Kähler–Einstein Metrics with Mixed Singularities

• Kähler–Einstein metrics are Kähler metrics on complex manifolds whose Ricci curvature is proportional to the metric, incorporating both cone and Poincaré (cusp) singularities along prescribed divisors.
• The Monge–Ampère formulation recasts the curvature condition into a solvable PDE using smooth approximations, finite-energy solutions, and a priori estimates to ensure convergence.
• Vanishing theorems derived from this framework have significant implications in birational geometry, demonstrating the non-existence of certain global symmetric differentials on varieties with ample K_X+D.

A Kähler–Einstein (KE) metric on a complex manifold is a Kähler metric whose Ricci curvature is proportional to the metric itself, that is, $\operatorname{Ric}(\omega) = \lambda\,\omega$ for some constant $\lambda$. Classical results cover the existence, uniqueness, and structure of these metrics in the compact smooth setting, but recent work has extended the theory to allow prescribed singularities along divisors, especially of mixed Poincaré and cone type. This leads to a rich framework that encompasses analytic, geometric, and algebro-geometric aspects, particularly when considering effective $\mathbb{R}$-divisors with simple normal crossing (SNC) support and coefficients in $[1/2,1]$.

1. Geometric Setting: Mixed Poincaré and Cone Singularities

Let $X$ be a compact Kähler manifold of complex dimension $n$, and let $D = \sum_{i=1}^N a_i D_i$ be an effective $\mathbb{R}$-divisor with simple normal crossing support, with each coefficient $a_i \in [1/2,1]$. The divisor can be decomposed as $D = D_{\mathrm{klt}} + D_{\mathrm{lc}}$ where $D_{\mathrm{klt}}$ collects components with $a_i<1$ and $D_{\mathrm{lc}}$ those with $a_i=1$. The key assumption is that the twisted canonical class $K_X + D$ is ample, i.e., $c_1(K_X+D)$ contains a Kähler form.

The boundary behavior of the desired metric is determined by the coefficients $a_i$. Along components $D_j$ with $a_j<1$, the metric should have cone singularities of angle $2\pi\beta_j$, with $\beta_j=1-a_j\in [0,1/2]$; for $a_k=1$, one requires Poincaré (cusp) singularities.

In local holomorphic coordinates $(z_1,\ldots,z_n)$ near the SNC divisor, the model mixed singularity metric takes the explicit form: $$\omega_{\mathrm{mod}} = \sum_{j=1}^r \frac{i\,dz_j\wedge d\bar z_j}{|z_j|^{2(1-\beta_j)}} + \sum_{k=r+1}^{r+s} \frac{i\,dz_k\wedge d\bar z_k}{|z_k|^2 (\log |z_k|^2)^2} + \sum_{\ell=r+s+1}^n i\,dz_\ell\wedge d\bar z_\ell,$$ with $$D|_{U} = \sum_{j=1}^r (1-\beta_j)\{z_j=0\} + \sum_{k=r+1}^{r+s}\{z_k=0\}$$, organized by cone and cusp behavior. For a KE metric $\omega$ on $X_0=X\setminus \mathrm{Supp}(D)$, the "mixed Poincaré–cone growth" is defined by $$C^{-1}\omega_{\mathrm{mod}} \leq \omega \leq C\omega_{\mathrm{mod}}$$ for some $C>0$ in a neighborhood of any point of Supp($D$).

2. Kähler–Einstein Equation and Monge–Ampère Formulation

The negative curvature case of interest seeks a Kähler form $\omega$ on $X_0$ satisfying

$\operatorname{Ric}(\omega) = -\omega + [D],$

where $[D]$ denotes the current of integration on $D$. Introducing a smooth reference form $\omega_0 \in c_1(K_X + D)$ and local defining sections $s_i$ for $D_i$ (with single logarithmic branches), the equation is reduced to a scalar complex Monge–Ampère equation: $$(\omega_0 + i\partial\bar\partial \varphi)^n = e^{-\varphi - \psi}\,\omega_0^n,$$ with

$$\psi = \sum_{i\in \mathrm{klt}} a_i \log|s_i|^2 + \sum_{i\in \mathrm{lc}} \log|s_i|^2 + \mathrm{smooth}.$$

Here, $\varphi$ is a $\omega_0$-plurisubharmonic potential, with finite energy (i.e., $\varphi \in \mathcal{E}^1(X, \omega_0)$), and the right-hand side prescribes the singularity structure. The existence and uniqueness problem is thus recast as one for a global Monge–Ampère equation with prescribed singularities and volume normalization.

