Conference: New Developments in Singularity Theory

Phillip Griffiths, Institute of Advanced Study and University of Miami: Normal functions and their invariants

Normal functions and their infinitesimal invariants are basic Hodge theoretic invariants of a pair (X,Z) where X is a smooth projective variety and Z is an algebraic subvariety or algebraic cycle in X. They have both a structure and applications to examples, and in this talk we will give an exposition of some aspects of the theory with emphasis on the singularity structure and some open issues.

Benjamin Gammage, Harvard University

Aline Zanardini, École Polytechnique Fédérale de Lausanne: cAn singularities revisited

I will report on joint work with N. Adaloglou and F. Pasquotto concerning certain invariants of  quasihomogeneous cAn singularities. I will discuss some aspects of the birational geometry of these threefold singularities, which are encoded in the so-called symplectic cohomology of their Milnor fibres.

Eduardo de Lorenzo Poza, KU Leuven & Basque Center for Applied Mathematics (BCAM)

Helge Ruddat, University of Stavanger: The twistor space of the universal unfolding of the A_n singularity and beyond

If the Milnor fiber is a configuration of spheres, we can use a quiver Q to describe it. We consider the case where Q is of finite mutation-type, for example the A_n quiver for the A_n singularity. Associated to Q are two interesting complex manifolds:

(1) the space of stability conditions for the derived CY3 category of the Ginzburg algebra associated to the quiver,

(2) the complex cluster Poisson variety.
Each of these manifolds is constructed from the combinatorics of the quiver, though in very different ways.

Tom Bridgeland and I introduce an "interpolation" of these complex manifolds: we give a construction of a complex manifold together with a submersion to the complex plane which we call the stability twistor space. The fiber over the origin is a finite quotient of the space of stability conditions whereas every other fiber is an etale cover of the cluster Poisson variety associated to the quiver. The twistor space fiber can be identified with the universal unfolding of the singularity for which the quiver describes the Milnor fiber.

Giancarlo Urzúa, Pontificia Universidad Católica de Chile: Degenerations of del Pezzo surfaces with only Wahl singularities"

A degeneration of a complex nonsingular del Pezzo surface with only log-terminal singularities (=quotient singularities in dimension 2) can be thought of as a degeneration with only Wahl singularities. These singularities are the log-terminal singularities that admit a rational homology smoothing (i.e. smoothing with Milnor number equal to 0). We classify all of these degenerations, which extends what is known for the complex projective plane through Markov numbers. I will explain the classification and some applications. This is a joint work with Juan Pablo Zúñiga (UC Chile).

Javier Fernández de Bobadilla, Basque Center for Applied Mathematics (BCAM): A'Campo spaces and their applications

To any normal crossing degeneration one can associate its A'Campo space.it replaces the central fibre by a radius 0 copy of the Milnor fibration. A'Campo permored this construction topologically and used it to produce monodromy with special dynnamics, which allowed to compute monodromy zeta function. We have extended A'Campo construction to the symplectic category. Applications by now are symplectic monodromy with special dynnamics (leading to a proof of Zariski multiplicity conjecture families with constant Milnor number), and SYZ fibrations. Joint with T.Pelka.

Tim Gräfnitz, Leibniz University Hannover: Mutations and deformations of Gorenstein toric varieties via log resolutions

Deformations of Gorenstein toric varieties are intimately linked to the notion of mutations. This connection is somewhat inspired by mirror symmetry. After summarizing the relevant notions, conjectures and known results, I will present a new approach using logarithmic geometry, as well as some first results in this direction that may serve as a proof of concept. This is joint work with Alessio Corti and Helge Ruddat.

Leonardo Cavenaghi, Campinas I: F-bundles

We shall quickly go through the concept of F-bundles, aiming to explain the A-model F-bundle necessary to the definition of Hodge atoms.

