arXiv research

Method resolves 4D symplectic orbifolds using complex geometry.

Smooth resolutions found for quotient of R^2 by infinite discrete groups.

We use the techniques of integration of Poisson manifolds into symplectic Lie groupoids to build symplectic resolutions (= desingularizations) of the closure of a symplectic leaf. More generally, we show how Lie groupoids can be used to lift singularities, in particular when one imposes a compatibility condition with a…

We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups.

Projective resolves symplectic Steinberg module for number rings.

Study symplectic cohomology of certain singularities using homological mirror symmetry.

We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symple…

We associate to each symplectic $4$-orbifold $X$ a canonical smooth symplectic resolution $π: \tilde{X}\rightarrow X$, which can be done equivariantly if $X$ comes with a symplectic $G$-action by a finite group. Moreover, we show that the resolutions of the symplectic $4$-orbifolds $X/G$ and $\tilde{X}/G$ are in the sa…

We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symple…

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformati…

Some Poisson structures do admit resolutions by symplectic manifolds of the same dimension. We give examples and simple conditions under which such resolutions can not exist.

We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme.…

Floer theory constructs filtrations on quantum cohomology for symplectic manifolds.

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

Constructing compact non-Kähler manifolds with and without the Hard Lefschetz Condition

We resolve Spin(7)-orbifolds using algebraic and symplectic techniques.

Simplified presentation of symplectic fillings of lens spaces.

We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian spaces can be interpreted as Lagrangians in the adjoint quotient, this allows one to …

We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit algorithm inspired by the minimal model program for complex 3-dimensional algebraic vari…

In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we also demonstrate that every such a minimal symplectic fillin…

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…

In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a …

New Stein fillings found for non-weighted homogeneous singularities.

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…

Suppose that $C=(C_1,..., C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,ω)$ with connected, negative definite intersection graph $Γ_C$. We show that by replacing an appropriate neighborhood of $\cup C_i$ with a smoothing $W_S$ of a normal surface singularity $(S, 0)$ w…

New symplectic annular Khovanov homology connects knot theory to Floer homology.

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…

A Calabi-Yau orbifold is locally modeled on C^n/G where G is a finite subgroup of SL(n, C). In dimension n=3 a crepant resolution is given by Nakamura's G-Hilbert scheme. This crepant resolution has a description as a GIT/symplectic quotient. We use tools from global analysis to give a geometrical generalization of the…

Study intersection cohomology and Lagrangian fibrations in symplectic varieties.

New method recovers hyperkähler metrics from twistor models.

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

Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to Kähler manifolds with c_1 = 0. The construction beg…

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids …

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

Using the Penrose transform, we construct analogues of the BGG (Bernstein-Gelfand-Gelfand) resolutions in certain singular infinitesimal characters, in the holomorphic geometric setting, over the Lagrangian Grassmannian. We prove the exactness of the constructed complex over the big affine cell.

We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi-Yau surfaces from complex surfaces of general type, as well as from rational and ruled s…

We find all $P$-resolutions of quotient surface singularities (especially, tetrahedral, octahedral, and icosahedral singularities) together with their dual graphs, which reproduces Jan Steven's list [Manuscripta Math. 1993] of the numbers of $P$-resolutions of each singularities. We then compute the dimensions and Miln…

Let S be a K3 surface that admits a non-symplectic automorphism $ρ$ of order 3. We divide $S\times \mathbb{P}^1$ by $ρ\timesψ$ where $ψ$ is an automorphism of order 3 of $\mathbb{P}^1$. There exists a threefold ramified cover of a partial crepant resolution of the quotient that is a Calabi-Yau orbifold. We compute the …

Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to…

New proof shows unique symplectic fillings for certain surface singularity links.

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered co…

Computes $L_\infty$-algebroid for linear foliations on vector spaces.

We prove that any symplectic Fano $6$-manifold $M$ with a Hamiltonian $S^1$-action is simply connected and satisfies $c_1 c_2(M)=24$. This is done by showing that the fixed submanifold $M_{\min}\subseteq M$ on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a $2$-sphere or a po…

Book teaches how Lagrangian torus fibration base geometry can be read off.

We pursue the symplectic description of toric Kahler manifolds. There exists a general local classification of metrics on toric Kahler manifolds equipped with Hamiltonian two-forms due to Apostolov, Calderbank and Gauduchon(ACG). We derive the symplectic potential for these metrics. Using a method due to Abreu, we rela…

The paper constructs Ricci-flat Kähler manifolds with specific decay properties.

This thesis contains work which appeared in several papers. Additionally to the results in the papers it contains a detailed introduction and some further proofs and remarks. The dissertation gives a description of the topology and symplectic and algebraic geometry of Hitchin's hyperkaehler moduli space M of rank 2 Hig…