# Semi-stable Higgs bundles, Spring 2019

Introduction. In this seminar we work through (part of) a paper of Dona Arapura proving a type of Kodaira-Saito vanishing for semi-stable Higgs bundles. This is fun because it uses characteristic p methods and in particular Ogus-Vologodsky correspondence.

Organizational:

1. This semester the students will give the lectures.
2. Please email me if you want to be on the associated mailing list.
3. Time and place: Room 407, Fridays 10:30 -- 12:00 AM.
4. First short organizational meeting: Friday, January 25 at 10:30 AM in Room 407. Everybody interested please attend.

Schedule: (things will still move around, etc)

• 2/01 Carl Lian, Lecture 1
• 2/08 Dmitrii Pirozkov, Lecture 2
• 2/15 Noah Olander, Lecture 3
• 2/22 Johan de Jong, Lecture 4
• 3/01 Raymond Cheng, Lectuer 5
• 3/08 ??
• 3/15 Qxiao Ma, Lecture 7
• 3/22 No lecture; spring recess
• 3/29
• 4/05
• 4/12
• 4/19
• 4/26
• 5/03

Lectures: Overall strategy. Understand the material in Section 2 of Arapura's paper [A] in positive characteristic with D = 0. Make sure to work with coherent sheaves (and not just vector bundles) at all times.

1. Introduction to the material. Talk about Deligne-Illusie overall and in particular how to prove Cor 2.11 using the splitting in the derived category.
2. Theorem 6 of [A]. Give the equivalence of categories between nilpotent Higgs sheaves of exponent less than p and the category of nilpotent flat sheaves of exponent less than p. This was first proved by Ogus and Vologodsky, but we will follow the proof as given in [LSZ].
3. Theorem 8 part (1) of [A]. Show that under the correspondence in the first talk, semi-stable objects correspond to semi-stable objects. This is Corollary 5.10 of [L3]; give a direct proof.
4. Theorem 5 of [A] which is the same as Theorem 5.5 of [L3]. Review Langton's method and then give the proof of the theorem roughly as in Langer's paper. Explain clearly what the output of the argument is.
5. Theorem 7 of [A] which is the same as Corollary 2.27 of [OV]. Let (E, theta) and (V, nabla) correspond via the equivalence of the first lecture. Explicitly produce an isomorphism between their de Rham complexes working affine locally and glueing. (This will require some actual thought to do.)
6. Corollary 2.4 of [A]. This is straightforward except that the statement of Lemma 2.5 has to be modified in order for it to make sense.
7. Boundedness results for semistable sheaves in positive characteristic. Explain enough so it can be used in the next lecture. See [L4].
8. Theorem 8 part (2) of [A]. You can look at the paper [LSYZ] for a proof.
9. Theorem 1 of [A]. Having done all the work above this should now be OK.
10. Applications of the theorem.

References:

[LSZ] Nonabelian Hodge theory in positive characterstic via exponential twisting, by Guitang Lan, Mao Sheng, Kang Zuo. arxiv link, doi link

[A] Kodaira-Saito vanishing via Higgs bundles in positive characteristic, by Donu Arapura. published link, arxiv link

[LSYZ] Semistable Higgs bundles of small ranks are strongly Higgs semistable, by Guitang Lan, Mao Sheng, Yanhong Yang, Kang Zuo arxiv link

[OV] Nonabelian Hodge Theory in Characteristic p, by A. Ogus and V. Vologodsky Ogus webpage, numdam link, doi link

[L1] The Bogomolov-Miyaoka-Yau inequality for logarithmic surfaces in positive characteristic, by Adrian Lander doi link

[L2] Bogomolov's inequality for Higgs sheaves in positive characteristic, by Adrian Langer doi link

[L3] Semistable modules over Lie algebroids in positive characteristic, by Adrian Langer published link

[L4] Semistable sheaves in positive characteristic, by Adrian Langer doi link This paper has an erratum, see doi link

[S] Logarithmic nonabelian Hodge theory in characteristic p, by Daniel Kenneth Schepler proquest link

[DI] Relevements modulo p^2 et decomposition du complexe de de Rham by Pierre Deligne and Luc Illusie IAS link