abstracts

Intersection bundles and the Cappell--Miller torsion for Riemann surfaces

In these lectures, we will report on joint work with R. Wentworth on an intersection theoretic approach to complex Chern--Simons bundles. This is based on the theory of intersection bundles initiated by Deligne, combined with a variant of the theory of Chern--Simons secondary classes. We apply our results to show that the Cappell--Miller torsion for flat vector bundles on Riemann surfaces fulfills a Riemann--Roch type formula. 

Lecture 1. Intersection bundles and Deligne--Riemann--Roch.

Intersection bundles are certain liftings of fiber integrals of characteristic classes, to the level of line bundles. They were invented by Deligne and further developed by Elkik. We will present this formalism for families of Riemann surfaces, and formulate Deligne's Riemann--Roch statement in this framework. 

Lecture 2. Intersection connections and canonical extensions.

The formalism of intersection bundles can be enriched over metrics and Chern connections. Building on this, we will explain how to incorporate non-necessarily flat connections, and explain the curvature formulas in this context. 

Lecture 3. Relative, complex, Chern--Simons bundles. 

A natural setting of application of the intersection connections formalism are moduli spaces of flat vector bundles on Riemann surfaces. We will show how this gives raise to an intersection theoretic approach to complex Chern--Simons line bundles.

Lecture 4. Cappell--Miller torsion and Deligne--Riemann--Roch.

We will conclude these lectures by upgrading the Deligne--Riemann--Roch isomorphism to flat vector bundles. In this context, we will show that the holomorphic Cappell--Miller torsion is the natural replacement of the Quillen metric in the hermitian setting.

Towards the comparison of higher torsion invariants

The aim of this minicourse is to explain some recent results concerning the comparison of the higher topological and analytical torsions. These higher torsion invariants are family versions of the classical Franz-Reidemeister and Ray-Singer torsions of a flat bundle over a manifold. In the case of a single manifold, the celebrated result of Cheeger-Müller/Bismut-Zhang gives a comparison formula between the two torsions, and a similar result is expected in the family setting. There are mainly two approaches to this problem: one due to Igusa, which uses axiomatization, and one due to Bismut-Goette, which uses Morse theory.

After giving some background, we will explain how Igusa's approach, along with results of various authors, can be used to compare the higher torsions associated with a trivial bundle. Building of this result, we explain how to use Bismut-Goette's approach to prove a relative comparison formula, which links a renormalized version of higher torsions, in which we have removed the torsion of a trivial bundle. This renormalization allow to simplify some analytic difficulties arising when trying to extend the work of Bismut-Goette. 

This minicourse is based on joint works with Junrong Yan, Yeping Zhang and Jialin Zhu.

Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck

The aim of this mini-course is to provide a comprehensive introduction to our recent book "Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck" co-authored with J.-M. Bismut and Z. Wei.

Lecture 1. Superconnections and Chern-Weil theory 

We recall some basic construction on Quillen’s superconnection and the associated Chern-Weil theory. In the context of complex geometry, we introduce the anti-holomorphic superconnection and explore the corresponding Bott-Chern/Chern-Weil theory. 

Lecture 2. Coherent sheaves and anti-holomorphic superconnections 

We show how the anti-holomorphic superconnection provides a global resolution for coherent sheaves and construct the Bott-Chern cohomology valued Chern character for coherent sheaves. 

Lecture 3. RRG for immersions and the uniqueness of Chern characters 

Given an immersion $i: X\to Y$  and a coherent sheaf $F$ on $X$, we establish the RRG formula for the Chern character of the direct image $i_* F$. We provide an axiomatic description of our Chern characters. 

Lecture 4. Grauert’s direct Image theorem and RRG for projections 

For a projection $\pi : M \times S \to S$ and a coherent sheaf $F$ on $M \times S$, we outline how elliptic theory provides a new proof of Grauert’s direct image theorem, ensuring the coherence of the higher direct image  $R^{\bullet}\pi_* F$. We discuss the obstruction that arises in establishing the RRG formula for projections within elliptic theory. Finally, we introduce the hypoelliptic Laplacian and, without getting into analytical details, explain how it provides the RRG formula for projections. 

