Castelnuovo's bound and rigidity in almost complex geometry

  title={Castelnuovo's bound and rigidity in almost complex geometry},
  author={Aleksander Doan and Thomas Walpuski},
  journal={Advances in Mathematics},

Bifurcations of embedded curves and an extension of Taubes' Gromov invariant to Calabi-Yau 3-folds

We define an integer-valued virtual count of embedded pseudoholomorphic curves of two times a primitive homology class and arbitrary genus in symplectic Calabi–Yau 3-folds, which can be viewed as an

The Gopakumar-Vafa finiteness conjecture

TheGopakumar–Vafa conjecture predicts that the BPS invariants of a symplectic 6–manifold, defined in terms of the Gromov–Witten invariants, are integers and all but finitely many vanish in every

Invariant probability measures from pseudoholomorphic curves I

We introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odddimensional smooth manifolds using

An elementary alternative to ECH capacities.

  • M. Hutchings
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 2022
The embedded contact homology (ECH) capacities are a sequence of numerical invariants of symplectic four-manifolds that give (sometimes sharp) obstructions to symplectic embeddings. These capacities

Conjugate linear perturbations of Dirac operators and Majorana fermions

We study a canonical class of perturbations of Dirac operators that are defined in any dimension and on any Hermitian Clifford module bundle. These operators generalize the 2-dimensional Jackiw–Rossi

Counting embedded curves in symplectic 6-manifolds

Based on computations of Pandharipande, Zinger proved that the Gopakumar-Vafa BPS invariants $\mathrm{BPS}_{A,g}(X,\omega)$ for primitive Calabi-Yau classes and arbitrary Fano classes $A$ on a


We study a canonical class of perturbations of Dirac operators that are defined in any dimension and on any Hermitian Clifford module bundle. These operators generalize the 2-dimensional Jackiw–Rossi



Gromov’s Compactness Theorem for Pseudo-holomorphic Curves

Mikhail Gromov introduced pseudo-holomorphic curves into symplectic geometry in 1985. This book aims to present in detail the original proof for Gromov's compactness theorum for pseudo-holomorphic

Equivariant Brill-Noether theory for elliptic operators and super-rigidity of $J$-holomorphic maps

The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of

Transversality and super-rigidity for multiply covered holomorphic curves

We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of

Existence of symplectic surfaces

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the

Automatic transversality and orbifolds of punctured holomorphic curves in dimension four

We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of

Lectures on Symplectic Field Theory

This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for PhD students given in 2015-16. It covers the essentials of the analytical theory of punctured

Gromov’s compactness theorem for pseudo holomorphic curves

We give a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary.

Regularity Theory for 2-Dimensional Almost Minimal Currents II: Branched Center Manifold

We construct a branched center manifold in a neighborhood of a singular point of a 2-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of