44 results found
Corti A, Filip M, Petracci A, 2022, Mirror symmetry and smoothing Gorenstein toric affine 3-folds, Facets of Algebraic Geometry: A Collection in Honor of William Fulton's 80th Birthday, Editors: Aluffi, Anderson, Hering, Mustata, Payne, Publisher: Cambridge University Press, Pages: 132-163
We state two conjectures that together allow one to describe the set of smoothing components of a Gorenstein toric affine 3-fold in terms of a combinatorially defined and easily studied set of Laurent polynomials called 0-mutable polynomials. We explain the origin of the conjectures in mirror symmetry and present some of the evidence.
Coates T, Corti A, da Silva G, 2022, On the Topology of Fano Smoothings, Springer Proceedings in Mathematics and Statistics, Vol: 386, Pages: 135-156, ISSN: 2194-1009
Suppose that X is a Fano manifold that corresponds under Mirror Symmetry to a Laurent polynomial f, and that P is the Newton polytope of f. In this setting it is expected that there is a family of algebraic varieties over the unit disc with general fiber X and special fiber the toric variety defined by the spanning fan of P. Building on recent work and conjectures by Corti–Hacking–Petracci, who construct such families of varieties, we determine the topology of the general fiber from combinatorial data on P. This provides evidence for the Corti–Hacking–Petracci conjectures, and verifies that their construction is compatible with expectations from Mirror Symmetry.
Corti A, Gugiatti G, 2021, Hyperelliptic integrals and mirrors of the Johnson-Kollár del Pezzo surfaces, Transactions of the American Mathematical Society, Vol: 374, Pages: 8603-8637, ISSN: 0002-9947
For all integers k > 0, we prove that the hypergeometric functionIbk(α) = X∞ j=0 (8k + 4)j [divided by] !j! (2j)! (2k + 1)j!2 (4k + 1)j! αjis a period of a pencil of curves of genus 3k+1. We prove that the function Ibk is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X8k+4 ⊂ P(2, 2k + 1, 2k + 1, 4k + 1). Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X were first constructed by Johnson and Koll´ar. The feature of these surfaces that makes our mirror construction especially interesting is that | − KX| = |OX(1)| = ∅. This means that there is no way to form a Calabi–Yau pair (X, D) out of X and hence there is no known mirror construction for X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.
Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirrordescription for the big equivariant quantum cohomology of toric Deligne-Mumfordstacks. More precisely, we prove that the big equivariant quantum D-module of atoric Deligne-Mumford stack is isomorphic to the Saito structure associated tothe mirror Landau-Ginzburg potential. We give a GKZ-style presentation of thequantum D-module, and a combinatorial description of quantum cohomology as aquantum Stanley-Reisner ring. We establish the convergence of the mirrorisomorphism and of quantum cohomology in the big and equivariant setting.
Coates T, Corti A, Iritani H, et al., 2019, Some applications of the mirror theorem for toric stacks, Publisher: INT PRESS BOSTON, INC
Corti A, Heuberger L, 2016, Del Pezzo surfaces with 1/3 (1,1) points, Manuscripta Mathematica, Vol: 153, Pages: 71-118, ISSN: 1432-1785
We classify non-smooth del Pezzo surfaces with 13(1,1) points in 29 qG-deformation families groupedinto six unprojection cascades (this overlaps with work of Fujita and Yasutake ), we tabulate their biregular invariants, we give good model constructions for surfaces in all families as degeneracy loci in rep quotient varieties, and we prove that precisely 26 families admit qG-degenerations to toric surfaces. This work is part of a program to study mirror symmetry for orbifold del Pezzo surfaces .
Corti A, Kaloghiros A, 2016, The Sarkisov program for Mori fibred Calabi-Yau pairs, Algebraic Geometry, Vol: 3, Pages: 370-384, ISSN: 2214-2584
We prove a version of the Sarkisov program for volume preserving birational maps of Mori fibred Calabi–Yau pairs valid in all dimensions. Our theorem generalises the theorem of Usnich and Blanc on factorisations of birational maps of (C×)2 that preserve the volume form dx x ∧ dy y.
