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International audience; In this article, we study topological properties of Voisin's punctual Hilbert schemes of an almost-complex fourfold X. In this setting, we compute their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on X, which are shown to be canonical in K-theory.

International audience; In this article, we study the rational cohomology rings of Voisin's punctual Hilbert schemes $X^{[n]}$ associated to a symplectic compact fourfold $X$. We prove that these rings can be universally constructed from $H^*(X,\mathbb{Q})$ and $c_1(X)$, and that Ruan's crepant resolution conjecture holds if $c_1(X)$ is a torsion class. Next, we prove that for any almost-complex compact fourfold $X$, the complex cobordism class of $X^{[n]}$ depends only on the cobordism class of $X$.

We show that Lagrangian submanifolds in six-dimensional nearly K��hler (non K��hler) manifolds and in twistor spaces $Z\sp{4n+2}$ over quaternionic K��hler manifolds $Q\sp{4n}$ are minimal. Moreover, we will prove that any Lagrangian submanifold $L$ in a nearly K��hler manifold $M$ splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly K��hler part of $M$ and the second factor is Lagrangian in the K��hler part of $M$. Using this splitting theorem we then describe Lagrangian submanifolds in nearly K��hler manifolds of dimensions six, eight and ten., 19 pages

We study the problem of existence of stationary disks for domains in almost complex manifolds. As a consequence of our results, we prove that any almost complex domains which is a small deformations of a strictly linearly convex domain $D \subset C^n$ with standard complex structure admits a singular foliation by stationary disks passing through any given internal point. Similar results are given for foliation by stationary disks through a given boundary point, Comment: 22 pages

The regularity up to the boundary of a proper holomorphic map between strictly pseudoconvex bounded domains D and D ′ of C has been widely studied when n ≥ 2 and is now as well understood as the one-dimensional case. A continuous map F : D → D ′ is said to be proper if F −1(K) is compact for every compact set K in D ′. If D and D ′ have C r -boundaries (r ≥ 2) then such a map F has a C r−1/2-extension to the boundary, and this is the maximal regularity [4; 20] that can be expected. Various authors have contributed to this result. We just mention Fefferman, who proved in 1974 that if D and D ′ have smooth boundaries and if F is a biholomorphism, then F extends smoothly to the boundary [9]. We refer to the survey of Forstneric [10] for a thorough history. Our aim here is to study the boundary behavior of proper pseudoholomorphic maps between strictly pseudoconvex regions in almost complex manifolds. In order to establish an analogue of the known result in the complex situation, we will need to adapt objects and tools specific to the integrable case. Note, for example, that there is no longer a notion of an analytic set and that the Jacobian of a pseudoholomorphic map is not pseudoholomorphic. Our method uses pseudoholomorphic discs. Originally introduced by E. Bishop, this method has provided geometric proofs of various versions of Fefferman’s theorem [17; 25] even in the almost complex case [5; 12]. We consider the following situation. Let D be a bounded domain in some smooth (real) manifold, and let J be an almost complex structure of class C1 on D that is smooth in D. Throughout this paper, we will say that (D, J ) is a strictly pseudoconvex region if D is defined by {ρ < 0}, where ρ is a C 2-regular defining function that is strictly J-plurisubharmonic on D. We say (D, J ) is a strictly pseudoconvex region of class C r when ρ and J are at least of class C . In the complex situation, the regularity is thus the regularity of the boundary. The first step of our proof is to derive the Holder 1 2 -continuous extension, which comes from a sized estimate of the set of regular values and from estimates of the Kobayashi metric. To obtain more regularity, the main obstacle compared with the biholomorphic case is obviously the existence of critical points. Thus we begin with studying the locus of all these points. For the complex case, Pinchuk [19; 20]

