Claire Voisin, born on 4 March 1962, is a highly distinguished French mathematician specializing in algebraic geometry, and is widely recognized as a leading international expert in the field.
Throughout her career, Voisin has made significant contributions to several key areas of algebraic geometry and related fields, including the Kodaira problem, algebraic cycles, the Hodge conjecture, Green’s conjecture, the rationality problem, and mirror symmetry.
Her exceptional work has been recognized with numerous prestigious awards, including the European Mathematical Society Prize (1992), the Ruth Lyttle Satter Prize in Mathematics (2007), the Clay Research Award (2008), the CNRS Gold Medal (2016), the Shaw Prize (2017) and the Crafoord Prize (2024).
In addition to her awards, Voisin has been an invited speaker at the International Congress of Mathematicians (ICM) in 1994 and a plenary speaker in 2010. She also held the chair of algebraic geometry at the Collège de France from 2015 to 2020. Her contributions have earned her membership in several esteemed scientific societies, such as the Académie des Sciences, the Royal Society of London, and the National Academy of Sciences (US).

Lee: Thank you, Claire, for accepting our invitation to be interviewed. I know it's a long week, with jet lag and everything else.

Lin: You look more nervous than Claire.

Lee: Haha! May we start by asking Claire to give our readers a self introduction. In particular, can you (Claire) tell us why you chose to be a research mathematician, and what influenced your decision?

Voisin: I'm Claire Voisin. I am 62 and I work as a researcher in algebraic geometry. I have had the CNRS position since 1986. In fact, I was very lucky because I was 24 when I got that position. That means that starting from that time, I had a full-time researcher position at CNRS. And most of my career has been done at this institution, CNRS, except for a parenthesis of 4 years where I have been appointed as a professor at Collège de France. That was between 2016 and 2020. In fact, this is extremely honorific. Collège de France is a very famous institution. French people are very proud about it because it is a historic place which has been founded in the 16th century under the King François I. The foundation of Collège de France was part of this extraordinary period called humanism, where intellectual life was considered as a priority. Collège de France is a very beautiful institution and when I have been called there, I was extremely proud and honored. But I must say that after a while, I felt that it's absolutely not the right place for me.

Lee: Why is that?

Voisin: Maybe the main reason is that CNRS is a big institution where you are anonymous. You are just an employee number. This obscurity, for me, was more propitious, favorable to do mathematics. When I was at Collège de France, I was more in the limelight. I had to answer interviews and I had to bear this feeling that I am an important person. That was too much for me. Finally, I actually feel better being in obscurity. So I made a decision to leave Collège. Making this decision of leaving a position which is very honorific was not totally obvious. But I absolutely did not regret because when I came back to CNRS, actually I felt much better. And since I did very good mathematics, I have no regret. So there was this parenthesis. Apart from this, my career has been completely linear, I think, just following the promotion after promotion that you have when you are an employee in some place. And for how I went to mathematics, the French system has some good aspect that I didn't have to make a choice at an early age and I come from an easier period when we, the young people, did not worry too much about their future. I absolutely never asked myself what I will do in the future. In my whole studies, I always followed my interests and of course, I believe that was made possible because the system worked quite well for me.

