The Batyrev-Borisov construction associates a to dual pair of nef-partitions
$\\Delta=\\Delta_1+\\dotsb+\\Delta_c$ and $\\nabla=\\nabla_1+\\dotsb+\\nabla_c$ a pair
of Calabi-Yau complete intersections
$(Y_{\\Delta_1,\\dotsc,\\Delta_c},Y_{\\nabla_1,\\dotsc,\\nabla_c})$ in Gorenstein Fano
toric varieties $(X_\\Delta,X_\\nabla)$. These Calabi-Yau varieties are singular
in general. Batyrev and Nill have developed a generating function $\\Est$ for the
stringy Hodge numbers of Batyrev-Borisov mirror pairs. This function depends
solely on the combinatorics of the nef-partitions and, under this framework,
Batyrev-Borisov mirror pairs pass the stringy topological mirror symmetry test
$\\hst^{p,q}(Y_{\\Delta_1,\\dotsc,\\Delta_c})=\\hst^{d-p,q}(Y_{\\nabla_1,\\dotsc,\\nabla_c})$.
Recently, Aspinwall and Plesser have defined the notion of a complete
non-reflexive mirror pair $(\\scrA,\\scrB)$ and used this notion to study
Calabi-Yau complete intersections in non-Gorenstein toric varieties. Complete
mirror pairs generalize the notion of a dual pair of almost reflexive Gorenstein
cones $(\\sigma,\\sigma^\\bullet)$ developed by Mavlyutov to propose a
generalization of the Batyrev-Borisov mirror construction. The only known
example of either of these two notions is the complete intersection of a quintic
and a quadric in $\\PP_{211111}^5$. We construct $2152$ distinct examples of
complete mirror pairs and $1077$ distinct examples of dual pairs of almost
reflexive Gorenstein cones. Additionally, we propose a generalization of Batyrev
and Nill's stringy $E$-function, called the na\\\"{i}ve stringy $E$-function
$\\gEst$, that is well-defined for complete mirror pairs.
","abstract_html":"<p>The Batyrev-Borisov construction associates a to dual pair of nef-partitions</p><p>\\Delta=\\Delta1+\\dotsb+\\Deltac and \\nabla=\\nabla1+\\dotsb+\\nablac a pair</p><p>of Calabi-Yau complete intersections</p><p>(Y\\Delta1,\\dotsc,\\Deltac,Y\\nabla1,\\dotsc,\\nablac) in Gorenstein Fano</p><p>toric varieties (X\\Delta,X\\nabla). These Calabi-Yau varieties are singular</p><p>in general. Batyrev and Nill have developed a generating function $\\Est$ for the</p><p>stringy Hodge numbers of Batyrev-Borisov mirror pairs. This function depends</p><p>solely on the combinatorics of the nef-partitions and, under this framework,</p><p>Batyrev-Borisov mirror pairs pass the stringy topological mirror symmetry test</p><p>\\hstp,q(Y\\Delta1,\\dotsc,\\Deltac)=\\hstd-p,q(Y\\nabla1,\\dotsc,\\nablac).</p><p>Recently, Aspinwall and Plesser have defined the notion of a complete</p><p>non-reflexive mirror pair $(\\scrA,\\scrB)$ and used this notion to study</p><p>Calabi-Yau complete intersections in non-Gorenstein toric varieties. Complete</p><p>mirror pairs generalize the notion of a dual pair of almost reflexive Gorenstein</p><p>cones (σ,σ\\bullet) developed by Mavlyutov to propose a</p><p>generalization of the Batyrev-Borisov mirror construction. The only known</p><p>example of either of these two notions is the complete intersection of a quintic</p><p>and a quadric in \\PP2111115. We construct $2152$ distinct examples of</p><p>complete mirror pairs and $1077$ distinct examples of dual pairs of almost</p><p>reflexive Gorenstein cones. Additionally, we propose a generalization of Batyrev</p><p>and Nill's stringy $E$-function, called the na\\"{i}ve stringy $E$-function</p><p>$\\gEst$, that is well-defined for complete mirror pairs.</p>","abstract_has_math":true,"creators":["Fitzpatrick, Brian David"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Aspinwall, Paul S"],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017","date_published":"2017","updated_at":"2026-07-24T02:06:59Z","subjects":["Mathematics","calabi yau","mirror symmetry","toric geometry"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10161/14531","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Aspinwall, Paul S"]},{"key":"dc:creator","label":"Author","values":["Fitzpatrick, Brian David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-05-16T17:28:47Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-05-16T17:28:47Z"]},{"key":"dc:date.issued","label":"Date","values":["2017"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","calabi yau","mirror symmetry","toric geometry"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10161/14531"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Batyrev-Borisov construction associates a to dual pair of nef-partitions
$\\Delta=\\Delta_1+\\dotsb+\\Delta_c$ and $\\nabla=\\nabla_1+\\dotsb+\\nabla_c$ a pair
of Calabi-Yau complete intersections
$(Y_{\\Delta_1,\\dotsc,\\Delta_c},Y_{\\nabla_1,\\dotsc,\\nabla_c})$ in Gorenstein Fano
toric varieties $(X_\\Delta,X_\\nabla)$. These Calabi-Yau varieties are singular
in general. Batyrev and Nill have developed a generating function $\\Est$ for the
stringy Hodge numbers of Batyrev-Borisov mirror pairs. This function depends
solely on the combinatorics of the nef-partitions and, under this framework,
Batyrev-Borisov mirror pairs pass the stringy topological mirror symmetry test
$\\hst^{p,q}(Y_{\\Delta_1,\\dotsc,\\Delta_c})=\\hst^{d-p,q}(Y_{\\nabla_1,\\dotsc,\\nabla_c})$.
