arXiv research

Study of a basic Hitchin equation on Sasakian 3-folds, showing hyperKähler metric.

The paper studies symplectic structures on character varieties of Sasakian threefolds.

We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds.…

The $7$-dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$-structure $\varphi$ induced by the Calabi-Yau $3$-orbifold basic geometry. We disti…

The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.

Optimizes bounds for threefold singularity volumes.

Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.

The study classifies holomorphic projective connections on complex threefolds.

K-stability proven for a specific type of Fano threefold.

We study the variety of Poisson structures and compute Poisson cohomology for two families of Fano threefolds - smooth cubic threefolds and the del Pezzo quintic threefold. Along the way we reobtain by a different method earlier results of Loray, Pereira and Touzet in the special case we are considering.

Study noncommutative deformations of Calabi-Yau threefolds.

The study examines non-Kähler threefolds with specific metrics and finds they are quasi-bundles over surfaces.

New research confirms Kähler-Einstein metrics for all Fano threefolds of degree 22.

The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A Corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.

We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact Kähler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to Kähler threefolds. In the appendix of arXiv:0805.2192,…

We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds …

Study on balanced Hermitian threefolds with parallel Bismut torsion.

Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.

We investigate geometric invariants of the one parameter family of Mukai threefolds that admit $\mathbb C^{*}$ action. In particular we find the invariant divisors in the anticanonical system, and thus establish a bound on the log canonical thresholds. Furthermore we find an explicit description of such threefolds in t…

Kähler-Ricci flow shows type II singularity on Fano threefolds.

Study immersions of Sasakian manifolds into Sasakian space forms.

Study shows Futaki invariant vanishes on most Fano threefolds.

Found a stable 3D shape with specific properties.

We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds wi…

Complex manifolds can only map to curves, restricting Clemens threefolds and $S^6$.

We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real 6-manifold fibres smoothly over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three.

We study the problem of existence of Kähler--Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree $22$ that admit a faithful action of the multiplicative group $\mathbb{C}^\ast$. We prove that, except possibly two explicitly described cases, all such smooth Fano threefolds are Kähler--…

The study classifies Kähler threefolds with special fiber bundles.

We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.

We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP…

We construct balanced metrics on the family of non-Kähler Calabi-Yau threefolds that are obtained by smoothing after contracting $(-1,-1)$-rational curves on Kähler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of $k\geq 2$ copies of $S^3\time…

Spinorially constructs Sasakian and 3-Sasakian structures in arbitrary dimensions.

The object of this paper is to study the invariant submanifolds of Sasakian generalized-Sasakian-space-form. Here, we obtain some equivalent conditions for an invariant submanifold of a Sasakian generalized-Sasakian-space-forms to be totally geodesic.

Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.

We show that the contact reduction can be specialized to Sasakian manifolds. We link this Sasakian reduction to Kähler reduction by considering the Kähler cone over a Sasakian manifold. We present examples of Sasakian manifolds obtained by $S^1$ reduction of standard Sasakian spheres.

Study classifies quasi-Sasakian 3-manifolds, linking them to Sasakian and co-Kähler structures.

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrang…

We prove that for a compact Kähler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every Kähler threefold of Kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modifi…

Defines generalized Sasakian structures in contact geometry.

Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.

Study on lightlike geometry in indefinite Sasakian statistical manifolds.

Constructs special Lagrangian 3-spheres in non-Kähler compact threefolds.

Study special Lagrangian sections in Calabi-Yau threefolds, showing stability conditions imply isomorphism to special Lagrangians.

Study para-Sasakian φ-symmetric spaces using Boothby-Wang fibration.

In their papers published in 1993 and 1994, by expressing certain physical quantity in two distinct ways, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable equivalence between Ray-Singer analytic torsion and elliptic instanton numbers for Calabi-Yau threefolds. After their discovery, in a paper published in 2008, …

In this paper, we give an expression and some estimates of the curvature tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this paper, an extra restriction of the limiting Hodge structure for the degenera…

Sasakian immersions prove Sasaki-Ricci solitons are η-Einstein with rational constants.

We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric…