Obstruction bundle for spaces with Kuranishi structure

Question

In the symplectic topology view on Gromov-Witten-Invariants some authors use what they call a Kuranishi structure on the moduli of stable maps. These were introduced by Fukaya and Ono and are also used in their big book on Fukaya categories.They are also used a lot in recent papers by Joyce.

The key feature of the Kuranishi structure is that there locally exists a homeomorphism or a diffeomorphism, depending on whether you follow Fukaya or Joyce, to a zero set of a section of the "obstruction bundle".

On the algebraic geometry side there also is something involving the term obstruction, namely the perfect obstruction theory on the moduli space of maps. This is a morphism [E_-1 -> E₀] -> L_X, where L is the cotangent complex.

Here's my question: To what does the obstruction bundle on the symplectic topology side correspond on the algebraic side?

There are three candidates I can think of:

• E_-1

• the kernel of E_-1 -> E₀

• the thing the kernel of E_-1 -> E₀ maps surjectively to, mostly called the T²

The definition of the obstruction bundle on the symplectic side is a finite dimensional supspace of the cokernel of the linearized operator of the pseudoholomorphic curve equation, p.978 of Fukaya-Ono:"Arnold Conjecture and Gromov Witten Invariant." As a die hard algebraic geometer, that's just to hard to digest...

Answer 1

Here's a view of the symplectic side of the bridge.

The Kuranishi model (see Donaldson-Kronheimer, The geometry of four-manifolds, ch. 4) goes like this. You're interested in a (moduli) space $M$ cut out as $\psi^{-1}(0)$, for some smooth but nonlinear map of Banach spaces, $\psi \colon (E,0) \to (F,0)$ such that $\delta:=D_0\psi$ is a Fredholm operator (finite dim kernel and cokernel). That means that $\delta$ is "almost" an isomorphism, and Kuranishi's principle is that one can construct a non-linear map $\kappa \colon \ker(\delta) \to \mathrm{coker}(\delta)$, such that $\kappa(0)=0$ and $D_0 \kappa =0$, and (locally near $0$) a homeomorphism $M \to \kappa^{-1}(0)$. This gives a finite-dimensional model of $M$. "Kuranishi structures" are a formalism in which one can say that $M$ is everywhere-locally given as the zeros of maps like $\kappa$.

In the case of genus 0 GW theory, $M$ is the moduli space of (say) parametrized pseudo-holomorphic maps from $S^2$ to an almost complex manifold $X$; $\psi$ is a non-linear Cauchy-Riemann operator, and, for a pseudo-holomorphic map $u\in M$, $D_u \psi$ is a linearized C-R operator - the $(0,1)$-part of a covariant derivative acting on sections of $u^\ast TX$. Its kernel can be identified with the holomorphic sections $H^0(S^2,u^\ast TX)$ of the holomorphic structure on the vector bundle $u^\ast TX$ defined by the C-R operator. Its cokernel is isomorphic to $H^1(S^2,u^\ast TX)$. If you're lucky, you have a Zariski-smooth moduli space $M$ whose Zariski tangent space at $u$ is $\ker D_u\psi$. In this case, one could take $\kappa=0$, and then the spaces $\mathrm{coker} (D_u\psi)$ form a vector bundle $Obs \to M$, which is what symplectic geometers usually call the obstruction bundle. One can now try to divide everything by $Aut(S^2)$, and quite possibly get into orbi-mathematics.

In the integrable case, still with non-singular but excess-dimensional $M$, I could write $Obs$ as $R^1\pi_* \Phi^*T_X$, where $T_X$ is the holomorphic tangent sheaf, $\Phi$ the evaluation map $\mathbb{P}^1\times M \to X$, and $\pi$ the projection to $M$. At this point, if what I've said is accurate, you and other algebraic geometers out there are better placed than me to answer your question. Is it your $(E_{-1})^\vee$?

Of course, you didn't really want to assume $M$ smooth. A place where these deformation theories are compared in generality is Siebert's 1998 paper Algebraic and symplectic Gromov-Witten invariants coincide.

Answer 2

By the way, the notion of Kuranishi structure is an attempt to formalize ideas that had been used in the gauge theory literature somewhat eariler. See for example Friedman and Morgan's book Smooth Four Manifolds and Complex Surfaces. In that book the obstruction bundle idea is used in the definition of the Donaldson invariants. The obstruction bundle idea for studying nonlinear PDE problems of course goes back at least to Kuranshi. Taubes used this idea to study solutions to the Yang-Mills equations near bubbling in the 1982 paper "Self-dual Yang-Mills connections on non-self-dual 4-manifolds" in way that lead to the modern use.