arXiv research

In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply con…

Estimates symplectic Calabi-Yau equation using Cheng-Yau method.

Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.

Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.

Study finite group actions on symplectic Calabi-Yau 4-manifolds with non-zero first Betti number.

Stability of Type IIA flow ensures Kähler properties of Calabi-Yau 3-folds.

We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C$ independent of $B$ via the Hodge duality $*$. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of nonc…

We study type one generalized complex and generalized Calabi--Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the generalized complex structure, the twisting class. We prove that in a compact, type on…

Generalized Calabi-Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are b…

Constructs projective moduli spaces for Calabi-Yau pairs.

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

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

The paper explores complex geometries of 3-forms on symplectic 6-manifolds.

New groups can't be fundamental groups of symplectic Calabi-Yau manifolds.

Elliptic surfaces have unique Lefschetz pencils and Calabi-Yau diffeomorphisms.

In this paper we study the geometry of manifolds with vector cross product and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and co…

Generalizing the construction of the Maslov class for a Lagrangian embedding in a symplectic vector space, we prove that it is possible to give a consistent definition of this class for any Lagrangian submanifold of a Calabi-Yau manifold. Moreover, we prove that this class can be represented by the contraction of the K…

Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to Kähler manifolds with c_1 = 0. The construction beg…

The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.

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

We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.

In this paper we summarize our recent work in the construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties and the symplectic Strominger-Yau-Zaslow conjecture, together with some new development. It is submittded to the Proceedings of the Conference in Symplectic Geometry and Mirror S…

Continues work on derived manifolds and symplectic schemes, constructing virtual classes.

Symplectic 4-manifolds with Kodaira dimension zero can be viewed as symplectic Calabi-Yau surfaces. We are able to completely determine their Betti numbers by proving two general results on quaternionic vector bundles.

Auxiliary equations improve bounds in symplectic geometry.

Let (M, ω) be a compact symplectic 4-manifold with a compatible almost complex structure J. The problem of finding a J-compatible symplectic form with prescribed volume form is an almost-Kähler analogue of Yau's theorem and is connected to a programme in symplectic topology proposed by Donaldson. We call the correspond…

This paper pursues the study of the Calabi-Yau equation on certain symplectic non-Kaehler 4-manifolds, building on a key example of Tosatti-Weinkove in which more general theory had proved less effective. Symplectic 4-manifolds admitting a 2-torus fibration over a 2-torus base are modelled on one of three solvable Lie …

Let S be a K3 surface that admits a non-symplectic automorphism $ρ$ of order 3. We divide $S\times \mathbb{P}^1$ by $ρ\timesψ$ where $ψ$ is an automorphism of order 3 of $\mathbb{P}^1$. There exists a threefold ramified cover of a partial crepant resolution of the quotient that is a Calabi-Yau orbifold. We compute the …

Study explores unstable 3-forms on Calabi-Yau 3-folds.

A generalized Calabi-Yau structure is a geometrical structure on a manifold which generalizes both the concept of the Calabi-Yau structure and that of the symplectic one. In view of a result of Lin and Tolman in generalized complex cases, we introduce in this paper the notion of a generalized moment map for a compact L…

We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and except…

We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then give a new construction of an infinite family of symplectic Calabi-Yau surfaces with first Betti number b_1=2,3, along wi…

We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete …

The classical Kaehler potential is a real-valued function (KP) such that one can determine a Kaehler (symplectic) structure by differentiating KP. We define a mirror Kaehler potential on Calabi-Yau 3-folds, a real-valued function (MKP) such that one can determine a complex structure by differentiating MKP.

Study stability of pseudo-Kähler and neutral Calabi-Yau manifolds, finding stability in 2D but failing in higher dimensions.

We prove that the Calabi-Yau equation can be solved on the Kodaira-Thurston manifold for all given $T^2$-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic four-manifolds with compatible but non-integrable almost complex structures.

In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in t…

In this paper we state an analog of Calabi's conjecture proved by Yau. The difference with the classical case is that we propose deformation of the complex structure, whereas the complex Monge--Ampère equation describes deformation of the Kähler (symplectic) structure.

Extends Gromov invariant to Calabi-Yau 3-folds.

In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.

R.C.McLean showed that the moduli space of nearby submanifolds of a smooth, compact, orientable special Lagrangian submanifold L in a Calabi-Yau manifold X is a smooth manifold and its tangent space at L is identified with the space of harmonic one forms on L. In this paper, we will extend this result from Calabi-Yau m…

Study symplectic 4-orbifolds with vanishing canonical class, finding new structures and resolutions.

We discuss a conjecture of Donaldson on a version of Yau's Theorem for symplectic forms with compatible almost complex structures and survey some recent progress on this problem. We also speculate on some future possible directions, and use a monotonicity formula for harmonic maps to obtain a new local estimate in the …

We give two applications of the Aleksandrov-Bakelman-Pucci estimate to the Calabi-Yau equation on symplectic four-manifolds. The first is solvability of the equation on the Kodaira-Thurston manifold for certain almost-Kahler structures assuming $S^1$-invariance, extending a result of Buzano-Fino-Vezzoni. The second is …

The paper classifies and studies symplectic and contact properties of circular spherical divisors.

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…

Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these…