Related papers: The Algebraic Rational Blow-Down

Rational curves on Hilbert schemes of points on $K3$ surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up…

The natural sum operation for symplectic manifolds is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n-2)-submanifold V. Given a similar pair (Y,W) with a symplectic…

The standard (Berezin-Toeplitz) geometric quantization of a compact Kaehler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal covering space, provided that the lift of the…

Let (M, g, omega) be a compact, almost-Kaehler Einstein 4-manifold of negative star-scalar curvature. Then (M, omega) is a MINIMAL symplectic 4-manifold of general type. In particular, M cannot be differentiably decomposed as a connected…

In this paper, we are mainly concerned with the blow-up algebras of the secant varieties of balanced rational normal scrolls. In the first part, we give implicit defining equations of their associated Rees algebras and fiber cones.…

We prove that an integrable system over a symplectic manifold, whose symplectic form is covariantly constant w.r.t. the Gauss-Manin connection, carries a natural hyper-symplectic structure. Moreover, a special Kaehler structure is induced…

We construct explicit maximal symplectic packings of minimal rational and ruled symplectic 4-manifolds by few balls in a very simple way.

We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus $g\geq…

The aim of this paper is to present a construction of smooth rational surfaces in projective fourspace with degree 12 and sectional genus 13. The construction is based on exterior algebra methods, finite field searches and standard…

Using McMullen's rational surface automorphisms, we construct projective rational manifolds of higher dimension admitting automorphisms of positive entropy with arbitrarily high number of Siegel disks and those with exactly one Siegel disk.

We present a new proof of a result due to Taubes: if X is a closed symplectic four-manifold with b_+(X) > 1+b_1(X) and with some positive multiple of the symplectic form a rational class, then the Poincare dual of the canonical class of X…

Motivated by the construction of H. Endo and Y. Gurtas, changing a positive relator in Dehn twist generators of the mapping class group by using lantern substitutions, we show that 4-manifold $K3#2\CPb$ equipped with the genus two Lefschetz…

In a previous work, we proved that each minimal symplectic filling of any oriented lens space, viewed as the singularity link of some cyclic quotient singularity and equipped with its canonical contact structure, can be obtained from the…

A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly…

We present natural and general ways of building Lie groupoids, by using the classical procedures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The…

We establish various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify…

We compute the double complex of smooth complex-valued differential forms on projective bundles over and blow-ups of compact complex manifolds up to a suitable notion of quasi-isomorphism. This simultaneously yields formulas for 'all'…

A method of constructing a class of bihamiltonian structures is presented. Elements of this class are generalizations of the so-called bihamiltonian structures of general position on odd-dimensional manifolds. The method consists in a…

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $\iota:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_{\iota}$. In this paper, we study…

We give new rational blowdown constructions of exotic CP^2#n(-CP^2) (5\leq n\leq 9) without using elliptic fibrations. We also show that our 4-manifolds admit handle decompositions without 1- and 3-handles, for 7\leq n\leq 9. A strategy for…