Simon Felten, Oxford University: The logarithmic smoothness of non-d-semistable normal crossing schemes

Let V/ℂ be a normal crossing scheme. When V is d-semistable, i.e., ℰxt¹(Ω¹_V,𝒪_V) ≅ 𝒪_D for the singular locus D = Sing(V), there is a global log smooth structure over the standard log point S₀. When
ℰxt¹(Ω¹_V,𝒪_V) is only globally generated, we can still find a well-behaved log smooth structure over the standard log point S₀; however, it is not globally defined but defined outside a log singular locus Z ⊂ V.
Up to now, we considered the thus-constructed object as a partial log scheme (V, V – Z,ℳ), i.e., a log scheme whose log structure ℳ is defined only on an open subset. This has a number of drawbacks, for example when
studying (classical as well as derived) deformations of or log curves on (V, V – Z, ℳ).
In this talk, I propose a formalism to extend the log smooth structure across Z as a sharp lax log structure. This sharp lax log structure admits a chart, is locally isomorphic to the spectrum of a (sharp) lax log ring, and turns out to have the infinitesimal lifting property in the category of integral sharp lax log schemes—hence we may say that our extension is log smooth.
The same formalism can also be applied to the positive and simple toric log Calabi­­­–Yau spaces of the Gross–Siebert program. In this setting, the infinitesimal lifting property holds in general only up to codimension 3, corresponding to the fact that the general fiber of a toric degeneration can have singularities in codimension 4.

Felipe Espreafico, Sorbonne Université: Refined Donaldson-Thomas Invariants and Milnor Fibres

Aiming to understand the relation to other "refined invariants », we explain how to obtain a quadratic, A^1-version of Donaldson-Thomas invariants from the motivic refinements first introduced in Kontsevich-Soibelman. Following ideas from Behrend, Bryan and Szendroi, we provide predictions for these invariants in a few simple examples, mainly the computation of DT invariants of A^3. Over the real numbers, we recover invariants that were already considered in Physics by Krefl and Walcher. We also discuss, during the construction, refinements of the Euler characteristic of the Milnor Fibre for critical loci and the so-called Behrend functions. Our main goal is to draw relationships with the literature, including works of Levine, Denef and Loser, Azouri, Pepin-Lehaulleur, Srinivas, Comte and Fichou, among others. We end by posing some further questions on the topic. This is joint work with Johannes Walcher. We also comment on joint work in progress with Ran Azouri.

Diego Matessi, Università degli Studi di Milano: Lagrangian fibrations on Calabi-Yau hypersurfaces I

In joint work with Mak-Ruddat-Zharkov, we prove the existence of Lagrangian torus fibrations on Calabi-Yau hypersurfaces in toric Fano manifolds given by a reflexive polytope. The result is motivated by the Strominger-Yau-Zaslow conjecture which predicts the existence of these fibrations on Calabi-Yau manifolds near large complex structure limits. In this talk I will outline the main set up of the construction. The idea is to replace the ordinary algebraic equation, with a new one involving "ironing coefficients" and a convex potential which have the effect of breaking the manifold in local models. Over these models we apply the Liouville flow technique in the style of Evans-Mauri.

Leonardo Cavenaghi, Campinas II: Hodge atoms

We shall define Hodge atoms, discuss the first invariants, and present some applications.

Ilia Zharkov, Kansas State University:  Lagrangian fibrations on Calabi-Yau hypersurfaces II

Matej Filip, University of Ljubljana: Deformations and mutations

We establish a correspondence between one-parameter deformations of an affine Gorenstein toric variety X, determined by a polytope P, and mutations of a Laurent polynomial f with Newton polytope P. In dimension three, we show how to construct components of the deformation space of X via mutations.

Andrew Hanlon, University of Oregon: Topological toric mirror monodromies

Mirror symmetry predicts an action by the fundamental group of a stringy Kahler moduli space on the derived category of an algebraic variety. For a toric variety, a model for this space is understood, but constructing the action is a still an open problem in general. We will see that in the case of the $A_{n-1}$ singularity, this action can be studied via a moduli space of Legendrians isotopic to the FLTZ Legendrian and that we recover a braid group action first discovered by Seidel and Thomas. This talk is based on joint work with Michela Barbieri and Jeff Hicks.

