  # Moduli, Algebraic Cycles, and Invariants

## Periodic Reporting for period 1 - MACI (Moduli, Algebraic Cycles, and Invariants)

Reporting period: 2018-09-01 to 2020-02-29

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.
There has been enormous progress in the directions of the original proposal.
The main research highlights/breakthroughs in the first 18 months are:

1) Pixton’s formula for the Picard stack [direction(i)]: 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.

2) The holomorphic anomaly for many CY geometries [direction(iv)].
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 result concerns CP2 relative
to the elliptic cubic by Bousseau, Fan, Guo, and Wu.

3) Virasoro constraints for stable pairs in the stationary toric case [direction (v)].
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).

Here, the direction numbers match research directions proposed in MACI.
In all three areas (1)-(3) described above in the progress description, the results
go well beyond the state of the art before (covering directions (i),(iv), and (v) of
the MACI proposal). Fundamental results in directions (ii) and (iii) will come in the future.