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The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate parabolic bundles (i.e., vector bundles with flags at finitely many points)? Said differently, how do parabolic bundles arise in nature?

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Parabolic bundles were introduced in the 70's by Mehta and Seshadri in the set up of a Riemann surface with cusps. They were trying to generalize the Narasimhan-Seshadri correspondence on a compact Riemann surface (between polystable bundles of degree $0$ and unitary representations of the fundamental group). In the non-compact case, they were able to determine the missing piece of data - partial flags and weights at each cusp. They established what is now called the Mehta-Seshadri correspondence. Then they proceeded to study the moduli space.

Mehta, V. B.; Seshadri, C. S. Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248 (1980), no. 3, 205–239.

http://link.springer.com/article/10.1007%2FBF01420526

Since then, the definition of a parabolic bundle has been clarified (tensor product with the initial definition is not really computable for instance) and generalized. This is a long story starting with C.Simpson, I.Biswas, and many authors. The upshot is that given a scheme $X$, a Cartier divisor $D$, and an integer $r$, there is a one to one tensor (and Fourier-like) equivalence between parabolic vector bundles on $(X,D)$ with weights in $\frac{1}{r}\mathbb Z$ and standard vector bundles on a certain orbifold $\sqrt[r]{D/X}$, the stack of $r$-th roots of $D$ on $X$. So one can turn your question in: why are these orbifolds natural ? They were first introduced by A.Vistoli in relation with Gromov-Witten theory. They also turned out to be related to the section conjecture (rational points of stack of roots are Grothendieck's packets in his anabelian letter to Faltings). So parabolic sheaves - and stack of roots - are ubiquitous. They are also very strongly related to logarithmic geometry.

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This paper by Agnihotri and Woodward uses a Narasimhan-Seshadri correspondence between parabolic bundles and unitary connections to determine the possible spectrum of a product of two (special) unitary matrices of known spectrum. They start with a triple of unitary matrices with product $1$, N-S relate that to bundles on $\mathbb P^1$ with parabolic structure at three points, classify those bundles as maps of the $\mathbb P^1$ into a Grassmannian, and end up at quantum Schubert calculus of Grassmannians. Maybe not the most obviously natural source of parabolic bundles, but a wonderful application.

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They arise in analytic theory of differential equations with regular singularities, Riemann Hilbert problem and Painleve equations.

MR1924757 Biswas, Indranil A criterion for the existence of a flat connection on a parabolic vector bundle. (English summary) Adv. Geom. 2 (2002), no. 3, 231–241.

MR1488348 Arinkin, D.; Lysenko, S. On the moduli of SL(2)-bundles with connections on P1∖{x1,⋯,x4}. Internat. Math. Res. Notices 1997, no. 19, 983–999.

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