LEADER 05122nam 22006253 450 001 9910915703703321 005 20231110220602.0 010 $a1-4704-7167-1 035 $a(MiAaPQ)EBC29379017 035 $a(Au-PeEL)EBL29379017 035 $a(CKB)24267884500041 035 $a(EXLCZ)9924267884500041 100 $a20220721d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aTheta Functions on Varieties with Effective Anti-Canonical Class 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$dİ2022. 215 $a1 online resource (122 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.278 311 08$aPrint version: Gross, Mark Theta Functions on Varieties with Effective Anti-Canonical Class Providence : American Mathematical Society,c2022 9781470452971 327 $aCover -- Title page -- Introduction -- Chapter 1. The affine geometry of the construction -- 1.1. Polyhedral affine pseudomanifolds -- 1.2. Convex, piecewise affine functions -- Chapter 2. Wall structures -- 2.1. Construction of ? -- 2.2. Monomials, rings and gluing morphisms -- 2.3. Walls and consistency -- 2.4. Construction of \foX^{?} -- Chapter 3. Broken lines and canonical global functions -- 3.1. Broken lines -- 3.2. Consistency and rings in codimension two -- 3.3. The canonical global functions _{ } -- 3.4. The conical case -- 3.5. The multiplicative structure -- Chapter 4. The projective case -theta functions -- 4.1. Conical affine structures -- 4.2. The cone over a polyhedral pseudomanifold -- 4.3. Theta functions and the Main Theorem -- 4.4. The action of the relative torus -- 4.5. Jagged paths -- Chapter 5. Additional parameters -- 5.1. Twisting the construction -- 5.2. Twisting by gluing data -- Chapter 6. Abelian varieties and other examples -- Appendix A. The GS case -- A.1. One-parameter families -- A.2. The universal formulation -- A.3. Equivariance -- A.4. The non-simple case in two dimensions -- Bibliography -- Back Cover. 330 $a"We show that a large class of maximally degenerating families of n-dimensional polarized varieties comes with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n-(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities [Gross and Siebert, From real affine geometry to complex geometry (2011)] and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K3 surfaces [Gross, Pandharipande and Siebert, The tropical vertex ; Gross, Hacking and Keel, Mirror symmetry for log Calabi-Yau surfaces (2015); Gross, Hacking, Keel, and Siebert, Theta functions and K3 surfaces (In preparation)]. For trivial wall structures on the n-torus we retrieve the classical theta functions. We anticipate that wall structures can be constructed quite generally from maximal degenerations. The construction given here then provides the homogeneous coordinate ring of the mirror degeneration along with a canonical basis. The appearance of a canonical basis of sections for certain degenerations points towards a good compactification of moduli of certain polarized varieties via stable pairs, generalizing the picture for K3 surfaces [Gross, Hacking, Keel, and Siebert, Theta functions and K3 surfaces (In preparation)]. Another possible application apart from mirror symmetry may be to geometric quantization of varieties with effective anti-canonical class"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aFunctions, Theta 606 $aSurfaces, Algebraic 606 $aMirror symmetry 606 $aCalabi-Yau manifolds 606 $aAlgebraic geometry -- Surfaces and higher-dimensional varieties -- Mirror symmetry$2msc 606 $aAlgebraic geometry -- Surfaces and higher-dimensional varieties -- Calabi-Yau manifolds$2msc 606 $aAlgebraic geometry -- Surfaces and higher-dimensional varieties -- Fano varieties$2msc 615 0$aFunctions, Theta. 615 0$aSurfaces, Algebraic. 615 0$aMirror symmetry. 615 0$aCalabi-Yau manifolds. 615 7$aAlgebraic geometry -- Surfaces and higher-dimensional varieties -- Mirror symmetry. 615 7$aAlgebraic geometry -- Surfaces and higher-dimensional varieties -- Calabi-Yau manifolds. 615 7$aAlgebraic geometry -- Surfaces and higher-dimensional varieties -- Fano varieties. 676 $a515/.984 676 $a515.984 686 $a14J33$a14J32$a14J45$2msc 700 $aGross$b Mark$067537 701 $aHacking$b Paul$0755862 701 $aSiebert$b Bernd$01778981 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910915703703321 996 $aTheta Functions on Varieties with Effective Anti-Canonical Class$94302240 997 $aUNINA