Hodge Bundle Research Articles

BCOV invariants of Calabi–Yau manifolds and degenerations of Hodge structures

Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve counting on a Calabi--Yau manifold is related to a conjectured invariant, only depending on the complex structure of the mirror, and built from Ray--Singer holomorphic analytic torsions. To this end, extending work of Fang--Lu--Yoshikawa in dimension 3, we introduce and study the BCOV invariant of Calabi--Yau manifolds of arbitrary dimension. To determine it, knowledge of its behaviour at the boundary of moduli spaces is imperative. We address this problem by proving precise asymptotics along one-parameter degenerations, in terms of topological data and intersection theory. Central to the approach are new results on degenerations of $L^2$ metrics on Hodge bundles, combined with information on the singularities of Quillen metrics in our previous work.

On the compactification of the Drinfeld modular curve of level [formula omitted

Let p be a rational prime and q a power of p. Let n be a non-constant monic polynomial in Fq[t] which has a prime factor of degree prime to q−1. In this paper, we define a Drinfeld modular curve Y1Δ(n) over A[1/n] and study the structure around cusps of its compactification X1Δ(n), in a parallel way to Katz-Mazur's work on classical modular curves. Using them, we also define a Hodge bundle over X1Δ(n) such that Drinfeld modular forms of level Γ1(n), weight k and some type are identified with global sections of its k-th tensor power.

Hodge and Prym Tau Functions, Strebel Differentials and Combinatorial Model of $${\mathcal {M}}_{g,n}$$

The goal of the paper is to apply the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on Strebel differentials. These line bundles are tensor products of the determinants of the Hodge or Prym vector bundles with the standard tautological line bundles $$\mathcal {L}_j$$ , and the sections are constructed in terms of tau functions. The combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials where the phase of each tau-function provides a section of a circle bundle. The phase of the ratio of the Prym and Hodge tau functions gives a section of the $$\kappa _1$$ -circle bundle. By evaluating the increment of the phase around co-dimension 2 sub-complexes, we identify the Poincare dual cycles to the Chern classes of the corresponding line bundles: they are expressed explicitly as combination of Witten’s cycle $$W_{5} $$ and Kontsevich’s boundary. This provides combinatorial analogues of Mumford’s relations on $${\mathcal {M}}_{g,n}$$ and Penner’s relations in the hyperbolic combinatorial model. The free homotopy classes of loops around $$W_{5} $$ are interpreted as pentagon moves while those of loops around Kontsevich’s boundary as combinatorial Dehn twists. Throughout the paper we exploit the classical description of the combinatorial model in terms of Strebel differentials, parametrized in terms of period, or homological coordinates; we show that they provide Darboux coordinates for the symplectic structure introduced by Kontsevich. We also express the latter as the intersection pairing in the odd homology of the canonical double cover.

System of Hodge bundles and generalized opers on smooth projective varieties

Let k be an algebraically closed field of any characteristic. Let X be a polarized irreducible smooth projective algebraic variety over k. We give criterion for semistability and stability of system of Hodge bundles on X. We define notion of generalized opers on X, and prove semistability of the Higgs bundle associated to generalized opers. We also show that existence of partial oper structure on a vector bundle E together with a connection ∇ over X implies semistability of the pair (E,∇).

On the rank of the flat unitary summand of the Hodge bundle

Let f : S → B f\colon S\to B be a nonisotrivial fibered surface. We prove that the genus g g , the rank u f u_f of the unitary summand of the Hodge bundle f ∗ ω f f_*\omega _f , and the Clifford index c f c_f satisfy the inequality u f ≤ g − c f u_f \leq g - c_f . Moreover, we prove that if the general fiber is a plane curve of degree ≥ 5 \geq 5 , then the stronger bound u f ≤ g − c f − 1 u_f \leq g - c_f-1 holds. In particular, this provides a strengthening of bounds proved by M. A. Barja, V. González-Alonso, and J. C. Naranjo and by F. F. Favale, J. C. Naranjo, and G. P. Pirola. The strongholds of our arguments are the deformation techniques developed by the first author and by the third author and G. P. Pirola, which display here naturally their power and depth.

Effective Divisors in the Projectivized Hodge Bundle

Abstract We compute the class of the closure of the locus of canonical divisors in the projectivization of the Hodge bundle ${\mathbb{P}}\overline{{\mathcal{H}}}_g$ over $\overline{{\mathcal{M}}}_g$, which have a zero at a Weierstrass point. We also show that the strata of canonical and bicanonical divisors with a double zero span extremal rays of the respective pseudoeffective cones.

Family of curves with large unitary summand in the Hodge bundle

In this note, we construct non-isotrivial families of curves of genus $$g\ge 2$$ , where the rank of the unitary summand contained in the Hodge bundle can be as large as $$(2g+1)/3$$ , and hence disprove Xiao’s conjecture for the unitary rank.