We study the approximation of J-holomorphic maps continuous to the boundary from ma domain in the complex plane into an almost complex manifold by maps J-holomorphic to the boundary, giving partial results in the non-integrable case. For the integrable case, we study arcs in complex manifolds and establish the existence of neighborhoods biholomorphic to open sets in Euclidean space for several classes of arcs. As an application, we obtain $\mathcal{C}^k$ approximation of holomorphic maps continuous to the boundary into complex manifolds by maps holomorphic to the boundary, provided the boundary is nice enough., Comment: Based on the author's doctoral dissertation/

In this paper, firstly, we obtain the higher order vertical and complete lifts of type (1,1) and type (0,2) by means of lifted geometric structures on extended complex manifolds. Then using higher order lifting theory, we generalize the classification of almost Hermitian manifolds M to the extended complex manifolds kM. © 2003, EUT Edizioni Universita di Trieste.

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. As an application, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach-Lie groups., 13 pages

In this paper, we will obtain the (r, s) order completevertical lifts and horizontal lifts of order higher of the complex structures on complex manifold M to the canonical extensions. © 2004 EUT Edizioni Universita di Trieste.

We study about lifts on extended complex manifold. More clearly, we will obtain higher order vertical and complete lifts of differentiable elements on complex manifold M to the extended complex manifold kM. © 2004 EUT Edizioni Universita di Trieste.

Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z) such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by d = (c_1(s)^2 -3sigma(X) -2chi(X))/4, where s is a certain spin^C structure naturally associated to w., Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper20.abs.html

We present existence results for certain singular 2-dimensional foliations on 4-manifolds. The singularities can be chosen to be simple, e.g. the same as those that appear in Lefschetz pencils. There seems to be a wealth of such creatures on most 4-manifolds. In certain cases, one can prescribe surfaces to be transverse or be leaves of these foliations. The purpose of this paper is to offer objects, hoping for a future theory to be developed on them. For example, foliations that are taut might offer genus bounds for embedded surfaces (Kronheimer's conjecture)., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-43.abs.html

We define a nondegenerate Monge-Amp\`ere structure on a 6-dimensional manifold as a pair $(\Omega,\omega)$, such that $\Omega$ is a symplectic form and $\omega$ is a 3-differential form which satisfies $\omega\wedge\Omega=0$ and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau structure and we study its integrability from the point of view of Monge-Amp\`ere operators theory. The result we prove appears as an analogue of Lychagin and Roubtsov theorem on integrability of the almost complex or almost product structure associated with an elliptic or hyperbolic Monge-Amp\`ere equation in the dimension 4. We study from this point of view the example of the Stenzel metric on the cotangent bundle of the sphere $S^3$., Comment: 14 pages, accepted for publication in Letters in Mat. Physics

By using moving frames and directred digraphs, we study invariant (1,2)-symplectic structures on complex flag manifolds. Let $F$ be a flag manifold with height $k-1$. We show that there is a $k$-dimensional family of invariant (1,2)-symplectic metrics of any parabolic structure on $F$. We also prove any invariant almost complex structure $J$ on $F$ with height 4 admits an invariant (1,2)-symplectic metric if and only if $J$ is parabolic or integrable.

Let (R, J) be the space R equipped with an almost complex structure J . We recall that J is a differentiable map R → GL(4,R) such that J(p) = −Id, for every p ∈ R. Let M be a differentiable hypersurface in R. For every p ∈ M the tangent hyperplane TpM contains a (unique) J-invariant plane T J p M . The distribution of planes p → T J p M is called the Levi distribution on M , and M is said to be J-Levi flat whenever p → T J p M is integrable. In view of Frobenius Theorem M is then foliated by regular surfaces on which J induces an integrable almost complex structure. Consequently M is foliated by complex curves, whose complex structure is in general different from that induced by J0, the standard one. Let J = J0. The problem of finding a Levi flat hypersurface with a prescribed boundary Γ has been extensively studied by methods of the geometric theory of several complex variables (cf. [BG], [BK], [A], [S], [K], [CS], [ST]). A different approach is found in [SlT] where the boundary problem for Levi flat graphs is reduced to a Dirichlet problem for a nonlinear, second order, elliptic degenerate operator L, the so called Levi operator (see (??)