Periodic Reporting for period 4 - TRANSHOLOMORPHIC (New transversality techniques in holomorphic curve theories)
Reporting period: 2023-03-01 to 2023-08-31
A common paradigm in symplectic topology (and also in neighboring areas such as differential topology) is that one discovers deep geometric phenomena by studying the topological properties of spaces of solutions to certain partial differential equations (PDEs), determined by the underlying geometric 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 built-in symmetries that cannot be perturbed away, except by abstract methods that destroy a certain amount of meaningful structure.
The motivating goal of this project was 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 other fields where analogous technical issues arise. The ideal long-term 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 understanding would have a large impact on the field of symplectic topology, where transversality difficulties have impeded progress on many difficult problems 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 -- and in principle on all fields of pure and applied mathematics, in which the delicate balance between symmetry and genericity is a theme with universal relevance.
The original impetus for the project was provided by a paper of the PI that proved a long-standing conjecture on "super-rigidity": the statement is roughly that on certain types of 6-dimensional symplectic manifolds, the maximal degree of transversality conceivable is actually attained under generic assumptions. That paper established the approach of decomposing a solution space into disjoint smooth "strata" determined by transversality conditions: if one can determine the dimensions of these strata with enough precision, then 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 functions 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 motivating goal of this project was 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 other fields where analogous technical issues arise. The ideal long-term 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 understanding would have a large impact on the field of symplectic topology, where transversality difficulties have impeded progress on many difficult problems 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 -- and in principle on all fields of pure and applied mathematics, in which the delicate balance between symmetry and genericity is a theme with universal relevance.
The original impetus for the project was provided by a paper of the PI that proved a long-standing conjecture on "super-rigidity": the statement is roughly that on certain types of 6-dimensional symplectic manifolds, the maximal degree of transversality conceivable is actually attained under generic assumptions. That paper established the approach of decomposing a solution space into disjoint smooth "strata" determined by transversality conditions: if one can determine the dimensions of these strata with enough precision, then 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 functions 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.
A gap in the PI's paper on the super-rigidity conjecture was discovered in Summer 2018, and considerable effort in the first year of the project was devoted toward filling that gap, which was both successful and fruitful in other respects, e.g. it led to important clarifications on a local property of linear differential operators known as "Petri's condition," which we now understand as playing a fundamental role in the study of elliptic PDEs subject to symmetry conditions.
In the mean time, several other subprojects were undertaken, including the work of three Ph.D. students on the following problems:
(1) Developing an equivariant transversality theory for closed holomorphic curves in symplectic manifolds equipped with symplectic actions by finite 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.
In parallel, three postdocs worked on the following problems:
(4) Using stratification techniques to develop a new theory of virtual fundamental classes for moduli spaces.
(5) 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.
(6) Applications of transversality techniques to the study of contact manifolds.
All of the subprojects in the list above encountered delays due to the Covid pandemic, one consequence of which was that the three PhD students had not yet finished their research when the grant reached its end date. As of the final report, the current status is that the results of project (1) are in the final stages of being written up as a dissertation, to be defended in late Summer 2024, and these results should be published in a journal soon after. Projects (2) and (3) remain work in progress, but with predictable outcomes that we expect to see published within the next few years. Subprojects (4)-(6) also remain work in progress, though some interim results were achieved already within the framework of projects (5) and (6), leading e.g. to published papers on holomorphic curves with Lagrangian boundary conditions, exact orbifold fillings of contact manifolds, and tightness of contact manifolds in higher dimensions. Progress on project (4) has been hampered by the technical issues mentioned in the next paragraph, but it remains as prominent motivation in the ongoing research program of the PI.
One of the subprojects originally envisioned for the first year of the grant 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 seminar talk in 2020, two junior colleagues at Princeton (S. Bai and M. Swaminathan) began working on it independently of our group, and they made impressive progress that led eventually to a 100-page publication solving the wall-crossing problem in the presence of doubly-covered holomorphic curves. The limitation of their work to covers of degree 2 is the result of an apparent technical roadblock that also impacted our project: further progress seems to require a "higher-order" version of Petri's condition for generic Cauchy-Riemann operators, and the methods developed so far do not suffice to establish this condition. We consider it nonetheless to be a solvable problem, and a high priority in the future research of the PI, as it would 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 subprojects were undertaken, including the work of three Ph.D. students on the following problems:
(1) Developing an equivariant transversality theory for closed holomorphic curves in symplectic manifolds equipped with symplectic actions by finite 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.
In parallel, three postdocs worked on the following problems:
(4) Using stratification techniques to develop a new theory of virtual fundamental classes for moduli spaces.
(5) 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.
(6) Applications of transversality techniques to the study of contact manifolds.
All of the subprojects in the list above encountered delays due to the Covid pandemic, one consequence of which was that the three PhD students had not yet finished their research when the grant reached its end date. As of the final report, the current status is that the results of project (1) are in the final stages of being written up as a dissertation, to be defended in late Summer 2024, and these results should be published in a journal soon after. Projects (2) and (3) remain work in progress, but with predictable outcomes that we expect to see published within the next few years. Subprojects (4)-(6) also remain work in progress, though some interim results were achieved already within the framework of projects (5) and (6), leading e.g. to published papers on holomorphic curves with Lagrangian boundary conditions, exact orbifold fillings of contact manifolds, and tightness of contact manifolds in higher dimensions. Progress on project (4) has been hampered by the technical issues mentioned in the next paragraph, but it remains as prominent motivation in the ongoing research program of the PI.
One of the subprojects originally envisioned for the first year of the grant 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 seminar talk in 2020, two junior colleagues at Princeton (S. Bai and M. Swaminathan) began working on it independently of our group, and they made impressive progress that led eventually to a 100-page publication solving the wall-crossing problem in the presence of doubly-covered holomorphic curves. The limitation of their work to covers of degree 2 is the result of an apparent technical roadblock that also impacted our project: further progress seems to require a "higher-order" version of Petri's condition for generic Cauchy-Riemann operators, and the methods developed so far do not suffice to establish this condition. We consider it nonetheless to be a solvable problem, and a high priority in the future research of the PI, as it would potentially open the way toward a much wider application of ideas from singularity theory toward moduli spaces of holomorphic curves.