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# Workshop on Fibration and Degeneration in Calabi-Yau Geometry

## June 24, 2024 @ 8:00 am - June 26, 2024 @ 5:00 pm

**Workshop on Fibration and Degeneration in Calabi-Yau Geometry**

**Dates:** June 24-26, 2024

**Location: **Harvard CMSA, 20 Garden Street, Cambridge, MA 02138

**Organizer:** Chuck Doran, Harvard CMSA

Calabi-Yau manifolds occupy a central place in geometry. Their critical role as the cut-case between basic Fano building blocks and the zoo of General Type manifolds is key to the wide variety of important applications of Calabi-Yau geometry to theoretical physics. In turn, ideas from theoretical physics, such as Mirror Symmetry, help shape investigations in Calabi-Yau geometry

This workshop focuses on a structural feature of Calabi-Yau geometry identified a decade ago by Doran, Harder, and Thompson. It is an organizing principle that conjecturally underlies any and all constructions of mirror pairs of Calabi-Yau manifolds. Put simply, the DHT Mirror Symmetry slogan is: “Degeneration is mirror to fibration.”

**Confirmed Speakers:**

- David Favero (University of Minnesota)
- Andrew Harder (Lehigh University)
- Jesse Huang (University of Alberta)
- Mohsen Karkheiran* (University of Alberta)
- Matt Kerr* (Washington University in St. Louis)
- Thorsten Schimannek* (Utrecht University)
- Michael Schultz (Virginia Tech)
- Alan Thompson (Loughborough University)
- Fenglong You (University of Nottingham & ETH Zurich)

**= via Zoom*

**Schedule**

**Monday, June 24, 2024**

9:30 – 10:00 am: Breakfast

10:00 – 11:00 am

**Speaker:** Alan Thompson, Loughborough University

**Title:** Mirror symmetry for fibrations and degenerations of K3 surfaces

**Abstract:** I will describe recent progress, joint with Luca Giovenzana, on the DHT problem for K3 surfaces. I will give an lattice-theoretic definition for when a Tyurin degeneration of K3 surfaces and an elliptically-fibred K3 surface, with an appropriate splitting of the base, form a mirror pair. I will then explain how this definition is compatible with lattice polarised mirror symmetry for K3 surfaces and with Fano-LG mirror symmetry for (quasi) del Pezzo surfaces. The upshot will be a concrete statement of the DHT conjecture for K3 surfaces.

12:00 – 1:00: Lunch

1:00 – 2:00 pm

**Speaker:** David Favero, University of Minnesota

**Title:** Homotopy Path Algebras and Resolutions

**Abstract:** A homotopy path algebra is like a directed version of the group ring on a fundamental group. One can imagine a directed graph (quiver) embedded in a topological space and considering the path algebra up to homotopy. Alternatively, one can think of homotopy classes of directed paths in a stratified topological space. I will introduce homotopy path algebras and describe their connections to mirror symmetry and resolutions of coherent sheaves on toric varieties.

3:00 – 4:00 pm

**Speaker:** Andrew Harder, Lehigh University

**Title: **Tropical Hodge theory for hypersurfaces and Clarke duality

**Abstract: **Results of Itenberg, Katzarkov, Mikhalkin, and Zharkov (IKMZ) show that if a projective variety admits a smooth tropicalization, then there is a collection of sheaves on its tropicalization that can be used to compute its Hodge numbers. However, smooth tropicalizations fail to exist even in the case of toric hypersurfaces. In work with Sukjoo Lee, we show that for any toric hypersurface, an analogue of IKMZ’s result holds. I’ll discuss this sheaf, and how this allows us to prove that Clarke dual pairs of Landau-Ginzburg models satisfy a particular Hodge number duality. This is a vast generalization of work of Batyrev and Borisov from the 90s.

4:00 – 4:30 pm: Coffee/Tea

**Tuesday, June 25, 2024**

9:30 – 10:00 am: Breakfast

10:00 – 11:00 am

**Speaker:** Matt Kerr, Washington University in St. Louis

**Title:** Hypergeometric families and Beilinson’s conjectures

**Abstract:** I will describe the construction of motivic cohomology classes on hypergeometric families of Calabi-Yau 3-folds using Hadamard convolutions. These are analogous to elements of the Mordell-Weil group for families of elliptic curves, and produce solutions to certain inhomogeneous Picard-Fuchs equations. This is part of a joint project with Vasily Golyshev in which we numerically verify Beilinson’s conjectures in some new cases.