Voisin: At 17, I got a scholarship. I left the family house. My father was an engineer who lost his job in I think, maybe 1973, so there was not that much money. Also I come from a family of 12 children, so life at home was not that easy. Thanks to this, I got a scholarship and that was very good for me because I was completely free to do what I liked. I didn't have to ask money from my parents. With the scholarship, I went to classes préparatoires. Then I entered École Normale Supérieure and I got this small salary that we have in these grandes écoles in France. It is not a big salary, but it is already something generous because it tells you, “we like the fact that you study and please do your best.” So finally I always felt very free to follow my interest. At some point in classes préparatoires, I discovered very serious mathematics. In Classes préparatoires, the teaching is good, very substantial. But still, at that time I felt maybe from the viewpoint of depth or creativity or the general idea, there were still too many récipes; one had to learn to apply a certain finite number of tools to work on exercises, to learn theorems. So for me, the school was okay, but I didn't see the depth and how I can make it mine. At that time I didn't know that I wanted to be a mathematician. But of course I liked it; I was good at that. At that time, after I entered the École Normale Supérieure, I thought that I wanted to study philosophy because I thought there would be deeper insight into a deeper life. And finally, when you do more and more mathematics, you really see that the ideas are deep; you understand the importance of definition from the inside. And there is a whole intellectual construction that is very deep and you see also that your mind is fully involved. What I want to say is that up to a certain point, it was mostly about being good at that and good at school and just doing what I was supposed to do. Finally more and more it was really about discovering beautiful ideas. So I will not say there was a point where I told myself now I want to be a mathematician, but I was more and more interested and there was absolutely no obstacle because at École Normale Supérieure, I had this salary. After that, very quickly I got this CNRS position, so I never had to ask the question what I am going to be. I was there before I formulated the question.

Lee: If I understand you correctly, you said when you entered École Normale Supérieure, you were thinking of maybe doing philosophy. Then during that period gradually you were thinking more and more about mathematics and discovered mathematics for yourself.

Voisin: Absolutely. For example, up to a certain point, I thought mathematics is something that is a finite construction, maybe like chess. And I am absolutely not interested into games of any kind. I am a very serious person. I was really more interested into how our brain can produce abstract concepts. So I wanted to really put my hand on the source of the idea and the way mathematics was taught.

Lee: I have this impression that mathematics in France has an elevated status compared with mathematics in other cultures. Is that true? I know a French engineer who really wanted to be a mathematician, but he gave up mathematics for engineering because he did not think he was good enough. He said if he had the chance, he would love to do mathematics and assured me that it was a general attitude among a majority of the French population.

Voisin: It's true, but at the same time, it is a very delicate subject. Maybe you don't read the newspaper Le Monde. They published recently a paper explaining that the children in France are among the last ones in the ranking of success in math at school.

Lee: I see.

Voisin: Maybe we are not exactly last, but inside a group of wealthy countries OECD, we do very poorly. So there is a big question about that because at the same time, of course, we have a very high level of mathematical tradition and many Fields medalists for the size of the country. I know by experience that the teaching in classes préparatoires is really excellent and that you can really become an excellent mathematician being in France. At the same time, if you consider what happens at school (up to high school), the situation is, I would say, close to a disaster.

Lee: Is this a recent phenomenon?

Voisin: In this same paper, they argued by a second derivative argument that our decline started to ...

Lin: Reach the minimum?

Voisin: Now we are in a very bad situation. I think that it is at least for 20 years or maybe more, 30 years, that we have seen that problem. I think that one problem is the bad level of teachers in elementary school, bad level in mathematics. This is probably due to -- relative to what you mentioned at the beginning -- the fact that mathematics is highly valued. And the consequence is that people who accept to be teachers in elementary school do not have a good level in mathematics because otherwise they could get a better salary by doing another job. In fact most of them have better background in literature, history, human sciences. So there is this reason. The other reason is that there has been a complete lack of coherence in the way of organizing the teaching of mathematics. Let us speak about primary school, generally between 7 and 10 or 11 for example. The big question is: Should we give priority to just practice with numbers or solving problems ? The conclusion of this incoherence is that,what we call the program, what each teacher has to teach in a given level, has changed every 5 years since maybe the 60’s. This is also a reason for a bad level of teachers. They don't know what to teach and what they teach does not correspond to the way they learned mathematics themselves.

Lin: Have you ever thought of giving some suggestion to the government? Since in France ...

Voisin: I think the only suggestion that I would make is that they give a better salary to the teachers in primary school and high school because if you give a good salary to people, then the people who have a good background will be interested. You can certainly be a teacher because you like it, because it's a wonderful situation, it's a beautiful way of earning your salary. But the problem is that if the salary is so small that it is ridiculous, who wants to do this?

