## Final Report Summary - GEOMODULI (Geometry of moduli spaces of bundles over curves)

The researcher C. Pauly has worked on several problems related to the geometry of moduli spaces of vector bundles.

A central theme of the investigation is the notion of oper introduced in the Geometric Langlands Program and applied to the universe of Frobenius geometry (positive characteristic) and Hitchin's moduli space of Higgs bundles. Based on a recent article, joint with Kirti Joshi (University of Arizona, USA), published in Advances in Mathematics, one extends the notion of oper to opers of higher types. It is shown that

opers of higher types which are nilpotent resp. dormant, i.e. have nilpotent resp. zero p-curvature, are finite, which parallels one of the main results for nilpotent resp. dormant opers. On the other hand it is observed that in the higher rank set-up the involved Quot-schemes which parameterize subsheaves of the Frobenius direct images have a dimension which strictly exceeds their expected dimension. This is quite

different from the line bundle case (classical opers). A conjectural formula for the dimension of the loci of stable Frobenius destabilized rank-r vector bundles is given and proved in the rank 2 case. This poject was carried out mainly during a visit of Kirti Joshi (June 2016) to the

Laboratoire Dieudonné in Nice. The final version of the article will be available shortly. It should be noted that dormant opers and their moduli have attracted some attention from several Japanese mathematicians (S. Mochizuki, Y. Wakabayashi).

A project on a similar topic was undertaken in collaboration with Adrian Langer (University of Warsaw) during his visit at the Laboratoire Dieudonné in June 2015. The project consists in extending the deformation of the Hitchin system in positive caracteristic, which was worked out in a previous paper by Laszlo and Pauly in the curve case, to a smooth projective variety of arbitrary dimension defined over a field of positive characteristic. The main step consists in checking that the coefficients of the characteristic polynomial are flat for the canonical connection. Then, the project moves on to investigating the possibility of generalizing the notion of opers to higher dimensions.

Jointly with Thomas Baier (Lisbon), Michele Bolognesi (Montpellier) and Johan Martens (Edinburgh), C. Pauly investigated the Verlinde spaces for rank 2 and level 4, which is the space of global sections of the fourth power of the determinant line bundle over the moduli space

of semistable rank-2 vector bundles over a smooth projective curve. The reason for looking at this particular case is an unsolved and intriguing problem on the monodromy of the Hitchin/WZW connection on the associated sheaf of conformal blocks or Verlinde bundle, as well as a result due to W. Oxbury and C. Pauly giving an isomorphism with abelian theta functions of level 3 on Prym varieties. Some partial results on the Hitchin connection have been obtained, giving evidence that the monodromy should be finite in this particular case.

During the period May - August 2017 Ana Peon was visiting the Laboratoire Dieudonné as a post-doctoral researcher. The following

result was show: a vector bundle is very stable, i.e. the bundle has no non-zero nilpotent Higgs fields, if and only if the restriction of the Hitchin map to the cotangent space at the point defined by the vector bundle in the moduli space is a proper map. This result and its corollaries have given rise to a preprint.

Moreover, the C* -action on the nilpotent cone in the moduli space of Higgs bundles was investigated. Based on work of T. Hausel, P. Gothen and J. Heinloth, one expects to arrive at a clear understanding of the two limit points (zero and infinity) of this C*-action in terms of the underslying vector bundle and its associated filtrations (Harder-Narasimhan and iterated kernels/images).

During the period 2013 - 2017 C. Pauly supervised a PhD student Hacen Zelaci. The subject of the thesis was a generalization of the classical notion of Prym variety to higher rank vector bundles, i.e. the moduli spaces of anti-invariant vector bundles over a smooth projective curve equipped with an involution. The analogue of conformal blocks for twisted affine loop algebras was also studied in connection with these moduli spaces of anti-invariant bundles.

Jointly with two post-docs (Ana Peon and Marco Antei), the researcher C. Pauly organized a 3-days school on the fundamental group scheme in May 2014 and a one-week conference on Higgs bundles and related topics in May 2017 (for the details, see section 2).

