14
$\begingroup$

I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see (proof) if that space is Kahler or not.

I'm mainly interested in moduli spaces of geometric structures on surfaces: Teichmuller (ok, that's I know is Kahler), Moduli space of complex projective structures (is this space kahler? where can I find a reference?), et cetera...

Is there a standard (or not) reference for this kind of problems?

$\endgroup$
8
$\begingroup$

The corresponding Kahler metric is called "Weil-Petersson metric"; it is often constructed using infinite-dimensional determinants of the corresponding Laplace operators. The standard reference is a sequence of papers by Bismut-Gillet-Soule: J.Bismut, H.Gillet and C.Soule, Analytic torsion and holomorphic determinant bundles, I,II,III, Comm. Math Phys., 115, 1988, 49-78, 79-126, 301-351; see also http://arxiv.org/abs/math/0406408, http://arxiv.org/abs/math/0312172, Leon A. Takhtajan, Lee-Peng Teo, "Weil-Petersson metric on the universal Teichmuller space I, II".

$\endgroup$
  • $\begingroup$ Weil-Petersson metric is a canonical metric on moduli space of "stable" varieties. We need to have some notion of stability . For instanse mouli space of Kahler Einstein manifolds $\endgroup$ – user21574 Apr 24 '16 at 21:03
6
$\begingroup$

The Weil-Petersson metric is not complete metric in general but in the case of abelian varieties and $K3$ surfaces, the Weil-Petersson metric turns out to be equal to the Bergman metric of the Hermitian symmetric period domain, hence is in fact complete K\"ahler Einstein metric. Weil and Ahlfors showed that the Weil-Petersson metric is a Kahler metric and later Gang Tian gave a different proof for it also

There is a belief due to Fujiki-Tian saying that a moduli space with canonical metric (K\"ahler Einstein metric, cscK,...)is likely to be quasi-projective.

Recently Chi Li http://arxiv.org/pdf/1502.06532.pdf prove that there is a canonical continuous Hermitian metric on the Tian's CM-line bundle over the proper moduli space $M$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson metric. Which Weil-Petersson metric is a Kahler metric

Let $\pi:X \to S$ be a projective family of canonically polarized varieties. Equip the relative canonical bundle $K_{X/S}$ with the hermitian metric that is induced by the fiberwise K\"ahler-Einstein metrics. The log Weil-Petersson form is equal, up to a numerical factor, to the fiber integral

$$\omega_{WP}=\int_{X_s}c_1\left(K_{X/S}\right)^{n+1}=\left(\int_{X_s}|A|_{\omega_s}^2\right)ds\wedge d\bar s$$

$A$ represents the Kodaira-Spencer class of the deformation,

There exists a Kahler metric on moduli space of Calabi-Yau varieties we call it again Weil Petersson metric and can be written as Ricci curvature of direct image of relative Line bundle.

Take holomorphic fiber space $\pi:X\to B$ and assume $\Psi_y$ be any local non-vanishing holomorphic section of Hermitian line bundle $\pi_*(K_{X/B}^l)$, then the Weil-Petersson (1,1)-form on a small ball $N_r(y)\subset B$ can be written as

$$\omega_{WP}=-\sqrt{-1}\partial_y\bar{\partial_y}\log \left((\sqrt{-1})^{n^2}\int_{X_y}(\Psi_y\wedge \overline{\Psi_y})^{\frac{1}{l}}\right)$$ Note that $\omega_{WP}$ is globally defined on $B$

Fujiki-Schumacher considered the moduli space of Kahler manifolds admitting constant scalar curvature Kahler (cscK) metrics. They proved that the natural Weil-Petersson metric is always Kahler by interpreting it as the Chern curvature of a determinant line bundle equipped with Quillen metric.

$\endgroup$
2
$\begingroup$

The moduli space of holomorphic normal projective connections is an affine space, or empty, as it is identified with the collection of all holomorphic 1-cocycles whose coboundary is a suitable ``traceless Atiyah class'' of the tangent bundle. The paper

Robert Molzon and Karen Pinney Mortensen, The Schwarzian derivative for maps between manifolds with complex projective connections, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3015–3036. MR 1348154 (96j:32028) 27, 55

is perhaps the best introduction to the theory of complex projective connections. I don't know a good reference for the relation to the Atiyah class, though it appears somewhere in the work of Kobayashi. The moduli space of flat holomorphic projective connections on a compact complex manifold is a complex subvariety, not known to be smooth.

$\endgroup$
  • $\begingroup$ On Riemann surfaces, all holomorphic projective connections are normal and flat, hence the common confusion in the literature as to what the definition of ``projective structure'' is on higher dimensional manifolds. $\endgroup$ – Ben McKay Apr 25 '16 at 9:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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