### ICTP Summer School and Conference on Hodge Theory

Thursday, July 01st, 2010 | Author: Konrad Voelkel

Writing from my very last day at the ICTP in Miramare (Trieste), Italy, I thought it's time to summarise some impressions, as promised. First, some remarks on the ICTP and sightseeing around Miramare (which might be useful to future visitors), then I will comment on the Summer School and finally the *Conference on Hodge Theory and Related Topics*.

### The ICTP

- The concept of the ICTP, founded 1964 by Nobel laureate Abdus Salam, seems to be, very broadly said, to give researchers in physics and mathematics from third-world-countries opportunities (including money) for research and learning with scientists from developed countries. To accomplish this goal, they have short-time visiting scientists (for about 6 months) and host a lot of summer schools, conferences and workshops in many different areas. Participants from not-that-much-developed countries get funding for housing and food, in the ICTP Guest Houses and restaurants. In my opinion, this works well, since I met several mathematicians from developing countries (like India or the USA ;-) ).
- The buildings I've heard of are: Leonardo Da Vinci Building, Enrico Fermi Building, Adriatico Guest House, Galileo Galilei Guest House. Just some small hints for anyone who intends to go to the ICTP:
- In the Leonardo Building main lecture hall, there is only one point in the whole room where you can find outlets. So, if you intend to use a laptop the whole day, find this spot (above, in the centre) and stick to it :-)
- The coffee machine in the Leonardo Building (near the toilets, ground floor) isn't that bad, if you choose the sugar level to be less than the default value of 6/8 points (more than 1/8 is already too sweet for me, and I like sweet coffee).
- The library in the Leonardo Building is a very nice place to hang out and wander off.
- In the Adriatico Guest House,
**there is a vending machine for beer**. It costs only 60 cent and isn't that bad. The problem is: if you come late, it's empty.

- The food in the Leonardo restaurant at the ICTP was not as good as you might expect from Italy. Especially in the evening, you get exactly the same as for lunch - but now it's probably cold and you have less choice. Therefore, a lot of people went to the next city to have dinner (which is, of course, more expensive).

### Sightseeing around Miramare, Trieste and Venice

- Miramare: there is the famous Miramare castle, surrounded by an artificial (yet beautiful) park, which is a must-see (i.e. exotic birds, fishes, turtles, palm trees, small pathways etc.).
- Trieste: I have no idea what I saw, I just walked across the city randomly. It is a nice, small city with a lot of restaurants. The sea side is not that beautiful but the buildings often look like imported from Vienna.
**I totally recommend eating ice cream**all the time. - Walking from Miramare to Trieste: Probably a bad idea. I did it three times, but I have to admit that the last third of the way, which is about 30 minutes long, is not beautiful at all. If you don't know where to go, you'll end up walking down a big street with high walls to the left and the right side for the whole 30 minutes. It is impossible to walk this last part along the sea. The first part, speaking of the hour it takes to walk from Miramare to Barcola, is nice, although the way along the sea is also the way along a big street. Anyway, there is a bus.
- Grotta Gigante, a huge cave, probably (at least officially) the largest cave that you can visit as ordinary tourist, in the world. And it is really gigantic! I've never been to any cave before and it was impressive.
- Venezia - only 2 hours and 10 EUR away, very nice (I guess there is no need to say more).

#### Miramare park main gate, near the ICTP

### Summer School

The summer school took two and a half week, where the first week was devoted to (recall?) very basic material: the first lectures defined manifolds and varieties and rushed over to de Rham cohomology, Kähler forms, sheaves, schemes and by the end of the first week the lectures about variations of Hodge structures and mixed Hodge structures reached the limit of what I knew already before (since I used my train ride to skim through Voisin's book).

Now seems to be a good time to thank Andreas Höring for having teached classical geometry of Kähler manifolds in Paris so well.

Since the lectures of Migliorini and de Cataldo on the Hodge theory of maps assumed knowledge of Abelian categories and spectral sequences, I still don't understand why the summer school started with these basic courses. I can not imagine that anyone who understood any of the lectures in the second week, could have possibly needed to learn what a manifold or a variety is. Maybe there are some hidden motives behind this, I haven't asked the organisers.

