Explicit derived structures for Drinfeld's two instructive moduli problems: the conjectural bracket on $\mathrm{Ext}^{\bullet-1}(I_Y,O_Y)$ and the quadratic extension on the derived moduli of abelian varieties

In his 1988 letter to Schechtman (published 2014), Drinfeld initiated derived deformation theory: he asked for a general theory in which a deformation problem over a field of characteristic 0 is controlled by a differential graded Lie algebra, and posed the duality questions that became Koszul duality for operads. The programmatic part of the letter was subsequently resolved by derived algebraic geometry and by the theorem that differential graded Lie algebras are equivalent to formal moduli problems in characteristic 0. What no later work settles is the explicit computational content of the letter's two 'instructive examples'. First, for a subscheme $Y$ of a smooth variety $M$ with ideal sheaf $I_Y$, the tangent complex of the derived quot scheme has cohomology $\mathrm{Ext}^i(I_Y,O_Y)$, but the explicit differential graded Lie (or $L_\infty$) structure on $\bigoplus_i\, \mathrm{Ext}^{i-1}(I_Y,O_Y)$ has never been determined; the letter conjectures the bracket $[a,b]=a\circ x\circ b-b\circ x\circ a$ in terms of the canonical class $x\in\mathrm{Ext}^1(O_Y,I_Y)$ of the ideal sequence, and asks for a proof or refutation. Second, on the moduli stack of (rigidified) abelian varieties the controlling differential graded Lie algebra is abelian, so the derived enhancement is encoded in a sheaf $\mathcal{O}=\mathcal{O}_0\oplus\mathcal{O}_1\oplus\cdots$ of supercommutative graded algebras, locally free over $\mathcal{O}_0$, and the letter asks for an explicit description of the extension $0\to\Lambda^2\mathcal{O}_1\to\mathcal{O}_2\to\mathcal{O}_2/\Lambda^2\mathcal{O}_1\to0$, the first quadratic invariant of that structure. The problem is to make these two derived enhancements explicit beyond their tangent complexes.