Voisin: I don't work with children. Of course I had children, I did mathematics with them but I thought it's absolutely not obvious to do mathematics in a constructive way with young children. I have 5 children and the last was born when I was 35, and so when I was around 35 or 40, I did some mathematics with them. Naively, because I like abstract things, I thought that the most interesting thing to discuss with my children was the logic, for example set theory, explaining what is an equivalence relation, what you can do with this. I discovered this was absolutely not good pedagogically. My children were much better at the same age if I, for example, turned them into solving the problems of 2 affine equations with 2 variables. You go to the shop and you buy a cake and 2 croissants and another time you buy a croissant and 3 cakes, at the end, you have to know the price of the cake and the croissant. My children were much better even at a very young age in solving these things. They were quite good in mathematics. However, I think if the children are too young, they do not have the ability in terms of the abstraction. What I want to say is that as a professional of mathematics, I would be the worst person to advise people about what to teach.

Lee: You said in France, there is the CNRS system allowing you to concentrate on research without teaching duties and with minimal other duties. This seems to be quite unique in France. I do not see any other similar system with this scale...

Voisin: I think there are some imitations like in Belgium.

Lin: FNRS.

Voisin: And I think also in Italy, there is something that looks a little similar, but not exactly like this.

Lee: In France this is done in a large scale. A lot of people are employed by CNRS. In other places, you might have selected few who have similar positions.

Voisin: CNRS is a democratic place and they hire beginners. It is true that it is a remarkable institution. It was created in a moment when France was doing better in the 60s, with the more generous viewpoints of the society. We are very lucky to have it. And I must say also that this is also very good to attract women to mathematics because we are still for a long time in a society where women are supposed to do more for their home duties and raising children. In case with my husband, we always share the home duties very equally, but the fact is that there are the pregnancies. Also mathematics is extremely competitive even if one doesn't like that word. It’s a fact. If you lose 9 months for a pregnancy and then another year for the baby and you accumulate this with 2 or 3 children, then it is almost impossible to stay in the course. For me, I consider that I have been extremely fortunate. I could never stop doing mathematics and that was made possible essentially because I was at CNRS. It would have been too much time consuming just to go to the university. So I was staying at home doing my research at home, which saved enough time and energy for me.

Voisin: I think that it is something that people try to do a lot to help women should think about. My feeling is that sometimes they simply do not do what they should do. They say they will help women by giving them priority in hiring. But after they get the position, the women have the feeling that they have been hired because they are women and it is not very pleasant. The very simple and practical things like giving them more freedom, more research time, are more helpful. This is really what is essential for women and it is true that the female French mathematical school is very good. We have many female mathematicians of very good level and if you look most of them went to CNRS.

Lin: I think another related problem is how to have more women students. For instance, in our math department, this year, only 2 of the new students are women. This is also problematic.

Voisin: We are faced with the same problem.

Lin: If there are no students, there are no researchers in the future.

Voisin: The same problem in France. In some sense, the problems do not begin at an advanced stage. They appear from the beginning. The girls, they are disappearing from the courses. It is true that in France, mathematics is considered as a sort of, how to say, it is used as a selection tool. It is presented from the beginning, even in elementary school and primary school, as something which is competitive. And already there’s difficult start for girls. But in fact, the situation is even worse. There are some clichés about the ability of girls in science and especially in mathematics. These clichés are about what are the best qualities of a girl. I think that at a very young age the girls get the idea of how to please their parents or society. It is terrible. I have grandchildren. I see my granddaughter who is 8 and she already has this.

Voisin: Let me come back to what I said at the beginning, the fact that I need obscurity to do mathematics. Because when you do research in mathematics, you really need to put all the world away and need to concentrate inside you. This is completely transverse, opposite to the idea to face the society around, to please people. I believe that in the education of girls they are more pushed toward being pleasant to look. Even this idea that they have to look beautiful is a nightmare. That means that they already depend on people looking at them. For me, it is about how they think about themselves.

Lin: Would you share this opinion with your children or children-in-law?

Voisin: Of course. You mean how I was as a mother?

Lin: Yeah, as a mother.

