arXiv research

Proves a tropical version of Clemens-Schmid sequence for tropical varieties.

Numerical experiments support conjecture about opers and nonabelian Hodge.

Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.

Proves the Hodge conjecture for complex projective manifolds.

Disproves Fedotov's conjecture on higher-order Shephard inequalities.

Study of section conjecture analogues over complex numbers and Kodaira fibrations.

Study uses Zilber-Pink conjecture and dynamical methods to solve rigidity problems.

Anabelian geometry reformulated using Hodge theory for hyperbolic curves.

Proves Kato manifolds satisfy Hodge decomposition.

The paper develops $L^2$-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.

Lecture notes on advanced geometry equations and subvarieties.

This paper generalizes L2 cohomology theory for complex manifolds.

The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. I advertise a class of abelian four-folds due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which c…

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons a…

We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, fi…

We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bo…

Study actions of mapping class groups on surface representations, proving finite image for certain representations.

Researchers prove hot spots conjecture for Gaussian spaces.

The paper extends a conjecture about Euler characteristics of perverse sheaves on certain Kähler manifolds.

The paper shows that certain geometric structures remain unchanged under specific twists.

In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymp…

Extends Onsager's conjecture to Besov spaces on manifolds with boundary.

Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homog…

Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.

Proves cohomology theorems for tropical varieties.

We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that ca…

We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations a…

Uniformizes Hodge structures, proving Lyapunov exponents and log-Anosov monodromy.

The paper proves a conjecture about the positivity of the Euler characteristic for complex projective manifolds.

We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G_C-bundles with trivial Chern classes where G_C is the complexified grou…

Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.

We study one parameter degenerations of complex projective manifolds by introducing certain type of Hodge metrics coming from the pluricanonical forms. We show that degenerations with at most canonical singularities are all in the finite distance boundary of moduli spaces. We also propose the converse to be true in the…

T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irr…

We describe a conjectural formula via intersection numbers for the Masur-Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to compu…

We make a computational study to know what kind of isospectralities among lens spaces and lens orbifolds exist considering the Hodge--Laplace operators acting on smooth $p$-forms. Several evidenced facts are proved and some others are conjectured.

We construct real polarizable Hodge structures on the reduced leafwise cohomology of Kähler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's Kählerian analogue of the Weil conjectures carries over as well. Generalizing a construction of …

We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycl…

Carlson and Toledo conjectured that any infinite fundamental group $Γ$ of a compact Kähler manifold satisfies $H^2(Γ,\R)\not =0$. We assume that $Γ$ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ($\C$-VHS) on the Kähler manif…

We study some fundamental properties of real rectifiable currents and give a generalization of King's theorem in characterizing currents defined by positive real holomorphic chains. Our proof uses Siu's semicontinuity theorem and largely simplifies King's proof. A consequence of this result is a sufficient condition fo…

In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety $X$ can be connected to the generating function for Gromov-Witten invariants of $X$ by a series of differential operators $\{ L_m \mid m \geq 1 \}$ after a suitable change…

This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of …

We fix integers $k> 0$ and $n>0$. For a $k$-punctured Riemann surface $Σ\setminus \{ p_1,\ldots,p_k \}$ and a $k$-tuple $\boldsymbolμ=(μ^1,\ldots,μ^k)$ of partitions of $n$, we can define the character variety of type $\boldsymbolμ$. In this paper, we consider the case where $Σ=\mathbb{P}^1$ and $\boldsymbolμ$ is indiv…

We show that a conjectural extension of a fixed point formula in Arakelov geometry implies results about a tautological subring in the arithmetic Chow ring of bases of abelian schemes. Among the results are an Arakelov version of the Hirzebruch proportionality principle and a formula for a critical power of $\hat c_1$ …

This thesis explores algebraic cycles and moduli spaces over real numbers.

Machine learning predicts topological properties of Calabi-Yau manifolds.

In his recent work \cite{Y1}, X. Yang proved a conjecture raised by Yau in 1982 (\cite{Yau82}), which states that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. In this note, we prove that any compact Hermitian manifold $X$ with positive real bisectional curvature, its hod…

We propose a new approach to the Mirror Symmetry Conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to t…