
Smooth and Rough Positive Currents
with V. Tosatti
[
arXiv
]
Abstract ±
We study the different notions of semipositivity for (1,1) cohomology classes on K3 surfaces.
We first show that every big and nef class (and every nef and rational class) is semiample, and in particular it contains a smooth semipositive representative.
By contrast, a result of Cantat and Dupont implies that there exist irrational nef classes with no closed positive current representative which is smooth outside a proper analytic subset. We use this to answer negatively two questions of the secondnamed author.

The algebraic hull of the KontsevichZorich cocycle
with A. Eskin
and A. Wright
[
arXiv
]
Abstract ±
We compute the algebraic hull of the KontsevichZorich cocycle over any \(GL^+_2(R)\) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

Counting special Lagrangian fibrations in twistor families of K3 surfaces
[
arXiv
]
Abstract ±
The number of closed billiard trajectories in a rationalangled polygon grows quadratically in the length.
This paper gives an analogue on K3 surfaces, by considering special Lagrangian tori.
The analogue of the angle of a billiard trajectory is a point on a twistor sphere, and the number of directions admitting a special Lagrangian torus fibration with volume bounded by V grows like \(V^{20}\).

Families of K3 surfaces and Lyapunov exponents
Accepted to Israel Journal of Mathematics
[
arXiv
]
Abstract ±
Consider a family of K3 surfaces over a hyperbolic curve (i.e. Riemann surface).
Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number.
One proof uses the KugaSatake construction, which reduces the question to Hodge structures of weight 1.
A second proof uses integration by parts.
The case of maximal Lyapunov exponent corresponds to modular families, given by the Kummer construction on a product of isogenous elliptic curves.

Quaternionic covers and monodromy of the KontsevichZorich cocycle in orthogonal groups
with G. Forni
and C. Matheus
Accepted to Journal of the European Mathematical Society (JEMS)
[
arXiv
]
Abstract ±
We give an example of a Teichmüller curve which contains, in a factor of its monodromy, a group which was not observed before.
Namely, it has Zariski closure equal to the group \(\operatorname{SO}^*(6) \) in its standard representation;
up to finite index, this is the same as \(\operatorname{SU}(3,1)\) in its second exterior power representation.
The example is constructed using origamis (i.e. squaretiled surfaces).
It can be generalized to give monodromy inside the group \( \operatorname{SO}^*(2n)\) for all n, but in the general case the monodromy might split further inside the group.
Also, we take the opportunity to compute the multiplicities of representations in the (0,1) part of the cohomology of regular origamis, answering a question of MatheusYoccozZmiaikou.

Zero Lyapunov exponents and monodromy of the KontsevichZorich cocycle
Duke Mathematics Journal 166, no. 4 (2017), 657706
[
arXiv

journal
]
Abstract ±
We describe the situations in which the KontsevichZorich cocycle has zero Lyapunov exponents.
Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints.
We also describe the real Lie groups which can appear in the monodromy of the KontsevichZorich cocycle.
The number of zero exponents is then as small as possible, given its monodromy.

Splitting mixed Hodge structures over affine invariant manifolds
Annals of Mathematics
vol. 183 (2016), Issue 2, pp. 681713
[
arXiv

journal

Blog post
]
Abstract ±
We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties.
The result is deduced from a generalization of a theorem of Möller.
Namely, we prove that the image of a certain twisted AbelJacobi map lands in the torsion of a factor of the Jacobians.
This statement can be viewed as a splitting of certain mixed Hodge structures.

Semisimplicity and rigidity of the KontsevichZorich Cocycle
Inventiones Mathematicae September 2016, Volume 205, Issue 3, pp 617–670
[
arXiv

journal

Blog post
]
Abstract ±
We prove that invariant subbundles of the KontsevichZorich cocycle respect the Hodge structure.
In particular, we establish a version of Deligne semisimplicity in this context.
This implies that invariant subbundles must vary polynomially on affine manifolds.
All results apply to tensor powers of the cocycle and this implies that the measurable and realanalytic algebraic hulls coincide.
We also prove that affine manifolds parametrize Jacobians with nontrivial endomorphisms.
Typically a factor has real multiplication.
The tools involve curvature properties of the Hodge bundles and estimates from random walks.
In the appendix, we explain how methods from ergodic theory imply some of the global consequences of Schmid's work on variations of Hodge structures.
We also derive the KontsevichForni formula using differential geometry.

On Höldercontinuity of Oseledets subspaces
with V. Araujo
and A. Bufetov
Journal of the London Mathematical Society (JLMS) February 2016, Volume 93, Issue 1 , pp. 194–218
[
arXiv

journal
]
Abstract ±
For Hölder cocycles over a Lipschitz base transformation, possibly noninvertible, we show that the subbundles given by the Oseledets Theorem are Hoeldercontinuous on compact sets of measure arbitrarily close to 1.
The results extend to vector bundle automorphisms, as well as to the KontsevichZorich cocycle over the Teichmüller flow on the moduli space of abelian differentials.
Following a recent result of ChaikaEskin, our results also extend to any given Teichmüller disk.