Periodic Reporting for period 4 - MACI (Moduli, Algebraic Cycles, and Invariants)
Période du rapport: 2023-03-01 au 2023-08-31
Algebraic geometry is the study of varieties -- the zero sets of polynomial equations in several variables. The subject has a central role in mathematics with connections to number theory,
representation theory, and topology. Moduli questions in algebraic geometry concern the behavior of varieties as the coefficients of the defining polynomials vary. At the end of the
20th century, several basic links between the algebraic geometry of moduli spaces and path integrals in quantum field theory were made. The virtual fundamental class plays an essential
role in these connections. I propose to study the algebraic cycle theory of basic moduli spaces. The guiding questions are: What are the most important cycles? What is the structure of the
algebra of cycles? How can the classes of geometric loci be expressed? The virtual fundamental class and the associated invariants often control the answers. A combination of
virtual localization, degeneration, and R-matrix methods together with new ideas from log geometry will be used in the study.
Most of the basic moduli spaces in algebraic geometry related to varieties of dimension at most 3 -- including the moduli of curves, the moduli of maps, the moduli of surfaces, and
the moduli of sheaves on 3-folds -- will be considered. The current state of the study of the algebraic cycle theory in these cases varies from rather advanced (for the moduli of curves)
to much less so (for the moduli of surfaces). There is a range of rich open questions which I will attack: Pixton's conjectures for the moduli of curves, the structure of the ring of Noether-Lefschetz
loci for the moduli of K3 surfaces, the holomorphic anomaly equation in Gromov-Witten theory, and conjectures governing descendents for the moduli of sheaves. The dimension 3 restriction
is often necessary for a good deformation theory and the existence of a virtual fundamental class.
representation theory, and topology. Moduli questions in algebraic geometry concern the behavior of varieties as the coefficients of the defining polynomials vary. At the end of the
20th century, several basic links between the algebraic geometry of moduli spaces and path integrals in quantum field theory were made. The virtual fundamental class plays an essential
role in these connections. I propose to study the algebraic cycle theory of basic moduli spaces. The guiding questions are: What are the most important cycles? What is the structure of the
algebra of cycles? How can the classes of geometric loci be expressed? The virtual fundamental class and the associated invariants often control the answers. A combination of
virtual localization, degeneration, and R-matrix methods together with new ideas from log geometry will be used in the study.
Most of the basic moduli spaces in algebraic geometry related to varieties of dimension at most 3 -- including the moduli of curves, the moduli of maps, the moduli of surfaces, and
the moduli of sheaves on 3-folds -- will be considered. The current state of the study of the algebraic cycle theory in these cases varies from rather advanced (for the moduli of curves)
to much less so (for the moduli of surfaces). There is a range of rich open questions which I will attack: Pixton's conjectures for the moduli of curves, the structure of the ring of Noether-Lefschetz
loci for the moduli of K3 surfaces, the holomorphic anomaly equation in Gromov-Witten theory, and conjectures governing descendents for the moduli of sheaves. The dimension 3 restriction
is often necessary for a good deformation theory and the existence of a virtual fundamental class.
There has been great progress in the directions of the original proposal concerning basic moduli spaces in algebraic geometry
related to varieties of dimension at most 3 (including the moduli of curves, the moduli of maps, the moduli of surfaces, and
the moduli of sheaves on 3-folds). There is a range questions considered include: Pixton's conjectures for the moduli of curves,
the structure of the ring of Noether-Lefschetz loci for the moduli of K3 surfaces, the holomorphic anomaly equation in Gromov-Witten
theory, and conjectures governing descendents for the moduli of sheaves.
The main research highlights/breakthroughs are:
i) Pixton’s formula for the Picard stack: with Bae, Holmes, Schmitt, and Schwarz, we define a universal twisted double ramification
cycles on the Artin stack of curves with a line bundle. The main result is an exact formula for the universal DR cycle in the
tautological ring in the shape of Pixton's formula for the standard DR cycle. The result may be viewed as a universal Abel-Jacobi
calculation on the Picard stack. There are a number of consequences including the proof of all the previous
conjectures for the calculations of the fundamental classes of the spaces of meromorphic/holomorphic differentials in terms of
Pixton's formula.
With Molcho and Schmitt, we opened the study of the double ramification cycles in the logarithmic tautological ring on the
moduli space of curves. The paper was then followed by a calculation of the double ramification cycle via a new version of
Pixton's formula with Holmes, Molcho, Pixton, and Schmitt. The result is a breakthrough (and introduces several new ideas
and tools to the subject including the systematic use of stability conditions on the universal Picard for the calculation).
An outcome of our work (as developed by Ranganathan) is a complete determination of the log Gromov-Witten theory of
toric varieties relative to the full toric boundary.
ii-iii) We have made significant progress in the study of the virtual classes of moduli of sheaves on surfaces. An important
advance was Oprea-Pandharipande "Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics"
where exact formulas for the general type case were given.
With Oprea and Canning, we have completed the calculation of the Chow ring of the moduli space of K3 surfaces of degree 2.
