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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.

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  • $\begingroup$ 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, 2018 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, 2018 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, 2018 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, 2018 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, 2018 at 12:08

1 Answer 1

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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.

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  • $\begingroup$ Yes, I mean deformations keeping $X$ fixed but allowing $C,f$ to vary. $\endgroup$
    – Mohan Swaminathan
    Sep 7, 2018 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, 2018 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, 2018 at 17:21

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