Related papers: On Donaldson's 4-6 question

We introduce a direct generalization of the Weinstein conjecture to closed, Lichnerowicz exact, locally conformally symplectic manifolds, (for short $\lcs$ manifolds). This conjectures existence of certain 2-curves in the manifold, which we…

Let Sigma be a smooth complex curve, and let S be the product ruled surface Sigma \times CP^1. We prove a correspondence conjectured by Donaldson between finite energy U(2)-instantons over the cylinder Sigma \times S^1 \times R, and rank 2…

Let $G$ be a finite subgroup of $\mathrm{SU}(4)$ whose elements have age not larger than one. In the first part of this paper, we define $K$-theoretic stable pair invariants on the crepant resolution of the affine quotient $\mathbb{C}^4/G$,…

Famous work of Bridgeland and Smith shows that certain moduli spaces of quadratic differentials are isomorphic to spaces of stability conditions on particular 3-Calabi-Yau triangulated categories. This result has subsequently been…

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b_+ > 1 satisfy the duality Gr(A) = +/- Gr(K-A), where K is Poincare dual to the canonical class. Extending joint work with Simon Donaldson in math.SG/0012067,…

We prove the transformation formula of Donaldson-Thomas (DT) invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and…

Kreck's modified surgery gives an approach to classifying smooth $2n$-manifolds up to stable diffeomorphism, i.e. up to connected sum with copies of $S^n \times S^n$. In dimension 4, we use a combination of modified and classical surgery to…

In this article, we show that, at least for non-simply connected case, there exist an infinite family of nondiffeomorphic symplectic 4-manifolds with the same Seiberg-Witten invariants. The main techniques are knot surgery and a covering…

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide…

We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct…

We introduce marked relative Pandharipande-Thomas (PT) invariants for a pair $(X,D)$ of a smooth projective threefold and a smooth divisor. These invariants are defined by integration over the moduli space of $r$-marked stable pairs on…

Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds,…

For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the…

We extend a result of M. Katz on conformal systoles to all four-manifolds with b^+=1 which have odd intersection form. The same result holds for all four-manifolds with b^+=1 with even intersection form and which are symplectic or satisfy…

We study a conjectural relationship among Donaldson-Thomas type invariants on Calabi-Yau 3-folds counting torsion sheaves supported on ample divisors, ideal sheaves of curves and Pandharipande-Thomas's stable pairs. The conjecture is a…

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a…

We verify the conjecture formulated in math.AG/0111298 for suspension singularities of type $g(x,y,z)= f(x,y)+z^n$, where $f$ is an irreducible plane curve singularity. More precisely, we prove that the modified Seiberg-Witten invariant of…

We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V_5 and V_22 up to degree 3. We use torus localization for GW invariants computation, and use…

We use degeneration formula to study the change of stable pair invariants of 3-folds under blow-ups and obtain some closed blow-up formulae. Related results on Donaldson-Thomas invariants are also discussed. Our results give positive…

The Jones-Witten invariants can be generalized for non-singular smooth vector fields with invariant probability measure on 3-manifolds, giving rise to new invariants of dynamical systems [22]. After a short survey of cohomological field…