I prove that face rings of general triangulated spheres and manifolds
satisfy the Hard Lefschetz theorem for suitable (and in particular
generic) Artinian reductions, generalizing the classical version for
rationally smooth projective toric varieties greatly beyond the setting
of Kähler structures. This has several applications to combinatorial
topology, in particular:
- It resolves the g-conjecture, that is, we can now
fully characterize the possible face numbers of simplicial rational
homology spheres, resolving the g-conjecture of McMullen in full
generality and generalizing Stanley's earlier proof for simplicial
polytopes.
- It resolves a conjecture of Grünbaum, Kalai and Sarkaria, and
generalizes a result going back to Descartes and Euler: We prove that
for a simplicial complex ” that embeds into 2d-dimensional euclidean
space, the number of d-dimensional simplices exceeds the number of
(d1)-dimensional simplices by a factor of at most d+2. This in
particular implies a generalization of the celebrated crossing number
inequality of Ajtai,Chvátal, Leighton, Newborn and
Szemerédi to higher dimensions.
***the local version is more verbose; since the paper develops quite a
bit of techniques, and non-experts are quite interested in the results,
many people have questions. Mathematically, there is no change since
the first arxiv version but occasionally, if more than one person asks
the same question, I add some more wordy explanations. So if you have
questions, let me know.
**** I updated the local version after teaching a course about it in
spring of 2020. It contains also the derivation of a conjecture of
Kuehnel, and fixes some bibtex issues
See also this
post on Gil Kalai's blog about it, and even a post in Korean
(by Sang-il Oum). I am teaching a minicourse at Sapienza about it April
19, see the link here.
See also here for
an overview over the proof (intended as a first read if you find ~80
pages intimidating), as well as some alternative roads to some steps of
the same. It was written as a short exposition of the main ideas after
the minicourse at Sapienza. You can also have a look at the Hadamard lectures at IHES I gave about this topic.
At some point, Stavros Papadakis and Vasiliki Petrotou figured out an
amazing simplification of a key step in the program. This led us to
jointly figure out further simplifications and generalizations, see for
instance this for the latest. Moreover, you can check here for an expository article with Geva Yashfe that explained one of the innovations of the article, the partition complex.
On the flip
approach to the Lefschetz property
This short note examines
the conditions imposed on the Artinian reduction for the Lefschetz
property to hold. It provides a rather simple obstruction for the naive
"flip" approach to work in general to prove Lefschetz property, and
gives a quick check to determine fatal issues with arguments in the
context of this problem (though it did not keep people, including
myself, from trying this approach). See for instance arxiv:2001.06594 for a
recent attempt at using this approach, though it seems to fail all the
same.
- The resolution of the semistable
reduction conjecture and log-smoothness over valuation rings
these are actually two papers, one with
Gaku Liu and
Michael Temkin (
arxiv:1810.03131)
, and one in addition with
Igor Pak (
arxiv:1806.09168).
This
resolves a conjecture of Abramovic and Karu, and establishes the
existence of polystable/semistable modifications for log-varieties over
valuation rings/surjective morphisms of complex projective varieties,
generalizing classical results of Kempf, Knudsen, Mumford, Saint-Donat
and Waterman to log-varieties. In particular, the combinatorial core of
the argument is a generalization of the celebrated
Knudsen-Mumford-Waterman result concerning unimodilar triangulations of
lattice polytopes to Cayley polytopes; the construction is elementary,
but rather intricate. My coauthor, friend and former postdoc Gaku
realized that it was very powerful, and resolved some further questions
- Many
projectively unique polytopes
Inventiones Math, with G.M. Ziegler (arxiv:1212.5812)
We construct an infinite family of 4-polytopes whose realization spaces
have dimension smaller or equal to 96. This in particular settles a
problem going back to Legendre and Steinitz: To bound the dimension of
the realization space of a polytope in terms of its $f$-vector.
Moreover, we derive an infinite family of combinatorially distinct
69-dimensional polytopes whose real==ization is unique up to projective
transformation. This answers a problem posed by Perles and Shephard in
the sixties. The construction is rather cool, as it relies on
discretizing a certain partial differential equation, and solving it,
and is inspired by the rigidity of higher-dimensional conformal
geometry. The tools developped in this paper are rather rich, and the
result has recently found application in the theory of slack ideals.
