MPS Conference on Higher Dimensional Geometry, October 2023

Meng Chen
Fudan University

A lifting principle for canonical stability indices of varieties of general type

For any integer \(n>0\), the \(n\)-th canonical stability index \(r_n\) is defined to be the smallest positive integer so that the \(r_n\)-canonical map \(\Phi_{r_n}\) is stably birational onto its image for all nonsingular projective \(n\)-folds of general type. In this talk, I will explain the main steps towards proving the following conjecture(=”lifting principle”): \(r_n\) equals to the maximum of the set of those canonical stability indices of smooth projective \((n+1)\)-folds with sufficiently large canonical volumes. This is a joint work with Hexu Liu.
 

Antonella Grassi
University of Pennsylvania

Local, global, local-to-global “principles”and elliptic Calabi-Yau threefolds with certain singularities

We present  local, global and local-to-global properties of threefolds with certain singularities, in particular criteria for these threefolds to be rational homology manifolds and to satisfy rational Poincaré duality. The motivation comes from physics of string theory. We motivate and state a conjecture on the extension of Kodaira’s classification of singular fibers on relatively minimal elliptic surfaces to the class of birationally equivalent relatively minimal elliptically fibered varieties. We prove results  when X is an elliptic Calabi-Yau.
 

Chen Jiang
Fudan University

Numerically trivial automorphisms of hyperkähler 4-folds

An automorphism on a smooth projective variety is said to be numerically trivial if it induces a trivial action on the cohomology ring. A well-known result says that there is no numerically trivial automorphism for a K3 surface. I will report our recent progress on the absence of numerically trivial automorphisms of hyperkähler 4-folds.
 

Yongnam Lee
Korea Advanced Institute of Science and Technology/IBS Center for Complex Geometry

Smooth specialization of hypersurfaces in projective manifolds

In this presentation, we give a structure theorem for projective manifolds \(W_0\) with the property of admitting a one parameter deformation where a general fiber \(W_t\) is a smooth hypersurface in a projective smooth manifold \(Z_t\). Their structure is the one of special iterated univariate coverings, which we call normal type. We give an application to the case where \(Z_t\) is a projective space, respectively an abelian variety. We also give a characterizaton of smooth ample hypersurfaces in abelian varieties and describe an irreducible connected component of their moduli space. This talk is mainly based on joint work with Fabrizio Catanese.
 

John Lesieutre
Pennsylvania State University

More pathologies of the volume function

I will report on joint work with Valentino Tosatti and Simion Filip in which we show by example that for a pseudoeffective divisor D and ample A, the volume function vol(D+tA) for small values of t can exhibit various pathological behaviors.
 

Emanuele Macri
Université Paris-Saclay

Deformations of stability conditions

Bridgeland stability conditions have been introduced about 20 years ago, with motivations both from algebraic geometry, representation theory and physics. One of the fundamental problem is that we currently lack methods to construct and study such stability conditions in full generality. In this talk I would present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari and Zhao. As application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces.
 

Michael McQuillan
Università degli Studi di Roma Tor Vergata

Functorial resolution of singularities

The talk will give a general framework for resolution of singularities whether of varieties, resp. foliations, by way of a flow determined by an ideal, resp. a derivation, on the space of valuations of a complete regular local ring. In particular the convergence of this flow implies a wholly functorial solution of the resolution problem in question.
 

Alex Perry
University of Michigan

Hyperkähler varieties from CY2 categories

Hyperkähler varieties form one of the three building blocks for projective varieties with trivial canonical bundle. Their classification is widely open, but there is growing evidence that the problem is closely related to the classification of 2-dimensional Calabi-Yau categories, with the most optimistic hope being that any hyperkähler variety arises from a suitable moduli space of objects in such a category. I will discuss this circle of ideas, including work in progress with Arend Bayer, Laura Pertusi, and Xiaolei Zhao completing this program for hyperkähler varieties of Kummer type.
 

Nick Shepherd-Barron
King’s College London

Generic Torelli with denominators for elliptic surfaces.

It is well known that a very general curve of genus at least 5 is determined by its polarised rational Hodge structure. We prove an analogous result for elliptic surfaces.
 

Bernd Siebert
The University of Texas at Austin

Intrinsic Mirror Symmetry

One of the fundamental questions of mirror symmetry asks “How broadly do mirrors exist”? In the talk I will survey joint work with Mark Gross giving canonical candidate mirror varieties in the two basic cases (1) maximally unipotent degenerations of Calabi-Yau manifolds and (2) pairs (X,D) with trivial logarithmic canonical bundle. The construction is known or conjectured to reproduce all known algebraic-geometric mirror pairs.The coordinate rings of the intrinsic mirror families have module generators given by contact orders of curves with the irreducible components of the central fibers and with D, respectively. The structure coefficients are given by a variant of logarithmic Gromov-Witten invariants admitting negative contact orders, recently defined in joined work with Dan Abramovich and Qile Chen. Many interesting questions arise concerning the properties of intrinsic mirror families, including applications to moduli problems and classification theory.
 

