# A good review of a paper on Kähler-Einstein metrics with prescribed singularities

I am glad to be able to use this blog to highlight some of the really good reviews that appear in MathSciNet.  There are many ways for a review to be helpful.  We offer some advice to reviewers in our Guide to Reviewers.  Generally, a good review describes the context of the article (or book), the main results, and possibly compares it to other work.  This review by Hans-Joachim Hein of a paper by Campana, Guenancia, and Păun has all these qualities.

MR3134683
Campana, Frédéric(F-NANC-IE); Guenancia, Henri(F-PARIS6-IMJ); Păun, Mihai(F-NANC-IE)
Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. (English, French summary)
Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 6, 879–916.
32Q25 (14E05 53C55)

This paper establishes a version of S.-T. Yau’s solution of the Calabi conjecture [Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411; MR0480350 (81d:53045)] in the setting of Kähler metrics with conical singularities along normal crossing divisors. As an application, vanishing theorems for spaces of logarithmic holomorphic tensor fields are obtained, answering questions of the first author in [J. Inst. Math. Jussieu 10 (2011), no. 4, 809–934; MR2831280 (2012g:32027)].
To give an indication of the Calabi-Yau type results in this paper, let $X$ be a compact Kähler manifold and let $D = \sum a_i D_i$ be an effective $\Bbb{R}$-divisor with simple normal crossings in $X$ such that $a_i \in (0,1)$. Then, under the additional technical assumption that $a_i \in [\frac{1}{2},1)$, results of the following type are obtained: If $c_1(K_X + D) = 0$, then every Kähler class on $X$ contains a unique solution $\omega$ to the twisted Kähler-Einstein equation ${\rm Ric}(\omega) = 2\pi[D]$ such that $\omega$ is smooth away from $D$ and has conical singularities of cone angle $2\pi(1 – a_i)$ along $D_i$ in the sense of being locally bounded above and below by appropriate model Kähler forms. This includes analogous statements in the cases where $c_1(K_X +D)$ is either strictly positive or strictly negative (under additional stability type assumptions in the latter case).
These results are proved with the help of an $\epsilon$-regularization process: A singular background Kähler metric $\omega_0$ with the right conical singularities is written as the limit as $\epsilon \to 0$ of a family $\omega_\epsilon$ of smooth Kähler metrics; then complex Monge-Ampère equations with background metric $\omega_\epsilon$ are solved, with estimates on the solutions that are independent of $\epsilon$. The critical step here is to control the $\omega_\epsilon$-Laplacian of the solution. This is done using Yau’s method, relying crucially on the assumption that $a_i \in [\frac{1}{2},1)$ in order to guarantee that the holomorphic bisectional curvature of $\omega_\epsilon$ remains uniformly bounded from below as $\epsilon \to 0$.
The applications to holomorphic tensor fields take the following form: If $c_1(K_X + D) > 0$ then $H^0(X, T^r_s(X|D)) = 0$ for $r \geq s+1$, and if $c_1(K_X + D) < 0$ then $H^0(X,T_s^0(X|D)) = 0$ for $s \geq 1$. Here $r$ and $s$ denote the numbers of $T_X$ and $T_X^*$ factors, respectively, and “$|D$” refers to a natural condition of prescribed poles or zeros along $D$, which turns out to be equivalent to uniform boundedness in the metric sense with respect to a singular Kähler metric with the right conical singularities along $D$. Given the conically singular Kähler metrics with prescribed Ricci curvature produced above, these vanishing theorems can be proved by applying the Bochner method. (The necessary Bochner formulas, and the resulting vanishing theorems in the smooth case, can already be found in [K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton Univ. Press, Princeton, NJ, 1953; MR0062505 (15,989f)], although they do not seem to be very widely known except in the special case where $r + s = 1$.) Due to the singularities of the metric along $D$, it is necessary to work with a careful choice of a “logarithmic” cutoff function.
Let us mention that the Calabi conjecture part of this paper was solved by different methods in [S. Brendle, Int. Math. Res. Not. IMRN 2013, no. 24, 5727–5766; MR3144178] when $D$ is smooth, and in [T. D. Jeffres, R. Mazzeo and Y. A. Rubinstein, “Kähler-Einstein metrics with edge singularities”, preprint, arXiv:1105.5216] when $D$ is smooth but the cone angle is an arbitrary real number in $(0,2\pi)$. See [R. R. Mazzeo and Y. A. Rubinstein, C. R. Math. Acad. Sci. Paris 350 (2012), no. 13-14, 693–697; MR2971382] for an announcement of an extension of the latter work to the simple normal crossings case. Also, H. Guenancia and M. Păun were recently able to remove the $a_i \in [\frac{1}{2},1)$ condition from all of the results of the paper under review [“Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors”, preprint, arXiv:1307.6375].
Reviewed by Hans-Joachim Hein 