Sheaves, Quasimaps, Maps, Covers

Abstract

Let $X$ be a smooth variety, let $S$ be a smooth surface, let $C_{g,N} rightarrow overline{M}_{g,N}$ be a universal curve over a moduli space of stable marked curves and let $(C,mathbf{x})$ be a marked nodal curve. <br /> In the first part of the thesis, comprised of two chapters, we develop the theory of quasimaps to a moduli space of stable sheaves $M$ on $S$. A quasimap is a map to the moduli space of all sheaves (not necessarily stable), generically mapping to $M$. For each $epsilon in mathbb{R}_{>0}$, there exists a stability condition for quasimaps, termed $epsilon$-stability. Moduli spaces of $epsilon$-stable quasimaps interpolate between moduli spaces of stable maps to $M$ and moduli spaces of stable sheaves of the relative geometry $Stimes C_{g,N} rightarrow overline{M}_{g,N}$, the two being the moduli spaces of $epsilon$-stable quasimaps for extremal values of $epsilon$. Using Zhou's theory of calibrated tails, we prove wall-crossing formulas, which therefore relate Gromov--Witten invariants of $M$ and relative Donaldson--Thomas invariants of $Stimes C_{g,N} rightarrow overline{M}_{g,N}$. <br /> In the second part of the thesis, we introduce a stability condition for maps from nodal curves to $Xtimes C$ relative to $Xtimes mathbf {x}$ for each $epsilon in mathbb{R}_{leq 0}$, termed $epsilon$-admissibility. Moduli spaces of $epsilon$-admissible maps interpolate between moduli space of twisted stable maps to an orbifold symmetric product $[X^{(n)}]$ and stable maps to the relative geometry $Xtimes C_{g,N} rightarrow overline{M}_{g,N}$. Using Zhou's theory of calibrated tails, we prove wall-crossing formulas, which therefore relate orbifold Gromov--Witten invariants of $[X^{(n)}]$ and relative Gromov--Witten invariants of $Xtimes C_{g,N} rightarrow overline{M}_{g,N}}$. <br /> The main result of the thesis is establishment of correspondences between different enumerative theories, using aforementioned wall-crossings. In particular, we prove the wall-crossing part of Igusa cusp form conjecture; higher-rank/rank-one Donaldson--Thomas wall-crossing for some threefolds $Stimes C$; Donaldson--Thomas/Pandharipande--Thomas wall-crossing for some threefolds $Stimes C$. We show that the quantum cohomology of $S^{[n]}$ is determined by relative Pandharipande--Thomas theory of $Stimes mathbb{P}^1$ for del Pezzo and K3 surfaces. Finally, we express crepant resolution conjecture for the pair $S^{[n]}$ and $[S^{(n)}]$ in terms of Gromov--Witten/Pandharipande--Thomas correspondence for $S times C$, thereby proving 3-point genus-0 crepant resolution conjecture, if $S$ is a toric del Pezzo surface.