The quantum period of a variety X is a generating function for certainGromov-Witten invariants of X which plays an important role in mirror symmetry.In this paper we compute the quantum periods of all 3-dimensional Fanomanifolds. In particular we show that 3-dimensional Fano manifolds with veryample anticanonical bundle have mirrors given by a collection of Laurentpolynomials called Minkowski polynomials. This was conjectured in joint workwith Golyshev. It suggests a new approach to the classification of Fanomanifolds: by proving an appropriate mirror theorem and then classifying Fanomirrors. Our methods are likely to be of independent interest. We rework theMori-Mukai classification of 3-dimensional Fano manifolds, showing that each ofthem can be expressed as the zero locus of a section of a homogeneous vectorbundle over a GIT quotient V/G, where G is a product of groups of the formGL_n(C) and V is a representation of G. When G=GL_1(C)^r, this expresses theFano 3-fold as a toric complete intersection; in the remaining cases, itexpresses the Fano 3-fold as a tautological subvariety of a Grassmannian,partial flag manifold, or projective bundle thereon. We then compute thequantum periods using the Quantum Lefschetz Hyperplane Theorem ofCoates-Givental and the Abelian/non-Abelian correspondence ofBertram-Ciocan-Fontanine-Kim-Sabbah.
Anderson G, Buck D, Coates T, et al., 2015, Drawing in Mathematics From Inverse Vision to the Liberation of Form, LEONARDO, Vol: 48, Pages: 439-448, ISSN: 0024-094X
© The Authors 2015. We prove a Givental-style mirror theorem for toric Deligne-Mumford stacks χ. This determines the genus-zero Gromov-Witten invariants of χ in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen-Ruan orbifold cohomology of χ.
Akhtar M, Coates T, Corti A, et al., 2015, Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces, Proceedings of the American Mathematical Society, Vol: 144, Pages: 513-527, ISSN: 1088-6826
We state a number of conjectures that together allow one to classify a broadclass of del Pezzo surfaces with cyclic quotient singularities using mirrorsymmetry. We prove our conjectures in the simplest cases. The conjecturesrelate mutation-equivalence classes of Fano polygons with Q-Gorensteindeformation classes of del Pezzo surfaces.
Corti A, Haskins M, Nordstroem J, et al., 2015, G(2)-manifold and associative submanifolds via semi-fano 3-folds, Duke Mathematical Journal, Vol: 164, Pages: 1971-2092, ISSN: 0012-7094
We construct many new topological types of compact G2G2-manifolds, that is, Riemannian 77-manifolds with holonomy group G2G2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 33-folds built from semi-Fano 33-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 77-manifolds completely; we find that many 22-connected 77-manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G2G2-metrics. Many of the G2G2-manifolds we construct contain compact rigid associative 33-folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G2G2-metrics. By varying the semi-Fanos used to build different G2G2-metrics on the same 77-manifold we can change the number of rigid associative 33-folds we produce.
Corti A, Smith I, 2015, Conifold transitions and Mori theory (vol 12, pg 767, 2005), MATHEMATICAL RESEARCH LETTERS, Vol: 23, Pages: 733-734, ISSN: 1073-2780
Corti A, Haskins M, Nordström J, et al., 2013, Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds, Geometry & Topology, Vol: 17, Pages: 1955-2059, ISSN: 1465-3060
We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–foldsstarting with (almost) any deformation family of smooth weak Fano 3–folds. Thisallow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previouslyonly a few hundred ACyl Calabi–Yau 3–folds were known. We pay particularattention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. SemiFano3–folds satisfy stronger cohomology vanishing theorems and enjoy certaintopological properties not satisfied by general weak Fano 3–folds, but are far morenumerous than genuine Fano 3–folds. Also, unlike Fanos they often contain P1s withnormal bundle O.1/˚ O.1/,giving rise to compact rigid holomorphic curves inthe associated ACyl Calabi–Yau 3–folds.We introduce some general methods to compute the basic topological invariants ofACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a smallnumber of representative examples in detail. Similar methods allow the computationof the topology in many other examples.All the features of the ACyl Calabi–Yau 3–folds studied here find application in where we construct many new compact G2 –manifolds using Kovalev’s twistedconnected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano3–folds are particularly well-adapted for this purpose.
