Related papers: Mukai flops and Pl\"ucker type formulas for hyper-…

Jieao Song recently conjectured a formula for the class of a Lagrangian plane on a hyperk\"ahler variety of $K3^{[n]}$-type in terms of the class of a line on it. We give a proof of this conjecture if the line class is primitive.

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…

Let $X$ be a hyperk\"ahler variety, and let $Z\subset X$ be a Lagrangian subvariety. Conjecturally, $Z$ should have trivial intersection with certain parts of the Chow ring of $X$. We prove this conjecture for certain Hilbert schemes $X$…

We extend the formula for the Chern classes of blow-ups of algebraic varieties due to Porteous and Lascu-Scott, and of symplectic and complex manifolds due to Geiges and Pasquotto, to the blow-ups of almost complex manifolds. Our approach…

We study the intersection theory of punctured pseudoholomorphic curves in $4$-dimensional symplectic cobordisms. We first study the local intersection properties of such curves at the punctures. We then use this to develop topological…

The goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and…

We present a survey of some results and questions related to the notion of scalar curvature in the setting of symplectic supermanifolds.

In this paper we study the Lagrangian Floer theory over $\Z$ or $\Z_2$. Under an appropriate assumption on ambient symplectic manifold, we show that the whole story of Lagrangian Floer theory in \cite{fooo-book} can be developed over $\Z_2$…

We study tropicalization of symplectic flag varieties with respect to the Pl\"ucker embedding. We identify a particular maximal prime cone in this tropicalization by explicitly giving its facets. For every interior point of this maximal…

This paper is a sequel to math.AG/0203287. A generalization of the Mukai flop has been studied by E. Markman. Here we call it a stratified Mukai flop. In this paper, we observe that, for a stratified Mukai flop: $X \to \bar{X} \leftarrow…

Let k > 2. We show that if a closed orientable 2k-manifold K, with Euler characteristic not equal to -2, admits an exact Lagrangian immersion into complex Euclidean 2k-space with one transverse double point and no other self-intersections,…

Motivated by the physical concept of special geometry two mathematical constructions are studied, which relate real hypersurfaces to tube domains and complex Lagrangean cones respectively. Me\-thods are developed for the classification of…

In this paper, we establish a truncated non-integrated defect relation for meromorphic mappings from a complete K\"ahler manifold into a projective variety intersecting a family of hypersurfaces located in subgeneral position, where the…

We review the quiver descriptions of symplectic and hyperk\"ahler implosion in the case of SU(n) actions. We give quiver descriptions of symplectic implosion for other classical groups, and discuss some of the issues involved in obtaining a…

We give necessary and sufficient conditions for a Lagrangian submanifold of a K\"ahler manifold to be biharmonic. Furthermore, we classify biharmonic PNMC Lagrangian submanifolds in the complex space forms.

Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality \cite[Theorem 0.1]{Langer} we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.

Germs of tubular neighborhood embeddings for submanifolds N of manifolds M are in one-one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of `normal forms results' for geometric…

We determine multiplication and convolution topological algebras for classes of $\omega$-ultradifferentiable functions of Beurling type. Hypocontinuity and discontinuity of the multiplication and convolution mappings are also investigated.

A subvariety of a complex projective space has a well-known dual variety, which is the set of its tangent hyperplanes. The purpose of this paper is to generalise this notion for a subvariety of a quite general partial flag variety. A…

We study symplectic structures on K\"ahler surfaces with p_g = 0. We give an example of a projective surface which admits a symplectic structure which is not compatible with any K\"ahler metric.