Leonardo Cavenaghi, Campinas III: G-equivariant Hodge atoms

We shall introduce the concept of G-equivariant Hodge atoms and provide applications. Time permitting, we shall explain how Chen-Ruan cohomology enhances the invariants in the theory, possibly leading to the development of a "Mendelev table" for G-equivariant birational types.

Enrique Becerra & Ernesto Lupercio, CINVESTAV: From Elliptic Genera to Spectra: A Vertex Algebra Approach to Isolated Singularities

The Steenbrink spectrum of an isolated hypersurface singularity encodes subtle Hodge–theoretic information about its Milnor fiber. In this talk I will explain how this spectral data naturally arises as the semiclassical limit of an elliptic genus associated to the singularity.
Our approach relies on constructing a twisted version of the chiral de Rham complex, viewed as a quantization of the classical twisted de Rham complex. The partition function of this vertex algebra reproduces the elliptic genus of the corresponding Landau–Ginzburg model.
This framework extends the construction of Borisov and Libgober, which was restricted to weighted homogeneous singularities defined via toric orbifold methods, to arbitrary isolated singularities in a purely algebraic way. Conceptually, it provides a bridge between Hodge theory and the modular geometry of conformal field theory.

Enrique Becerra/Ernesto Lupercio II: From Elliptic Genera to Spectra: A Vertex Algebra Approach to Isolated Singularities

The Steenbrink spectrum of an isolated hypersurface singularity encodes subtle Hodge–theoretic information about its Milnor fiber. In this talk I will explain how this spectral data naturally arises as the semiclassical limit of an elliptic genus associated to the singularity.
Our approach relies on constructing a twisted version of the chiral de Rham complex, viewed as a quantization of the classical twisted de Rham complex. The partition function of this vertex algebra reproduces the elliptic genus of the corresponding Landau–Ginzburg model.
This framework extends the construction of Borisov and Libgober, which was restricted to weighted homogeneous singularities defined via toric orbifold methods, to arbitrary isolated singularities in a purely algebraic way. Conceptually, it provides a bridge between Hodge theory and the modular geometry of conformal field theory.

Cheuk Yu Mak, University of Sheffield: Lagrangian fibrations on Calabi-Yau hypersurfaces III

This is a continuation of the talks by Matessi and Zharkov. In this talk, we describe the symplectic hypersurface. As a result of our construction, the symplectic hypersurface admits a covering by local pieces which have partial linearity among their overlaps. This allows us to run Evans-Mauri's Liouville flow technique inductively. We will also explain the features of the resulting Lagrangian torus fibration. 

Renato Viana, University of São Paulo: Open-string Quantum Lefschetz formula

Let Y be a symplectic divisor of X, \omega. In the Kahler setting, Givental's (closed-string) Quantum Lefschetz formula relates certain Gromov-Witten invariants (encoded by the G function) of X and Y. Given an Lagrangian L in (Y, \omega|Y), we can lift it to a Lagrangian L' in neighbourhood NY \subset X. We will introduce the notion of the potential of a Lagrangian, which encodes information of Maslov index 2 J-holomorphic disks with boundary on it. From the work of Biran-Khanevski, we can extract a formula for when (X,L',Y,L) forms a monotone tuple (we will define this notion), and the minimal Chern number of Y is 2. We generalise the formula in this setting when we allow the minimal Chern number to be 1. We can use this to show the existence of infinitely many Lagrangian tori in CP^n, Quadrics, Cubics, and other symplectic manifolds, among other results. Following the work of Tonkonog on gravitational descendants, we recover an explicit Quantum Lefschetz formula appearing in the work of Coates-Corti-Galkin-Kasprczyk. Interestingly, their formula applies in a different context--specifically when X is toric -- which neither contains nor is contained in the monotone tuple setting. Motivated by this, we introduce an alternative set of hypotheses, typically satisfied when X degenerates to a toric manifold, under which a broader open-string Quantum Lefschetz formula applies. The differences between these sets of hypotheses will be discussed. This is joint work with Luis Diogo, Dmitry Tonkonog and Weiwei Wu.

