Table Of Content

Strangeduality andsymmetryof singularities Arnold’sstrange duality Orbifold Strange duality and symmetry of singularities Landau-Ginzburg models (joint work with Atsushi Takahashi) Invertiblepolynomials Diagonalsymmetries Objective Orbifoldcurves Dolgachevnumbers StringyEulernumber Cuspsingularities Institutfu¨rAlgebraischeGeometrie withgroupaction LeibnizUniversitätHannover Gabrielovnumbers Spectrum Mirrorsymmetry Liverpool, June 21, 2012 Strangeduality Varianceofthe spectrum Examples Directionsfor furtherresearch (cid:73) C.T.C. Wall: A note on symmetry of singularities. Bull. London Math. Soc. 12, 169–175 (1980) Wall 75 Strangeduality andsymmetryof singularities Arnold’sstrange duality Orbifold Landau-Ginzburg models Invertiblepolynomials (cid:73) —, C.T.C. Wall: Kodaira singularities and an extension Diagonalsymmetries Objective of Arnold’s strange duality. Compositio Math. 56, Orbifoldcurves 3–77 (1985). Dolgachevnumbers StringyEulernumber Cuspsingularities withgroupaction Gabrielovnumbers Spectrum Mirrorsymmetry Strangeduality Varianceofthe spectrum Examples Directionsfor furtherresearch Wall 75 Strangeduality andsymmetryof singularities Arnold’sstrange duality Orbifold Landau-Ginzburg models Invertiblepolynomials (cid:73) —, C.T.C. Wall: Kodaira singularities and an extension Diagonalsymmetries Objective of Arnold’s strange duality. Compositio Math. 56, Orbifoldcurves 3–77 (1985). Dolgachevnumbers StringyEulernumber (cid:73) C.T.C. Wall: A note on symmetry of singularities. Cuspsingularities withgroupaction Bull. London Math. Soc. 12, 169–175 (1980) Gabrielovnumbers Spectrum Mirrorsymmetry Strangeduality Varianceofthe spectrum Examples Directionsfor furtherresearch Classification of singularities Strangeduality andsymmetryof singularities f(x) = f(x ,...,x ) complex polynomial with f(0) = 0 1 n Arnold’sstrange and isolated singularity at 0 ∈ Cn, i.e. duality (cid:16) (cid:17) gradf(x) = ∂f (x),..., ∂f (x) (cid:54)= 0 for x (cid:54)= 0, |x| < ε. Orbifold ∂x1 ∂xn Lmaonddealsu-Ginzburg X := f−1(0) hypersurface singularity Invertiblepolynomials Diagonalsymmetries Objective V. I. Arnold (1972, 1973, 1975): Orbifoldcurves Dolgachevnumbers (cid:73) 0-modal (simple) singularities: ADE StringyEulernumber (cid:73) unimodal singularities: Cuspsingularities withgroupaction (cid:73) simple elliptic singularities Gabrielovnumbers Spectrum (cid:73) T : f(x,y,z)=xp+yq+zr +axyz, a∈C∗, p,q,r Mirrorsymmetry 1 + 1 + 1 <1 (cusp singularities) Strangeduality p q r Varianceofthe (cid:73) 14 exceptional singularities spectrum (cid:73) bimodal singularities Examples Directionsfor (cid:73) 8 bimodal series furtherresearch (cid:73) 14 exceptional singularities Arnold’s strange duality (1) Strangeduality andsymmetryof singularities 14 exceptional unimodal singularities related to Schwarz triangular groups Arnold’sstrange duality Γ(α ,α ,α ) ⊂ PSL(2;R) 1 2 3 Orbifold Landau-Ginzburg (cid:73) Dolgachev numbers Dol(X) = (α ,α ,α ), models 1 2 3 Invertiblepolynomials π , π , π angles of hyperbolic triangle Diagonalsymmetries α1 α2 α3 Objective (cid:73) Gabrielov numbers Gab(X) = (γ ,γ ,γ ), Orbifoldcurves 1 2 3 Dolgachevnumbers StringyEulernumber Cuspsingularities withgroupaction Gabrielovnumbers Spectrum Mirrorsymmetry Coxeter-Dynkin diagram Strangeduality Varianceofthe spectrum Arnold’s strange duality: X ↔ X∗ Examples (cid:73) Dol(X) = Gab(X∗) Directionsfor furtherresearch (cid:73) Gab(X) = Dol(X∗) Arnold’s strange duality (2) Strangeduality andsymmetryof singularities Name Dol(X) Gab(X) Dual Arnold’sstrange E 2,3,7 2,3,7 E duality 12 12 E 2,4,5 2,3,8 Z Orbifold 13 11 Landau-Ginzburg E 3,3,4 2,3,9 Q models 14 10 Invertiblepolynomials Z 2,3,8 2,4,5 E Diagonalsymmetries 11 13 Objective Z 2,4,6 2,4,6 Z 12 12 Orbifoldcurves Z 3,3,5 2,4,7 Q Dolgachevnumbers 13 11 StringyEulernumber Q10 2,3,9 3,3,4 E14 Cuspsingularities withgroupaction Q 2,4,7 3,3,5 Z 11 13 Gabrielovnumbers Q 3,3,6 3,3,6 Q Spectrum 12 12 Mirrorsymmetry W 2,5,5 2,5,5 W 12 12 Strangeduality Varianceofthe W13 3,4,4 2,5,6 S11 spectrum S 2,5,6 3,4,4 W Examples 11 13 S 3,4,5 3,4,5 S Directionsfor 12 12 furtherresearch U 4,4,4 4,4,4 U 12 12 E.