Voisin: Some of our children are mathematicians…My husband is also a mathematician. He is a high level mathematician. We both have a sort of deep priority in intellectual life. I think that we really conveyed to our children this idea that intellectual achievement constructs your personality. This is not true only of mathematics, not true only for those of our children, who are now professional mathematicians. I have never been very obsessed by the success of my children at school, because when I was myself a child, being good at school was not my priority. I just wanted to learn and to progress. I was not very much inside the system. I never encouraged my children to just to follow, to just try to be successful. It was more about the fact that, in the activities we had, thinking was very important. For example, I remember, it was not necessarily about mathematics. You see, before doing mathematics, you have to master the language. You know how to construct rigorous sentences, not only the grammar, but also logically. I remember we did a lot of it. It was for fun in some sense, but a lot of games are about the language. For example, we were walking, someone, maybe me or one of my children, told a story, and in this story, there was a contradiction. We had to find the contradiction. Or another game which was, I think, in some sense closer to mathematics. Your know these stories that you read for your children, Perrault, Andersen, the Walt Disney stories, etc. So then I took one of these famous stories and I told more or less the same story, but changed all the details. The 3 pigs became 3 sisters. And all the details, the context, the situation around had been changed. But the story, how it developed and the conclusion are more of less the same. The children had to find which story it was. That was a lot of fun, these things, but in fact, we were very close to mathematics. You know, in mathematics, I think, Poincaré said that mathematics is the art of giving the same name to things that look different, but work the same way. And in some sense, when we did these games around the language, we were closer to mathematics than to literature.

Lin: And also, many people might have already asked you this question. As a mathematician, you are a successful mathematician. Also, outside mathematics, so family-wise, also successful. So how to reach this balance between mathematics and life? Could you share something about this with us?

Voisin: In fact, I have been asked this question many times. I always give the same answer. For me, having this family -- it was a big family -- of course, took me a lot of time. But I think that it also helped me a lot because I always found it hard psychologically to do mathematics. There are very bad periods, periods of very low inspiration, periods where you are less in love with mathematics. And in these periods, I felt, I would say, unwell with respect to my job. And I would say that my family, that was something to the contrary, something where I always felt rather strong and doing well. So, I consider that my family made my personality stronger, while, for me, the mathematics is a little bit destroying because you are faced with failure and doubt. Also, you are faced with so many very clever people. So it's definitely, for me, very hard. And it's not so easy to feel that you are in the right place when you do mathematics. So in some sense, I got more confidence from the family side than from the mathematics side.

Lee: You have many children, right?

Voisin: Yes, five.

Lee: It’s a lot of effort to raise them well. But you said you also get a lot of satisfaction and confidence from these children.

Voisin: Yes, I think that this gave me the balance that I needed. Otherwise, I think that…I think that I was very lucky. I think that this really made me a more balanced person. I don't think I was strong enough psychologically just to do mathematics and just to devote my whole life to that.

Lin: In daily life, if not mathematics, what other things would you enjoy doing?

Voisin: I read a lot. I read novels, I re-read classical novels that I read in the past, but also some new things. So now maybe being older, I also read sometimes biographies or essays. Otherwise, I like to paint, also to picture, also some clay modeling. When my children were still at home I played music with them. All of them played music very nicely. I followed them in music, but I started too late and I was not totally satisfied. You know, I started the violin at 35 or something like that. And I never got a beautiful sound, which is probably the most important thing.

Voisin: In any case, there are a lot of things that I like to do. At the moment I don't play music anymore because I no longer have children to play with. But I still do some manual activity. I like to walk also. I don't have so many hobbies but I am very good in a game called Bilboquet. Maybe you don't even know the name.

Voisin: This is ridiculous, but it was a classical game. I think one of the sons of Catherine de Medici, maybe Charles IX was very fond of it. At that time, bilboquet was very fashionable. To play the bilboquet, you have a sort of stick, and then there is a ball, which is related to the stick by a cord. And you lift it and you have to catch...

Lin: In Japan, I think there is something similar called kendama.

Voisin: For some reason, I am very good at that. It's just a joke.

Lin: Do you like to sometimes think about mathematics while walking?

