Related papers: Higher jet evaluation transversality of $J$-holomo…

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of…

For a given coorientable contact manifold $(M,\Xi)$ with contact distribution $\Xi$, we consider its contact forms $\lambda$ with $\ker \lambda = \Xi$, and the associated contact triads $(M,\lambda, J)$. For a generic choice of contact form…

We study the topology of the configuration spaces $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j [C(n,w)]$ behaves in qualitatively different…

We suggest a general method of computation of the homology of certain smooth covers $\hat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces $\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we consider moduli spaces…

We prove a generalization of Thom's transversality theorem. It gives conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M)\ra J^r(D,N)$ is generically (for $f:M\ra N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply…

We study the relation between $J$-anti-invariant $2$-forms and pseudoholomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in…

Let $(X, \omega, c_X)$ be a real symplectic 4-manifold with real part $R X$. Let $L \subset R X$ be a smooth curve such that $[L] = 0 \in H_1 (R X ; Z / 2Z)$. We construct invariants under deformation of the quadruple $(X, \omega, c_X, L)$…

Hamiltonian quantization of an integral compact symplectic manifold M depends on a choice of compatible almost complex structure J. For open sets U in the set of compatible almost complex structures and small enough values of Planck's…

We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we…

The fundamental properties of $J$-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a $J$-holomorphic map in terms of its energy. The cylinder inequality stipulates and…

We study the restrictions of rank 2 semistable vector bundles E on P^2 to conics. A Grauert-Mulich type theorem on the generic splitting is proven. The jumping conics are shown to have the scheme structure of a hypersurface J_{2} in P^5 of…

This paper addresses two questions related to mapping problems. In the first part of the paper, we discuss some recent results regarding the $2$-jet determination for biholomorphisms between smooth (weakly) pseudoconvex hypersurfaces; in…

We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $\mathfrak{p}$ and a volume form $\sigma$ on an oriented surface $M$ equip the total space…

A symplectic or orthogonal bundle $V$ of rank $2n$ over a curve has an invariant $t(V)$ which measures the maximal degree of its isotropic subbundles of rank $n$. This invariant $t$ defines stratifications on moduli spaces of symplectic and…

We establish a quantitative relationship between mixed de Rham classes and the geometric complexity of metric connections with totally skew torsion on product manifolds where both factors are compact oriented surfaces. For any…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

The construction of manifold structures and fundamental classes on the (compactified) moduli spaces appearing in Gromov-Witten theory is a long-standing problem. Up until recently, most successful approaches involved the imposition of…

Given a smooth and projective curve C and a smooth and projective toric variety X, we first describe a compactification of the space of morphisms from C to X representing a fixed homology class, and after we study the intersection theory on…

Let $\mathcal{M}_{C}(2, 0)$ be the moduli space of semistable rank two and degree zero Higgs bundles on a smooth complex hyperelliptic curve $C$ of genus three. We prove that the quotient of $\mathcal{M}_{C}(2, 0)$ by a twisted version of…

Using the theory of exterior differential systems, we study the existence of germ of pseudo-holomorphic disk in a real analytic hypersurface locally defined in a complex manifold equipped with J a real analytic almost complex structure. The…