[Math] Why is Gauss credited with this connection

ag.algebraic-geometryho.history-overview

Let $\pi : X \to B$ be a family of compact Kähler manifolds over a smooth base $B$. We then have a local system $\mathcal R^k \pi_* \mathbb Z$ (for your favorite $k$) of abelian groups over $B$, whose fiber over a point $b$ is the cohomology group $H^k(X_b, \mathbb Z)$.

We can tensor this system by $\mathcal O_B$ and obtain a holomorphic vector bundle $E^k \to B$. This bundle is equipped with a flat connection $\nabla$, that is induced by the exterior derivative $d$ in local coordinates. This connection is called the Gauss-Manin connection of the bundle $E^k$.

Now: Why do we call this the Gauss-Manin connection, when it seems that nothing that Gauss could have worked on relates to it?

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The short answer is we call it the Gauss–Manin connection because that's what Grothendieck called it. The name is attributed to Grothendieck in two early, seminal pieces: namely, Katz's thesis and a subsequent article of his ("On the differentiation of De Rham cohomology classes with respect to parameters", Katz & Oda).

If you can read (some) French, look through Yves Andre's chapter in "Geometric Aspects of Dwork Theory". If not, check through the start of this arXiv paper (through the first couple paragraphs of 1.2).

The slightly less short answer is the one Gerhard Paseman alluded to above; quoting from the aforelinked arXiv paper ("Towards a nonlinear Schwarz’s list", Philip Boalch):

One reason hypergeometric equations are interesting is that they provide the simplest explicit examples of Gauss–Manin connections. Indeed this is one reason Gauss was interested in them: he observed that the periods of a family of elliptic curves satisfy a (Gauss) hypergeometric equation. (The modern interpretation of this is as the explicit form of the natural flat connection on the vector bundle of first cohomologies over the base of the family of elliptic curves, written with respect to the basis given by the holomorphic one forms—and their derivatives—on the fibres.) Nowadays there is still much interest in such linear differential equations “coming from geometry”.

Thus the nonlinear analogue of the Gauss hypergeometric equation should be the explicit form of the simplest nonabelian Gauss–Manin connection (i.e the explicit form of the natural connection on the bundle of first nonabelian cohomologies of some family of varieties). The simplest interesting case corresponds to taking the universal family of four punctured spheres and taking cohomology with coefficients in $\mathrm{SL}_2(\mathbb{C})$ (one needs a non-trivial family of varieties with nonabelian fundamental groups).

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[Math] analytic vs. algebraic Gauss-Manin connection

The two constructions are compatible.

Your first definition of the Gauss-Manin connexion is $ DR^{-1} (R f_* \mathbb{C}_X ) $. Here $DR : D^b_{hr}(\mathcal{D}_X) \to D^b_c( \mathbb{C}_X )$ and $DR(\mathcal{M}) = \omega_X \otimes^L_{D_X} \mathcal{M}$ is the analytic de Rham complex. This is an equivalence by the Riemann-Hilbert correspondance. It sends an $\mathscr{O}$-coherent $\mathcal{D}$-module (i.e. a vector bundle with an integrable connexion) to a local system (i.e. a locally constant sheaf). The inverse functor sends a locally constant $V$ to the vector bundle $\mathscr O_X \otimes_{\mathbb{C}} V$ together with the only connexion so that $V$ is the local system of horizontal sections in $(\mathscr O_X \otimes_{\mathbb{C}} V,\nabla)$.

Your second definition is a special case of the direct image $\mathcal{H}^n(f_+\mathscr O_X)$ in the sense of D-modules for $f$ smooth. The (algebraic or analytic) de Rham complex on $X$ is filtered by $$ L^r\Omega_X^\bullet = f^*\Omega_Y^r \otimes \Omega_X^{\bullet-r} $$ This induces a spectral sequence $$ R^pf_*(Gr_L^q \Omega_X^\bullet) \Rightarrow R^{p+q}f_*(\Omega_X^\bullet) $$ But $Gr_L^q \Omega_X^\bullet = \Omega^q_S \otimes \Omega_{X/S}^{\bullet-q}$ and the Gauss-Manin connexion can be interpreted as the differential $$ R^nf_*\Omega_{X/S} \to \Omega^1_S \otimes R^{n}f_*\Omega_{X/S} $$ in the spectral sequence. Now the analyfication functor is compatible with inverse and direct images of $\mathcal{O}$-modules and it sends $\Omega^i_X$ to $\Omega^i_{X^{an}}$ so it sends one spectral sequence to the other. This shows that the analytic and algebraic Gauss-Manin connexions are compatible.

It remains to prove is that $DR$ is compatible with direct images $$ DR f_+ \mathcal{M} \overset{\sim}{\to} Rf_* DR \mathcal{M} $$ (for $f$ a smooth morphism of complex analytic varieties and $\mathcal{M} =\mathscr O_X$ a regular connexion).

For $f:X\to S$ an open immersion this is a theorem of Deligne (cf. Borel IV.6.1) and it is actually equivalent to regularity (by a theorem of Mebkhout I think).

