Complex Geometry C. Herbert Clemens

Total Page：16

File Type：pdf, Size：1020Kb

Complex Geometry C. Herbert Clemens Professor 1976 A. B. 1961 Holy Cross College Ph. D. 1966 University of California, Berkeley Assistant Professor, Columbia University, 1973-75 Phone: (801) 581-5275 [email protected] Activities and Awards at the secondary, undergraduate, graduate and research level. Invited Speaker International Congress of Mathematicians (1974, 1986) Professor Clemens’s research contributions University of Utah Distinguished Research Award (1983) include solution, with Phillip Griffiths, of a Member of Committee on Mathematics long-standing conjecture as to whether any solution space to a system of polynomial equations which admits a many-to-one Research Interest parametrization in fact admits a one-to-one parametrization. He later showed that it is Professor Clemens career combines a strong not in general true that the equivalence interest in research in the mathematical group of subspaces of a solution space to a area of algebraic geometry with a lifelong system of polynomial equations can be gen- interest in undergraduate and precollege erated by finitely many subspaces, and some mathematics education and in nurturing years after that he showed that solution sets mathematics and mathematics education in to a single equation in general do not contain the developing world. He spent five years as curves of low genus. His current research a assistant professor and associate professor interests are centered on deformation theo- at Columbia University in New York City ry, that is, the study of the obstructions to before joining the Mathematics Department continuously deforming subspaces when one of the University of Utah in 1975. Other changes the coefficients in the polynomial educational activities since coming to Utah equations defining a solution space. It has include five years as director of the Institute been shown by physicists that these for the Theory and Application of questions are important to unified field Mathematics, an arm of the Mathematics theories. Department dedicated to university-wide and community outreach, and, since 1994, math coordinator for the ‘Ndahoo’aah Selected Publications Program, combining the teaching of tradi- 1. (joint with H. Kley) Counting curves which move tional culture with math and computer with threefolds. J. Alg. Geom. 9 (2000), 175-200. preparation in the predominantly Navajo high schools of southeastern Utah. 2. Homological equivalence, modulo algebraic equivalence, is not finitely generated Publ. Math. I.H.E.S. 58 Professor Clemens is also currently (1983), 231-250. Secretary/Treasurer of the Commission for 3. (joint with P. Griffiths), The intermediate Jacobian Development and Exchanges of the of the cubic threefold, Annals of Mathematics 95 International Mathematical Union and (1972) 281-356. chairs the Steering Committee of the Park 4. (joint with J. Kollár and S. Mori) Higher City Mathematics Institute, an Institute for Dimensional Complex, Geometry. Astérisque 166,1988. Advance Study sponsored summer program 5 Scrapbook of Complex Curve Theory. Plenum Press, of mathematics and mathematics teaching 1980. www.math.utah.edu/~clemens.