Chengxi Wang
University of California, Los Angeles

Fano varieties with extreme behavior

It is attractive to classify Fano varieties with various types of singularities that originated from the minimal model program. For a Fano variety, the Fano index is the largest integer m such that the anti-canonical divisor is Q-linearly equivalent to m times some Weil divisor. For Fano varieties of various singularities, I show the Fano indexes can grow double exponentially with respect to the dimension. Those examples are also conjecturally optimal and have a close connection with Calabi-Yau varieties of extreme behavior.
 

De-Qi Zhang
National University of Singapore

Structures theorems and applications of non-isomorphic surjective endomorphisms of smooth projective threefolds

Let f: X to X be a non-isomorphic surjective endomorphism of a smooth projective threefold X. We prove that any birational minimal model program becomes f-equivariant after iteration, provided that f is delta-primitive. Here by ‘delta-primitive’ we mean that there is no f-equivariant dominant rational map X to Y to a positive lower-dimensional projective variety Y such that the first dynamical degree delta_f remains unchanged. We further determine the building blocks of f. As the first application, we prove the Kawaguchi-Silverman conjecture (about the equality of dynamic degree and arithmetic degree) for every non-isomorphic surjective endomorphism of a smooth projective threefold. As the second application, we reduce the Zariski dense orbit conjecture for f to a terminal threefold with only f-equivariant Fano contractions.

Air and Train
For individuals in Groups A and B the foundation will arrange and pay for round-trip travel from their home city to the conference.

All travel and hotel arrangements must be booked through the Simons Foundation’s preferred travel agency. Travel arrangements not booked through the preferred agency must be pre-approved by the Simons Foundation and a reimbursement quote must be obtained through the foundation’s travel agency. Travel specifications can be provided by clicking the registration link above.

Personal Car
Personal car trips over 250 miles each way require prior approval from the Simons Foundation via email.

The James NoMad Hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage, therefore it is encouraged that participants walk or take public transportation to the Simons Foundation.

Hotel
Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of four nights at The James NoMad Hotel. Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link above.

The James NoMad Hotel
22 E 29th St
New York, NY 10016
(between 28th and 29th Streets)
https://www.jameshotels.com/new-york-nomad/

For driving directions to The James NoMad, please click here.

Attendance
In-person participants and speakers are expected to attend all meeting days. Partial participation is permitted so long as the individual fully attends the first day, which is typically Thursday for two-day meetings. Participants receiving hotel and travel support wishing to arrive on meeting days which conclude at 2:00 PM will be asked to attend remotely.

COVID-19 Vaccination
Individuals accessing Simons Foundation and Flatiron Institute buildings must be fully vaccinated against COVID-19.

Entry & Building Access
Upon arrival, guests will be required to show their photo ID to enter the Simons Foundation and Flatiron Institute buildings. After checking-in at the meeting reception desk, guests will be able to show their meeting name badge to re-enter the building. If you forget your name badge, you will need to provide your photo ID.

The Simons Foundation and Flatiron Institute buildings are not considered “open campuses” and meeting participants will only have access to the spaces in which the meeting will take place. All other areas are off limits without prior approval.

If you require a private space to conduct a phone call or remote meeting, please contact your meeting manager at least 48-hours ahead of time so that they may book a space for you within the foundation’s room reservation system.

Guests
Meeting participants are required to give 24 hour advance notice of any guests meeting them at the Simons Foundation either before or after the meeting. Outside guests are discouraged from joining meeting activities, including meals.

Ad hoc meeting participants who did not receive a meeting invitation directly from the Simons Foundation are discouraged.

Children under the age of 18 are not permitted to attend meetings at the Simons Foundation. Furthermore, the Simons Foundation does not provide childcare facilities or support of any kind. Special accommodations will be made for nursing parents.

Reimbursement
Individuals in Groups A & B will be reimbursed for meals and local expenses including ground transportation. Expenses should be submitted through the foundation’s online expense reimbursement platform after the meeting’s conclusion.

Expenses accrued as a result of meetings not directly related to the Simons Foundation-hosted meeting (a satellite collaboration meeting held at another institution, for example) will not be reimbursed by the Simons Foundation and should be paid by other sources.

Below are key reimbursement takeaways; a full policy will be provided with the final logistics email circulated approximately 2 weeks prior to the meeting’s start.

Meals
The daily meal limit is $125 and itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses.

• Meals taken on travel days are reimbursable.
• Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner) are not reimbursable.
• If a meal was not provided on a meeting day, dinner for example, that expense is reimbursable.
• Meals taken on days not associated with Simons Foundation-coordinated events are not reimbursable.
• Minibar expenses are not reimbursable
• Meal expenses for a non-foundation guest are not reimbursable.
• Group meals consisting of fellow meeting participants paid by a single person will be reimbursed up to $65 per person per meal and the amount will count towards each individual’s $125 daily meal limit.

Ground Transportation
Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport) as well as local transportation. While in NYC, individuals are encouraged to use public transportation and not use taxi, Uber or Lyft services.

Registration and Travel Assistance
Ovation Travel Group
[email protected]
(917) 408-8384 (24-Hours)
www.ovationtravel.com

Meeting Questions and Assistance
Emily Klein
Assistant Manager, Events, MPS, Simons Foundation
[email protected]
(646) 751-1262