Background

Classical deformation theory studies families of geometric objects over Artinian bases: the first-order deformations of an object form a vector space, and obstructions to extending them live in a second vector space. The organizing principle formulated in the letter (the Deligne–Drinfeld principle) is that in characteristic 0 a deformation problem is controlled by a differential graded (DG) Lie algebra $\mathfrak{g}$ — a chain complex of vector spaces carrying a degree-zero Lie bracket compatible with the differential. Deformations over a commutative DG Artinian base $B$ are then described by the solutions of the Maurer–Cartan equation $\,d\omega+\tfrac{1}{2}[\omega,\omega]=0$ for $\omega\in\mathfrak{g}\otimes B$, taken up to gauge equivalence (the equivalence generated by the adjoint action of the degree-zero elements). The letter proves the adjoint pair realizing the Koszul duality between DG Lie algebras and differential graded cocommutative coalgebras: a DG Lie algebra $\mathfrak{g}$ is sent to its Chevalley complex $C^\ast(\mathfrak{g})$, the commutative differential graded algebra obtained as the symmetric algebra on the shifted dual $\mathfrak{g}^{\vee}[-1]$ with differential built from the bracket, and this correspondence is essentially an equivalence. Drinfeld also allows higher multilinear operations $\Lambda^n\mathfrak{g}\to\mathfrak{g}$ governed by coherent identities, which he calls a 'Sugawara–Lie algebra' — the structure now called an $L_\infty$-algebra (or sh-Lie algebra); his term did not survive, and Stasheff's historical survey records the 1988 correspondence as an early development of $L_\infty$-algebras. The letter's question 'where is this written down?' is answered by Drinfeld's own 2014 comment pointing to Quillen's rational homotopy theory (1969), where the same correspondence appears for connected chain Lie algebras, and by Hinich's 1997 paper making the homological algebra of homotopy algebras systematic. The letter's question (3), whether there is a general concept of duality of operads (an operad being a symmetric sequence encoding a class of algebraic structures with multilinear operations), was resolved by Koszul duality for operads: Ginzburg–Kapranov (1994) constructed the Koszul dual of a binary quadratic operad and proved that the operads Com (commutative), Lie and Ass (associative) are Koszul, which entails the acyclicity of the bar complexes governing the letter's 'Sugawarized' structures; the Loday–Vallette monograph treats this in full. What the letter calls the 'moduli hyperspace' and 'hyper-ringed spaces' corresponds to what is now called a derived stack: an enhancement of an algebraic stack whose structure sheaf takes values in commutative DG algebras (or simplicial rings), with a tangent complex $\mathbb{T}$ whose cohomology groups $H^i$ encode the $i$-th order infinitesimal structure (classically only $H^0$ and, for the obstruction theory, $H^1$ are visible). Derived algebraic geometry (Toën's 2014 survey) developed this framework, and Pridham constructed derived moduli stacks of (polarised) schemes and sheaves. The culminating theorem, due independently to Lurie (in his ICM 2010 address on moduli problems for ring spectra) and Pridham, states that over a field of characteristic 0 the homotopy theory of DG Lie algebras is equivalent to that of formal moduli problems (functors on augmented Artinian commutative DG algebras satisfying a descent condition); in particular a derived enhancement determines, and is locally determined by, a controlling $L_\infty$-structure on the shifted tangent complex, unique up to quasi-isomorphism. Toën's Bourbaki exposé 1111 documents this theorem and explicitly credits the letter as the origin of the whole program. The theorem is an existence and uniqueness statement: it provides no explicit model. The letter itself illustrates the gap with two examples whose explicit structures the later literature does not compute. Example (a): the Hilbert scheme of a smooth variety $M$ parametrizes closed subschemes $Y\subset M$; the tangent space at $Y$ is $\mathrm{Hom}(I_Y,O_Y)$ for the ideal sheaf $I_Y$, and the higher analogues are the $\mathrm{Ext}^i(I_Y,O_Y)$. Ciocan-Fontanine–Kapranov constructed the derived quot scheme (enhancing the Quot functor classifying quotients of a fixed coherent sheaf; the quotient $\mathcal{O}_M\to\mathcal{O}_Y$ case enhances the Hilbert scheme) and computed its tangent complex: $H^i$ at $[\mathcal{O}_M\to\mathcal{O}_Y]$ is $\mathrm{Ext}^i(I_Y,O_Y)$. Toën records the subtlety that the Hilbert-side and quot-side DG Lie algebras (models built from the conormal bundle versus from the ideal sheaf) are not quasi-isomorphic even though they determine the same functor on classical (non-DG) Artinian rings — so which derived enhancement is intended is itself part of the problem. Example (b): an abelian variety $A$ is a projective group variety; its tangent bundle is trivial and its translation-invariant vector fields commute, so the DG Lie algebra controlling the deformations of the marked pair $(A,p)$ is quasi-isomorphic to its cohomology with zero bracket (the deformation theory of abelian varieties is unobstructed and the moduli stack is smooth). Since the Chevalley complex of an abelian DG Lie algebra is a free graded-commutative algebra with zero differential, the structure sheaf $\mathcal{O}$ of the derived moduli ('hyperspace') is, locally on the moduli stack, a free supercommutative graded algebra over $\mathcal{O}_0$: $\mathcal{O}=\mathcal{O}_0\oplus\mathcal{O}_1\oplus\mathcal{O}_2\oplus\cdots$, with $\mathcal{O}_1$ identifying with (the shifted dual of) the tangent sheaf and $\mathcal{O}_2/\Lambda^2\mathcal{O}_1$ with (the shifted dual of) the bundle of obstruction groups $H^2(A,\Theta_A)$. The letter asks for the sheaf $\mathcal{O}$ explicitly, in particular for the extension of locally free sheaves $0\to\Lambda^2\mathcal{O}_1\to\mathcal{O}_2\to\mathcal{O}_2/\Lambda^2\mathcal{O}_1\to0$ (an extension is a short exact sequence of sheaves; extensions of a fixed pair of locally free sheaves are classified, up to isomorphism, by a single cohomology class in an $\mathrm{Ext}^1$ group). This extension is the first quadratic invariant of the derived enhancement beyond the classical smooth moduli stack. Pridham's study of the derived moduli of abelian schemes computes the homotopy groups of the tangent complex (recovering unobstructedness), but not the multiplicative structure of $\mathcal{O}$ or the extension class.

