nLab D-geometry

This entry is about the concept in the algebraic geometry of jet bundles and differential equations. For the concept of geometry as seen by D-branes see at D-brane geometry.

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Idea

The term $\mathcal{D}$-geometry refers to a synthetic formulation of differential geometry with a focus on the geometry of de Rham spaces. A quasicoherent sheaf of modules over a de Rham space is called a D-module, short for module over the algebra of differential operators, which is where the term “D-geometry” derives from.

Related entries

• regular differential operator, ordinary differential equation

• connection, flat connection, local system, D-module, perverse sheaf

• holonomic D-module, meromorphic connection, characteristic variety

• diffiety, Grothendieck connection, crystal, de Rham space

• synthetic differential geometry, infinitesimal object

• Kähler differential, tangent category, tangent (∞,1)-category, cotangent complex, deformation theory, formally smooth morphism, jet bundle, jet space

• Stokes phenomenon, wall crossing

• microlocal analysis, algebraic microlocalization

• Riemann-Hilbert problem, Fuchsian equation

• algebraic approaches to differential calculus

• regular differential operator in noncommutative geometry

• holonomic quantum field, Hodge theory, algebraic analysis

References

• Alexander Beilinson and Vladimir Drinfeld, Chiral Algebras

  chapter 2, Geometry of D-schemes (pdf)

• Jacob Lurie, Notes on crystals and algebraic $\mathcal{D}$-modules (2010) [pdf]