(In addition to solving an old problem, this work won me the
European Prize in Combinatorics in 2015)
A
rnau and I had some amazing collaborations over the years on realization spaces of polytopes, so let me highlight one: The
celebrated universality theorem
of Mnëv proves, among other things, that every semialgebraic set
defined over the integers appears as the realization space of some
polytope. This was an amazing achievement.
Our contribution: Well, there was a conjecture in his paper that the
same should be true for open semialgebraic sets, and simplicial
polytopes. In other words, the general position version of his theorem.
This turned out to be much harder, but we managed and thereby solved
his conjecture.
- Hodge
theory for combinatorial geometries
Annals of Mathematics, with June Huh and Eric Katz (publication version)
The
characteristic polynomial of a matroid is a fundamental and mysterious
invariant of matroids with many problems surrounding it. Among the most
resilient problems is a conjecture of Rota, Heron and Welsh proposing
that the coefficients of the characteristic polynomial are log-concave.
We prove this conjecture by relating it to, and then establishing a,
Hodge theory on certain Chow rings associated to general matroids.
Ultimately, once you realize that the proof is essentially an adaption
of a technique developped and/or used earlier several times (by
Elias-Williamson, Karu and others) and original credit for developping
the program that now is combinatorial Hodge Theory should go to the amazing work of Peter McMullen, it becomes a tad less exciting, but it was certainly very surprising back then.
(See also Matt Baker's blog
entry about this work. June and I won the New Horizons Prize for
this paper in 2018).
- Filtered geometric
lattices and Lefschetz Section Theorems over the tropical semiring
2014, with Anders Bjoerner (arxiv:1401.7301)
This proves a conjecture of Mikhalkin and Ziegler, and
prove a Lefschetz section theorem for "smooth" (that is, locally
matroidal) tropical varieties. The paper went through several
iterations, where we noticed smaller new results but also improved
presentation, as this draws from several different areas and people
were not familiar with, for instance, stratified Morse theory. At some
point, I
also noticed it contained a mistake: that the Lefschetz theorem is true
with respect to integral tropical Hodge groups. In the end, the proof
and theorem with respect to Mikhalkin's Hodge theory works only in
characteristic 0, and I added a counterexample in positive
characteristic. The paper is, to best of my knowledge, correct in this
form, though it is also somewhat obsolete and I have not submitted it
again: because of the work on the Rota conjecture, we now know that the
hard Lefschetz theorem holds as well (this implication was figured out
by Amini and Piquerez). Still, there are several results in this paper
that may be useful to the reader, and the methods too: one of the
applications is to prove that complete intersections in polyhedral
geometry have nice topology, see the note below (published here).
- Relative
Stanley--Reisner theory and Upper Bound Theorems for Minkowski sums
Publications mathématiques de l'IHÉS, with Raman
Sanyal (arxiv:1405.7368)
Here I extend and formalize some of the
machinery of Stanley-Reisner complexes to relative complexes,
especially manifolds, and apply this to solve the upper bound problem
for Minkowski sums. This was quite useful for several later
developments in the area, and contains several results beyond just
Minkowski sums. I also later discovered that one could also prove an
analogous lower bound using much the same methods (prodded by Yue Ren
who asked me about it). You can find it here, and Yue´s followup here.
- The
Hirsch conjecture holds for normal flag complexes
2013, Math. of Operations Research, with B.
Benedetti (arxiv.org:1303.3598)
We use the basic fact that locally convex sets of
small intrinsic diameter
in CAT(1) spaces are convex to prove the following result: Every flag
and normal simplicial complex satisfies the nonrevisiting path
conjecture, and in particular the diameter bound conjectured by Hirsch
for all polyhedra. Furthermore, this paper contains a combinatorial
proof of mine of the same result that previously appeared on G.
Kalai's Blog (though I found the geometric proof before
the combinatorial one.) It is also rather cool as a standalone result.