Gauss - Manin Connectivity

Karl Friedrich Gauss

With a bundle whose fibers are smooth manifolds (or smooth algebraic varieties ), we can associate some bundle with a flat connection , called the Gauss – Manin connection .

Definition

Let be $Y\to X$ ${\ displaystyle Y \ to X}$ $Y\to X$ - bundle whose layers $Y_{x}$ ${\ displaystyle Y_ {x}}$ $Y_{x}$ - smooth manifolds. Consider a vector bundle $E\to X$ ${\ displaystyle E \ to X}$ $E\to X$ with layers $E_{x}=H_{\mathrm {dR} }^{k}(Y_{x})$ ${\ displaystyle E_ {x} = H _ {\ mathrm {dR}} ^ {k} (Y_ {x})}$ $E_{x}=H_{\mathrm {dR} }^{k}(Y_{x})$ . In other words, hang instead of each layer of it $k$ ${\ displaystyle k}$ $k$ cohomology de Rama . By Eresmann’s theorem, , smooth bundles are locally trivial, so that in a sufficiently small neighborhood along the base, one can identify the layers with each other and proclaim smooth sections $E$ ${\ displaystyle E}$ $E$ sections that correspond to smooth variations of the cohomology class under trivialization. Strictly speaking, we did not define a bundle, but only a bundle , but it really will be a bundle of sections of the bundle.

Yuri Ivanovich Manin

For simplicity, suppose for a moment that the layers are compact. The de Rham cohomology of a compact manifold is isomorphic to singular cohomology $H^{k}(Y_{x},\mathbb {Z} )\otimes \mathbb {R}$ ${\ displaystyle H ^ {k} (Y_ {x}, \ mathbb {Z}) \ otimes \ mathbb {R}}$ $H^{k}(Y_{x},\mathbb {Z} )\otimes \mathbb {R}$ thus in each layer $E_{x}$ ${\ displaystyle E_ {x}}$ $E_{x}$ there is a lattice of integer cohomologies that smoothly depends on the point $x$ ${\ displaystyle x}$ $x$ . The Gauss – Manin connection is defined as the connection with respect to which the local sections at each point taking values in this integer lattice are flat.

A description of the Gauss – Manin connection through flat sections provides a convenient way to visualize it, however, for its existence, the existence of an integer structure on cohomology is absolutely not necessary. She admits the following description. Choose in the bundle $Y\to X$ ${\ displaystyle Y \ to X}$ $Y\to X$ Eresmann connectivity $H\subset TY$ ${\ displaystyle H \ subset TY}$ $H\subset TY$ . If $s\in \Gamma (E)$ ${\ displaystyle s \ in \ Gamma (E)}$ $s\in \Gamma (E)$ - some section, it can be implemented by a set of closed forms $\sigma _{x}\in \Omega ^{k}(Y_{x})$ ${\ displaystyle \ sigma _ {x} \ in \ Omega ^ {k} (Y_ {x})}$ $\sigma _{x}\in \Omega ^{k}(Y_{x})$ . Eresmann's chosen connection allows him to continue to a single form $\sigma \in \Omega ^{k}(Y)$ ${\ displaystyle \ sigma \ in \ Omega ^ {k} (Y)}$ $\sigma \in \Omega ^{k}(Y)$ , redefining it in directions transversal to layers, by the condition $\iota _{h}\sigma =0$ ${\ displaystyle \ iota _ {h} \ sigma = 0}$ $\iota _{h}\sigma =0$ for all $h\in H$ ${\ displaystyle h \ in H}$ $h\in H$ . Note that this form does not have to be closed. Define the Gauss - Manin connection $\nabla$ ${\ displaystyle \ nabla}$ $\nabla$ in this way: $(\nabla _{v}s)_{x}=\left[\left(\mathrm {Lie} _{\widetilde {v}}\sigma \right)|_{Y_{x}}\right]\in H_{\mathrm {dR} }^{k}(Y_{x})$ ${\ displaystyle (\ nabla _ {v} s) _ {x} = \ left [\ left (\ mathrm {Lie} _ {\ widetilde {v}} \ sigma \ right) | _ {Y_ {x}} \ right] \ in H _ {\ mathrm {dR}} ^ {k} (Y_ {x})}$ $(\nabla _{v}s)_{x}=\left[\left(\mathrm {Lie} _{\widetilde {v}}\sigma \right)|_{Y_{x}}\right]\in H_{\mathrm {dR} }^{k}(Y_{x})$ . Here $v$ ${\ displaystyle v}$ $v$ Is an arbitrary vector field at the base, and ${\widetilde {v}}$ ${\ displaystyle {\ widetilde {v}}}$ ${\widetilde {v}}$ - its raising with the help of Eresmann's connection, that is, a section $H$ ${\ displaystyle H}$ $H$ , when projecting to the base, passing into $v$ ${\ displaystyle v}$ $v$ . Checking that this is a well-defined connection (that is, such a Lie derivative will be closed in the restriction on the layers, and this operation satisfies the Leibniz identity) is not difficult; it’s a little harder to show that it does not depend on the choice of Eresmann's connection.

