# Quasi-projectivity of the moduli space of Kahler-Einstein Fano varities and vanishing Lelong number

Chi Li, Xiaowei Wang, Chenyang Xu proved the Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds. My question is about when central fibre $X_0$ along Kahler-Einstein Fano fibration $X\to B$ has log-terminal singularities. Then the moduli space of singular Kahler-Einstein Fano fibres is Quasi-projective(An old conjecture due to Gang Tian)?

In fact, if we prove that the Lelong number of singular hermitian metric $(L_{CM},h_{WP})$ corresounding to Weil-Petersson current on virtual CM-bundle has vanishing Lelong number, then $L_{CM}$ is nef and if we know the nefness of $L_{CM}$ then due to a result of E.Viehweg, we can prove that the moduli space of Kahler-Einstein Fano manifolds $\mathcal M$ is quasi-projective and its compactification $\mathcal M$ is projective.

( we know if $(L,h)$ be a positive , singular hermitian line bundle , whose Lelong numbers vanish everywhere. Then $L$ is nef) But Gang Tian recently proved that CM-bundle $L_{CM}$ is positive

But by the same theorem 2 and Poroposition 6, of the paper, Georg Schumacher and Hajime Tsuji, Quasi-projectivity of moduli spaces of polarized varieties, Annals of Mathematics,159(2004), 597–639

by assuming diameter of fibers $X_t$ is uniform bounded (which is my question)we can show that the Weil-Petersson current on mduli space of Fano Kahler-Einstein arieties has zero Lelong number(in smooth case it is easy).

Or Theorem 3.4 http://www.mathematik.uni-marburg.de/~schumac/doubar.pdf

A theorem of Donaldson-Sun states that if $X_t$ are Kahler-Einstein metric with negative Ricci curvature with uniform diameter bound, then the central fiber is normal and klt singularities at worst. In view of the moduli theory of canonically polarized varieties, limit of fibers should have canonical singularities.

when fibres are polarized Calabi-Yau varities, then the diameter of fibres is uniform bounded iff the central fibre $X_0$ has canonical singularities

$$diam(X_t,\omega_t)\leq 2+C\int_{X_t}\Omega_t\wedge\bar\Omega_t$$

My question is , $X_t$ are Kahler-Einstein metric with positive Ricci curvature with uniform diameter bound, then the central fiber is normal and has log-terminal singularities at worst and vise versa? In fact by knowing this fact we can construct a canonical Weil-Petersson metric on moduli space of Kahler-Einstein Fano varieties with mild singularities

• If you are using Schumacher-Tsuji, please read carefully the follow-on literature about that article. – Jason Starr Aug 15 '16 at 12:13
• Are you aware of Koll'ar's paper about Schumacher-Tsuji? – Jason Starr Aug 15 '16 at 12:23
• No, I am not aware about that paper, can you send me?. – user21574 Aug 15 '16 at 12:27
• J'anos Koll'ar, Non-Quasi-Projective Moduli Spaces, Annals of Math. 164 (2006), pp. 1077-1096, arxiv.org/pdf/math/0501294v2.pdf – Jason Starr Aug 15 '16 at 12:34
• Thank you. But Koll'ars counter-example is nothing to do with my question. – user21574 Aug 15 '16 at 12:50