arXiv research

Study lattices in specific Lie groups for geometric structures.

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiformes, carrying a flat left invariant linear connection anf often a left invariant symplectic form. As a consequence we obtain an infinity of, non homeomorphic, compact affine or symplectic manifolds.…

Study compact symplectic solvmanifolds' hard Lefschetz property.

Maximal representations in symplectic lattices proven for most cases.

Study shows no deformation retractions for certain symplectic lattices.

The paper constructs a star product on a symplectically reduced phase space for a lattice gauge model.

Spin Lefschetz fibrations can represent any group and lattice point.

Smooth but not symplectic embeddings of rational balls in complex projective plane found.

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.

We obtain an algorithmic construction of the isotropy lattice for a lifted action of a Lie group $G$ on $TM$ and $T^*M$ based only on the knowledge of $G$ and its action on $M$. Some applications to symplectic geometry are also shown.

Every homomorphism from finite index subgroups of a universal lattices to mapping class groups of orientable surfaces (possibly with punctures), or to outer automorphism groups of finitely generated nonabelian free groups must have finite image. Here the universal lattice denotes the special linear group G=SL_m(Z[x1,..…

This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., compact manifolds that are the quotient of some Lie group G with a left invariant contact structure and a uniform lattice subgroup. We re-examine Alexander's criteria for existence of lattices on solvable…

In this paper we show that a certain solvable Lie group constructed in a paper by Benson and Gordon has no lattices. This result answers (in the negative way) a question posed by several authors in the context of symplectic geometry. The main theorem is proved with the use of rational homotopy theory.

We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in general with the Lie algebra one) and a minimal model. We show that some of these …

Flows on (or variations of) discrete curves in $\R^2$ give rise to flows on a subalgebra of functions on that curve. For a special choice of flows and a certain subalgebra this is described by the Toda lattice hierachy. In the paper it is shown that the canonical symplectic structure on $\R^{2N}$, which can be interpre…

New symplectic structures found on complex manifolds without Kähler structures.

In this paper we construct a family of simply connected, spin, non-complex, symplectic 4-manifolds which cover all but finitely many allowed lattice points $(χ, c)$ lying in $0 \leq c \leq 8.76χ$. Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on $(2n+1)(S^2 \times S^2)$…

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

New method for quantizing symplectic manifolds with Lagrangian bundles.

The Kodaira-Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov-Witten invariants which count pseudoholomorphic tori in the Kodaira-Thurston manifold. For a fixed symplectic form the Gromov-Witten invariant is trivial so we consider the twistor family of left-in…

Hexagonal diagrams link complex curves in $\mathbb{CP}^2$ to minimal genus surfaces.

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is eas…

The transcendental Hodge lattice of a projective manifold $M$ is the smallest Hodge substructure in $p$-th cohomology which contains all holomorphic $p$-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manif…

A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $π^+$ corresponding to $\om^+$ is po…

A compact solvmanifold of completely solvable type, i.e. a compact quotient of a completely solvable Lie group by a lattice, has a Kähler structure if and only if it is a complex torus. We show more in general that a compact solvmanifold $M$ of completely solvable type endowed with an invariant complex structure $J$ ad…

Study of Lagrangian Engel structures on symplectic 4-manifolds.

Study LCS structures of the second kind on Lie algebras.

Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability…

The embedded contact homology (ECH) of a 3-manifold with a contact form is a variant of Eliashberg-Givental-Hofer's symplectic field theory, which counts certain embedded J-holomorphic curves in the symplectization. We show that the ECH of T^3 is computed by a combinatorial chain complex which is generated by labeled c…

Study shows bi-Hamiltonian structure for polygon evolutions in centro-affine space.

The paper disproves a conjecture about 3D manifolds using even lattice points.

Constructs symplectic structures from rational functions on fans.

The study constructs symplectic solvmanifolds satisfying the hard-Lefschetz condition.

New findings show some symplectic solvmanifolds fail hard-Lefschetz condition.

To the integral symplectic group Sp(2g,Z) we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the Siegel space of genus g has the homotopy type of a bouquet of (g-2)-spheres. This…

We encode the variation structure of a quasihomogeneous polynomial with an isolated singularity as introduced by Nemethi in a set of spectral flows of the signature operator on the Milnor bundle by varying global elliptic boundary conditions in a specific way using the quasihomogeneous circle action on the Brieskorn la…

Study LCS structures on Lie algebras of type I, proving trivial Morse-Novikov cohomology and constructing solvmanifolds.

Study symplectic fillings of lens spaces, focusing on virtually overtwisted contact structures.

We present a geometric construction of central S^1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central S^1-extensions of the universal cover o…

The study examines harmonic forms on almost Hermitian manifolds and complex surfaces.

For a Lie group $G=\R^{n}\ltimes_φ\R^{m}$ with the semi-simple action $φ:\R^{n}\to {\rm Aut}(\R^{m})$, we show that if $Γ$ is a finite extension of a lattice of $G$ then $K(Γ, 1)$ is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group $Γ$ satisfies the hard Lefschetz proper…

Study lens spaces' definite fillings, classifying those with specific inequalities.

We prove that a positive allowable Lefschetz fibration, PALF in short, admits a structure of exact Lefschetz fibration in the sense of Seidel \cite{Se08}. If the two-fold first Chern class of the total space is zero, we obtain the Fukaya-Seidel category. We prove that the derived Fukaya-Seidel category of PALF is indep…

The paper studies representations of braid groups via curves and finds conditions for their Zariski closure and arithmeticity.

We consider three families of lattices on the oscillator group $G$, which is an almost nilpotent not completely solvable Lie group, giving rise to coverings $G \to M_{k, 0} \to M_{k, π} \to M_{k, π/2}$ for $k\in \Z$. We show that the corresponding families of four dimensional solvmanifolds are not pairwise diffeomorphi…

The existence or non-existence of Einstein metrics on 4-manifolds with non-trivial fundamental group and the relation with the underlying differential structure are analyzed. For most points $(n,m)$ in a large region of the integer lattice, the manifold $n\mathbb C \mathbb P^2\#m \overline{\mathbb C \mathbb P^2}$ is sh…

Planar multilinks prove rational singularities in surface geometry.

The paper proves geometrical finiteness for automorphism groups of K3 surfaces and related varieties.