For $f:X\to S$ proper this is done in (Borel VIII.15): analyfication induces the natural transformation $DR f_+ \to Rf_* DR $ and it is an isomorphism because of the projection formula. In the case $\mathcal{M} =\mathscr O_X$ you can make this a little bit more concrete. It is enough to prove that the natural transformation is an isomorphism on the fibers. But by proper base change, this means you can suppose $S$ is a point so this is equivalent to $H^n_{dR}(X^{an}) \to H^{n}(X^{an};\mathbb{C})$ being an isomorphism which is the Poincaré Lemma.

Flat Holomorphic Connections – Existence Criteria

In the affine algebraic case there is always an algebraic connection: Let $A$ be a commutative unital ring and let $E$ be a finite rank projective $A$-module. There is the Atiyah sequence

$$ 0 \rightarrow \Omega^1_A \otimes E \rightarrow J^1(E) \rightarrow^{\pi} E \rightarrow 0$$

and since $E$ is projective and $\pi$ surjective, it follows $\pi$ has an $A$-linear split $s: E \rightarrow J^1(E)$.The splitting $s$ corresponds to a connection

$$\nabla: E \rightarrow \Omega^1_A \otimes E$$

and $\nabla$ is "seldom" a flat algebraic connection. There is an explicit formula for the curvature $R_{\nabla}$ of a connection/covariant derivative $\nabla$: Let

$$p: A^n \rightarrow E$$

be a surjection and let $u_1,..,u_n$ be a basis for $F:=A^n$ as $A$-module. Let $s$ be an $A$-linear splitting of $p$

Let $\phi:=s \circ p: F \rightarrow F$ be the idempotent of $E$ corresponding to $p,s$ and let $\phi:=(a_{ij})$ with $a_{ij}\in A$ be the matrix of $\phi$ in the basis $u_i$. Define for any pair $z,x\in \operatorname{Der}(A)$ the matrix

$$L1.\text{ }M:= [(z(a_{ij})), (x(a_{ij}))] \in A^{n \times n}\cong \operatorname{End}_A(F).$$

You may prove that $M $ induce a map $M^*\in \operatorname{End}_A(E)$ and the following formula holds:

Theorem 1: $R_{\nabla}(z,x):= M^* \in \operatorname{End}_A(E)$.

From formula L1 it follows $\nabla$ is seldom a flat connection. Hence in the affine situation it is easy to give explicit examples of non-flat algebraic connections: You must calculate a splitting $s$ of $p$ and the idempotent element $\phi$.

In the projective case it follows $E$ may not have a connection - there is always a cohomology class $a(E)$ which is zero iff $E$ has a connection. If $X$ is a complex projective algebraic manifold, there is a correspondence between flat connections on finite rank vector bundles on $X$ and finite dimensional complex representations of the topological fundamental group $\pi(X)$ of $X$. Given a flat algebraic connection $(E,\nabla)$ on $X$ it follows $E^{\nabla}$ is a local system of finite dimensional complex vector spaces on $X$, and to $E^{\nabla}$ you get a finite dimensional complex representation

$$\rho: \pi(X) \rightarrow GL(V).$$

This correspondence (the "Riemann-Hilbert correspondence") is an "equivalence of categories" in an appropriate sense (this is "vague"). Hence you consider the category of pairs $(E,\nabla)$ where $E$ is a finite rank vector bundle on $X$ with a flat connection $\nabla$, and "maps of connections". You also consider the category of finite dimensional complex representations of $\pi(X)$ and maps of representations. Hence there is "as many" flat connections as representations of the topological fundamental group. This is a well developed theory (originating in some papers of Weil I believe, many people have contributed to this study) from the 40s and 50s. In the below mentioned book you will find this further developed in the framework of "holonomic D-modules" - this is a well developed theory. The book also gives many references.

In the affine situation there is always a space

$$C:=\operatorname{Hom}_A(\operatorname{Der}(A), \operatorname{End}_A(E))$$

of connections, and if $\Omega^1_A$ is a finite rank projective $A$-module you get

$$ C\cong \Omega^1_A \otimes \operatorname{End}_A(E),$$

hence the "parameter-space of connections" is a finite rank vector bundle on $A$. Hence given a connection $\nabla$ with curvature given by Theorem 1, you may add a potential $\phi \in C$ to get a new connection $\overline{\nabla}:=\nabla + \phi$. Hence if you want to study the problem if $E$ has a flat algebraic connection you must study the "moduli space" $C$. Similar for $X$ - this again is a well devleoped theory - "moduli spaces of connections". If you consider the "set of potentials" $\phi \in C$ with the property that the curvature is zero

$$L2.\text{ }R_{\overline{\nabla}}=0$$

you get a subvariety $M^{fl}(E) \subseteq \mathbb{V}(C^*)$ parametrizing flat connections on $E$, and in high dimension and low rank, the system of equations defining $M^{fl}(E)$ will be "overdetermined". Hence it is not clear if $E$ has a flat algebraic connection in general. Hence in the affine algebraic situation there is always a connection which is non-flat in general by formula L1, and it is an open problem to determine if $E$ has a flat algebraic connection. You must study the subvariety you get from equation L2.

