complex geometry

Contents

Idea

For a ringed space $(X, \mathcal{O}_X)$ there is its Picard group of invertible objects in the category of $\mathcal{O}_X$-modules. When $X$ is a projective integral scheme over $k$ the Picard group underlies a $k$-scheme, this is the Picard scheme $Pic_X$. This scheme varies in a family as $X$ varies in a family. From this starting point one can naturally generalize to more general relative situations.

Often one considers just the connected component $Pic_X^0$ of the neutral element in $Pic_X$, and often (such as in the discussion below, beware) it is that connected component (only) which is referred to by “Picard scheme”. The difference between the two is measured by the quotient $Pic_X/Pic_X^0$, which is called the Néron-Severi group of $X$. Though at least for $X$ an algebraic curve, $Pic_X^0$ goes by a separate name: it is the Jacobian variety of $X$.

The completion of the Picard scheme at its neutral element (hence either of $Pic_X$ or $Pic_X^0$) is the formal Picard group.

Definition

The Picard variety of a complete smooth algebraic variety $X$ over an algebraically closed field parametrizes the Picard group of $X$, more precisely the set of classes of isomorphic invertible quasicoherent sheaves with vanishing first Chern class.

The Picard scheme is a scheme representing the relative Picard functor $Pic_{X/S}: (Sch/S)^{op}\to Set$ by $T\mapsto Pic(X_T)/f^*Pic(T)$. In this generality the Picard functor has been introduced by Grothendieck in FGA, along with the proof of representability. An alternate form of this functor in terms of the derived functor of $f_*$ is $Pic_{X/S}(T)=H^0(T, R^1f_{T*}\mathcal{O}_{X_T}^*)$.

Note we must work with the relative functor because the global Picard functor $Pic_X(T)=Pic(X_T)$ has no hope of being representable as it is not even a sheaf. Consider any non-trivial invertible sheaf in $Pic(X_T)$. This becomes trivial on some cover $\{T_i\to T\}$, so $Pic(X_T)\to \prod Pic(X_{T_i})$ is not injective.

Representability

For this section suppose $f:X\to S$ is s separated map, finite type map of schemes. Many general forms of representability have been proven several of which are given in FGA explained. Here we list several of the common forms:

• Suppose $\mathcal{O}_S\to f_*\mathcal{O}_X$ is universally an isomorphism (stays an isomorphism after any base change), then we have a comparison of relative Picard functors $Pic_{X/S}\hookrightarrow Pic_{X/S, zar}\hookrightarrow Pic_{X/S, et}\hookrightarrow Pic_{X/S, fppf}$. They are all isomorphisms if $f$ has a section.

• If $Pic_{X/S}$ is representable by a scheme, then by descent theory for sheaves it is representable by the same scheme in all the topologies listed above. In general, representability gives representability in a finer topology (of the ones listed).

• If $Pic_{X/S}$ is representable then a universal sheaf $\mathcal{P}$ on $X\times Pic_{X/S}$ is called a Poincaré sheaf. It is universal in the following sense: if $T\to S$ and $\mathcal{L}$ is invertible on $X_T$, then there is a unique $h:T\to Pic_{X/S}$ such that for some $\mathcal{N}$ invertible on $T$ we get $\mathcal{L}\simeq (1\times h)^*\mathcal{P}\otimes f_T^*\mathcal{N}$.

• If $f$ is (Zariski) projective, flat with integral geometric fibers then $Pic_{X/S, et}$ is representable by a separated and locally of finite type scheme over $S$.

• Grothendieck’s Generic Representability: If $f$ is proper and $S$ is integral, then there is a nonempty open $V\subset S$ such that $Pic_{X_V/V, fppf}$ is representable and is a disjoint union of open quasi-projective subschemes.

• If $f$ is a flat, cohomologically flat in dimension 0, proper, finitely presented map of of algebraic spaces, then $Pic_{X/S}$ is representable by an algebraic space locally of finite presentation over $S$.

Picard Stack

The Picard stack $\mathcal{Pic}_{X/S}$ is the stack of invertible sheaves on $X/S$, i.e. the fiber category? over $T\to X$ is the groupoid of line bundles on $X_T$ (not just their isomorphism classes). (Hence it is the Picard groupoid equipped with geometric structure).

If $X$ is proper and flat, then $\mathcal{Pic}_{X/S}$ is an Artin stack since $\mathcal{Pic}_{X/S}=\mathcal{Hom}(X, B\mathbb{G}_m)$ is the Hom stack which is Artin.

Note the following “failure” of the relative Picard scheme: points on $Pic_{X/S}$ do not parametrize line bundles. The low degree terms of the Leray spectral sequence give the following exact sequence $H^1(X_T, \mathbb{G}_m)\to H^0(T, R^1f_*\mathbb{G}_m)\to H^2(T, \mathbb{G}_m)\to H^2(X_T, \mathbb{G}_m)$, but as noted above $Pic_{X/S}(T)=H^0(T, R^1f_*\mathbb{G}_m)$, so we see exactly when a $T$-point comes from a line bundle it is when that point maps to $0$ in this sequence.

This gives us an obstruction theory lying in $H^2(T, \mathbb{G}_m)$ for a point corresponding to a line bundle. If $Pic_{X/S}$ is representable we could take $T=Pic_{X/S}$ to find a universal obstruction. Intuitively this is because the Picard stack is the right object to look at for the moduli problem of line bundles over $X$. The Picard scheme is the $\mathbb{G}_m$-rigidification of the Picard stack.

The natural map $\mathcal{Pic}_{X/S}\to Pic_{X/S}$ is a $\mathbb{G}_m$-gerbe. But isomorphism classes of $\mathbb{G}_m$-gerbes over $T$ are in bijective correspondence with $H^2(T, \mathbb{G}_m)$ and so the above map could be thought of as a geometric realization of the universal obstruction class.

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$Calabi-Cau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

References

Specifically on the Picard stack:

Revised on June 13, 2014 04:42:46 by Urs Schreiber (145.116.130.115)