Local index theory and the R genus

The R genus was obtained by Gillet, Soulé and Zagier in [1] as a conjectural defect to the Todd class, in their program to establish a Riemann-Roch theorem in arithmetic geometry, by putting together the holomorphic torsions of the projective spaces. Conjecturally, the R genus had to appear in the computation of the behavior of Quillen metrics under embeddings.

In [2], as part of a program to describe the behavior of Quillen metrics under embeddings, we made a construction of characteristic classes via local index theoretic data, in which the full R genus comes out by computations involving a family of harmonic oscillators along the fibers of a holomorphic vector bundle. This result was used in our paper with Lebeau [3], where the comparison formula for Quillen metrics was established.

The construction is based on the following observation: the proof of the index theorem for classical Dirac operators can be expressed in terms of objects that appeared (later!) in the corresponding result for family of Dirac operators. In other words, the index theorem for families is already encoded in the index theorem for one single Dirac operator. The same observation is also valid in the case of Kähler manifolds.

In our talk, we will review the essential steps that led to the results in [2]. We believe these steps are still relevant.

In a later twist to the story, in a later paper [4], Bost has shown how the result of Gillet, Soulé and Zagier can be deduced from our main result with Lebeau [3].

References

[1] H. Gillet and C. Soulé. Analytic torsion and the arithmetic Todd genus. Topology, 30(1):21–54, 1991. With an appendix by D. Zagier.

[2] J.-M. Bismut. Koszul complexes, harmonic oscillators, and the Todd class. J. Amer. Math. Soc., 3(1):159–256, 1990. With an appendix by the author and C. Soulé.

[3] J.-M. Bismut and G. Lebeau. Complex immersions and Quillen metrics. Inst. Hautes Études Sci. Publ. Math., (74):ii+298 pp. (1992), 1991.

[4] J.-B. Bost. Analytic torsion of projective spaces and compatibility with immersions of Quillen metrics. Internat. Math. Res. Notices, (8):427–435, 1998.

A Deligne--Riemann--Roch isometry for flat unitary vector bundles over modular curves.

Consider a family of compact Riemann surfaces, as well as holomorphic vector bundles on them. If all of these are endowed with smooth metrics, one can relate two objects of different nature by an isometry: the determinant line bundle with the Quillen metric, and some intersection bundles. This is a consequence of the work of Deligne and Bismut--Gillet--Soulé. However, the smoothness assumption on the metrics is crucial, and a bit restrictive to study situations which are interesting from the standpoint of number theory.

One such setting is that of modular curves, with flat unitary holomorphic vector bundles coming from unitary representations of Fuchsian groups of the first kind. In this case, the natural metric are not smooth, as they present singularities at certain points. Freixas i Montplet and von Pippich were able to overcome this complication for the trivial line bundle, and prove a Deligne--Riemann--Roch isometry.

In this talk, we will see how to extend this result to a more general class of flat unitary vector bundles over modular curves. Possible applications of this result include the computation of the first non-zero derivative at 1 of the Selberg zeta function and of the Weil--Petersson norm of weight 1 modular forms with Nebentypus. Beyond these, the strategy employed here can be adapted to deal with other situations where metrics present singularities, like certain toric varieties and higher-dimensional modular varieties.

Birational geometry and BCOV invariants.

BCOV invariants are certain real-valued invariants of Calabi-Yau manifolds defined via analytic torsion. I will report on my joint work with Yeping Zhang proving that the BCOV invariants are birational invariants for Calabi-Yau manifolds. Our method takes inspiration from the theory of motivic integration and allows us to generalize the construction of the BCOV invariants, as well as it birational invariance, to some extent in the setting of mildly singular varieties.