Lino Grama, Campinas: New look at Milnor spheres

We explore the interplay between Spherical T-duality, exotic spheres, and the generalized log transform, revealing new connections in geometric topology and complex geometry. Spherical T-duality generalizes classical T-duality by replacing the circle group $\mathrm{U}(1)$ with the 3-sphere $\mathrm{S}^3$ or $\mathrm{SU}(2)$, relating $\mathrm{SU}(2)$-bundles equipped with degree-7 cohomology
cocycles. A striking class of examples involves the 7-dimensional homotopy spheres $\Sigma^7$, whose product with $\mathrm{S}^1$
exhibits distinct holomorphic structures under spherical T-duality,
contrasting with classical Hopf manifolds such as $\mathrm{S}^3 \times
\mathrm{S}^1$.

Semon Rezchikov, IHES & Barnard College: Topological Hochschild Homology, Equivariant Floer Theory and p-Curvature.

I will discuss an interpretation of the p-curvature of the Getzler-Gauss-Manin connection in terms of a natural operation on equivariant Hochschild homology. The symplectic interpretation of this result (following work of Lee and Chen) identifies the p-curvature of the quantum connection of a nondegenerate Calabi Yau symplectic manifold with the quantum Steenrod operation. Some of the (seemingly abstract) algebraic structures arising in this argument should have a direct geometric interpretation in terms of computations in equivariant Floer homology, which I will discuss at the end of the talk.

Enrique Becerra, CINVESTAV III: From Elliptic Genera to Spectra: A Vertex Algebra Approach to Isolated Singularities

The Steenbrink spectrum of an isolated hypersurface singularity encodes subtle Hodge–theoretic information about its Milnor fiber. In this talk I will explain how this spectral data naturally arises as the semiclassical limit of an elliptic genus associated to the singularity.
Our approach relies on constructing a twisted version of the chiral de Rham complex, viewed as a quantization of the classical twisted de Rham complex. The partition function of this vertex algebra reproduces the elliptic genus of the corresponding Landau–Ginzburg model.
This framework extends the construction of Borisov and Libgober, which was restricted to weighted homogeneous singularities defined via toric orbifold methods, to arbitrary isolated singularities in a purely algebraic way. Conceptually, it provides a bridge between Hodge theory and the modular geometry of conformal field theory.

Enrique Becerra/Ernesto Lupercio IV: From Elliptic Genera to Spectra: A Vertex Algebra Approach to Isolated Singularities

The Steenbrink spectrum of an isolated hypersurface singularity encodes subtle Hodge–theoretic information about its Milnor fiber. In this talk I will explain how this spectral data naturally arises as the semiclassical limit of an elliptic genus associated to the singularity.
Our approach relies on constructing a twisted version of the chiral de Rham complex, viewed as a quantization of the classical twisted de Rham complex. The partition function of this vertex algebra reproduces the elliptic genus of the corresponding Landau–Ginzburg model.
This framework extends the construction of Borisov and Libgober, which was restricted to weighted homogeneous singularities defined via toric orbifold methods, to arbitrary isolated singularities in a purely algebraic way. Conceptually, it provides a bridge between Hodge theory and the modular geometry of conformal field theory.

Enrique Becerra/Ernesto Lupercio V: From Elliptic Genera to Spectra: A Vertex Algebra Approach to Isolated Singularities

The Steenbrink spectrum of an isolated hypersurface singularity encodes subtle Hodge–theoretic information about its Milnor fiber. In this talk I will explain how this spectral data naturally arises as the semiclassical limit of an elliptic genus associated to the singularity.
Our approach relies on constructing a twisted version of the chiral de Rham complex, viewed as a quantization of the classical twisted de Rham complex. The partition function of this vertex algebra reproduces the elliptic genus of the corresponding Landau–Ginzburg model.
This framework extends the construction of Borisov and Libgober, which was restricted to weighted homogeneous singularities defined via toric orbifold methods, to arbitrary isolated singularities in a purely algebraic way. Conceptually, it provides a bridge between Hodge theory and the modular geometry of conformal field theory.