Corti A, Lazic V, 2013, New outlook on the Minimal Model Program, II, MATHEMATISCHE ANNALEN, Vol: 356, Pages: 617-633, ISSN: 0025-5831
Coates T, Corti A, Galkin S, et al., 2012, Mirror Symmetry and Fano Manifolds, Vol: n/a
Corti A, Golyshev V, 2011, Hypergeometric equations and weighted projective spaces, SCIENCE CHINA-MATHEMATICS, Vol: 54, Pages: 1577-1590, ISSN: 1674-7283
Corti A, Kaloghiros A, Lazic V, 2011, Introduction to the Minimal Model Program and the existence of flips, Bull. London Math. Soc.
The first aim of this note is to give a concise, but complete andself-contained, presentation of the fundamental theorems of Mori theory - thenonvanishing, base point free, rationality and cone theorems - using modernmethods of multiplier ideals, Nadel vanishing, and the subadjunction theorem ofKawamata. The second aim is to write up a complete, detailed proof of existenceof flips in dimension n assuming the minimal model program with scaling indimension n-1.
Corti A, 2010, Finite generation of adjoint rings after Lazic: an introduction, CLASSIFICATION OF ALGEBRAIC VARIETIES, Pages: 197-220
Coates T, Corti A, Lee Y-P, et al., 2009, The quantum orbifold cohomology of weighted projective spaces, ACTA MATHEMATICA, Vol: 202, Pages: 139-193, ISSN: 0001-5962
Coates T, Corti A, Iritani H, et al., 2009, COMPUTING GENUS-ZERO TWISTED GROMOV-WITTEN INVARIANTS, DUKE MATHEMATICAL JOURNAL, Vol: 147, Pages: 377-438, ISSN: 0012-7094
Corti A, 2009, Three equivalent conjectures on the birational geometry of Fano 3-folds, PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS, Vol: 264, Pages: 45-47, ISSN: 0081-5438
Corti A, 2007, 3-fold flips after Shokurov, Flips for 3-folds and 4-folds, ISBN: 9780198570615
This chapter gives a concise, complete, and pedagogical proof of existence of 3-fold flips according to Shokurov. In particular, the foundation of the theory of b-divisors, algebras of rational functions, and Shokurov's asymptotic saturation property are developed systematically from first principles.
Corti A, McKernan J, Takagi H, 2007, Saturated mobile b-divisors on weak del Pezzo klt surfaces, Flips for 3-folds and 4-folds, ISBN: 9780198570615
This chapter studies some of the simple examples of saturated mobile b-divisor. In particular, a classification is given of saturated mobile bdivisors on del Pezzo surfaces. The examples reveal a surprisingly intricate structure theory and suggest further conjectures and applications in higher dimensions.
Corti A, 2007, Flips for 3-folds and 4-folds, ISBN: 9780198570615
The minimal model program in algebraic geometry is a conjectural sequence of algebraic surgery operations that simplifies any algebraic variety to a point where it can be decomposed into pieces with negative, zero, and positive curvature, in a similar vein as the geometrization program in topology decomposes a three-manifold into pieces with a standard geometry. The last few years have seen dramatic advances in the minimal model program for higher dimensional algebraic varieties, with the proof of the existence of minimal models under appropriate conditions, and the prospect within a few years of having a complete generalization of the minimal model program and the classification of varieties in all dimensions, comparable to the known results for surfaces and 3-folds. This edited collection of chapters, authored by leading experts, provides a complete and self-contained construction of 3-fold and 4-fold flips, and n-dimensional flips assuming minimal models in dimension n-1. A large part of the text is an elaboration of the work of Shokurov, and a complete and pedagogical proof of the existence of 3-fold flips is presented. The book contains a self-contained treatment of many topics that could only be found, with difficulty, in the specialized literature. The text includes a ten-page glossary.
Coates T, Corti A, Iritani H, et al., 2007, The Crepant Resolution Conjecture for Type A Surface Singularities
Coates T, Corti A, Iritani H, et al., 2007, Computing Genus-Zero Twisted Gromov-Witten Invariants
Corti A, Hanamura M, 2007, Motivic decomposition and intersection Chow groups II, Pure and Applied Mathematics Quarterly, Vol: 3, Pages: 181-203, ISSN: 1558-8599
Coates T, Corti A, Iritani H, et al., 2006, Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples
Coates T, Corti A, Lee Y-P, et al., 2006, The Quantum Orbifold Cohomology of Weighted Projective Space
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