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Classification of smooth horizontal Schubert varieties

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Abstract

We show that the smooth horizontal Schubert subvarieties of a rational homogeneous variety G / P are homogeneously embedded cominuscule , and are classified by subdiagrams of a Dynkin diagram. This generalizes the classification of smooth Schubert varieties in cominuscule G / P.

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Notes

  1. There is a typo in [14, Section 2.7.1]; see Sect. 4.4 for the corrected statement.

  2. See, in particular, Step 3 of [17, Section 7.3]; in the case under consideration, \(t = 1\) and \(i_t = \mathtt {i}\).

  3. In Hodge theory, rational homogeneous varieties arise as the compact duals of period domains (and, more generally, Mumford–Tate domains). In that context, a submanifold M is horizontal if and only if it satisfies Griffiths’ transversality condition. The set \({\mathcal {C}}_o\) of lines makes two distinct and unrelated appearances in Hodge theory: it may arise as the kernel of the Griffiths–Yukawa coupling (a differential invariant associated with a variation of Hodge structures), and as the compact dual associated with certain “boundary components” that arise when studying degenerations of Hodge structure.

  4. There is a typo in [14, Section 2.7.1]; on p. 87, S and \(S'\) should be swapped. This is corrected for in the discussion above.

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Authors and Affiliations

  1. Department of Mathematics, Washington University in St. Louis, One Brookings Drive, Campus Box 1146, St. Louis, MO, 63130-4899, USA

    Matt Kerr

  2. Mathematics Department, Duke University, Box 90320, Durham, NC, 27708-0320, USA

    Colleen Robles

Authors
  1. Matt Kerr
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  2. Colleen Robles
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Correspondence to Colleen Robles.

Additional information

Kerr is partially supported by NSF grants DMS-1068974, 1259024, and 1361147. Robles gratefully acknowledges partial support from the NSF via grants DMS 1559592, 02468621, and 1361120. This work was undertaken while Robles was a member of the Institute for Advanced Study; she thanks the institute for a wonderful working environment and the Robert and Luisa Fernholz Foundation for financial support.

Appendix: Dynkin diagrams

Appendix: Dynkin diagrams

For the reader’s convenience we include in Fig. 2 the Dynkin diagrams of the complex simple Lie algebras. Recall that: each node corresponds to a simple root \(\alpha _i \in \mathcal {S}\); two nodes are connected if and only if and in this case the number if edges is (that is, i, j are ordered so that the inequality holds). Below, if \(G = B_r\), then \(r \geqslant 3\); and if \(G = D_r\), then \(r \geqslant 4\).

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Kerr, M., Robles, C. Classification of smooth horizontal Schubert varieties. European Journal of Mathematics 3, 289–310 (2017). https://doi.org/10.1007/s40879-017-0140-x

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