arXiv research

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution stru…

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution…

Proves representability of complex semigroup systems.

Uniformizes varieties with log-canonical singularities using ball quotients.

The abstract discusses homological stability in topological moduli spaces.

This paper resolves symplectic orbifolds and applies it to finite group actions.

Study symplectic 4-orbifolds with vanishing canonical class, finding new structures and resolutions.

The `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution tower' which projects …

Paper proposes a new method for better super-resolution images.

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blo…

In this paper differential operators on various moduli spaces (e.g. of holomorphic vector bundles) are described in a canonical way in terms of the geometry of a certain distinguished completion of an appropriate configuration space.

We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Ka…

We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous …

Let $X_0$ be an affine variety with only normal isolated singularity $p$ and $π: X\to X_0$ a smooth resolution of the singularity with trivial canonical line bundle $K_X$. If the complement of the affine variety $X_0\backslash\{p\}$ is the cone $C(S)=\Bbb R_{>0}\times S$ of an Einstein-Sasakian manifold $S$, we shall p…

The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.

Let K be an algebraically closed field of characteristic zero, endowed with a complete nonarchimedean norm. Let X be a K-rigid analytic variety and Σa semianalytic subset of X. Then the closure of Σin X with respect to the canonical topology is again semianalytic. The proof uses Embedded Resolution of Singularities.

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing an…

Simplified presentation of symplectic fillings of lens spaces.

Paper proves Whitney stratified spaces can be given a conically smooth structure.

Develops Poisson and Dirac manifolds of compact types with applications.

Solves Tian's stabilization problem for toric Fano manifolds.

The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly pseudoconvex domains in a complex manifold. Such a manifold carries a complete Kähler-Einstein metric if and only if its canonical bundle is positive. We consider the restricted case in which the CR structure on $\partial M$ is normal. In this case M…

The paper constructs ALF Calabi-Yau metrics on specific manifolds.

The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup G of SU(2) and the geometry of the minimal resolution of the quotient of the affine plane by G. In this paper we discuss a possible generalization of the McKay correspondence to the case when G is replace…

We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toric singularities (already found by Goldstein), proper versions of these fibrations, and fibrations on flat deformations of c…

Study excess logarithmic residues for foliations to bound invariant hypersurfaces and test log canonicity.

Refined asymptotics of scalar-flat ALE four-manifolds

The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the…

Building upon ideas of Hironaka, Bierstone-Milman, Malgrange and others we generalize the inverse and implicit function theorem (in differential, analytic and algebraic setting) to sets of functions of larger multiplicities (or ideals). This allows one to describe singularities given by a finite set of generators or by…

The paper characterizes symplectic fillings of Seifert 3-manifolds using rational blowdowns.

The paper classifies minimal symplectic fillings of small Seifert 3-manifolds.

Develops a new volatility model for prediction markets.

New Stein fillings found for non-weighted homogeneous singularities.

New steady Kähler-Ricci solitons found on orbifolds.

The article studies Ricci-flat metrics on complex projective space.

CPOPT-Net predicts sparse client actions in banking using tensor decomposition and neural networks.

We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero-Gattazzo surface is the …

This paper deals with the question of J.Morava on existence of canonical complex cobordism class of singular submanifold. We present several solutions of this question for $X_r(ξ)$ -- the set of points where $\dimξ-r+1$ generic sections of a complex vector bundle $ξ$ are linearly dependent. The corresponding complex co…

Solves non-Abelian Rainich problem for SU(2) gauge fields.

The paper studies symplectic operations on Stein fillings of Brieskorn singularities.

I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…

Develops a new volatility model for prediction markets.

Study D3-brane solutions on resolved C^3/Γ singularities, proving metric conjecture.

Solves complex equation for specific geometric solitons.

Constructs a unique Levi-Civita connection for generalised metrics.

Uniform bounds prove connection between Kähler metrics and RCD spaces.

The study proves that certain surface singularities are planar only for $A_n$-singularities.