Conference: Spencer Fest

Jose Seade, UNAM: Baum-Bott residues and the connectivity of the singular set of holomorphic foliations.

In the late 1960s, Bott established a vanishing theorem that revealed topological obstructions to the integrability of holomorphic distributions. This result led to the development of the Baum–Bott residues, powerful invariants associated with holomorphic foliations on complex manifolds and localized at their singular sets. In this talk, I will present recent joint work with O. Calvo-Andrade, M. Correa, and M. Jardim, in which we use Baum–Bott residues to show that, under suitable conditions, the connectivity of the singular set of a holomorphic distribution can itself be viewed as a topological obstruction to integrability—echoing Bott’s pioneering ideas. 

Maxim Kontsevich, IHES (Zoom)

Ljudmila Kamenova, Stony Brook University (Zoom): Lagrangian fibrations on hyperkaehler manifolds: rational sections, Neron models and multiplicities

Let M be a hyperkaehler (i.e., holomorphic symplectic) manifold, equipped with a Lagrangian fibration $p:M \rightarrow B$. We would explore some properties of the possible multiple fibers. Assuming that the fibration only has reduced fibers in codimension one, we prove that a certain twistor deformation $p' : M' \rightarrow B$ of $p:M \rightarrow B$ admits a meromorphic section. The various results in this talk are joint with F. Bogomolov. M. Verbitsky and F. Campana. 

Benjamin Bakker, University of Illinois Chicago: Baily—Borel compactifications of period images and the b-semiampleness conjecture

Building on previous work of Satake and Baily, Baily and Borel proved in 1966 that arithmetic locally symmetric varieties admit canonical projective compactifications whose graded rings of functions are given by automorphic forms.  Such varieties include moduli spaces of abelian varieties, and have rich algebraic and arithmetic geometry.  Griffiths suggested in 1970 that the same might be true for the image of any period map, which would provide canonical compactifications of many moduli spaces, including for instance those of Calabi--Yau varieties. In joint work with S. Filipazzi, M. Mauri, and J. Tsimerman, we confirm Griffiths' suggestion, and prove that the image of any period map admits a canonical functorial compactification.  We also show how the same techniques yield a resolution to an important conjecture in birational geometry, the b-semiampleness conjecture.  Both proofs crucially use o-minimal GAGA, and the latter application additionally uses results of Ambro and Kollar on the geometry of minimal lc centers.  

Eyal Markman, UMass Amherst: Semiregular secant sheaves and Weil classes on abelian varieties

We will emphasize the role of the semi-regularity theorem of Bloch and Buchweitz-Flenner in the proof of the algebraicity of the Weil classes of split Weil-type on abelian sixfolds with complex multiplication by an imaginary quadratic number field. The latter implies the Hodge
conjecture for abelian varieties of dimension less than or equal to 5. We will also discuss the expected role of the semi-regularity theorem in generalizations of the above result for higher dimensions and for CM fields.

Enrique Becerra, CINVESTAV: The Chiral Cyclic Complex of a Calabi-Yau Category

The Chiral De Rham complex (CDR) of a smooth complex algebraic variety is a sheaf of conformal vertex algebras that can be seen as a quantization of the classical De Rham complex. In the Calabi-Yau case, the CDR carries an N=2 superconformal structure, and its partition function recovers the elliptic genus of the variety.
In this talk, we introduce the Chiral Cyclic Complex as an extension of this construction to the setting of non-commutative geometry. For a given Calabi-Yau dg-category, we construct an N=2 conformal vertex algebra that deforms the periodic cyclic complex. We show how this framework allows for a natural definition of the elliptic genus in the non-commutative case, mirroring the classical theory and providing new invariants for Calabi-Yau categories.

Carlos Simpson, University of Nice (Zoom)

Xavier Gómez-Mont Ávalos, CIMAT: On Holomorphic Foliations with Singularities

In 1960 professors K. Kodaira and D. Spencer published the paper ”Multifoliate Structures” where they extended their previously developed deformation theory of complex manifolds. They restricted themselves to the non-singular case. I will show how to extend their deformation methods to holomorphic foliations with singularities in compact Kahler manifolds using hypercohomology of a sequence of sheaves on using Bott’s partial connection. I will also give some results on uniformization, equidistribution and Teichm¨uller spaces for holomorphic foliations by curves with singularities.

Rodolfo Aguilar Aguilar, CINVESTAV: Infinitesimal Methods and the Log Clemens Conjectures

In the classical theory of Calabi-Yau threefolds, the deformation theory of curves is governed by Hodge theory—a relationship materialized in the Clemens conjectures. In this talk, I will start by introducing infinitesimal Abel-Jacobi maps for smooth pairs $(X, Y)$. I will then introduce the notion of smooth $\mathbb{Q}$-log Calabi-Yau threefold pairs $(X, Y)$ and study their deformation theory.

