We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show,…
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Decomposes complex manifolds with trivial canonical bundle into homogeneous structures.
problem Decomposing complex manifolds with trivial canonical bundle into homogeneous structures.
method Using MMP and foliation theory, we prove a decomposition theorem and deduce properties of holomorphic geometric structures.
result Holomorphic geometric structures on are locally homogeneous away from an analytic subset of complex codimension at least two.
Let be a regular neighborhood of a negative chain of -spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let be a rational homology ball which is smoothly embedded in . Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequen…
We extend the framework of K-stability (Tian, Donaldson) to more general algebro-geometric setting, such as partial desingularisations of (fixed) singularities, (not necessarily flat) families over higher dimensional base and the classical birational geometry of surfaces. We also observe that "concavity" of the volume …
For any flat projective family $(\mX,\mL)\rightarrow C$ such that the generic fibre $\mX_η$ is a klt Q-Fano variety and $\mL|_{\mX_η}\sim_{Q}-K_{X_η}$, we use the techniques from the minimal model program (MMP) to modify the total family. The end product is a family such that every fiber is a klt Q-Fano variety. Moreov…
Log minimality proven for weak K-moduli compactifications of Calabi-Yau varieties.
problem Constructing weak K-moduli compactifications of Calabi-Yau varieties.
method Revisiting classical problem, proving log minimality of normalizations under conditions.
result Log minimality of weak K-moduli compactifications under certain conditions.
Defines plurisubharmonic metrics on hybrid spaces and proves their canonical extensions.
problem Defining and analyzing plurisubharmonic metrics on hybrid spaces.
method Introduces a class of plurisubharmonic metrics on hybrid spaces and proves their canonical extensions.
result Canonical plurisubharmonic extensions of metrics on hybrid spaces are continuous and can be described in terms of canonical models.