ag.algebraic geometry - Obstruction bundle for spaces with Kuranishi structure

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 $delta:=D_0psi$ 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 $uin M$, $D_u psi$ 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 $ker D_upsi$. In this case, one could take $kappa=0$, and then the spaces $mathrm{coker} (D_upsi)$ 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 $R^1pi_* Phi^*T_X$, where $T_X$ is the holomorphic tangent sheaf, $Phi$ the evaluation map $mathbb{P}^1times M to X$, 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.

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