Inspired by the symmetry of the absolute case, we establish a duality theorem for the twisted normal bundle of curves in these pairs. Specifically, for the cubic threefold, this infinitesimal analysis motivates a relative generalization of the Clemens conjecture regarding the finite number of rational curves and the structure of their normal bundles.

Finally, we discuss the recent resolution of these conjectures by A. Zahariuc. We will provide a sketch of the proof, which uses Gromov-Witten invariants, illustrating how the infinitesimal predictions made by the Kodaira-Spencer machinery are realized in global enumerative geometry.

Herb Clemens, Ohio State University

Robert Bryant, Duke University: Spencer sequences and special holonomy

The subject of special holonomy (essentially, the study of Riemannian metrics whose holonomy is a proper subgroup of the orthogonal group) was initiated by J.A. Schouten and É. Cartan in the 1920s, with Cartan announcing the first nontrivial examples of metrics in dimension 4 with holonomy SU(2).  In 1954, M. Berger produced a very restricted list of possible holonomy groups of locally irreducible, non-locally-symmetric Riemannian manifolds, but it took another 30 years before the local classification was complete.  Along the way, connections with D.C. Spencer’s ideas on deformations of geometric structures began to play an important role.  In this talk, I will highlight some appearances of Spencer cohomology in the geometry of manifolds with special holonomy, particularly the use of Spencer sequences to understand the higher order invariants of G-structures.

Robert Friedman, Columbia University

Laurent Meersseman, Université d'Angers

Giulia Saccà, Columbia University

Evening Dedicated to Donald Spencer

Marco Manetti, Sapienza University of Rome: An  algebraic framework for semi-regularity maps and obstructions to deformations

In 1959, Kodaira and Spencer proved that every semi-regular compact complex submanifold of codimension 1 has unobstructed embedded deformations. This result was generalized by Bloch (1972) to locally complete intersection (LCI) projective submanifolds and by Buchweitz and Flenner (2003) to coherent sheaves on projective manifolds. A submanifold or coherent sheaf is called semi-regular if a certain map, called the semi-regularity map, is injective. It is easy to prove that Kodaira–Spencer’s theorem holds in a stronger form: all obstructions to deformation lie in the kernel of the semi-regularity map. However, both the papers of Bloch and of Buchweitz–Flenner leave this question unanswered. Attempts to prove that semi-regularity maps annihilate obstructions have motivated research in deformation theory for several years, leading to various proofs under additional assumptions. Recently, a complete and positive answer has been given by Pridham and, independently, by Bandiera, Lepri, and Manetti.

In this talk, extending the approach of Bandiera–Lepri–Manetti, we define semi-regularity maps in the algebraic context of filtered curved Lie algebras and explain why each of them annihilates the obstructions of the associated deformation functor (work in progress).

Dror Varolin, Stony Brook University

Radu Laza, Stony Brook University

Alberto Verjovsky, UNAM: Uniformization and dynamics of laminations and foliations by Riemann surfaces. Some interesting new examples of holomorphic foliations on compact 4-dimensional complex manifolds.

We present several results about laminations and foliations by Riemann surfaces. In
particular we recall some aspects of the problem of the continuous and simultaneous uniformization (in the sense of K¨oebe and Poincar´e) of the leaves of a lamination (or foliation) by hyperbolic surfaces. In particular, we describe the Teichm¨uller theory of the universal hyperbolic lamination of Dennis Sullivan. Solenoidal surfaces are topological spaces which are locally homeomorphic to the product of a Cantor set with an open subset of R 2. In a sense they are “diffuse” versions of surfaces (seen at a long distance they look like surfaces). We will also describe the dynamical and ergodic properties of horocycle foliations associated with hyperbolic solenoidal surfaces (minimal and topological mixing examples).
Lastly, we will describe holomorphic and differentiable foliations by Riemannsurfaces a in compact 4-dimensional complex (non K¨ahler) 4-folds and give examples of some of their deformations, a topic developed by Donald Spencer.

Ernesto Lupercio, CINVESTAV

Mihai Păun, University of Illinois Chicago (Zoom)

William Goldman, University of Maryland: Deformations of Locally Homogeneous Structures

In 1936, Ch. Ehresmann formulated a classification problem for geometric structures locally modeled on homogneous spaces. These correspond to flat connections and are intimately related to the space of representations of the fundamental group, as well as to the deformation theory of complex structures pioneered by Kodaira and Spencer. In this talk I will describe the general theory as well as specific examples of the classification problem and its relation to the classification of Riemann surfaces.