Recently, Aspinwall and Plesser have defined the notion of a complete
non-reflexive mirror pair $(\\scrA,\\scrB)$ and used this notion to study
Calabi-Yau complete intersections in non-Gorenstein toric varieties. Complete
mirror pairs generalize the notion of a dual pair of almost reflexive Gorenstein
cones $(\\sigma,\\sigma^\\bullet)$ developed by Mavlyutov to propose a
generalization of the Batyrev-Borisov mirror construction. The only known
example of either of these two notions is the complete intersection of a quintic
and a quadric in $\\PP_{211111}^5$. We construct $2152$ distinct examples of
complete mirror pairs and $1077$ distinct examples of dual pairs of almost
reflexive Gorenstein cones. Additionally, we propose a generalization of Batyrev
and Nill's stringy $E$-function, called the na\\\"{i}ve stringy $E$-function
$\\gEst$, that is well-defined for complete mirror pairs.
"]},{"key":"dc:title","label":"Title","values":["Complete Mirror Pairs and Their Naive Stringy Hodge Numbers"]}]}],"canonical_facts":{"dc:contributor.advisor":["Aspinwall, Paul S"],"dc:creator":["Fitzpatrick, Brian David"],"dc:date.accessioned":["2017-05-16T17:28:47Z"],"dc:date.available":["2017-05-16T17:28:47Z"],"dc:date.issued":["2017"],"dc:description.abstract":["The Batyrev-Borisov construction associates a to dual pair of nef-partitions
$\\Delta=\\Delta_1+\\dotsb+\\Delta_c$ and $\\nabla=\\nabla_1+\\dotsb+\\nabla_c$ a pair
of Calabi-Yau complete intersections
$(Y_{\\Delta_1,\\dotsc,\\Delta_c},Y_{\\nabla_1,\\dotsc,\\nabla_c})$ in Gorenstein Fano
toric varieties $(X_\\Delta,X_\\nabla)$. These Calabi-Yau varieties are singular
in general. Batyrev and Nill have developed a generating function $\\Est$ for the
stringy Hodge numbers of Batyrev-Borisov mirror pairs. This function depends
solely on the combinatorics of the nef-partitions and, under this framework,
Batyrev-Borisov mirror pairs pass the stringy topological mirror symmetry test
$\\hst^{p,q}(Y_{\\Delta_1,\\dotsc,\\Delta_c})=\\hst^{d-p,q}(Y_{\\nabla_1,\\dotsc,\\nabla_c})$.
Recently, Aspinwall and Plesser have defined the notion of a complete
non-reflexive mirror pair $(\\scrA,\\scrB)$ and used this notion to study
Calabi-Yau complete intersections in non-Gorenstein toric varieties. Complete
mirror pairs generalize the notion of a dual pair of almost reflexive Gorenstein
cones $(\\sigma,\\sigma^\\bullet)$ developed by Mavlyutov to propose a
generalization of the Batyrev-Borisov mirror construction. The only known
example of either of these two notions is the complete intersection of a quintic
and a quadric in $\\PP_{211111}^5$. We construct $2152$ distinct examples of
complete mirror pairs and $1077$ distinct examples of dual pairs of almost
reflexive Gorenstein cones. Additionally, we propose a generalization of Batyrev
and Nill's stringy $E$-function, called the na\\\"{i}ve stringy $E$-function
$\\gEst$, that is well-defined for complete mirror pairs.
"],"dc:identifier.uri":["https://hdl.handle.net/10161/14531"],"dc:subject":["Mathematics","calabi yau","mirror symmetry","toric geometry"],"dc:title":["Complete Mirror Pairs and Their Naive Stringy Hodge Numbers"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T02:06:59Z"}