Dmitry Kaledin

This definition of a Gauss-Manin connection is elegantly formulated in terms of differentially graded algebras . This allows us to transfer the definition of the Gauss - Manin connection to noncommutative geometry : Getzler ^[1] , and Kaledin ^[2] constructed the Gauss-Manin connection on periodic cyclic homology.

Application

Gauss - Manin connection in the first cohomology of a family of elliptic curves with equations $x^{3}+y^{3}+z^{3}=\lambda xyz$ ${\ displaystyle x ^ {3} + y ^ {3} + z ^ {3} = \ lambda xyz}$ over the punctured Riemann sphere parameterized by the complex parameter $\lambda$ ${\ displaystyle \ lambda}$ , defines a differential equation, known as the Picard-Fuchs equation . Gauss considered a similar equation for a family of curves $y^{2}=x(x-1)(x-\lambda )$ ${\ displaystyle y ^ {2} = x (x-1) (x- \ lambda)}$ ; a general description of such equations in the case when the base is an algebraic curve was given by Manin ^[3] , and in the general case by Grothendieck ^[4] . He belongs to the name “Gauss – Manin connection”, as well as an abstract algebraic-geometric description of this connection as one of the arrows in the Leray spectral sequence of for a suitable beam.

Vladimir Arnold

The Gauss - Manin connection is also used in symplectic geometry . Exactly let $Y\to X$ ${\ displaystyle Y \ to X}$ Is a bundle whose layers are Lagrangian tori. The tangent space to the base of such a bundle can be identified with a certain subspace in the space of sections of the normal bundle to the layer hanging above this point. But for a Lagrangian submanifold, the normal bundle is isomorphic to the cotangent, so that these sections define differential 1-forms on the layer. It turns out that these forms are closed, and their cohomology classes are all kinds of classes of the first cohomology of the layer. Thus, the tangent bundle to the base of the Lagrangian bundle is isomorphic to the bundle of the first cohomology of layers, and therefore has a canonical flat connection, Gauss – Manin connection. In mechanics, this statement has a corollary, known as the Liouville-Arnold theorem : for a Hamiltonian system having as many independent integrals in involution as there are degrees of freedom, equations of motion can be solved in quadratures. A holomorphic version of the Liouville-Arnold theorem defines a flat connection with monodromy $\mathrm {SL} (2n,\mathbb {Z} )\ltimes \mathbb {R} ^{2n}$ ${\ displaystyle \ mathrm {SL} (2n, \ mathbb {Z}) \ ltimes \ mathbb {R} ^ {2n}}$ outside some divisor on $\mathbb {C} \mathrm {P} ^{n}$ ${\ displaystyle \ mathbb {C} \ mathrm {P} ^ {n}}$ , the base of a holomorphic Lagrangian bundle on a hyperkähler manifold . The most obvious case when the total space is the K3-surface , the layers are elliptic curves, and the base is the Riemann sphere with 24 punctures, studied by Kontsevich and Soibelman ^[5] .

Notes

[1] [1]

[2] [math / 0702068v2] Cyclic homology with coefficients

[3] Algebraic curves over fields with differentiation

[4] On the de Rham cohomology of algebraic varieties

[5] [math / 0406564] Affine structures and non-archimedean analytic spaces

Gauss - Manin Connectivity

Definition

Application

Notes

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