My level is at the beginning of a second year master. I'm interested in the project of translating some features of differential geometry to algebraic geometry. I'd like to know if there is an equivalent of the notion of connection in algebraic geometry. I've heard about Grothendieck connections, but I did not know if it is the only possible way to abstract the notion of connecton, nor I found any accessible material on the web. Can someone help me?
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1$\begingroup$ Some references here ncatlab.org/nlab/show/Grothendieck+connection might be useful. $\endgroup$– QfwfqCommented Aug 30, 2020 at 12:05
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$\begingroup$ Give this page a try. $\endgroup$– MathCrawlerCommented Aug 30, 2020 at 12:16
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3$\begingroup$ If you can read french, I recommend Equations Différentielles à Points Singuliers Réguliers by Pierre Deligne, Springer Lecture Notes 163. The notion of connection in algebraic geometry is very clearly explained. $\endgroup$– abxCommented Aug 30, 2020 at 12:26
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4 Answers
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I think if you can read french the best source is Deligne book but there are several interesting articles by katz you can read for example:On the differentiation of de rham cohomology classes with respect to parameters, Algebraic solutions of differential equations (p-curvature and the hodge filtration),On the differential equations satisfied by period matrices.
there is also a beautiful article by Coleman where he introduces basic properties of connections and uses them to prove an interesting conjecture in number theory: Manin’s proof of the mordell conjecture over function fields.
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$\begingroup$ Hopefully, the beautiful article mentioned above is this one $\endgroup$ Commented Aug 30, 2020 at 14:52
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In high class topic which connects Differential Geometry to Algebraic geometry is about finding canonical metric by using Minimal Model Program. Finding generalized Einstein metric by using MMP.
The second concept is the study of degeneration of Kähler- Einstein Manifolds, which you must know a lot about Algebraic geometry and Geometric Analysis.
Gauss-Manin connection and Weil-Petersson Geometry on Moduli spaces and Mirror Symmetry. You need much to know about Algebraic geometry and Differential Geometry See Gunnar Thor Magnusson Thesis
Study of Quillen metrics on holomorphic determinants, give a connects Differential Geometry to Algebraic geometry
https://link.springer.com/article/10.1007/BF01466774
There is branch in Mathematics which is called Analytical Algebraic Geometry which connects Differential Geometry to Algebraic geometry
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2$\begingroup$ How does this answer the question? The OP was specifically asking about connections ($\nabla$), no? $\endgroup$– QfwfqCommented Aug 30, 2020 at 12:03
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1$\begingroup$ See this thesis, theses.fr/2016LIL10170 $\endgroup$– user160903Commented Aug 30, 2020 at 12:15
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$\begingroup$ Ok now the answer has been edited. I still think the first two paragraphs may not be particularly on topic (general complex differential geometry rather than specifically connections), but okay, now it's an answer to the OP (or contains one). -- Re the last paragraph, are you referring to complex analytic geometry or "algebraic analysis" in the sense of D-modules etc? $\endgroup$– QfwfqCommented Aug 30, 2020 at 13:07
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$\begingroup$ Riemann–Hilbert correspondence . See the lecture series of Zoghman in YouTube. You can see also RH correspondance and Bridgeland stability conditions. $\endgroup$– user160903Commented Aug 30, 2020 at 14:11
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9$\begingroup$ Oh, someone read my thesis. That’s unexpected. $\endgroup$ Commented Aug 30, 2020 at 17:43
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One possible way to define connections is to use the language of sheaves and differential forms; see e.g. Chap. III, Definition 1.5, on p. 70 of [1]. Since these notions are equally well at one's disposal within the category of schemes, the notion of a connection on vector bundles can be defined for Algebraic Geometry along entirely analogous lines. For this, see e.g. this paper of Brian Osserman.
[1] Wells, R.O., Differential Analysis on Complex Manifolds (Graduate Texts in Mathematics 65), Third Edition. Springer 2008.
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You should also have a look into Atiyah's 'Complex analytic connections in fibre bundles'. Even though he considers complex manifolds you can gain a lot insights from reading this paper.
