arXiv research

The paper connects algebraic geometry to threefold theories.

We have developed a mathematical theory of the topological vertex--a theory that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in hep-th/0305132 on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Cala…

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

Developing deformation theory for Calabi-Yau 3-folds with boundary.

New non-Kähler 3-folds constructed via log conifold transitions.

We introduce a homology theory whose Euler characteristics counts ASD bundles over four dimensional co-associative submanifolds in (almost) G_2 manifolds. As a TQFT, in relative situations, we have the Fukaya-Floer category of Lagrangians intersection in the moduli space of special Lagrangian submanifolds in CY threefo…

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 local third Chern class for point singularities on threefolds.

For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, we will compare the order of the Alexander invariant at t=1 and one of Ruelle-Selberg L-function at s=0. Our result may be considered as a geometric analog of the Iwasawa main conjecture in the algebraic …

For a local system on a compact hyperbolic threefold, under a cohomological assumption, we will show that the order of its twisted Alexander polynomial and of the Ruelle L function at $s=0$ coincide. Moreover we will show that their leading constant are also identical. These results may be considered as a solution of a…

Lecture notes on conifold transitions between Calabi-Yau manifolds.

In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact Kähler t…

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.

In arXiv:1008.1018 it is shown that a given stable vector bundle $V$ on a Calabi-Yau threefold $X$ which satisfies $c_2(X)=c_2(V)$ can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explic…

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.

We survey contributions of Robert MacPherson to the theory of arithmetic groups. There are two main areas we discuss: (i) explicit reduction theory for Siegel modular threefolds, and (ii) constructions of compactifications of locally symmetric spaces. The former is joint work with Mark McConnell, the latter with Lizhen…

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,…

Study on balanced Hermitian threefolds with parallel Bismut torsion.

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 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 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 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 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…

Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.

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…

In alignment with a programme by Donaldson and Thomas [DT], Thomas [Th] constructed a deformation invariant for smooth projective Calabi-Yau threefolds, which is now called the Donaldson-Thomas invariant, from the moduli space of (semi-)stable sheaves by using algebraic geometry techniques. In the same paper [Th], Thom…

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…

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal…

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

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

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

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…