Related papers: Remarks on Mukai threefolds admitting $C^{*}$ acti…
We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we effectively determine the existence of Kahler-Ricci solitons for those manifolds via…
This paper is motivated by the question of how motivic Donaldson--Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi--Yau threefolds, defined by quivers with homogeneous…
We show that the Elliott invariant is a classifying invariant for the class of $C^*$-algebras that are simple unital infinite dimensional inductive limits of sequences of finite direct sums of building blocks of the form $$ \{f\in…
We study the topology of closed, simply-connected, 6-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by $SU(2)$ or $SO(3)$. We show that their Euler characteristic agrees with that of the known…
We study discrete, cocompact, isometric actions of groups on Hadamard spaces, and the induced actions on ideal boundaries. For a class of groups generalizing fundamental groups of three-dimensional graph manifolds, we find a set of…
In this paper we study modular tensor categories (braided rigid balanced tensor categories with additional finiteness and non-degeneracy conditions), in particular, representations of quantum groups at roots of unity. We show that the…
In this article we focus on multiplier Hermitian-Einstein metrics introduced by Mabuchi which include K\"ahler-Einstein metrics, K\"ahler-Ricci solitons and Mabuchi solitons as special cases. We also focus on KSM-manifolds, which are…
We identify and study a class of hyperbolic 3-manifolds (which we call Macfarlane manifolds) whose quaternion algebras admit a geometric interpretation analogous to Hamilton's classical model for Euclidean rotations. We characterize these…
We show exceptionality of certain families of non-quasismooth weighted hypersurfaces. In particular these admit K\"ahler-Einstein metrics. Our examples are produced by the monomials generating the complex deformations of orbifolds whose…
Two parameter families of plane conics are called nets of conics. There is a natural group action on the vector space of nets of conics, namely the product of the group reparametrizing the underlying plane, and the group reparametrizing the…
Given a Fano manifold $(X,\omega)$ we develop a variational approach to characterize analytically the existence of K\"ahler-Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically.…
In these notes, we carefully analyze the properties of the "ramified" Seiberg-Witten equations associated with supersymmetric configurations of the Seiberg-Witten abelian gauge theory with surface operators on an oriented closed…
Half a century ago Manin showed that given a number field $k$ and a rational prime $\ell$, there exists a uniform bound for the order of cyclic $\ell$-power isogenies between two non-CM elliptic curves over $k$. We generalize this to…
We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge…
We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using Geometric Invariant Theory and the anticanonical…
In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between…
I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit…
We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a…
In this paper we study holomorphic actions of the complex multiplicative group on complex manifolds around a singular (fixed) point. We prove linearization results for the germ of action and also for the whole action under some conditions…
We introduce and study $\mu$K-stability of polarized schemes with respect to general test configurations as an algebro-geometric aspect of the existence of $\mu$-cscK metrics, which is introduced in the paper arXiv:1902.00664 as a framework…