## Periodic Reporting for period 2 - TRANSHOLOMORPHIC (New transversality techniques in holomorphic curve theories)

Reporting period: 2020-03-01 to 2021-08-31

A common paradigm in symplectic topology (and also in neighboring areas such as differential topology) is that one proves deep theorems about symplectic structures by studying the topological properties of solution spaces of certain PDEs that depend on those structures. The most popular PDE used for this purpose in the symplectic setting is the nonlinear Cauchy-Riemann equation, whose solutions are called J-holomorphic curves. In order to use this tool effectively, one needs to prove that the solution spaces are "transversely cut out" so that they have a nice local geometric structure and their dimensions can be predicted. Unfortunately, this is often not the case, as the equation also has certain symmetries built in which cannot be perturbed away, except by abstract methods that destroy a certain amount of meaningful structure.

The motivating goal of this project is essentially to overcome the classic incompatibility between transversality and symmetry, in particular as it affects holomorphic curve techniques in symplectic topology, though we also hope that the methods developed in this project will inspire and inform similar developments in neighboring fields where analogous technical issues arise. The ideal objective is to attain a sufficiently comprehensive understanding of transversality in the presence of symmetry so that the the inevitable juxtaposition of both concepts in moduli problems would come to be viewed as a feature rather than a bug. Most immediately, this would have a large impact on the field of symplectic topology, where transversality difficulties have been a major impediment to progress for over 30 years. More generally, one can expect progress in this area to have a positive impact on other fields that interact with symplectic topology: notably differential topology and mathematical physics.

The impetus for this project was provided by a paper written by the PI in 2016 (and also heavily revised during the course of this project), which proved a long-standing conjecture stating that for generic choices of geometric data on certain types of 6-dimensional symplectic manifolds (i.e. Calabi-Yau 3-folds), holomorphic curves are always "super-rigid". This result can be summarized as saying that the maximal degree of transversality conceivable in the Calabi-Yau setting is actually attained for generic choices. That paper introduced the idea of decomposing a moduli space of holomorphic curves into disjoint strata determined by transversality conditions: if one can determine the dimensions of these strata with enough precision, one can prove in many cases that the subset on which non-ideal transversality conditions hold is empty. This idea has been used before in the study of singularities for smooth maps between finite-dimensional spaces, and our project could in some sense be understood as an attempt to extend classical singularity theory to the infinite-dimensional geometric setting in which the analysis of holomorphic curves naturally lives. The various concrete goals of the project also include: (1) extending the techniques in the super-rigidity paper to the more general setting of symplectic field theory (SFT), which involves punctured holomorphic curves in noncompact symplectic cobordisms between contact manifolds, and (2) exploring various applications of these techniques to open problems in the study of symplectic and contact manifolds.

The motivating goal of this project is essentially to overcome the classic incompatibility between transversality and symmetry, in particular as it affects holomorphic curve techniques in symplectic topology, though we also hope that the methods developed in this project will inspire and inform similar developments in neighboring fields where analogous technical issues arise. The ideal objective is to attain a sufficiently comprehensive understanding of transversality in the presence of symmetry so that the the inevitable juxtaposition of both concepts in moduli problems would come to be viewed as a feature rather than a bug. Most immediately, this would have a large impact on the field of symplectic topology, where transversality difficulties have been a major impediment to progress for over 30 years. More generally, one can expect progress in this area to have a positive impact on other fields that interact with symplectic topology: notably differential topology and mathematical physics.

The impetus for this project was provided by a paper written by the PI in 2016 (and also heavily revised during the course of this project), which proved a long-standing conjecture stating that for generic choices of geometric data on certain types of 6-dimensional symplectic manifolds (i.e. Calabi-Yau 3-folds), holomorphic curves are always "super-rigid". This result can be summarized as saying that the maximal degree of transversality conceivable in the Calabi-Yau setting is actually attained for generic choices. That paper introduced the idea of decomposing a moduli space of holomorphic curves into disjoint strata determined by transversality conditions: if one can determine the dimensions of these strata with enough precision, one can prove in many cases that the subset on which non-ideal transversality conditions hold is empty. This idea has been used before in the study of singularities for smooth maps between finite-dimensional spaces, and our project could in some sense be understood as an attempt to extend classical singularity theory to the infinite-dimensional geometric setting in which the analysis of holomorphic curves naturally lives. The various concrete goals of the project also include: (1) extending the techniques in the super-rigidity paper to the more general setting of symplectic field theory (SFT), which involves punctured holomorphic curves in noncompact symplectic cobordisms between contact manifolds, and (2) exploring various applications of these techniques to open problems in the study of symplectic and contact manifolds.

The main result achieved in the project so far is one that was (falsely) believed to have been established before it began: an important gap in the PI's paper on the super-rigidity conjecture was discovered in Summer 2018 and took nearly a year to be repaired. This effort was fruitful in a few ways: aside from filling the gap, it produced some important clarification on a local property of linear differential operators known as "Petri's condition", which we expect to play a crucial role in many future developments within the project, as well as potential applications of our ideas to problems separate from holomorphic curve theory. The revised version of the super-rigidity paper was resubmitted for publication in May 2019 and is still under peer review.

