Related papers: Toric K\"ahler metrics seen from infinity, quantiz…
We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics…
Abreu-Sena-Dias have constructed two distinct families of scalar-flat K\"ahler non-compact toric metrics using Donaldson's rephrasing of Joyce's construction in action-angle coordinates. In this paper and using the same set-up, we show that…
A toric degeneration in algebraic geometry is a process where a given projective variety is being degenerated into a toric one. Then one can obtain information about the original variety via analyzing the toric one, which is a much easier…
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and…
On a compact K\"ahler manifold $X$, Toeplitz operators determine a deformation quantization $(\operatorname{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ with separation of variables [10] with respect to transversal complex polarizations…
The Donaldson-Fujiki K\"ahler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of K\"ahler metrics as a moment map, can be lifted canonically to a hyperk\"ahler reduction.…
In this short note, we investigate the existence of orbifold K\"ahler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold K\"ahler-Einstein metric.…
Let G/H be a pseudo-Riemannian semisimple symmetric space. The tangent bundle T(G/H) contains a maximal G-invariant neighbourhood of the zero section where the adapted complex structure exists. Such neighbourhood is endowed with a canonical…
A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2\pi\Z^n$, is a smooth projective toric variety completely…
In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichm{\"u}ller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit sphere at each point in the tangent space…
Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by…
We study the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety with branching set the invariant divisor under the algebraic torus action. These are completions (compactifications) of the…
A tropical expansion is a degeneration of a toroidal embedding, induced by a polyhedral subdivision of its tropicalisation. Each irreducible component of a tropical expansion admits a collapsing map down to a stratum of the original…
We investigate the Hitchin hyperk\"ahler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as…
The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations…
In this paper we study a $k$-dimensional analytic subvariety of the complex algebraic torus. We show that if its logarithmic limit set is a finite rational $(k-1)$-dimensional spherical polyhedron, then each irreducible component of the…
In this article we study compact K\"ahler manifolds $X$ admitting non-singular holomorphic vector fields with the aim of extending to this setting the classical birational classification of projective varieties with tangent vector fields.…
We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the…
Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T^k. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their…
In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the previous cotangent…