8
$\begingroup$

Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb Z_m$ along $D$. Suppose now we have a Kähler orbifold metric $\omega$ on $(S,D,m)$ (with metric singularity along $D$). I would like to find a short proof or a reference for the following statement:

Statement. For arbitrary small $\varepsilon$-neighborhood $U_{\varepsilon}$ of $D$ in $S$ there is a smooth Kähler metric $\omega_{\varepsilon}$ on $S$ such that $\omega_{\varepsilon}$ coincides with $\omega$ in $S\setminus U_{\varepsilon}$.

PS. I will be happy, if the statement is proven in the following situation: $S$ is a ruled surface and $D$ is embedded in $S$ as a holomorphic section (i.e. it intersects all $\mathbb CP^1$-fibers in one point). Moreover if necessary one may assume that the Kähler orbi-metric is invariant under a holomorphic $S^1$-action on $S$ fixing $D$ pointwise.

PPS. I also removed the condition on $\omega_{\varepsilon}$ to be in the same cohomology class as $\omega$ (since it turned out to be irrelevant for me).

$\endgroup$
12
  • $\begingroup$ Did you try a very nice bump function and averaging? What is your definition of "orbi-singularity"? $\endgroup$ Apr 14, 2017 at 7:11
  • $\begingroup$ I did not try. Orbifold metric is a metric that can be obtained locally as a quotient by isometry. In the particular case that I consider it has the following local model: Take the ball $B_1$ in $\mathbb C^2$ i.e. $|z^2|+|w^2|\le 1$ take a Kahler metric on it, invariant under the linear $\mathbb Z_m$ action on $B_1$, fixing $z=0$. Take quotient. $\endgroup$
    – aglearner
    Apr 14, 2017 at 7:51
  • $\begingroup$ I know the (usual) definition of orbi-fold metric, but what is an "orbi-singularity." I don't know this word. To me, for example, the plane/$\mathbb{\mu}_n$ is smooth. Is that an orbi-singularity though? $\endgroup$ Apr 14, 2017 at 8:07
  • 1
    $\begingroup$ I think I can see how to do this, for metrics in rational cohomology classes, but I'm skeptical that the two metrics would have the same K\"{a}hler class. $\endgroup$
    – Craig
    Apr 17, 2017 at 20:40
  • 2
    $\begingroup$ Theorem 6.2 of arxiv.org/pdf/math/0412405.pdf , I read it long times ago , see also arxiv.org/pdf/math/0411522.pdf $\endgroup$
    – user21574
    Apr 25, 2017 at 18:13

1 Answer 1

1
$\begingroup$

Let $S=S^{sing}\cup S^{reg}$ take an orbifold resolution $\pi : \tilde S \to S$ with simple normal crossing exceptional divisor $E = \pi^{−1}(S^{sing})$ such that $\pi$ is an isomorphism over $S^{reg}$. then take $\omega_\epsilon=\pi^*\omega_0-\epsilon\theta_E$ as K\"ahler metric on $\tilde S $ in adiabatic classes(see Lemma 4.2.3 that why such metric $\omega_\epsilon$ is as Kahler metric in fibration case or Lemma1 for resolution case . Here $\omega_0$ is a positive closed $(1,1)$-current on $S$ and $\theta_E$ is a positive closed $(1,1)$-current on $E$)

Consider a smooth geometric orbifold given by $\mathbb Q$-divisor $$D=\sum_{j\in J}(1-\frac{1}{m_j})D_j$$ where $m_j\geq 2$ are positive integers and $\text{Supp}D=\cap_{j\in J}D_j$ is of normal crossings divisor. Let $\omega$ be any K\"ahler metric on $S$, let $C >0$ be a real number and $s_j\in H^0\left(S,\mathcal O_X(D_j)\right)$ be a section defining $D_j$. Consider the following expression

$$\omega_D=C\omega+\sqrt[]{-1}\sum_{j\in J}\partial\bar\partial |s_j|^{2/m_j}$$

If $C$ is large enough, the above formula defines a closed positive $(1,1)$ -current (smooth away from $D$). Moreover

$$\omega_D\geq \omega$$ in the sense of currents. Consider $\mathbb C^n$ with the orbifold divisor given by the equation

$$\prod_{j=1}^nz_j^{1-{1}/{m_j}}=0$$ (with eventually $m_j=1$ for some $j$). The sections $s_j$ are simply the coordinates $z_j$ and a simple computation gives

$$\omega_D=\omega_{eucl}+\sqrt[]{-1}\sum_{j=1}^n\partial\bar\partial |z_j|^{2/m_j}=\omega_{eucl}+\sqrt[]{-1}\sum_{j=1}^n\frac{dz_j\wedge d\bar {z_j}}{m_j^2|z_j|^{2(1-1/m_j)}}$$

See paper of Frédéric Campana; Mihai Păun Orbifold generic semi-positivity: an application to families of canonically polarized manifolds (Semi-positivité orbifolde : une application aux familles de variétés canoniquement polarisées) Vol. 65 no. 2 (2015), p. 835-861 Details

http://www.numdam.org/article/AIF_2015__65_2_835_0.pdf

$\endgroup$
5
  • 2
    $\begingroup$ Hassan, $S$ is a smooth complex surface, it has no singularities. Could you please change your answer accordingly? In the line three of your text you write without justification, that $\omega_{\epsilon}$ is Kaher. This does not make sense to me $\endgroup$
    – aglearner
    Apr 26, 2017 at 0:20
  • 3
    $\begingroup$ Also you don't say what is $\omega_0$, you don't say what is $\theta_E$, etc... $\endgroup$
    – aglearner
    Apr 26, 2017 at 0:22
  • 3
    $\begingroup$ $\omega_\epsilon$ is Kahler metric in adiabatic classes, I added a reference. It is known fact due to Tian-Yau $\endgroup$
    – user21574
    Apr 26, 2017 at 0:34
  • 3
    $\begingroup$ You may see Lemma1 of arxiv.org/pdf/1209.2198.pdf $\endgroup$
    – user21574
    Apr 26, 2017 at 1:50
  • 2
    $\begingroup$ I deleted my previous comment (since it was a bit aggressive) but I still think that in this answer you don't really address my question but write something (that I can not really understand) on an adjacent topic. $\endgroup$
    – aglearner
    May 15, 2017 at 20:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.