2018 Theses Doctoral

# K-theoretic enumerative geometry and the Hilbert scheme of points on a surface

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions.

We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kos\c{c}az and Vafa.

Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\C^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.

## Files

- Arbesfeld_columbia_0054D_14606.pdf application/pdf 558 KB Download File

## More About This Work

- Academic Units
- Mathematics
- Thesis Advisors
- Okounkov, Andrei
- Degree
- Ph.D., Columbia University
- Published Here
- April 26, 2018