3. Existence and Uniqueness: Analytic Techniques and A Priori Estimates

The solution strategy proceeds in several analytic steps:

• Reduction to finite-energy solutions: The desired negative curvature KE metric extends uniquely as a finite energy current solving the above Monge–Ampère equation. Uniqueness follows from the comparison principle for such currents (Guedj–Zeriahi).
• Regularization and continuity method: On the locus where the divisors with coefficient 1 are removed (the lc locus), one constructs a smooth Carlson–Griffiths type reference metric with Poincaré cusps along $D_{\mathrm{lc}}$, and smooth approximation metrics $\widehat{\omega}_\varepsilon$ regularizing the cone part. Regularized equations,

$$(\widehat{\omega}_\varepsilon + i\partial\bar\partial \varphi_\varepsilon)^n = e^{-\varphi_\varepsilon - \psi_\varepsilon}\,\widehat{\omega}_\varepsilon^n,$$

are solved via the Tian–Yau/Kobayashi theory for log pairs.

• A priori $C^0$ and Laplacian estimates: Uniform $C^0$ bounds (maximum principle, boundedness of the data), Laplacian bounds (via lower Ricci and bisectional curvature bounds), Evans–Krylov interior $C^{2,\alpha}$ and Schauder bootstrap yield smooth convergence of $\varphi_\varepsilon$ on compact subsets of $X_0$.
• Limit and characterization: The limiting metric $\omega = \omega_0 + i\partial\bar\partial \varphi$ solves $\operatorname{Ric}(\omega) = -\omega + [D]$ and has the required mixed Poincaré and cone singularities along $D$.

The key Laplacian estimate (Yau's method) is: if $(X,\omega)$ is complete, $\operatorname{Ric}(\omega)\geq -B$, and $u$ solves

$$(\omega + i\partial\bar\partial u)^n = e^{F+u} \omega^n,$$

with $u$ bounded, then

$$\sup_X|u| \leq \sup_X|F|,\qquad \operatorname{tr}_\omega(\omega + i\partial\bar\partial u) \leq C$$

for $C$ depending on $\sup |F|, \inf F, B, n$.

4. Geometric and Birational Consequences: Vanishing Theorems

Define the sheaf of $D$-tensors $T^r_s(X|D)$ as generated locally by

$$z^{\lceil (h_I - h_J)\cdot a \rceil} \partial_{z_{i_1} \otimes \cdots} \otimes dz_{j_1} \otimes \cdots,$$

where the multi-indices and rounding encode prescribed zeros and poles along $D$ according to the $a_i$.

Vanishing theorem: If $K_X + D$ is ample and all $a_i\in [1/2,1]$, for $r \geq s+1$,

$H^0(X, T^r_s(X|D)) = 0.$

The proof leverages the mixed Poincaré–cone KE metric: $D$-tensors are identified with global holomorphic tensors on $X_0$ bounded with respect to the KE metric (possibly twisted by a singular line bundle), and the Bochner formula shows negativity of the relevant curvature terms for $r \geq s+1$, up to vanishing error at the boundary.

This vanishing applies in birational geometry and orbifold settings, showing the non-existence of certain symmetric differentials on mildly singular pairs.

5. Comparison with Earlier Work and Context in Birational Geometry

The results for mixed Poincaré–cone singularities extend the known existence theory:

• Kobayashi and Tian–Yau's existence theorem for log pairs with all $a_i=1$ (pure Poincaré singularities).
• The pure conic case ($a_i<1$), with foundational work by Brendle, Mazzeo–Jeffres–Rubinstein, and Campana–Guenancia–Păun.

The mixed case treats divisors with both cone angles $\beta_i = 1-a_i \in [0,1/2]$ and components with Poincaré cusps, unifying and interpolating between these two types of singularities.

From the birational perspective, these vanishing theorems have implications for the structure of the sheaf of orbifold tensors, providing new tools in the study of moduli of varieties of general type and the non-existence of global symmetric differentials on certain pairs.

6. Open Problems and Further Directions

Several natural directions emerge:

• KE metrics with more general singularities: The existence and uniqueness theory for $K_X+D$ ample and more general coefficients, specifically for $a_i<1/2$ or $a_i>1$ (i.e., extending to divisors beyond Kawamata log terminal (klt) type, log canonical but not klt).
• Other Ricci curvature regimes: Positive or zero Ricci curvature with mixed singularities, i.e., spherical and Calabi–Yau cases with Poincaré–cone-type divisors.
• Fine boundary asymptotics: Polyhomogeneous expansions and regularity near the divisor, following analogous work in the pure cone case (e.g., Mazzeo–Jeffres–Rubinstein).

Each of these advances would further connect the partial differential equation (PDE) analysis of complex Monge–Ampère equations with birational, analytic, and metric aspects of varieties with boundary.

7. Summary Table: Mixed Poincaré–Cone Kähler–Einstein Metrics

Setting	Singularities	Existence/Uniqueness
$K_X+D$ ample, $a_i\in [1/2,1]$	Mixed Poincaré (cuspidal) and cone (angle $2\pi\beta_i$)	Yes; unique negative KE metric with prescribed mixed behavior along $D$

A plausible implication is the further extension of these mixed singularity techniques to the study of moduli and to analytic compactification problems in higher-dimensional algebraic geometry (Guenancia, 2012).