In the mean time, several other projects have been started, all of which will take at least another year before they can produce measurable results. Three Ph.D. students have been hired on the project and are working on the following problems:

(1) Developing an equivariant transversality theory for closed holomorphic curves in symplectic manifolds equipped with symplectic group actions (by finite or compact groups).

(2) Extending the index computations of the super-rigidity paper to the setting of punctured holomorphic curves in symplectic cobordisms, with an eye toward proving transversality and/or super-rigidity results for multiple covers in SFT.

(3) Developing the Fredholm and transversality theory of J-holomorphic curves in symplectic cobordisms under generic bifurcations of asymptotic data.

The three postdocs hired on the project have been working thus far on the following projects:

(1) Using stratification techniques to develop a new theory of virtual fundamental classes for moduli spaces.

(2) Proving an analogous stratification theorem for moduli spaces of holomorphic disks with Lagrangian boundary in the Calabi-Yau setting, with an eye toward "open" Gromov-Witten theory.

(3) Proving a conjecture of the PI regarding the "quasiflexible" class of contact structures on 3-manifolds.

One of the projects originally planned for the first year of this grant has turned out to be both more challenging and more interesting than expected: it is the study of wall-crossing phenomena for closed multiply covered holomorphic curves. After the PI mentioned this problem in an online talk at the Western Hemisphere Virtual Symplectic Geometry Seminar, two Ph.D. students at Princeton decided to attack it independently of our group, and they got quite far before running into an apparent roadblock: it seems that a "higher-order" version of Petri's condition for generic Cauchy-Riemann operators is needed in order to understand generic bifurcations in the presence of symmetry. The PI is currently working on proving this condition, which will potentially open the way toward a much wider application of ideas from singularity theory toward moduli spaces of holomorphic curves.

In the mean time, several other projects have been started, all of which will take at least another year before they can produce measurable results. Three Ph.D. students have been hired on the project and are working on the following problems:

(1) Developing an equivariant transversality theory for closed holomorphic curves in symplectic manifolds equipped with symplectic group actions (by finite or compact groups).

(2) Extending the index computations of the super-rigidity paper to the setting of punctured holomorphic curves in symplectic cobordisms, with an eye toward proving transversality and/or super-rigidity results for multiple covers in SFT.

(3) Developing the Fredholm and transversality theory of J-holomorphic curves in symplectic cobordisms under generic bifurcations of asymptotic data.

The three postdocs hired on the project have been working thus far on the following projects:

(1) Using stratification techniques to develop a new theory of virtual fundamental classes for moduli spaces.

(2) Proving an analogous stratification theorem for moduli spaces of holomorphic disks with Lagrangian boundary in the Calabi-Yau setting, with an eye toward "open" Gromov-Witten theory.

(3) Proving a conjecture of the PI regarding the "quasiflexible" class of contact structures on 3-manifolds.

One of the projects originally planned for the first year of this grant has turned out to be both more challenging and more interesting than expected: it is the study of wall-crossing phenomena for closed multiply covered holomorphic curves. After the PI mentioned this problem in an online talk at the Western Hemisphere Virtual Symplectic Geometry Seminar, two Ph.D. students at Princeton decided to attack it independently of our group, and they got quite far before running into an apparent roadblock: it seems that a "higher-order" version of Petri's condition for generic Cauchy-Riemann operators is needed in order to understand generic bifurcations in the presence of symmetry. The PI is currently working on proving this condition, which will potentially open the way toward a much wider application of ideas from singularity theory toward moduli spaces of holomorphic curves.

In addition to the six projects in progress mentioned above, we still hope to address the following open problems before the end of the grant:

(1) Using wall-crossing results to uncover extra structure in the Gromov-Witten invariants

(2) Developing a wall-crossing theory for multiple covers in SFT

(3) Applications of (2) toward the definition of cobordism maps and proving invariance in Embedded Contact Homology (independently of gauge theory)

(4) Applications to the study of hyperbolic 3-manifolds via their cotangent bundles

One of the most ambitious (though speculative) aims of the project is to find a higher-dimensional analogue of Hutchings's Embedded Contact Homology, which could be expected to have groundbreaking applications (as it has in dimension 3). We cannot call this an "expected result" since it is still too early to say whether it is a realistic goal, but this will become clearer after the completion of the second Ph.D. project listed above.

(1) Using wall-crossing results to uncover extra structure in the Gromov-Witten invariants

(2) Developing a wall-crossing theory for multiple covers in SFT

(3) Applications of (2) toward the definition of cobordism maps and proving invariance in Embedded Contact Homology (independently of gauge theory)

(4) Applications to the study of hyperbolic 3-manifolds via their cotangent bundles

One of the most ambitious (though speculative) aims of the project is to find a higher-dimensional analogue of Hutchings's Embedded Contact Homology, which could be expected to have groundbreaking applications (as it has in dimension 3). We cannot call this an "expected result" since it is still too early to say whether it is a realistic goal, but this will become clearer after the completion of the second Ph.D. project listed above.