Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = R\Gamma(C,df:T_C\to f^*T_X)$. Explicitly, this complex may be realized using the Dolbeault resolution of $T_C$ and $f^*T_X$. In this realization, there are three terms in this complex:

$L^0 = \Omega^0(C,T_C)$, $L^1 = \Omega^{0,1}(C,T_C)\oplus\Omega^0(C,f^*T_X)$ and $L^2 = \Omega^{0,1}(C,f^*T_X)$ with the differentials $L^0\to L^1$ and $L^1\to L^2$ given by a sum of pushforward by $df$ and the canonical $\bar\partial$ operator on a holomorphic vector bundle.

By some general philosophy (for example in the deformation theory book by Kontsevich-Soibelman), $L^\bullet$ should carry the structure of a differential graded Lie algebra (DGLA) such that the deformations of $f$ over a local Artin ring $(A,\mathfrak m)$ with residue field $\mathbb C$ can be seen as solutions $\omega\in L^1\otimes\mathfrak m$ to the Maurer-Cartan equation $d\omega + \frac12[\omega,\omega] = 0$ modulo the gauge action of $\exp(L^0\otimes\mathfrak m)$.

Can we realize the DGLA structure in this case explicitly? In particular, what is the explicit expression for the bracket $[\cdot,\cdot]:L^1\otimes L^1\to L^2$? I am able to see that the degree zero bracket $L^0\otimes L^0\to L^0$ should be simply the usual commutator Lie bracket of vector fields.

  • $\begingroup$ A natural guess is that $\Omega^{0}(f^* T_X) \otimes \Omega^{0} (f^* T_X) \to \Omega^{0,1}(f^*T_X)$ is given by pulling back a representative for the Kapranov bracket, while $\Omega^{0,1}(T_C) \otimes \Omega^0(T_X) \to \Omega^{0,1}(T_X)$ is induced by the Lie bracket of vector fields and the rest of the degree 1 map is zero. $\endgroup$ – Phil Tosteson Sep 7 '18 at 16:22
  • $\begingroup$ You can see that part of this is right by considering the fiber sequence of formal moduli problems which forgets the map to $f$. This corresponds to a short exact sequence of $dg$ lie algebras, where the quotient governs deformations of $C$ and the sub governs deformations of maps to $X$ where the curve is fixed. $\endgroup$ – Phil Tosteson Sep 7 '18 at 16:24
  • $\begingroup$ I'm not familiar with the Kapranov bracket. Could you point me to a reference where it is defined? Also, it would be nice if you can expand this comment into a more detailed answer. $\endgroup$ – Mohan Swaminathan Sep 7 '18 at 19:49
  • $\begingroup$ Kapranov shows in "Rozansky–Witten invariants via Atiyah classes" that the Atiyah class makes $T_X[-1]$ into a lie algebra in $D(X)$. The relationship of this Lie algebra to deformation theory is discussed here: mathoverflow.net/questions/143269. Unfortunately, I don't have enough expertise to give a detailed answer. $\endgroup$ – Phil Tosteson Sep 7 '18 at 21:21
  • $\begingroup$ arxiv.org/abs/math/0507287 and arxiv.org/abs/math/0601312 might be exactly what you're interested in, as well as the subsequent works of the authors; these two papers give an explicit $L_{\infty}$-structure on the cone of Kodaira-Spencer dg-Lie algebras for two manifolds and describe the corresponding Maurer-Cartan functor $\endgroup$ – Grisha Papayanov Nov 18 '18 at 12:08

Firstly, I assume you mean deformations of $C$ over $X$ ("deformations of $f$" is ambiguous, as it could mean fixing neither or both of $C$ and $X$).

The DGLA philosophy is then that there should exist some DGLA quasi-isomorphic to the explicit realisation of the complex $L$ you wrote down. It doesn't guarantee a DGLA structure on $L$ itself, though it will transfer a non-canonical $L_{\infty}$ structure.

In this case, you can reinterpret the problem as trying to deform $\mathcal{O}_C$ as a sheaf of $f^{-1}\mathcal{O}_X$-algebra. The DGLA you want should then be an explicit model for $\mathbf{R}\Gamma(C,\mathbf{R}\mathrm{Der}_{f^{-1}\mathcal{O}_X}(\mathcal{O}_C))$. At this point, you encounter the problem that free algebra resolutions and flabby sheaf resolutions don't interact well.

One explicit model is given by first forming the Harrison complex (or a natural analogue for holomorphic functions) $\mathrm{Harr}_{f^{-1}\mathcal{O}_X}(\mathcal{O}_C)$ (a sheaf of DGLAs), then take a nice open cover $\mathfrak{U}$ of $C$ and form a Cech complex $\check{C}(\mathfrak{U},\mathrm{Harr}_{f^{-1}\mathcal{O}_X}(\mathcal{O}_C))$, giving a cosimplicial DGLA. Then apply Thom-Whitney cochains to give a DGLA.

You'll find various related constructions in several works by Iacono, Manetti and Fiorenza, as well as Ciocan-Fontanine's derived Hilbert schemes and some of my early papers.

  • $\begingroup$ Yes, I mean deformations keeping $X$ fixed but allowing $C,f$ to vary. $\endgroup$ – Mohan Swaminathan Sep 7 '18 at 19:53
  • $\begingroup$ What would be the analogue of this in the non-integrable case, i.e., a pseudoholomorphic map to a symplectic manifold with a compatible almost complex structure? It's not clear to me how to generalize the explicit model described in the answer to this case. $\endgroup$ – Mohan Swaminathan Sep 10 '18 at 12:24
  • $\begingroup$ @Mohan In that case, you could probably look at something like derived derivations of $\Omega^{0,\bullet}_C$ as a sheaf of cdgas over $f^{-1}\Omega^{0,\bullet}_X$. $\endgroup$ – Jon Pridham Sep 10 '18 at 17:21

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.