12:00 – 1:00: Lunch

1:00 – 2:00 pm

**Speaker:** Fenglong You, University of Nottingham & ETH Zurich

**Title:** Theta functions in mirror symmetry

**Abstract:** To obtain a mirror of a Calabi—Yau manifold using Gross—Siebert’s intrinsic mirror symmetry, one considers a maximally unipotent monodromy degeneration of the Calabi—Yau and take proj of the degree zero part of a relative quantum cohomology ring associated with the degeneration. Theta functions form a canonical basis of the degree zero part of the relative quantum cohomology ring. Theta functions can also be defined in terms of punctured invariants of the broken line type. I will explain a variant of intrinsic mirror symmetry using orbifold invariants, theta functions for general snc pairs and a relation with the DHT conjecture.

3:00 – 4:00 pm

**Speaker:** Mohsen Karkheiran, University of Alberta

**Title:** Emergence of Heterotic-Type II duality from DHT conjecture

**Abstract: **The duality between Heterotic and Type IIA strings was conjectured in mid-90’s based on the properties of 4D N=2 field theories and solitonic strings in 6D. Here, we show that this duality can also emerge from the DHT conjecture. We assume both IIA and IIB strings are compactified over toric Calabi-Yau threefolds which admit K3-fibrations with arbitrary polarizations. Then by applying the Hori-Vafa mirror symmetry to the “pieces” of these Calabi-Yau manifolds, we will be able to derive the defining data for Heterotic strings. This approach works for any gauge group on the Heterotic side, and we will show how it can be practically useful to derive the Heterotic dual for any toric Calabi-Yau threefolds in Type IIA or F-theory.

4:00 – 4:30 pm: Coffee/Tea

**Wednesday, June 26, 2024**

9:30 – 10:00 am: Breakfast

10:00 – 11:00 am

**Speaker:** Thorsten Schimannek, Utrecht University

**Title:** Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds

**Abstract:** Smooth M_m-polarized K3-fibered Calabi-Yau (CY) 3-folds have been classified in [DHNT] and [KT] in terms of the choice of a generalized functional invariant (GFI) and, in the case m=1, a generalized homological invariant (GHI). The resulting geometries generally exhibit a small number of complex structure moduli greater or equal to two. I will start my talk by discussing a concrete choice of these invariants that realizes (almost all of) the geometries with exactly two moduli and describe the structure of the corresponding moduli spaces. The corresponding variations of Hodge structure are entirely determined by the regular periods, for which we obtain a generic expression in terms of m and three integers i,j,s. Using the form of this period and Batyrev-Borisov mirror symmetry I will then explicitly construct the corresponding mirror CY 3-folds with two Kaehler moduli and show consistency with the DHT conjecture. In the cases with s=0, the mirror CY 3-folds are again K3-fibered but with a 2m-polarization. The generic form of the periods allows us to derive generic modular expressions for the A-model topological string free energies and we argue that those are a consequence of a Tyurin degeneration of the GFI with the central fiber being an M_m-polarized K3.

The talk is based on work in progress with Charles Doran and Boris Pioline.

12:00 – 1:00: Lunch

1:00 – 2:00 pm

**Speaker:** Michael Schultz, Virginia Tech

**Title:** Mirror Symmetry from Irrationality Proofs and a Proposal for Local Invariants

**Abstract:** While Apéry’s original proof of the irrationality of ζ(3) stunned the mathematics community in 1978, subsequent generations of mathematicians (including a number of those at this workshop) have discovered geometric and modular structures underlying these irrationality proofs that are arguably even more striking. One such well known example are connections to modular pencils of elliptic curves and K3 surfaces and their Picard-Fuchs operators, which exhibit maximally unipotent monodromy. These objects are respectively mirror dual to anticanonical divisors in certain del Pezzo surfaces and Fano threefolds, and their Picard-Fuchs operators to the A-side connection on small quantum cohomology for these varieties. Although the Yukawa couplings calculated in classical mirror symmetry for elliptic curves and K3 surfaces are trivial, I will show in this talk how a blend of the perspectives above allows one to define “virtual” Yukawa couplings for these families that are not trivial. It will be proposed that the utility of this perspective is in computing local invariants related to the mirror, which recovers some known results in the literature and utilizes connections to work on the DHT conjecture and the twist construction of Doran & Malmendier.

3:00 – 4:00 pm

**Speaker:** Jesse Huang, University of Alberta

**Title: **An invitation to global toric mirror symmetry

4:00 – 4:30 pm: Coffee/Tea