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Yongbin Ruan

Publications and source records attributed to Yongbin Ruan.

At least 55 records · Page 3Linked to original sources
arXiv2008

Quantum Cohomology and Crepant Resolutions: A Conjecture

We give an expository account of a conjecture, developed by Coates--Corti--Iritani--Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold X to the quantum cohomology of a crepant resolution Y of X. We explore some consequences of this conjecture, showing that it implies versions of both the Cohomological Crepant Resolution Conjecture and of the Crepant Resolution Conjectures of Ruan and Bryan--Graber. We also give a "quantized" version of the conjecture, which determines higher-genus Gromov--Witten invariants of X from those of Y.

math.AG↗
arXiv2008

Geometry and analysis of spin equations

We introduce W-spin structures on a Riemann surface and give a precise definition to the corresponding W-spin equations for any quasi-homogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the solutions of W-spin equations when W is a non-degenerate, quasi-homogeneous polynomial whose variables all have weight (or fractional degree) wt(x_i) < 1/2. In particular, the compactness theorem holds for the A,D, and E superpotentials.

math.DG↗
arXiv2008

Positive divisors in symplectic geometry

In this paper, we gave some explicit relations between absolute and relative Gromov-Witten invariants. We proved that a symplectic manifold is symplectic rationally connected if it contains a positive divisor symplectomorphic to $P^n$.

math.SG↗
arXiv2007

Birational cobordism invariance of uniruled symplectic manifolds

A symplectic manifold $(M,ω)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.

math.SG↗
arXiv2007

Uniruled symplectic divisors

This is a paper devoted to the symplectic birational geometry program where many basic notions are defined in terms of genus 0 GW invariants. We show that the existence of a positive uniruled symplectic divisor often implies that the ambient manifold has a nonzero uniruled genus 0 GW invariant, hence is uniruled as well. This confirms a part of the dichotomy on uniruled symplectic divisors. In addition, it gives a rather general construction of uniruled symplectic manifolds, generalizing some beautiful results of McDuff.

math.SG↗
arXiv2005

Gerbes and twisted orbifold quantum cohomology

In this article, we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.

math.AG↗
arXiv2002

Stringy orbifolds

Recently, there is an explosive growth of activities to understand stringy properties of orbifolds. In this article, we survey some of recent developments.

math.AG↗
arXiv2001

Twisted Orbifold K-Theory

We use equivariant methods to establish basic properties of orbifold K-theory. We introduce the notion of twisted orbifold K-theory in the presence of discrete torsion, and show how it can be explicitly computed for global quotients.

math.AT↗
arXiv2001

Orbifold Gromov-Witten Theory

In this article, we introduce the notion of good map and use it to establish Gromov-Witten theory for orbifolds.

math.AG↗
arXiv2001

A New Cohomology Theory for Orbifold

Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality and has a natural ring structure. Some examples of orbifold cohomology ring are computed.

math.AG↗
arXiv2001

Stringy Geometry and Topology of Orbifolds

This is a survey article on the recent development of "stringy geometry and topology of orbifolds", a new subject of mathematics motivated by orbifold string theory.

math.AG↗
arXiv2000

Discrete torsion and twisted orbifold cohomology

In this article, we study the twisting procedure of orbifold cohomology. We introduce local system and construct twisted orbifold cohomology. Then, we generalize Vafa-Witten's notion of discrete torsion to general orbifold and examine its relation to local system. Finally, some examples are computed.

math.AG↗
arXiv2000

Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I

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 formula for the change of GW-invariants of any genus transform under flop and a general type I extremal transition. Other extremal transition will be handled in a subsequent paper.

math.AG↗
arXiv1999

Surgery, quantum cohomology and birational geometry

This is an expository article about the recent results on the relation between the naturality problem of quantum cohomology and birational geometry. We give a short survey on Li-Ruan's results. Some conjecture-proposals are discussed.

math.AG↗
arXiv1998

Virtual neighborhoods and pseudo-holomorphic curves

We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for nondegenerate Hamiltonian symplectomorphisms.

alg-geom↗
arXiv1996

Quantum cohomology of projective bundles over P^n

In this paper, we attempt to determine the quantum cohomology of projective bundles over the projective space P^n. In contrast to the previous examples, the relevant moduli spaces in our case frequently do not have expected dimensions. It makes the calculation more difficult. We overcome this difficulty by using excessive intersection theory.

math.AG↗