Voisin: Absolutely. It is my main reason to walk. Yes, really, I live in Paris, and I like to walk in Paris. Even if it is noisy, it puts me in the better mood to think about mathematics than if I am in the forest or in the nature, where I would look at the trees. But in Paris, I am just walking very quickly. Usually, I like to live with a problem in my mind. It is very pleasant. I think that walking helps, especially if you have really specific questions that you want to consider.

Lin: Even technical questions? Or you have some choice of...

Voisin: It is true that, I mean, if you have to make a hard computation, you are not going to do that. But recently, my husband asked me a small question about topology of surfaces. It was just topology of domains in the ${\mathbb{R}}^3$. He had a question about whether those which are bound to spheres can have just a cohomological characterization. I was very happy about this interruption of my own research. I started to walk in the street. I saw that it was a consequence of Poincaré duality for the manifold with boundary. It gave exactly the characterization that he wanted. My husband works in analysis. He makes a lot of computations, write, write, write. So, for him, doing mathematics is really being at his desk and making many computations. Of course, there is a part where he has to invent a strategy, but then, after that, the technical work. In algebraic geometry, it is very different. Most steps are rather conceptual. There are not so many very technical aspects. The technical aspects, sometimes you discover them when you are already writing the paper and you realize that one lemma is not so useful as you thought. In fact, it is true that I spend a lot of time doing mathematics away from my desk.

Lee: Going back to what you said, you started thinking about pursuing mathematics while you were in École Normale Supérieure. And then, at which point did you decide to do complex or algebraic geometry? Was that inspired by a certain teacher or did you just like it?

Voisin: In fact, it was not immediate because the years where I really discovered many different sorts of geometries were what we call M1, M2, Master 1 and 2, just before the thesis. In one of these two years I had and excellent course on Riemannian geometry by Marcel Berger, who died a few years ago. Riemannian geometry was very impressive. I had also an excellent course about Kähler geometry by Dan Burns and I was really struck when discovering Kähler geometry. Finally, I would say that originally I was considering more doing maybe symplectic geometry or even Riemannian geometry. For example, I went to discussing with Jean-Pierre Bourguignon who was professor at Ecole Polytechnique at that time. Finally, somebody at École Normale Supérieure told me that I should discuss this with Beauville. It was very good advice because if you do symplectic geometry, there are many very conceptual aspects, but analysis still plays an important role. Less than in Riemannian geometry, of course, but still analysis is important. For example, if you look at this incredible work by Donaldson about the existence of symplectic hypersurfaces, symplectic submanifolds of codimension 2 in a symplectic manifold, there is a lot of analysis to construct them. So, I should have done analysis, but I don't think I am good at that. So it was a very good advice from this person, Marie-France Vignéras, who told me that maybe algebraic geometry would be good for me. And indeed, I must say that in all the methods that I used, there was a lot of algebra. Even in Hodge theory, of course there are some analytical aspects in it, but if you look at the object, and this is still more true if you do the infinitesimal variation of Hodge structures, there is a lot of algebra. I like the geometry, the geometric structure, but it is true that when you do algebraic geometry, finally the tools are from algebra. And I think that was much more prudent for me, working in that field. But I arrived there not directly. I always remembered my original interest to Kähler geometry.

Lin: How did you learn algebraic geometry at that time?

Voisin: At that time, I read very nice books on commutative algebra, like Lang, Atiyah-Macdonald, Matsumura, which gave me a good background in commutative algebra, which is needed. Then I think I read Hartshorne, I spent one summer reading Hartshorne, doing the exercises. And then I read other books, but I thought that I learned a lot in Hartshorne. Later on, I read other books, for example, there are two books by Mumford. One of which is Introduction to Algebraic Geometry. Maybe that is not exactly the title, which was not totally convincing to me. For the other one, curiously, I felt I learned more algebraic geometry from it; it was his book on Abelian varieties. Maybe if you already had the Hartshorne, it doesn't bring you that much to read another introductory textbook. There were other books. Two books in particular are important: Arbarello-Cornalba-Griffiths-Harris and Barth-Peters-Van de Ven. The first one is about curves. Barth-Peters-Van de Ven is about surfaces. Sure, it's not general algebraic geometry, but in fact you learn a lot of general methods in these books. In the sense that Hartshorne was excellent for the whole introduction to scheme theory, cohomology of coherent sheaves, even more advanced things like flatness, characterization of flatness. For me, it was really an excellent book, with a sort of maybe deception that at the end of the book, you didn't see what an algebraic variety looks like and what you can do with it; it's more about general theory.