A central theme of the investigation is the notion of oper introduced in the Geometric Langlands Program and applied to the universe of Frobenius geometry (positive characteristic) and Hitchin's moduli space of Higgs bundles. Based on a recent article, joint with Kirti Joshi (University of Arizona, USA), published in Advances in Mathematics, one extends the notion of oper to opers of higher types. It is shown that

opers of higher types which are nilpotent resp. dormant, i.e. have nilpotent resp. zero p-curvature, are finite, which parallels one of the main results for nilpotent resp. dormant opers. On the other hand it is observed that in the higher rank set-up the involved Quot-schemes which parameterize subsheaves of the Frobenius direct images have a dimension which strictly exceeds their expected dimension. This is quite

different from the line bundle case (classical opers). A conjectural formula for the dimension of the loci of stable Frobenius destabilized rank-r vector bundles is given and proved in the rank 2 case. This poject was carried out mainly during a visit of Kirti Joshi (June 2016) to the

Laboratoire Dieudonné in Nice. The final version of the article will be available shortly. It should be noted that dormant opers and their moduli have attracted some attention from several Japanese mathematicians (S. Mochizuki, Y. Wakabayashi).

A project on a similar topic was undertaken in collaboration with Adrian Langer (University of Warsaw) during his visit at the Laboratoire Dieudonné in June 2015. The project consists in extending the deformation of the Hitchin system in positive caracteristic, which was worked out in a previous paper by Laszlo and Pauly in the curve case, to a smooth projective variety of arbitrary dimension defined over a field of positive characteristic. The main step consists in checking that the coefficients of the characteristic polynomial are flat for the canonical connection. Then, the project moves on to investigating the possibility of generalizing the notion of opers to higher dimensions.

Jointly with Thomas Baier (Lisbon), Michele Bolognesi (Montpellier) and Johan Martens (Edinburgh), C. Pauly investigated the Verlinde spaces for rank 2 and level 4, which is the space of global sections of the fourth power of the determinant line bundle over the moduli space

of semistable rank-2 vector bundles over a smooth projective curve. The reason for looking at this particular case is an unsolved and intriguing problem on the monodromy of the Hitchin/WZW connection on the associated sheaf of conformal blocks or Verlinde bundle, as well as a result due to W. Oxbury and C. Pauly giving an isomorphism with abelian theta functions of level 3 on Prym varieties. Some partial results on the Hitchin connection have been obtained, giving evidence that the monodromy should be finite in this particular case.

During the period May - August 2017 Ana Peon was visiting the Laboratoire Dieudonné as a post-doctoral researcher. The following

result was show: a vector bundle is very stable, i.e. the bundle has no non-zero nilpotent Higgs fields, if and only if the restriction of the Hitchin map to the cotangent space at the point defined by the vector bundle in the moduli space is a proper map. This result and its corollaries have given rise to a preprint.

Moreover, the C* -action on the nilpotent cone in the moduli space of Higgs bundles was investigated. Based on work of T. Hausel, P. Gothen and J. Heinloth, one expects to arrive at a clear understanding of the two limit points (zero and infinity) of this C*-action in terms of the underslying vector bundle and its associated filtrations (Harder-Narasimhan and iterated kernels/images).

During the period 2013 - 2017 C. Pauly supervised a PhD student Hacen Zelaci. The subject of the thesis was a generalization of the classical notion of Prym variety to higher rank vector bundles, i.e. the moduli spaces of anti-invariant vector bundles over a smooth projective curve equipped with an involution. The analogue of conformal blocks for twisted affine loop algebras was also studied in connection with these moduli spaces of anti-invariant bundles.

Jointly with two post-docs (Ana Peon and Marco Antei), the researcher C. Pauly organized a 3-days school on the fundamental group scheme in May 2014 and a one-week conference on Higgs bundles and related topics in May 2017 (for the details, see section 2).