Copy Link

Recommended publications

I. Overview of Activities, April, 2005-March, 2006 …
MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 …......……………………. 2 Innovations ………………………………………………………..... 2 Scientific Highlights …..…………………………………………… 4 MSRI Experiences ….……………………………………………… 6 II. Programs …………………………………………………………………….. 13 III. Workshops ……………………………………………………………………. 17 IV. Postdoctoral Fellows …………………………………………………………. 19 Papers by Postdoctoral Fellows …………………………………… 21 V. Mathematics Education and Awareness …...………………………………. 23 VI. Industrial Participation ...…………………………………………………… 26 VII. Future Programs …………………………………………………………….. 28 VIII. Collaborations ………………………………………………………………… 30 IX. Papers Reported by Members ………………………………………………. 35 X. Appendix - Final Reports ……………………………………………………. 45 Programs Workshops Summer Graduate Workshops MSRI Network Conferences MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 This annual report covers MSRI projects and activities that have been concluded since the submission of the last report in May, 2005. This includes the Spring, 2005 semester programs, the 2005 summer graduate workshops, the Fall, 2005 programs and the January and February workshops of Spring, 2006. This report does not contain fiscal or demographic data. Those data will be submitted in the Fall, 2006 final report covering the completed fiscal 2006 year, based on audited financial reports. This report begins with a discussion of MSRI innovations undertaken this year, followed by highlights
[Show full text]
Mori Dream Spaces and Birational Rigidity of Fano 3-Folds
Ahmadinezhad, H., & Zucconi, F. (2016). Mori dream spaces and birational rigidity of Fano 3-folds. Advances in Mathematics, 292, 410- 445. https://doi.org/10.1016/j.aim.2016.01.008 Peer reviewed version Link to published version (if available): 10.1016/j.aim.2016.01.008 Link to publication record in Explore Bristol Research PDF-document University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ MORI DREAM SPACES AND BIRATIONAL RIGIDITY OF FANO 3-FOLDS HAMID AHMADINEZHAD AND FRANCESCO ZUCCONI Abstract. We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces. 1. Introduction Rationality question for varieties covered by rational curves (e.g. unirational varieties) has been a fundamental problem in algebraic geometry, the answer to which has opened various subjects across the field. The first examples of unirational varieties that are irrational were found about the same time by Clemens and Griffiths [10], Iskovskikh and Manin [18], Artin and Mumford [3]. While the results of [10] and [3] indicate that the varieties under study (respectively, a smooth cubic 3-fold and some special singular quartic 3-folds in the weighted projective space P(1; 1; 1; 1; 2)) are irrational, the result in [18] shows that a smooth quartic 3-fold is not birational to any conic bundle or fibration into del 3 Pezzo surfaces, or any other Fano variety, including P , that is to say it is irrational in a strong sense.
[Show full text]
2008 Award for Distinguished Public Service
2008 Award for Distinguished Public Service The 2008 Award for Distinguished and continuation of the Park City/IAS Mathemat- Public Service was presented at the ics Institute. 114th Annual Meeting of the AMS in San Diego in January 2008. Biographical Sketch The Award for Distinguished Public Herbert Clemens earned his Ph.D. in 1966 from Service is presented every two years the University of California, Berkeley, under the to a research mathematician who has direction of Phillip A. Griffiths. He has taught at made a distinguished contribution to Columbia University, the University of Utah, and the mathematics profession during the preceding five years. The purpose of the Ohio State University, where he has been on the award is to encourage and recog- the faculty since 2002. He has served as director nize those individuals who contribute of the NSF Regional Geometry Institute, Park City, their time to public service activities UT, and chair of the Steering Committee for the IAS in support of mathematics. The award Park City Mathematics Institute. He was an invited Herbert Clemens carries a cash prize of US$4,000. speaker at the International Congress of Mathema- The Award for Distinguished Public ticians in 1974 and in 1986. His academic honors Service is made by the AMS Council include a Silver Medal from the Italian Mathemati- acting on the recommendation of a selection com- cal Society and a Laurea de honoris causa from the mittee. For the 2008 award the members of the Universita di Torino, among others. His research selection committee were: William J. Lewis, Carolyn R.
[Show full text]
Arxiv:1407.7478V1 [Math.AG]
THE STRUCTURE OF ALGEBRAIC VARIETIES JANOS´ KOLLAR´ Abstract. The aim of this address is to give an overview of the main questions and results of the structure theory of higher dimensional algebraic varieties. 1. Early history: Euler, Abel, Jacobi, Riemann Our story, like many others in mathematics, can be traced back at least to Euler who studied elliptic integrals of the form dx . Z √x3 + ax2 + bx + c The study of integrals of algebraic functions was further developed by Abel and Jacobi. From our point of view the next major step was taken by Riemann. Instead of dealing with a multi-valued function like √x3 + ax2 + bx + c, Riemann looks at the complex algebraic curve C := (x, y): y2 = x3 + ax2 + bx + c C2. ⊂ Then the above integral becomes dx ZΓ y for some path Γ on the algebraic curve C. More generally, a polynomial g(x, y) implicitly deﬁnes y := y(x) as a multi-valued function of x and for any meromorphic function h(u, v), the multi-valued integral h x, y(x) dx Z becomes a single valued integral arXiv:1407.7478v1 [math.AG] 28 Jul 2014 h x, y dx Z Γ for some path Γ on the algebraic curve C(g) := g(x, y)=0 C2. Substitutions ⊂ that transform one integral associated to a polynomial g1 into another integral associated to a g2 can be now seen as algebraic maps between the curves C(g1) and C(g2). Riemann also went further. As a simple example, consider the curve C deﬁned by (y2 = x3 + x2) and notice that (t3 t)2 (t2 1)3 + (t2 1)2.