Voisin: A similar approach can be done in Kähler geometry. I read in my Kähler period the book by André Weil, "Introduction to Kähler Manifolds" in French, an excellent book, where he does Hodge theory, the Kähler identities, and some of the Picard group. At the end of the book, you never met a single compact Kähler manifold. These books were absolutely excellent. Only later on, I read other books, more dealing with the objects from algebraic geometry. I think I also learned a lot at the same period from the big papers by Griffiths. He has a big series of papers in the theory of variation of Hodge structures. And there was the paper with Clemens-Griffiths, a long paper, where he does many general things, starting from a particular class of variety. It's about the cubic threefold, and already, there was this beautiful idea, that Hodge theory, that looks very transcendental, in fact has algebraic aspects which are inside algebraic geometry. Later on, I discovered that when you do general Hodge structures, the objects are of very transcendental nature. But also, there was this whole series of papers about infinitesimal variation of Hodge structures, and this is totally remarkable because the Hodge theory obviously has some transcendental aspects, because you have to consider your complex algebraic variety as a topological object, consider the underlying complex manifold that has a topology. So you consider its Betti cohomology, but Betti cohomology with integral coefficients is not part of algebraic geometry. This is really the transcendental aspect of the story. Betti cohomology with complex coefficients can be constructed from algebraic geometry, the algebraic de Rham cohomology.

Voisin: And in Hodge theory, if you just want the Hodge filtration, it is also coming from algebraic geometry. So in Hodge theory it's really a mixture of transcendental considerations, and also algebraic geometry considerations. When you do the infinitesimal theory of this, that is you look at how the Hodge filtration varies when you deform your variety, of course the topological type remains the same, you just deform the Hodge filtration. Then it's well known that the data that you get are completely from algebraic geometry. There is a lot of algebra to do with that, like the Torelli theorem. You recover the variety from its infinitesimal variation of Hodge structures, and finally from its Hodge structure itself. It was rather fascinating, all of these ideas were developed exactly when I started mathematics, because it was in the mid-80s. I read all of these papers, and the beginning of my mathematical life was very much about the infinitesimal variation of Hodge structure. I had discovered this in papers that were rather recent.

Lee: You also wrote many books, including the books about Hodge structures. Since you don't have the obligation to teach, how did you decide to write these books and maybe teach them at some point?

Voisin: I have taught advanced courses in algebraic geometry, although not every year. The book was a result of teaching a course in algebraic geometry over 2 years, the first part and the second part which was more advanced. Each course was about 40 hours, and then I typed the lecture. But I also sometimes gave courses in algebraic geometry without providing written notes, and the last time I taught a course was last year. It was about the topology of families of algebraic varieties, and the most important results in deformation theory and variations of Hodge structures, including the big results by Deligne about the topology of families, but also some infinitesimal variations of Hodge structures. I just do it because I think it's good for me not to do only research.

Lin: You also have many mathematics works. Which among them are your favorite?

Voisin: My three favorites, I would say, are my work on the Kodaira problem, then my work on stable irrationality by degeneration, and my recent work with Kollár. I like these 3 works because they are elegant and that was not that much work. It was just that there was an idea which was finally very efficient. For the first 2, it was really not much time to catch and exploit the idea. For the more recent work with Kollár, it took me more time, but it was different because I was so happy thinking about an easy problem. This was a classical problem from the 60s, easy to state, but finally you really needed a new idea to attack it. And it was really different, and I spent more time, maybe 6 months before concluding, finally thanks to the help of Kollár. I was so pleased working on a problem which was not sophisticated in nature. Otherwise, of course, my other works are often more technical. It's a fight, and it's not always so pleasant to even give talks, because of the technicalities. Of course, this is needed. It is the nature of the mathematics we are doing. But these 3 works, what was nice about them was the elegance and simplicity.