The organisation was very good, all course material was printed out in sufficient quantity, the lectures didn't take more time than expected, **there was coffee & cookies** in sufficient quantity twice a day and the overall working and learning atmosphere was fine. I missed problem sessions a little bit, but the intended problem sessions might have just turned out to be example sessions instead because of people requesting this. I was impressed that the organisers found a quick replacement for Claire Voisin, who couldn't come to Trieste. The replacement talks given by C. Schnell, F. Charles and M. Kerr were very good and understandable.

For the participants I recommend having a look at Charles Siegel's blog and website for lecture notes he has taken.

(photo licensed from Mike Scoltock under a Creative Commons Attribution-NonCommercial 2.0 Generic License)

### Conference

- D. Arapura:
*Beilinson-Hodge cycles on semiabelian varieties*; his joint paper with Kumar on the Beilinson-Hodge-Conjecture is related. This was my favourite talk! (By the way, see also his (unrelated) exposition of D-Modules and related Hodge Theory which I didn't know about before (thanks to Charles Siegel for pointing me to it)). - L. Illusie:
*Semistable reduction and vanishing theorems, after Lan and Suh*. - P. Griffiths:
*Hodge domains and automorphic cohomology*; there are some talk notes available. For the necessary background on Mumford-Tate groups, see Griffiths' lecture notes from the summer school. - M. Green:
*Vanishing of Chern Polynomials for Hodge Domains*. He introduced his talk with a joke along the lines of "It's my seventh talk since I came to the ICTP, and in many cultures, the number seven is special. After six days you should rest". The talk was about his SIGMA joint paper with Carlson and Griffiths. - C. Schnell:
*Néron models and Poincaré bundles*. It wasn't so much about Poincaré bundles than about mixed Hodge modules, Néron models and admissible normal functions. - G. Pearlstein:
*The locus of the Hodge classes in admissible variations of mixed Hodge structure*(joint work with Brosnan and Schnell). - H. Movasati:
*Automorphic functions for moduli of polarized Hodge structures*. He gave some intuition on modular forms as generating functions, then looked at the Hodge theory of elliptic curves, explained the notion of quasi/differential modular forms (the terminology seems to be unstable) and discussed some examples from physics.

- Doran:
*Normal forms for lattice polarized K3 surfaces and Siegel modular forms*(I had to skip this talk in favour of sleep - but I was told that it was good). - S. Usui:
*Neron Models in log mixed Hodge theory by weak fans*; slightly related preprint. - J. Carlson:
*Further speculation and progress on Hodge theory for cubic surfaces*(joint work with D. Toledo); related preprints. He introduced the talk with a story which concluded by "if you're confused, just keep going". - F. Charles:
*Remarks on the Lefschetz standard conjecture and hyperkähler varieties*, see his preprint on the topic. - L. Maxim:
*Characteristic classes of complex hypersurfaces*, see related paper. He introduced the virtual tangent bundle of a (possibly singular) hypersurface in a smooth manifold (the difference between tangent and normal bundle in K-Theory) and functorial homology characteristic classes (like Todd, L, Chern, but on homology); the general case are*Hirzebruch-type invariants*. He proceeded to express the (complicated) Brasselet-Schürmann-Yokura "Milnor-Hirzebruch"-classes in terms of virtual Milnor-Hirzebruch classes and invariants of the singularities. - M. Kerr:
*Mumford-Tate groups and the classification of Hodge structures*(more accurately, classification of Mumford-Tate subdomains, joint work with Griffiths and Green). - C. Siegel:
*The Schottky Problem*. He explained the well-known genus 3 and 4 cases and his approach to genus 5. - P. Dalakov:
*Deformations of the Hitchin section and DGLA's*.

- sadly, I missed the conference talks of E. Cattani, S. Cautis and L. Migliorini because I travelled back to Germany on Friday morning.

Again, Charles Siegel took notes for some of the conference talks, see here and here.

### Conclusion

I would be happy to go to the ICTP again in this life. Also, Hodge theory seems to be nice (at least parts of it).

small remark: I heard some rumour about a conference last year where they decided about the pronounciation of "Hartshorne". Clearly, the person is called Harts-horne, as several Australian mathematicians told me, but now the books name was decided to be Hart-shorne. Hilarious!