arXiv research

Uniform volume estimate for Kähler metrics in big cohomology classes.

Study convexity of Mabuchi functional in big cohomology classes.

Estimates Kaehler metrics' diameter in big cohomology classes.

Paper defines new stability and metrics for complex spaces.

We establish various stability results for solutions of complex Monge-Ampère equations in big cohomology classes, generalizing results that were known to hold in the context of Kähler classes.

Proves existence of Kähler-Einstein metrics in big cohomology classes.

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

Let $X$ be a compact Kähler manifold and $\{θ\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $θ$-plurisubharmonic functions with full mass a…

We show that the non pluripolar product of positive currents is a bimeromorphic invariant. Under some natural assumptions, we show that the (weighted) energy associated to big cohomology classes are also bimeromorphic invariants. We compare the weighted energy functionals of currents with respect to different cohomolog…

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and $θ$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,θ)$, making it a complete geodesic metric space.

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomolo…

We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-…

Suppose $(X,ω)$ is a compact Kähler manifold of dimension $n$, and $θ$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,θ)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog f…

Two Calabi-Yau theorems for Kähler manifold degenerations.

We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of a closed positive current in given big cohomology class. The solution is shown t…

New metric spaces for geodesic rays in cohomology classes.

The paper studies deformations of Kähler manifolds to normal bundles and restricted volumes of big classes.

Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.

The proof of the comparison principlein [EGZ11] is not complete. We provide here an alternative proof, valid in the ample locus of any big cohomology class, and discuss the resulting modifications.

To any $\mathfrak{g}$-manifold $M$ are associated two dglas $\operatorname{tot}\big(Λ^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big)$ and $\operatorname{tot} \big(Λ^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$, whose cohomologies $H_{\operatorn…

Study geodesic distances and convexity in contact sets.

Let $X$ be a compact Kähler unibranch complex analytic space of pure dimension. Fix a big class $α$ with smooth representative $θ$ and a model potential $φ$ with positive mass. We define and the study non-pluripolar products of quasi-plurisubharmonic functions on $X$. We study the spaces $\mathcal{E}^p(X,θ;[φ])$ of fin…

This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory, variational methods for big cohomology classes, semiclassical constructions of solutions …

New construction of Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.

We study families of complex Monge-Ampère equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\infty}$-estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the stud…

The paper proves boundedness of envelopes in complex manifolds.

Study on $m$-positivity in Kähler manifolds with new Monge-Ampère-type equation.

The paper proves vanishing bounded cohomology for various groups.

The paper defines a complete geodesic metric for high energy spaces in Kähler manifolds.

The big phase space, the geometric setting for the study of quantum cohomology with gravitational descendents, is a complex manifold and consists of an infinite number of copies of the small phase space. The aim of this paper is to define a Hermitian geometry on the big phase space. Using the approach of Dijkgraaf and …

In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, w…

Let $X$ be a compact Kähler manifold and $θ$ a smooth closed $(1,1)$-real form representing a big cohomology class $α\in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that any $θ$-plurisubharmonic function $\f$ can be approximated from above by a decreasing sequence…

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähle…

Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.

The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes …

We show how to use Bar-Natan's `divide and conquer' approach to computations to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.

Estimates for scalar curvature equations on Kähler manifolds with singularities.

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show th…

We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on Kähler manifolds away from the non-Kähler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of Kähler metrics on compact Kähler varieties away from the singular loc…

Study classifies mapping class groups with hyperbolic actions on infinite-type surfaces.

Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.

Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.

Paper calculates topological complexity of robot movement in narrow aisles.

Sufficient condition for log-continuity of complex Monge-Ampère solutions.

This thesis introduces big mapping class groups and their structure.