What does “control of a deformation problem” mean?

Is the expression “control of a deformation problem’ ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ever defined? Answer AttributionSource : Link , Question Author : Jim Stasheff , Answer Author : Community

Obstruction to lifting of global sections of invertible sheaves

Are there known examples of smooth projective hypersurface in $\mathbb{P}^3$, say $X$ and an invertible sheaf $L$ on $X$ with $H^0(X,L)>0$ satisfying the following property: There exists an infinitesimal deformation $X’$ of $X$ such that $L$ lifts as an invertible sheaf $L’$ on $X’$ (meaning $L’$ pulls back to $L$ on $X$) but there is … Read more

Deformations of the moduli space of ppav’s

Consider the complex algebraic moduli space X:=Ang of ppav’s of dimension g with some high enough level n structure (so that it represents the corresponding functor). Can one compute the dimension of the infinitesimal space of deformations to X (as a function of g and the level)? If V is a smooth projective variety, then … Read more

Deformation space and Kodaira-Spencer map of cyclic Galois coverings

This question concerns a statement from the paper by Ben Moonen “Special subvarieties arising from families of cyclic covers of the projective line. Documenta Math. 15 (2010)”, Lemma 5.5. ii). More precisely, Let C→T be a family of cyclic covers of P1C. Here T can be takec to be the big diagonal in (A1)N, i.e., … Read more

Local systems and locally free sheaves in families

Let Xt be a family of compact complex manifolds over the disk D⊂C. Formally, Xt is the fiber over t∈D of a proper, holomorphic submersion of a complex manifold X onto D. Suppose Et is a family of holomorphic vector bundles over Xt equipped with a family of holomorphic flat connections ∇t. If Vt denotes … Read more

Is complete intersection a open or closed property in Hilbert schemes

Fix an integer N, X a (smooth) complete intersection subvariety in PN. Denote by P the Hilbert polynomial of X (as a subvariety in PN). Consider the Hilbert scheme HilbP, parametrizing all subscheme in PN with Hilbert polynomial P. Let H be an irreducible component of HilbP containing the point corresponding to X. Then, 1) … Read more

DGLA related to the deformation of hopf Algebras

Recently I was considering Hopf algebras and Drinfeld’s twists. I stumbled upon a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by looking at the universal enveloping algebra of a Lie algebra(oid) and the Gerstenhaber structure induced by viewing them as (left invariant) polydifferential operators. Explicitely, … Read more

Building conilpotent coalgebras from co-square-zero-extensions

Let K be a field of char. 0. Given a chain complex X over K denote E(X) the co-square-zero-extension on X, i.e. the cocommutative non-counital dg-coalgebra structure on X with zero-comultiplication. We think of the E(X) as the most basic conilpotent cocommutative non-counital dg-coalgebras. Is it true that every conilpotent cocommutative non-counital dg-coalgebra can be … Read more

Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains

Let Y→M be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So Y and M are a complex manifold and each fiber is a relatively compact strictly pseuddoconvex (complex) domain with, maybe, different complex structures but all of them diffeomorphic. Let Y0 be the … Read more

Deformation of pairs (X,D) isotrovial along D

I have a log pair (X,D) which is purely log terminal and D is a projective Q-Cartier divisor (X may not be projective). Moreover, D is a variety of Fano type. Is there a space of finite type parametrizing first order deformation theory of the pair (X,D) which are isotrivial along D? i.e. the induced … Read more