Dissertation/Thesis Abstract

On the Gromov-Witten Theory of P1-Bundles Over Ruled Surfaces
by Lownes, Eric, Ph.D., University of Maryland, College Park, 2019, 87; 13812667
Abstract (Summary)

Let C be a smooth, connected, complex, projective curve of genus g and let D1,D2 be divisors of degree k1,k2 respectively. Let S be the decomposable ruled surface given by the total space of the following P1-bundle over C:

pC: P(OCOC(-D1))→C.

Let C0 be the locus of (1:0) in SP(OCOC(-D1)). Then let X be the threefold given by the total space of the following P1-bundle over S:

pS:P(OsOs(-E)) → S

where E=aC0+pC-1(D2). This determines an Ha-bundle over C where Ha is a Hirzebruch surface.

In this thesis we determine the equivariant Gromov-Witten partition function for all "section classes'' of the form s+m1f1 + m2f2 where s is a section of the map XC and f1,f2 are fiber classes, in the case that a=0,-1. A class is Calabi-Yau if KX⋅β=0. For a=0, the partition function of Calabi-Yau section classes is given by

[Special characters omitted]

where v1,v2 count the number of fibers and φ=2sinu/2. In the case that a=-1 the partition functions of Calabi-Yau section classes satisfy the following relations

Z(g|k1,k2)=Z(g|k1-2,k2+1)

Z(g|k1,k2)=I12 Z(g|k1-3,k2)+v2v2 Z(g|k1-4,k2)

Z(g|k1,k2)=-φ2v12v2-2Z(g-1|k1,k2)

+6φ4v12v2-1Z(g-2|k1,k2)+(256φ8v12v2+27φ8v14v2-2)Z(g-4|k1,k2)

which allow us to compute all the Calabi-Yau section class invariants from the following base cases:

g=0 1 2 3

k1=0 0 4 -φ2v12v2-2 12φ4v12v2-14v14v2-4

1 φ-2 0 φ2v12v2-1 16φ4v124v14v2-3

2 0 0 8φ2v12 64φ4v12v24v14v2-2

3 0 3v12 16φ2v12v24v14v2-1

As a corollary, we establish the Gromov-Witten/Donaldson-Thomas/Stable Pairs correspondence for the Calabi-Yau section class partition functions for these families of non-toric threefolds.

Indexing (document details)
Advisor: Gholampour, Amin
Commitee: Tamvakis, Harry, Brosnan, Patrick, Ramachandran, Niranjan, Gasarch, William
School: University of Maryland, College Park
Department: Mathematics
School Location: United States -- Maryland
Source: DAI-B 81/1(E), Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Mathematics
Keywords: Donaldson-Thomas theory, Enumerative geometry, Gromov-Witten theory, Ruled surfaces, Stable pairs
Publication Number: 13812667
ISBN: 9781085567459
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