(h_1\) but it is easy to give an estimate. Since \({\tilde{f}}\) vanishes at the origin we have \({\tilde{f}}=(f_1-h_0, f_2-h_0)=:(z_2 g_1, z_1 g_2)\), where \(g_1(z_2)\) and \(g_2(z_1)\) are holomorphic functions of one variable. Let \(G_1\) and \(G_2\) be the minimal extensions of \(g_1\) and \(g_2\), from \(V_1=\{z_1=0\}\) and \(V_2=\{z_2=0\}\), respectively. By the Ohsawa–Takegoshi theorem
Let \(H=z_2 G_1+z_1 G_2\). This is an extension of \({\tilde{f}}\) and
Hence
All in all we get the estimate for the minimal extension h of f,
When comparing this to the classical Ohsawa–Takegoshi theorem, let us first assume that \(a_0=f_1(0)=f_2(0)=0\), so we get
Translating this to the notation in the main text of this article, \(|f|^2/|z|^2=|u/\textrm{d}s|^2\), so this means that
which is exactly what the Ohsawa–Takegoshi theorem predicts. The problem with this estimate is that if the right-hand side is finite, it conversely forces \(a_0\) to be zero, so the ‘classical’ theorem deals only with this case. The purpose of the theorems in the introduction is to weaken the norm in the right-hand side so that this can become finite even if \(f=u\) does not vanish on the singular set.
Here one might hope that the quotient \( \frac{|f-h_0|^2}{|z|^2}\) in the right-hand side could be replaced by the squared norm of a derivative acting on f. Asymptotically as \(z\rightarrow 0\) on e.g., \(V_2\),
(the last equality follows as in the discussion leading to (A.1.3)). Again, thinking of the logarithm of the Bergman kernel, \(\log B_0\) as an approximation of \(\phi \), one is led to look for estimates in terms of \(\partial ^\phi f\), like in Theorem 1.7. Theorem 1.7, however, also contains a factor \(\log ^2(\max |z_j|^2)\), and we next give an example showing that something of this kind is necessary.
1.4 More Examples
We first give a counterexample (cf. [35]) to the most naive conjecture that the same estimate as for smooth varieties holds.
Claim 3: There is no universal constant, independent of the plurisubharmonic weight \(\phi \), such that the minimal extension satisfies
for all functions f that vanish at the origin.
To see this, take \(f=z_1\) when \(z_2=0\) and \(f=0\) when \(z_1=0\). Take \(\phi =\log |z_1-z_2|^2\). Any extension H must have the form \(H=z_1 G\). If H has finite norm, then \(G=0\) when \(z_1=z_2\). Hence H vanishes to second degree at the origin, which is not possible. This example also motivates the choice of weight \(e^{-\phi _L}\) in the last term of the right-hand side in formula (1.6.1). Indeed, replacing that by the more natural \(e^{-\phi _L/(1+\alpha )}\), the right-hand side would be finite for our choice of f, and as we have seen extension is not possible.
This same example also shows that it does not help to add the \(L^2\)-norm of the twisted derivative of f in the right-hand side. We use the notation
Claim 4: There is no universal constant, independent of the plurisubharmonic weight \(\phi \), such that the minimal extension satisfies
for all functions f that vanish at the origin.
Indeed, with the same choice of \(\phi \) and f as above, we have on \(V_2\) outside the origin
Since the weight \(\phi \) has a singularity at the origin we look at regular approximations. Let \(\phi _\varepsilon \) be the convolution of \(\phi \) with \((\pi \varepsilon ^2)^{-1}\chi _\varepsilon \), where \(\chi _\varepsilon \) is the characteristic function of the disk with radius \(\varepsilon \). Explicitly,
when \(|\zeta |<\varepsilon \) and \(\phi _\varepsilon (\zeta )=\log |\zeta |^2\) when \(|\zeta |\geqslant \varepsilon \). This gives a sequence of subharmonic functions decreasing to \(\phi \) on \(V_2\). We have
when \(|\zeta |<\varepsilon \). Hence
on \(V_2\). Since \(e^{-\phi _\varepsilon }\) is of size roughly \(\varepsilon ^{-2}\) when \(|\zeta |<\varepsilon \), the right-hand side in (A.3.5) stays bounded as \(\varepsilon \rightarrow 0\). Hence, if (A.3.5) held, we would again get an extension of finite norm in \(L^2(e^{-\phi })\), which we have seen is impossible. This motivates the logarithmic factor in the estimate (1.7.1) of Theorem 1.7.
It is easy to construct a compact analog of the example used to prove the claims 3 and 4 above, as we briefly indicate next.
Consider two transverse lines \(L_1\) and \(L_2\) in \(X:= {\mathbb {P}}^2\), and the adjoint bundle \(K_X+ L_1+ L_2+ {\mathcal {O}}(2)\). We define the metric h on \({\mathcal {O}}(2)\) by its weights
where \(f_i\) is the local expression of the section \(\sigma _i\) which defines the line \(L_i\).
We define the section u which equals \(\sigma _2\) on \(L_1\) and zero on \(L_2\). By Theorem 1.1 we can construct an extension \(U_\varepsilon \) of u for whose \(L^2\) norm is bounded by
which equals \(\int _{({\mathbb {C}}, 0)}\frac{1}{\varepsilon ^2+ |z|^2}\textrm{d}\lambda \). In particular, we see that the bound (1.6.1) cannot be improved by replacing the weight \(e^{-\varphi _L}\) with \(e^{-(1-\delta _0)\varphi _L}\), for any positive \(\delta _0\).
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Cao, J., Paun, M. & Berndtsson, B. On the Ohsawa–Takegoshi Extension Theorem. J Geom Anal 34, 25 (2024). https://doi.org/10.1007/s12220-023-01466-9
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DOI: https://doi.org/10.1007/s12220-023-01466-9