Example: If you let $n:=dim(A)\geq 10$ and $rk(E)=2$ you get an overdetermined system of equations defining $M^{fl}(E)$, hence in this case you may get an empty moduli space

$$M^{fl}(E)=\emptyset.$$

For such $E$ you always have a non-flat connection from equation L1 and Theorem 1. The system defining $M^{fl}(E)$ has $\binom{n}{2}$ equations and $rk(\Omega^1\otimes \operatorname{End}(E))=4n$. For $n>>0$ it follows this system does not "have a solution".

Question: "What about (1) ⟹ (2) and (3) ⟹ (4)?"

Answer: I believe it is an old conjecture that if $E$ has an algebraic/holomorphic connection, then $E$ has a flat algebraic /holomorphic connection, but I do not have a precise reference (maybe the paper of Atiyah from 1956 in TrAMS).

You'll find this statement

Citation: "Non-flat algebraic connections for bundles on complex projective manifolds are virtually non-existent (we know of none)"

in the introduction of

Spencer Bloch, Hélène Esnault, Algebraic Chern Simons Theory, American Journal of Mathematics, 119 Number 4 (1997) pp. 903–952, doi:10.1353/ajm.1997.0021, arXiv:alg-geom/9602001.

Note 1: In Borel's book page 226 you find the following construction: If $X \subseteq \mathbb{P}^n_{\mathbb{C}}$ is a smooth quasi projective algebraic variety and if $\omega^1_X$ is the canonical bundle of $X$, it follows the Lie derivative induce a right $D_X$-module structure on $\omega^1_X$. This does not imply there is a flat algebraic connection

$$\nabla: \omega^1_X \rightarrow \Omega^1_{X}\otimes \omega^1_X.$$

Since $D_X$ is a sheaf of non-commutative rings, there is no obvious relation between left $D_X$-modules and right $D_X$-modules. From a right $D_X$-module $E$ we get a left $D_X$-module $\omega^{-1}_X \otimes E$ and to the canonical bundle $\omega^1_X$ we get the trivial bundle $\mathcal{O}_X$. To a left $D_X$-module we get canonically a flat connection.

Note 2: If $X$ is a complex projective manifold and $E$ is an indecomposable finite rank vector bundle on $X$, it follows $\Gamma(X,\operatorname{End}(E))$ is a finite dimensional algebra over $\mathbb{C}$.

Borel, A. Algebraic D-modules. Perspectives in Mathematics, Vol. 2, Boston etc.: Academic Press, Inc., Harcourt Brace Jovanovich, Publishers. xii, 355 p.; $ 29.95; £25.00 (1987).

Note 3: There is the following general result: If $(L,a)$ is a Lie-Rinehart algebra and $(E,\nabla)$ is a connection, there is a non-abelian extension (the non-abelian Atiyah extension)

$$N1.\text{ } 0 \rightarrow \operatorname{End}_A(E) \rightarrow L(E,\nabla) \rightarrow L \rightarrow 0 $$

of Lie-Rinehart algebras which splits iff $E$ has a flat connection. There is a "cohomology set" $\operatorname{Ext}^1(L,(E,\nabla))$ classifying such non-abelian extensions. By definition

$$L(E,\nabla):=\operatorname{End}_A(E)\oplus L$$

with the following Lie product:

$$[(\phi,x),(\psi,y)]:=([\nabla(x),\psi]-[\nabla(y), \phi]+R_{\nabla}(x,y),[x,y])$$

for all $\phi, \psi\in \operatorname{End}_A(E)$ and $x,y\in L$. You must check that the sequence $N1$ splits iff there is an $A$-linear map $P: L \rightarrow \operatorname{End}_A(E)$ such that $\overline{\nabla}:=\nabla+P$ is a flat connection. A similar construction exists globally.

Hence given a holomorphic vector bundle $E$ on a complex projective manifold $X$, there are two obstructions: The Atiyah class $a(E)$ which is zero iff $E$ has a holomorphic (or algebraic) connection. The extension class $n(E)$ given by sequence $N1$ is zero iff $E$ has a flat holomorphic connection. The sequence $N1$ exists whenever $E$ has a holomorphic (or algebraic) connection.

Example: If $\mathbb{P}^n_k$ is complex projective space and if $\mathcal{O}(d)$ is the invertible sheaf with $d \geq 1$, it follows the Atiyah sequence

$$0 \rightarrow \Omega^1 \otimes \mathcal{O}(d) \rightarrow J^1(\mathcal{O}(d)) \rightarrow \mathcal{O}(d) \rightarrow 0$$

does not split, hence $\mathcal{O}(d)$ does not have a holomorphic (or algebraic) connection.

Any finite rank projective $A$-module $E$ with a non-zero class in $\operatorname{H}^2_{DR}(A)$ will have an algebraic connection and no flat algebraic connection. Hence the Atiyah sequence splits, but the sequence $N1$ does not split. I belive there is an explicit example in Loday's book "Cyclic Homology".

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[Math] analytic vs. algebraic Gauss-Manin connection

Flat Holomorphic Connections – Existence Criteria

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