Problem Statement

The problem consists of the two explicit computations posed as 'instructive examples' at the end of Drinfeld's 1988 letter to Schechtman, neither of which the later literature settles; both take place over a field of characteristic 0, the letter's setting. Part (a): let $M$ be a smooth variety and $Y\subset M$ a subscheme with ideal sheaf $I_Y$, so that the tangent space to the Hilbert scheme at $Y$ is $\mathrm{Hom}(I_Y,O_Y)$ and the higher analogues are $\mathrm{Ext}^i(I_Y,O_Y)$, the cohomology of the tangent complex of the derived quot scheme at the point $[\mathcal{O}_M\to\mathcal{O}_Y]$. Construct an explicit differential graded Lie algebra (equivalently, an $L_\infty$-algebra) structure, functorial in the closed embedding $Y\subset M$, on a model of $R\mathrm{Hom}(I_Y,O_Y)$ with the shift whose cohomology is $\bigoplus_i\,\mathrm{Ext}^{i-1}(I_Y,O_Y)$, controlling the deformations of $Y$ as a subscheme of $M$ over commutative DG Artinian bases; compute the induced bracket on cohomology; and in particular establish or refute the letter's conjectural formula

$$[a,b]=a\circ x\circ b-(-1)^{(i-1)(j-1)}\,b\circ x\circ a\quad\in\mathrm{Ext}^{i+j-1}(I_Y,O_Y)$$

for $a\in\mathrm{Ext}^{i-1}(I_Y,O_Y)$, $b\in\mathrm{Ext}^{j-1}(I_Y,O_Y)$, where $x\in\mathrm{Ext}^1(\mathcal{O}_Y,I_Y)$ is the class of the extension $0\to I_Y\to\mathcal{O}_M\to\mathcal{O}_Y\to0$ and the compositions are taken in the derived category of $\mathcal{O}_M$-modules (the letter writes $[a,b]=a\,x\,b-b\,x\,a$; the answer must pin down the grading, sign and composition conventions so that both sides are well defined on cohomology classes, and must verify that for $Y$ a point the bracket vanishes, as the letter notes). Part (b): on the moduli stack of abelian varieties with a marked point (the rigidification that kills the translation automorphisms), the DG Lie algebra controlling the deformations of a fiber is quasi-isomorphic to the abelian DG Lie algebra of translation-invariant vector-valued $(0,q)$-forms, so the structure sheaf $\mathcal{O}$ of the derived moduli is, locally, a free supercommutative graded algebra over $\mathcal{O}_0$, where $\mathcal{O}_0$ is the structure sheaf of the classical moduli stack, $\mathcal{O}_1$ identifies with (the shifted dual of) the tangent sheaf and $\mathcal{O}_2/\Lambda^2\mathcal{O}_1$ with (the shifted dual of) the bundle with fiber $H^2(A,\Theta_A)$. Describe this sheaf $\mathcal{O}$ explicitly and determine the extension of locally free sheaves

$$0\to\Lambda^2\mathcal{O}_1\to\mathcal{O}_2\to\mathcal{O}_2/\Lambda^2\mathcal{O}_1\to0,$$

in particular compute its extension class (or prove the extension splits, globally on the rigidified moduli stack or after pullback to a smooth cover). The common objective, exactly as in the source letter, is to make the derived enhancement of a moduli problem explicit beyond its tangent complex: part (a) asks for the linear structure (the bracket) on the Ext groups of an ideal sheaf, part (b) for the quadratic part of the structure sheaf on the moduli of abelian varieties. Both parts belong to the problem.

References

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