Remarks on BCOV invariants and degenerations of Calabi-Yau manifolds

For a one parameter family of Calabi-Yau threefolds, Green et al. (2009) expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV (Bershadsky-Cecotti-Ooguri-Vafa) invariants, studied by Fang et al. (2008). On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.

Singularities of metrics on Hodge bundles and their topological invariants

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds. We also describe corresponding invariants for more general degenerating families in the case of the Quillen metric.

Fujita decomposition over higher dimensional base

We generalize a result of Fujita, on the decomposition of Hodge bundles over curves, to the case of a higher dimensional base.

Compactification of strata of Abelian differentials

We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.

The algebraic hull of the Kontsevich–Zorich cocycle

We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL^+_2(R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

Tautological ring of strata of differentials

Strata of k-differentials on smooth curves parameterize sections of the k-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the $$\kappa $$ and $$\psi $$ classes of moduli spaces of pointed smooth curves along with the class $$\eta = \mathcal O(-1)$$ of the Hodge bundle. We show that if there is no pole of order k, then the tautological ring is generated by $$\eta $$ only, and otherwise it is generated by the $$\psi $$ classes corresponding to the poles of order k.

Eigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrations

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 Teichm\"uller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using Atiyah-Bott, Forni and M\"oller's works. We obtain several applications to Teichm\"uller dynamics conditional to the conjecture.

On exact correlation functions of chiral ring operators in 2d mathcal{N}=left(2, 2right) SCFTs via localization

We study the extremal correlation functions of (twisted) chiral ring operators via superlocalization in mathcal{N}=left(2, 2right) superconformal field theories (SCFTs) with central charge c ≥ 3, especially for SCFTs with Calabi-Yau geometric phases. We extend the method in arXiv: 1602.05971 with mild modifications, so that it is applicable to disentangle operators mixing on S2 in nilpotent (twisted) chiral rings of 2d SCFTs. With the extended algorithm and technique of localization, we compute exactly the extremal correlators in 2d mathcal{N}=left(2, 2right) (twisted) chiral rings as non-holomorphic functions of marginal parameters of the theories. Especially in the context of Calabi-Yau geometries, we give an explicit geometric interpretation to our algorithm as the Griffiths transversality with projection on the Hodge bundle over Calabi-Yau complex moduli. We also apply the method to compute extremal correlators in Kähler moduli, or say twisted chiral rings, of several interesting Calabi-Yau manifolds. In the case of complete intersections in toric varieties, we provide an alternative formalism for extremal correlators via localization onto Higgs branch. In addition, as a spinoff we find that, from the extremal correlators of the top element in twisted chiral rings, one can extract chiral correlators in A-twisted topological theories.

Existence of supersingular reduction for families of $K3$ surfaces with large Picard number in positive characteristic

We study non-isotrivial families of $K3$ surfaces in positive characteristic $p$ whose geometric generic fibers satisfy $\rho \ge 21 - 2h$ and $h \ge 3$, where $\rho$ is the Picard number and $h$ is the height of the formal Brauer group. We show that, under a mild assumption on the characteristic of the base field, they have potential supersingular reduction. Our methods rely on Maulik’s results on moduli spaces of $K3$ surfaces and the construction of sections of powers of Hodge bundles due to van der Geer and Katsura. For large $p$ and each $2 \le h \le 10$, using deformation theory and Taelman’s methods, we construct non-isotrivial families of $K3$ surfaces satisfying $\rho = 22 - 2h$.

Generalized μ-ordinary Hasse invariants

We give a short proof for the existence of μ-ordinary Hasse invariants for the good reduction special fiber of Shimura varieties of Hodge-type using the stack of G-zips introduced by Moonen–Wedhorn and Pink–Wedhorn–Ziegler. We give an explicit formula for the power of the Hodge bundle that admits a Hasse invariant. When G is a Weil restriction of G1, we relate the Ekedahl–Oort strata of G and those of G1.

Lyapunov exponents of the Hodge bundle over strata of quadratic differentials with large number of poles

We show an upper bound for the sum of positive Lyapunov exponents of any Teichmuller curve in strata of quadratic differentials with at least one zero of large multiplicity. As a corollary, it holds for any $SL(2,\mathbb R)$-invariant subspaces defined over $\mathbb Q$ in these strata. This proves Grivaux-Hubert's conjecture about the asymptotics of Lyapunov exponents for strata with a large number of poles in the situation when at least one zero has large multiplicity.

Hodge Integrals in FJRW Theory

We study higher-genus Fan–Jarvis–Ruan–Witten theory of any chain polynomial with any group of symmetries. Precisely, we give an explicit way to compute the cup product of Polishchuk and Vaintrob’s virtual class with the top Chern class of the Hodge bundle. Our formula for this product holds in any genus and without any assumption on the semi-simplicity of the underlying cohomological field theory.

Differential geometry of the mixed Hodge metric

We investigate properties of the mixed Hodge metric of a mixed period domain. In particular, we calculate its curvature and the curvature of the Hodge bundles. We also consider when the pull back metric via a period map is Kahler. Several applications in cases of geometric interest are given, such a for normal functions and biextension bundles.