Hebrew University topology and geometry seminar



May 27, 2015
1100-1235
Ross building, Seminar Room 63



Amitai Zernik

Hebrew University of Jerusalem

$A_\infty$ Localization Counts Holomorphic Discs in $(\mathbb{CP}^2,\mathbb{RP}^2)$




Abstract: If $M$ is a closed $S^1$ manifold, the Atiyah-Bott localization formula reduces the computation of certain integrals on $M$ to integration on the fixed points of $M$. This technique has found many applications; for example, Kontsevich has used it to compute the number of rational complex algebraic curves in $\mathbb{CP}^n$ passing through a given number of points.
Welschinger has defined a way to count real rational curves in $\mathbb{CP}^2$ passing through a given number of points. In this talk we will present a fixed point formula for this count, recast as a certain integral on the moduli of holomorphic discs in $(\mathbb{CP}^2,\mathbb{RP}^2)$ (this and all other necessary notions will be explained in the talk).
The central question is how to deal with contributions from the boundary of the moduli space. We use a unital twisted filtered $A_\infty$ algebra constructed from the moduli spaces. This algebra turns out to be homotopy equivalent to an algebra involving only the fixed point loci, and from this fact an explicit, diagrammatic recipe for computing the Welschinger numbers (and equivariant invariants generalizing them) emerges.
This is joint work with Jake Solomon.