Torus actions on small blow ups of CP^2

Welschinger invariants of Blow-ups of symplectic 4-manifolds

Using the degeneration technique, one studies the behavior of Welschinger invariants under the blow-up, and obtains some blow-up formulae of Welschinger invariants. One also analyses the variation of Welschinger invariants when replacing a pair of real points in the real configuration by a pair of conjugated points, and reproves Welschinger's wall crossing formula.

Circle and torus actions on equal symplectic blow-ups of CP^2

A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal sizes epsilon (where the size of CP^1 \subset CP^2 is one) admits an effective two dimensional torus action if k <= 3 and admits an effective circle action if (k-1)epsilon < 1. We show that these bounds are sharp if 1/epsilon is an integer.

Lagrangian Floer homology on symplectic blow ups

We show how to compute the Lagrangian Floer homology in the one-point blow up of the proper transform of Lagrangians submanifolds, solely in terms of information of the base manifold. As an example we present an alternative computation of the Lagrangian quantum homology in the one-point blow up of (CP^2,ω) of the proper transform of the Clifford torus.

Lagrangian blow-ups, blow-downs, and applications to real packing

Given a symplectic manifold (M, ω) and a Lagrangian submanifold L, we construct versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore, we show that if M admits an anti-symplectic involution ϕ and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an anti-symplectic involution on the blow-up as well. We derive a homological condition which determines when the topology of a real Lagrangian surface L = Fix(ϕ) changes after a blow down, and we use these constructions to study the real packing numbers and packing stability …

辛胀流形上的近复环面纤维化

Almost toric fibrations on symplectic blow ups

该研究针对带近复环面纤维化的辛4-流形，构造了近复环面胀形与辛胀形间的辛同胚，并证明在球嵌入与边界除子相容时，该同胚可保持诱导的辛对数典范除子。

The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by…