The result opens the door to further theorems and conjectures about the structure of the tautological rings of the moduli spaces
of surfaces. The Noether-Lefschetz geometry of the moduli of K3 surfaces led us to study related problems for the moduli space of
principally polarized abelian varieties. With Oprea and Canning, we have found the first geometric non-tautological class
(a Noether-Lefschetz locus) and proposed a conjectural theory for the virtual classes of Noether-Lefschetz
loci for the moduli of abelian varieties. A surprising outcome has been been a connection to Pixton's conjectures
for curves of compact type.
iv) We have studied the holomorphic anomaly equations for many Calabi-Yau geometries. After work with Lho which showed for the
first time how to prove the holomorphic anomaly equation from the R-matrix and the Givental-Teleman formula, there
has been a large number of results: for orbifold geometries, formal geometries, and local relative geometries. A beautiful
recent result concerns the complex projective plane relative to the elliptic cubic by Bousseau, Fan, Guo, and Wu.
Our ideas played a role in further developments by Janda, Ruan, and Jun Li on the holomorphic anomaly equation
for compact targets.
v) We have studied the Virasoro constraints for stable pairs in the stationary toric case. After finding a closed form for the
GW/PT descendent correspondent in the stationary toric case with Oblomkov and Okounkov, we have proved
the Virasoro constraints for stable pairs in the stationary toric case (with Oblomkov, Okounkov, and Moreira). In a
breakthrough by Bojko Lim, and Moreira, the Virasoro constraints for all moduli spaces of higher rank
stable sheaves on toric algebraic surfaces and curves of any genus have been proved. These results open a large
field of future study.
The direction numbers above match research directions proposed in MACI.
The progress on the grant has opened new lines of research in several directions discussed above.
related to varieties of dimension at most 3 (including the moduli of curves, the moduli of maps, the moduli of surfaces, and
the moduli of sheaves on 3-folds). There is a range questions considered include: Pixton's conjectures for the moduli of curves,
the structure of the ring of Noether-Lefschetz loci for the moduli of K3 surfaces, the holomorphic anomaly equation in Gromov-Witten
theory, and conjectures governing descendents for the moduli of sheaves.
The main research highlights/breakthroughs are:
i) Pixton’s formula for the Picard stack: with Bae, Holmes, Schmitt, and Schwarz, we define a universal twisted double ramification
cycles on the Artin stack of curves with a line bundle. The main result is an exact formula for the universal DR cycle in the
tautological ring in the shape of Pixton's formula for the standard DR cycle. The result may be viewed as a universal Abel-Jacobi
calculation on the Picard stack. There are a number of consequences including the proof of all the previous
conjectures for the calculations of the fundamental classes of the spaces of meromorphic/holomorphic differentials in terms of
Pixton's formula.
With Molcho and Schmitt, we opened the study of the double ramification cycles in the logarithmic tautological ring on the
moduli space of curves. The paper was then followed by a calculation of the double ramification cycle via a new version of
Pixton's formula with Holmes, Molcho, Pixton, and Schmitt. The result is a breakthrough (and introduces several new ideas
and tools to the subject including the systematic use of stability conditions on the universal Picard for the calculation).
An outcome of our work (as developed by Ranganathan) is a complete determination of the log Gromov-Witten theory of
toric varieties relative to the full toric boundary.
ii-iii) We have made significant progress in the study of the virtual classes of moduli of sheaves on surfaces. An important
advance was Oprea-Pandharipande "Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics"
where exact formulas for the general type case were given.
With Oprea and Canning, we have completed the calculation of the Chow ring of the moduli space of K3 surfaces of degree 2.
The result opens the door to further theorems and conjectures about the structure of the tautological rings of the moduli spaces
of surfaces. The Noether-Lefschetz geometry of the moduli of K3 surfaces led us to study related problems for the moduli space of
principally polarized abelian varieties. With Oprea and Canning, we have found the first geometric non-tautological class
(a Noether-Lefschetz locus) and proposed a conjectural theory for the virtual classes of Noether-Lefschetz
loci for the moduli of abelian varieties. A surprising outcome has been been a connection to Pixton's conjectures
for curves of compact type.
iv) We have studied the holomorphic anomaly equations for many Calabi-Yau geometries. After work with Lho which showed for the
first time how to prove the holomorphic anomaly equation from the R-matrix and the Givental-Teleman formula, there
has been a large number of results: for orbifold geometries, formal geometries, and local relative geometries. A beautiful
recent result concerns the complex projective plane relative to the elliptic cubic by Bousseau, Fan, Guo, and Wu.
Our ideas played a role in further developments by Janda, Ruan, and Jun Li on the holomorphic anomaly equation
for compact targets.
v) We have studied the Virasoro constraints for stable pairs in the stationary toric case. After finding a closed form for the
GW/PT descendent correspondent in the stationary toric case with Oblomkov and Okounkov, we have proved
the Virasoro constraints for stable pairs in the stationary toric case (with Oblomkov, Okounkov, and Moreira). In a
breakthrough by Bojko Lim, and Moreira, the Virasoro constraints for all moduli spaces of higher rank
stable sheaves on toric algebraic surfaces and curves of any genus have been proved. These results open a large
field of future study.
The direction numbers above match research directions proposed in MACI.
The progress on the grant has opened new lines of research in several directions discussed above.
In all directions (i)-(v)described above, the results go well beyond the state of the art before the grant. Many of the breakthroughs
were unexpected and have advanced the field in surprising ways.
were unexpected and have advanced the field in surprising ways.