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245 10 a| Theta functions on varieties with effective anti-canonical classh| [electronic resource] /c| Mark Gross, Paul Hacking, Bernd Siebert.;

260 a| Providence, RI :b| American Mathematical Society,c| 2022.;

300 a| xii, 103 pagesb| illustrationsc| 26 cm;

334 a| single unitb| mono0| http://id.loc.gov/vocabulary/issuance/mono;

353 a| bibliographyb| bibliography0| http://id.loc.gov/vocabulary/msupplcont/bibliography;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| Volume 278, Number 1367 (second of 6 numbers);

504 a| Includes bibliographical references.;

506 a| Available only to authorized users.;

520 a| "We show that a large class of maximally degenerating families of n-dimensional polarized varieties comes with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n-(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities [Gross and Siebert, From real affine geometry to complex geometry (2011)] and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K3 surfaces [Gross, Pandharipande and Siebert, The tropical vertex ; Gross, Hacking and Keel, Mirror symmetry for log Calabi-Yau surfaces (2015); Gross, Hacking, Keel, and Siebert, Theta functions and K3 surfaces (In preparation)]. For trivial wall structures on the n-torus we retrieve the classical theta functions. We anticipate that wall structures can be constructed quite generally from maximal degenerations. The construction given here then provides the homogeneous coordinate ring of the mirror degeneration along with a canonical basis. The appearance of a canonical basis of sections for certain degenerations points towards a good compactification of moduli of certain polarized varieties via stable pairs, generalizing the picture for K3 surfaces [Gross, Hacking, Keel, and Siebert, Theta functions and K3 surfaces (In preparation)]. Another possible application apart from mirror symmetry may be to geometric quantization of varieties with effective anti-canonical class"-- Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Functions, Theta0| http://id.loc.gov/authorities/subjects/sh85052352=| ^A40026;

650 0 a| Surfaces, Algebraic0| http://id.loc.gov/authorities/subjects/sh85130734=| ^A93432;

650 0 a| Mirror symmetry0| http://id.loc.gov/authorities/subjects/sh96008323=| ^A411739;

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650 7 a| Algebraic geometry -- Surfaces and higher-dimensional varieties -- Mirror symmetry2| msc;

650 7 a| Algebraic geometry -- Surfaces and higher-dimensional varieties -- Calabi-Yau manifolds2| msc;

650 7 a| Algebraic geometry -- Surfaces and higher-dimensional varieties -- Fano varieties2| msc;

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Theta functions on varieties with effective anti-canonical class / Mark Gross, Paul Hacking, Bernd Siebert.

Author/creator	Gross, Mark, 1965
Format	Electronic
Publication Info	Providence, RI : American Mathematical Society, 2022.
Description	xii, 103 pages illustrations 26 cm
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Functions, Theta Surfaces, Algebraic Mirror symmetry Calabi-Yau manifolds

Series	Memoirs of the American Mathematical Society, 0065-9266 ; Volume 278, Number 1367 (second of 6 numbers)
Abstract	"We show that a large class of maximally degenerating families of n-dimensional polarized varieties comes with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n-(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities [Gross and Siebert, From real affine geometry to complex geometry (2011)] and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K3 surfaces [Gross, Pandharipande and Siebert, The tropical vertex ; Gross, Hacking and Keel, Mirror symmetry for log Calabi-Yau surfaces (2015); Gross, Hacking, Keel, and Siebert, Theta functions and K3 surfaces (In preparation)]. For trivial wall structures on the n-torus we retrieve the classical theta functions. We anticipate that wall structures can be constructed quite generally from maximal degenerations. The construction given here then provides the homogeneous coordinate ring of the mirror degeneration along with a canonical basis. The appearance of a canonical basis of sections for certain degenerations points towards a good compactification of moduli of certain polarized varieties via stable pairs, generalizing the picture for K3 surfaces [Gross, Hacking, Keel, and Siebert, Theta functions and K3 surfaces (In preparation)]. Another possible application apart from mirror symmetry may be to geometric quantization of varieties with effective anti-canonical class"-- Provided by publisher.
Bibliography note	Includes bibliographical references.
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2022027936
ISBN	9781470452971 paperback
ISBN	pdf

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Theta functions on varieties with effective anti-canonical class / Mark Gross, Paul Hacking, Bernd Siebert.

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