arXiv research

Complete Calabi-Yau metrics on abelian fibrations over complex space.

Construct special Lagrangian fibrations on abelian varieties using retraction techniques.

The paper examines $L^2$-torsion in fibration cases relaxing standard conditions.

For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.

In this thesis we study asymptotic behavior of projective embeddings of abelian varieties and their amoebas. The projective embeddings are given by theta functions. It is known that a Lagrangian fibration of the abelian variety determines a basis of theta functions. After reviewing the relation from the viewpoint of ge…

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, w…

Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.

The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.

Solves a specific Calabi conjecture on special nilmanifolds.

Every fibration of a projective hyper-Kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypothes…

The study shows boundedness of certain fibered varieties in algebraic geometry.

In this paper we study some properties of almost abelian solvmanifolds using minimal models associated to a fibration. In particular we state a necessary and sufficient condition to formality and a method for finding symplectic strucures of this kind of solvmanifolds.

Prove long-time existence of pluriclosed flow on certain fibrations

Extends T-duality to more general fibrations, proving isomorphic cohomology and algebroids.

The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.

New groups from surface braids help create complex geometric shapes.

We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M5-brane fibers as target space. Mirror symmetry and various string and M-theory dualities involving G_2 manifolds may be incorporated into…

The paper proves fibration theorems for manifolds with almost nonnegative Ricci curvature.

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the re…

We examine Higgs bundles for non-compact real forms of SO(4,C) and the isogenous complex group SL(2,C)XSL(2,C). This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral data.

Describes spectral data for singular fibres of a specific Hitchin system.

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point…

Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.

We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in …

In this paper we will investigate torus actions on complete manifolds with calibrations. For Calabi-Yau manifolds M^2n with a Hamiltonian structure-preserving k-torus action we show that any symplectic reduction has a natural holomorphic volume form. Moreover Special Lagrangian (SLag) submanifolds of the reduction lift…

Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to…

The paper proves a structure theorem for compact Kähler manifolds with semi-positive holomorphic sectional curvature.

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type, in an iterated sense, with connecting fibrations and this structure descends to a resolution of the qu…

Study the monodromy and center-focus problems for rational maps defined by products of generic lines.

Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fix…

Let S be a closed surface with nonzero Euler characteristic. We prove the existence of an open neighborhood V of the identity map of S in the C^1-topology with the following property: if G is an abelian subgroup of Diff^1(S) generated by any family of elements in V then the elements of G have common fixed points. This …

Non-Abelian actions are resolved using equivariant K-theory and delocalized cohomology.

Log-conformal projective pairs restrict to simple geometric structures.

Study stabilizers of isotropic classes in rational 4-manifolds, finding diffeomorphisms that almost preserve Lefschetz fibrations.

Study of fibred faces of Thurston polyhedra for 2-component 2-bridge links

We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathem…

This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the bas…

We examine the action of the fundamental group $Γ$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles re…

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

Introduces new stability concept for Fano fibrations.

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.

Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism type…

A fibration of $\mathbb{R}^3$ by oriented lines is given by a unit vector field $V : \mathbb{R}^3 \to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationship…

Same genus-2 fibration structures for specific types found by different researchers.

Let S be an orientable surface with negative Euler characteristic, let ψ\in\Mod(S) be a mapping class of S, and let T_ψ be the mapping torus of ψ. We study the action of lifts of ψon the homology of finite covers of S via the torsion homology growth of towers of finite covers of T_ψ. We show that ψadmits a lift to a fi…