User c. jost - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:05:57Z http://mathoverflow.net/feeds/user/5631 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29772/differential-between-tangent-sheaves Differential between tangent sheaves C. Jost 2010-06-28T11:53:06Z 2010-06-28T15:45:03Z <p>Let $M$ be a smooth manifold. Its structure sheaf $\mathcal{O}_M$ is the sheaf of smooth real-valued functions. Together they form a ringed space $(M,\mathcal{O}_M)$. The tangent sheaf $\mathcal{T}_M$ is a sheaf of modules over the structure sheaf. It can be defined as the sheaf of derivations of the structure sheaf.</p> <p>A smooth map of manifolds $f: M \rightarrow N$ induces a morphism $df: \mathcal{T}_M \rightarrow f^*(\mathcal{T}_N)$ of $\mathcal{O}_M$-modules, where $f^*(\mathcal{T}_N)$ is the inverse image of $\mathcal{T}_N$. It is called the differential or pushforward of the map $f$.</p> <p>Does anyone have a reference for the definition of the differential? In which kind of textbook would this be explained? It seems to be somewhere between differential geometry and algebraic geometry but I could not find it in any textbook in neither of these areas.</p> <p>(I am not looking for the differential between tangent <em>bundles</em> which is explained in detail in every basic book on differential geometry.)</p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/22597/twisting-an-l-infinity-morphism-with-non-associated-maurer-cartan-elements Twisting an L-infinity-morphism with "non-associated" Maurer-Cartan elements C. Jost 2010-04-26T14:02:33Z 2010-04-27T14:29:57Z <h2>Background</h2> <p>Suppose we are given $L_\infty$-algebras $(g,Q)$ and $(g',Q')$ and an $L_\infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$. Furthermore, we have a Maurer-Cartan element $\pi$ of $(g,Q)$. One can twist $(g,Q)$ with the Maurer-Cartan element $\pi$ and obtains a new $L_\infty$-algebra that we call $(g,Q_\pi)$. Furthermore, we can construct a Maurer-Cartan element $\pi'$ of $(g',Q')$ by the formula</p> <p>$\pi' = \sum_{n=1}^\infty \frac{1}{n!} F_n(\pi, \ldots , \pi)$,</p> <p>where $F_n$ is the $\bigwedge^n g \rightarrow g'$-part of $F$. I don't know whether there is a (better) term, so I call $\pi$ and $\pi'$ associated Maurer-Cartan elements</p> <p>One can twist the morphism $F$ with the Maurer-Cartan elements $\pi$ and $\pi'$ and obtain an $L_\infty$-morphism $F_\pi$ from $(g,Q_\pi)$ to $(g',Q'_{\pi'})$. The references I found are <a href="http://arxiv.org/abs/math/0504420" rel="nofollow">Dolgushev: A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold</a> (section 2.4) and <a href="http://arxiv.org/abs/math/0502137" rel="nofollow">Yekutieli: Continuous and Twisted L_infinity Morphisms </a> (section 3).</p> <h2>Question</h2> <p>Given $L_\infty$-algebras $(g,Q)$ and $(g',Q')$, an $L_\infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$, a Maurer-Cartan elements $\pi$ of $(g,Q)$ <strong>and a Maurer-Cartan element $\omega$</strong>, $\omega\neq \pi'$, <strong>of $(g',Q')$</strong>. Can one construct an $L_\infty$-morphisms between $(g,Q)$ twisted with the Maurer-Cartan element $\pi$ and $(g',Q')$ twisted with the Maurer-Cartan element $\omega$, where the Maurer-Cartan elements are not "associated"? I.e. can one construct an $L_\infty$-morphism between $(g,Q_\pi)$ to $(g',Q'_\omega)$? </p> http://mathoverflow.net/questions/35127/why-does-the-algebraic-condition-of-flatness-on-the-structure-sheaves-give-a-goo Comment by C. Jost C. Jost 2010-08-11T08:25:13Z 2010-08-11T08:25:13Z Section II.3.4 in Eisenbud/Harris contains an elaborate introduction to flat families of schemes. Especially Proposition II.29 might help you. It says roughly that for a flat family + other conditions the fiber of a closed point 0 is the limit of the fibers of points b where b goes to 0. http://mathoverflow.net/questions/29772/differential-between-tangent-sheaves Comment by C. Jost C. Jost 2010-06-30T19:41:40Z 2010-06-30T19:41:40Z 1) Unfortunately, I could not find anything in Michor, although it gave me a reference for another fact some time ago. 2) Yes, everything seems to boil down to the identification of vector bundle morphisms and morphisms between locally free sheaves. This is luckily explained in the reference below. Thanks to you all! http://mathoverflow.net/questions/29772/differential-between-tangent-sheaves/29803#29803 Comment by C. Jost C. Jost 2010-06-30T19:38:00Z 2010-06-30T19:38:00Z There seems to be an explanation in both books, but I had difficulties understanding them without having read the whole book. But thanks anyway, and thanks for the algebraic definition of the differential! It gives another point of view, which is always good and it might be even necessary for me. http://mathoverflow.net/questions/29772/differential-between-tangent-sheaves/29795#29795 Comment by C. Jost C. Jost 2010-06-30T19:32:49Z 2010-06-30T19:32:49Z Yes, Ramanan treats the connection between morphisms of vector bundles and morphisms of locally free sheaves detailed and even applies it to tangent bundles and tangent sheaves. That was exactly what I was looking for. (Now I only would like to know who took the copy in our library without loaning it properly. But there is always google book search.) Thanks a lot! http://mathoverflow.net/questions/22597/twisting-an-l-infinity-morphism-with-non-associated-maurer-cartan-elements/22610#22610 Comment by C. Jost C. Jost 2010-04-27T08:43:56Z 2010-04-27T08:43:56Z Hm, it has not helped me yet. But thanks anyway, Peter! http://mathoverflow.net/questions/22597/twisting-an-l-infinity-morphism-with-non-associated-maurer-cartan-elements Comment by C. Jost C. Jost 2010-04-26T16:35:41Z 2010-04-26T16:35:41Z Unfortunately, I know too little about the formal manifold language of L-infinity algebras, so I did not get any intuition from there. Do you or anyone know a good introductory reference for this point of view? I only know the Kontsevich letter. (Maybe I should make this a separate question?)