Lin: Could we say that all these 3 works, one thing which is maybe something in common is algebraic cycles; algebraic cycles are something behind these 3?

Voisin: Yes but here, we are going to discuss my biggest frustration, which is the subject of algebraic cycles and Hodge theory. I didn't mention this was already the main subject of these papers by Griffiths, even if Griffiths didn't see everything. -- There were later some further very important inputs by Mumford and then Bloch. --- Progressively, we saw a picture where Hodge theory and algebraic cycles interact, and they are supposed to interact perfectly in the sense that the complexity of algebraic cycles is supposed to be reflected in the complexity of the Hodge structure. The Hodge structure is just an object from linear algebra, its complexity essentially is the length, we call this the level of the Hodge structure, it counts how many different $H^{p,q}$ you have in the decomposition. So it's a complexity that you can compute. But from this, if you believe the generalized Hodge conjecture by Grothendieck and much later the conjecture by Bloch and Beilinson, this Hodge theoretical complexity should dictate the size of the Chow groups.

Voisin: The simplest formulation typically is to take a smooth projective variety which has no non-zero holomorphic forms of positive degree. so $H^{i,0}=0$ is zero for i istrictly positive, this variety should have trivial $CH_0$ group; all the points are rationally equivalent. This is basically the Bloch conjecture. This conjecture still resists our best efforts. Ayoub announced a major big progress but he was still stuck at some point. For the same variety, we know that there are no $H^{i,0}$. The Hodge structures look like $H^{i,1}$ plus $H^{i-1,2}$ etc., but you do not have the extreme term. There is now this big conjecture by Grothendieck, which is a very important generalization of the Hodge conjecture, that tells you that the cohomology should be supported on a divisor. And, similarly -- you can imagine the generalization -- if your Hodge structures start only with $H^{i,2}$, you do not have $H^{i,0}$, $H^{i,1}$, you just have $H^{i,2}$, and continue, then the cohomology should be supported on a closed algebraic subset of codimension 2.

Voisin: There is zero progress on this conjecture. I find it is more important than the Hodge conjecture, because the problem with the Hodge conjecture is that finally there are comparatively few interesting varieties with a Hodge class. So for many algebraic varieties, we say the Hodge conjecture is true simply because there is no Hodge clas-ses except for the obvious ones. But if you look at the Grothendieck generalized Hodge conjecture, even for hypersurfaces, there are many examples where we do not know it. There are small degree hypersurfaces in projective space where we know how to compute this complexity, the level of the Hodge structures. This is thanks to Grif-fiths work. And there is no progress on the generalized Hodge conjecture in this case. This is my biggest frustration mathematically. This illustrates the fact that algebraic geometry is an incredible theoretical body of knowledge, built since the Grothendieck era, with later development, like K-theories, derived algebraic geometry. We have plenty of beautiful theories, but these theories do not give constructive tools. There are no construction tools for algebraic varieties. Curiously, about 10 years ago, there were these 2 conjectures that are the main conjectures in the field, the generalized Hodge conjecture and the generalized Bloch conjecture, that are both related to this shape of the Hodge structure. And typically for these low degree hypersurfaces, for which I would be very, very happy to know that they are true, I proved they are equivalent. That is, if you know the generalized Hodge conjecture, you also know the vanishing predicted by Bloch. So I find incredible being so stuck on a subject where I know exactly what is supposed to be true by these big equivalent conjectures. And even on these examples that I am quite familiar with, I am unable to solve them.

Lee: Thank you, Claire, for generously sharing your thoughts with us. Thank you, Hsueh-Yung. That’s all the time we have. Thank you very much.

• Yuan-Pin Lee is a faculty member at the Institute of Mathematics, Academia Sinica.
• Hsueh-Yung Lin is a faculty member at the Department of Mathematics, National Taiwan University.