-Wall extension Strangeduality andsymmetryof singularities Wall (1983): Classification of unimodal isolated singularities of complete intersections (ICIS) Arnold’sstrange duality 8 bimodal series ←→ 8 triangle ICIS in C4 Orbifold Landau-Ginzburg models quasihomogeneous heads related to Invertiblepolynomials Diagonalsymmetries quadrilateral groups Γ[α1,α2,α3,α4] Objective Orbifoldcurves Series Head Dol(X) Gab(X) Dual Dolgachevnumbers StringyEulernumber J3,k J3,0 2,2,2,3 2,3,10 J9(cid:48) Cuspsingularities Z Z 2,2,2,4 2,4,8 J(cid:48) withgroupaction 1,k 1,0 10 Gabrielovnumbers Q Q 2,2,2,5 3,3,7 J(cid:48) Spectrum 2,k 2,0 11 Mirrorsymmetry W W 2,3,2,3 2,6,6 K(cid:48) 1,k 1,0 10 Strangeduality W(cid:93) 2,2,3,3 2,5,7 L Vspaerciatnrucmeofthe 1,k 10 S S 2,3,2,4 3,5,5 K(cid:48) Examples 1,k 1,0 11 Directionsfor S(cid:93) 2,2,3,4 3,4,6 L furtherresearch 1,k 11 U U 2,3,3,3 4,4,5 M 1,k 1,0 11 Invertible polynomials (1) Strangeduality andsymmetryof singularities (cid:73) A quasihomogeneous polynomial f in n variables is Arnold’sstrange invertible duality Orbifold n n Landau-Ginzburg :⇐⇒ f(x1,...,xn) = (cid:88)ai (cid:89)xjEij mInovderetlibslepolynomials Diagonalsymmetries i=1 j=1 Objective Orbifoldcurves for some coefficients a ∈ C∗ and for a matrix E = (E ) Dolgachevnumbers i ij StringyEulernumber with non-negative integer entries and with detE (cid:54)= 0. Cuspsingularities 6 1 0 withgroupaction Gabrielovnumbers Ex.: f(x,y,z) = x6y +y3+z2, E = 0 3 0 Spectrum Mirrorsymmetry 0 0 2 Strangeduality Varianceofthe (cid:73) For simplicity: a = 1 for i = 1,...,n, detE > 0. spectrum i Examples (cid:73) An invertible quasihomogeneous polynomial f is Directionsfor non-degenerate if it has an isolated singularity at furtherresearch 0 ∈ Cn. Invertible polynomials (2) Strangeduality andsymmetryof singularities (cid:73) f is quasihomogeneous, i.e. there exist weights w ,...,w ∈ Q such that Arnold’sstrange 1 n duality Orbifold f(λw1x ,...,λwnx ) = λf(x ,...,x ) for all λ ∈ C∗. Landau-Ginzburg 1 n 1 n models Invertiblepolynomials Diagonalsymmetries (cid:73) Weights (w1,...,wn) defined by Objective Orbifoldcurves w  1 Dolgachevnumbers 1 StringyEulernumber . . E  ..  = .. Cwuitshpgsrionugpulaarcittiioesn w 1 Gabrielovnumbers n Spectrum Mirrorsymmetry (cid:73) Kreuzer-Skarke: A non-degenerate invertible polynomial Strangeduality Varianceofthe f is a (Thom-Sebastiani) sum of spectrum (cid:73) x1p1x2+x2p2x3+...+xmpm−−11xm+xmpm Examples Directionsfor (chain type; m≥1); furtherresearch (cid:73) x1p1x2+x2p2x3+...+xmpm−−11xm+xmpmx1 (loop type; m≥2). Berglund-Hu¨bsch transpose Strangeduality andsymmetryof singularities Arnold’sstrange duality Orbifold Landau-Ginzburg (cid:73) The Berglund-Hu¨bsch transpose fT is models Invertiblepolynomials Diagonalsymmetries fT(x1,...,xn) = (cid:88)n ai (cid:89)n xjEji . OOrbbjeifcotilvdecurves Dolgachevnumbers i=1 j=1 StringyEulernumber Cuspsingularities 6 0 0 withgroupaction Gabrielovnumbers Ex.: ET = 1 3 0, fT(x,y,z) = x6+xy3+z2 Spectrum Mirrorsymmetry 0 0 2 Strangeduality Varianceofthe spectrum Examples Directionsfor furtherresearch

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Gabrielov numbers. Spectrum. Mirror symmetry. Strange duality. Variance of the spectrum. Examples. Directions for further research. Strange duality and symmetry of singularities. (joint work with Atsushi Takahashi). Institut für Algebraische Geometrie. Leibniz Universität Hannover. Liverpool, Jun

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