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Register a new userRiemannian geometry of Kahler-Einstein currents III: compactness of Kahler-Einstein manifolds of negative scalar curvature
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Let $mathcal{K}(n, V)$ be the set of $n$-dimensional compact Kahler-Einstein manifolds $(X, g)$ satisfying $Ric(g)= - g$ with volume bounded above by $V$. We prove that after passing to a subsequence, any sequence ${ (X_j, g_j)}_{j=1}^infty$ in $mathcal{K}(n, V)$ converges, in the pointed Gromov-Hausdorff topology, to a finite union of complete Kahler-Einstein metric spaces without loss of volume. The convergence is smooth off a closed singular set of Hausdorff dimension no greater than $2n-4$, and the limiting metric space is biholomorphic to an $n$-dimensional semi-log canonical model with its non log terminal locus of complex dimension no greater than $n-1$ removed. We also show that the Weil-Petersson metric extends uniquely to a Kahler current with bounded local potentials on the KSBA compactification of the moduli space of canonically polarized manifolds. In particular, the coarse KSBA moduli space has finite volume with respect to the Weil-Petersson metric. Our results are a high dimensional generalization of the well known compactness results for hyperbolic metrics on compact Riemann surfaces of fixed genus greater than one.
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We show that a closed almost Kahler 4-manifold of globally constant holomorphic sectional curvature $kgeq 0$ with respect to the canonical Hermitian connection is automatically Kahler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern-Weil theory implies useful integral formulas, which are then combined with results from Seiberg--Witten theory.
Two results regarding Kahler supermanifolds with potential $K=A+Cthetabartheta$ are shown. First, if the supermanifold is Kahler-Einstein, then its base (the supermanifold of one lower fermionic dimension and with Kahler potential $A$) has constant scalar curvature. As a corollary, every constant scalar curvature Kahler supermanifold has a unique superextension to a Kahler-Einstein supermanifold of one higher fermionic dimension. Second, if the supermanifold is itself scalar flat, then its base satisfies the equation $$ phi^{bar ji}phi_{ibar j}=2Delta_0 S_0 + R_0^{bar ji}R_{0ibar j} - S_0^2, $$ where $Delta_0$ is the Laplace operator, $S_0$ is the scalar curvature, and $R_{0ibar j}$ is the Ricci tensor of the base, and $phi$ is some harmonic section on the base. Remarkably, precisely this equation arises in the construction of certain supergravity compactifications. Examples of bosonic manifolds satisfying the equation above are discussed.
We classify quadruples $(M,g,m,tau)$ in which $(M,g)$ is a compact Kahler manifold of complex dimension $m>2$ with a nonconstant function $tau$ on $M$ such that the conformally related metric $g/tau^2$, defined wherever $tau e 0$, is Einstein. It turns out that $M$ then is the total space of a holomorphic $CP^1$ bundle over a compact Kahler-Einstein manifold $(N,h)$. The quadruples in question constitute four disjoint families: one, well-known, with Kahler metrics $g$ that are locally reducible; a second, discovered by Berard Bergery (1982), and having $tau e 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kahler surface metrics; and a fourth family, present only in odd complex dimensions $mge 9$. Our classification uses a {it moduli curve}, which is a subset $mathcal{C}$, depending on $m$, of an algebraic curve in $R^2$. A point $(u,v)$ in $mathcal{C}$ is naturally associated with any $(M,g,m,tau)$ having all of the above properties except for compactness of $M$, replaced by a weaker requirement of ``vertical compactness. One may in turn reconstruct $M,g$ and $tau$ from this $(u,v)$ coupled with some other data, among them a Kahler-Einstein base $(N,h)$ for the $CP^1$ bundle $M$. The points $(u,v)$ arising in this way from $(M,g,m,tau)$ with compact $M$ form a countably infinite subset of $mathcal{C}$.
We derive some elliptic differential inequalities from the Weitzenbock formulas for the traceless Ricci tensor of a Kahler manifold with constant scalar curvature and the Bochner tensor of a Kahler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^infty $ pinching results are established to characterize Kahler-Einstein manifolds among Kahler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kahler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex space forms among Kahler manifolds with constant scalar curvature.
We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of $X$ over its canonical model $X_{can}$. This assumption on the Ricci curvature always holds when the Kodaira dimension of $X$ is $n$, $n-1$ or when the general fibre of $X$ over its canonical model is a complex torus. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to its canonical model when $X$ has Kodaira dimension $1$ with $K_X$ being semi-ample and the general fibre of $X$ over its canonical model being a complex torus. We also prove the Gromov-Hausdorff limit of collapsing Ricci-flat Kahler metrics on a holomorphically fibred Calabi-Yau manifold is unique and is homeomorphic to the metric completion of the corresponding twisted Kahler-Einstein metric on the regular part of its base.
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