Toeplitz-Fubini-Study forms and lowest eigenvalues of Toeplitz operators

For a Hermitian manifold equipped with a positive line bundle, Tian's approximation theorem states that, by considering the high tensor powers of this line bundle, the sequence of the induced Fubini-Study metrics via the Kodaira maps converges to the first Chern form of the line bundle. In this talk, we study the corresponding extensions of Tian's approximation theorem by combining the Kodaira maps with the Berezin-Toeplitz quantization, in particular with the twisting by the Toeplitz operators. Moreover, if the manifold is assumed to be compact, such a question is closely related to the study of the lowest eigenvalues of the Toeplitz operators with a given symbol. Finally, some estimates and further questions concerning the lowest eigenvalues of Toeplitz operators will be discussed.

Differential K-theory and Bismut-Cheeger eta forms

In the development of differential K-theory, the Bismut-Cheeger eta form plays a key role. In this talk, I will present a modified model of the differential K-group, building on the smooth K-theory framework developed by Bunke and Schick. I will discuss its equivariant generalization and explore key properties, with a particular focus on the localization property. This will lead to a localization formula for the Bismut-Cheeger eta forms. This work is a collaboration with Xiaonan Ma.

Refined torsion on singular spaces

In the 1990s Turaev generalised the topological (or Reidemeister) torsion of smooth manifolds, defining a complex valued invariant, whose absolute value yields back the Reidemeister torsion.

Analytic counterparts of the complex valued torsion of Turaev have been introduced by Braverman-Kappeller, Burghelea-Haller and Cappell-Miller in slightly different contexts. Comparison theorems between the thus defined analytic and topological invariants for smooth manifolds have been studied consequently.

The aim of this talk is to study refined torsions and comparison theorems in the context of singular spaces.

Scalar Curvature Lower Bound and Characteristic Number Estimate

Scalar curvature encodes the volume information of small geodesic balls within a Riemannian manifold, making it, to some extent, the weakest curvature invariant. This raises a natural question: what topological constraints does scalar curvature impose on manifolds? In this talk, we will show that for a Cartan-Hadamard manifold with a scalar curvature lower bound, its certain characteristic numbers can be controlled by its volume. This partially answers a conjecture by Gromov regarding characteristic number bounds on aspherical manifolds. This is joint work with Guoliang Yu.

Superconnection and family Bergman kernels

We establish an asymptotic version of Bismut's local family index theorem for the Bergman kernel associated with a fiberwise positive line bundle when the power tends to infinity. The key idea is to use the superconnection as in  the local family index theorem. In particular, we show the curvature operator of the associated direct image is a Toeplitz operator.

The (spherical) Mahler measure of the X-discriminant

Let P be a homogeneous polynomial in N+1 complex variables of degree d. The logarithmic Mahler Measure of P (denoted by m(P) ) is the integral of log|P| over the sphere in C^{N+1} with respect to the usual Hermitian metric and measure on the sphere.  Now let X be a smooth variety embedded in CP^N by a high power of an ample line bundle and let $\Delta$ denote a generalized discriminant of X wrt the given embedding, then $\Delta$ is an irreducible homogeneous polynomial in the appropriate space of variables.  

In this talk I will discuss work in progress whose aim is to find an asymptotic expansion of m(\Delta) in terms of the logarithm of the degree of the embedding.  The technical machinery required for this task was developed by Jean-Michel Bismut in several articles in the 1990's 

The twisted Ruelle zeta function and the Ray-Singer metric

In this talk, we will present the twisted dynamical zeta functions of Ruelle and Selberg, associated with non-unitary, arbitrary representations of the fundamental group of an odd-dimensional compact hyperbolic manifold. D. Fried conjectured a relation, under certain assumptions, between the Ruelle zeta function at zero and a spectral invariant, the analytic torsion. This is widely known as the Fried’s conjecture and such a relation relates a pure geometrical-dynamical object, the Ruelle zeta function, with a spectral and a topological invariant. We will present some results concerning this conjecture as well as some recent results concerning the relation between the twisted Ruelle zeta function and the Ray-Singer metric of the refined analytic torsion associated with the representation, as it is introduced by Braverman and Kappeler.

 Lichnerowicz theorem revisited 

A famous theorem of Lichnerowicz states that if a closed spin manifold carries a metric of positive scalar curvature, then its Hirzebruch A-hat genus vanishes. We will describe some advances arising from this classical result.