[Show full text]
From the AMS Secretary
From the AMS Secretary Society and delegate to such committees such powers as Bylaws of the may be necessary or convenient for the proper exercise American Mathematical of those powers. Agents appointed, or members of com- mittees designated, by the Board of Trustees need not be Society members of the Board. Nothing herein contained shall be construed to em- Article I power the Board of Trustees to divest itself of responsi- bility for, or legal control of, the investments, properties, Officers and contracts of the Society. Section 1. There shall be a president, a president elect (during the even-numbered years only), an immediate past Article III president (during the odd-numbered years only), three Committees vice presidents, a secretary, four associate secretaries, a Section 1. There shall be eight editorial committees as fol- treasurer, and an associate treasurer. lows: committees for the Bulletin, for the Proceedings, for Section 2. It shall be a duty of the president to deliver the Colloquium Publications, for the Journal, for Mathemat- an address before the Society at the close of the term of ical Surveys and Monographs, for Mathematical Reviews; office or within one year thereafter. a joint committee for the Transactions and the Memoirs; Article II and a committee for Mathematics of Computation. Section 2. The size of each committee shall be deter- Board of Trustees mined by the Council. Section 1. There shall be a Board of Trustees consisting of eight trustees, five trustees elected by the Society in Article IV accordance with Article VII, together with the president, the treasurer, and the associate treasurer of the Society Council ex officio.
[Show full text]
On Geometry of Fano Threefold Hypersurfaces 3
ON GEOMETRY OF FANO THREEFOLD HYPERSURFACES HAMID AHMADINEZHAD, IVAN CHELTSOV, JIHUN PARK Abstract. We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one. 1. Introduction End points of Minimal Model Program are either Mori fibre spaces or minimal models. In dimension three, Mori fibre spaces form three classes: Fano threefolds, del Pezzo fibrations over curves, and conic bundles over surfaces. They are Q-factorial with at worst terminal singulari- ties, and with relative Picard number one. The focus of this article is on Fano threefolds. They lie in finitely many deformation families (see [5] and [13]), and studying birational relations among them, as well as birational maps to other Mori fibre spaces is fundamental, as it sheds light to birational classification of rationally connected threefolds in general. For a Fano threefold X with terminal Q-factorial singularities, let A be a Weil divisor for which −KX = ιX A with maximum ιX ∈ Z>0. This integer ιX is known as the Fano index of X. It follows from [4] and [6] that there are precisely 130 families of Fano threefolds such that the Z-graded ring R(X, A)= H0 X, −mA Mm>0 has 5 generators. Denote these generators by x,y,z,t,w, respectively in degrees a0, a1, a2, a3, a4, and let the algebraic relation amongst them be f(x,y,z,t,w) = 0 of weighted degree d. This redefines X as a hypersurface in the weighted projective space P(a0, a1, a2, a3, a4) that is given by the quasi-homogeneous equation f(x,y,z,t,w)=0, 4 and the Fano index of X is computed via ιX = i=0 ai − d.
[Show full text]
List of Works by Alessio Corti
List of Works by Alessio Corti 11th February 2020 Recent arXiv articles 1. On the Topology of Fano Smoothings (with Tom Coates and Genival da Silva Jr), arXiv:1912.04383. 16 pp. 2. Hyperelliptic Integrals and Mirrors of the Johnson–Kollárdel Pezzo Surfaces (with Giulia Gugiatti), arXiv:1901.09026. 28 pp. Journal articles 3. Some Applications of the Mirror Theorem for Toric Stacks (with Tom Coates, Hiroshi Iritani and Hsian-Hua Tseng), Advances in Theoretical and Mathematical Physics 23, No. 3, 2019, 767{802 4. Hodge-theoretic mirror symmetry for toric stacks (with Tom Coates, Hiroshi Iritani and Hsian-Hua Tseng), J. Differential Geom. 114, No. 1, 2020, 41{115 5. Del Pezzo surfaces with 1=3(1; 1) points (with Liana Heuberger), Manuscripta Math. 153, No. 1{2, 2017, 71{118 6. Erratum to \Conifold transitions and Mori theory (with Ivan Smith), Math. Res. Lett. 23, No. 3, 2016, 733{734 7. The Sarkisov program for Mori fibred Calabi{Yau pairs (with Anne- Sophie Kaloghiros), Algebr. Geom. 3, No 3, 2016, 370{384 8. Quantum Periods for 3-Dimensional Fano Manifolds (with Tom Coates, Sergey Galkin and Alexander Kasprzyk), Geom. Topol. 20, no. 1, 2016, 103{256 1 9. Mirror symmetry and the classification of orbifold del Pezzo surfaces (with Mohammad Akhtar, Tom Coates, Liana Heuberger, Alexander Kasprzyk, Alessandro Oneto and Andrea Petracci), Proc. Amer. Math. Soc. 144, No. 2, 2016, 513{527 10. A Mirror Theorem for Toric Stacks (with Tom Coates, Hiroshi Iritani and Hsian-Hua Tseng), Comp. Math. 151, 2015, 1878{1912 11. G2-manifolds and associative submanifolds via semi-Fano 3-folds (with Mark Haskins, Johannes Nordströmand Tommaso Pacini), Duke Math.
[Show full text]
List of Works by Alessio Corti∗
List of Works by Alessio Corti∗ Recent arXiv articles 1. Hodge-Theoretic Mirror Symmetry for Toric Stacks (with Tom Coates, Hiroshi Iritani, and Hsian-Hua Tseng), arXiv:1606.07254. 58 pp., submitted to Journal fürdie reine und angewandte Mathematik (Crelle's Journal) 2. Del Pezzo surfaces with 1=3(1; 1) points (with Liana Heuberger), arXiv:1505.02092, 42 pp., to appear on Manuscripta Mathematica 3. The Sarkisov program for Mori fibred Calabi{Yau pairs (with Anne- Sophie Kaloghiros), arXiv:1504.00557, 12 pp., accepted for publica- tion on Algebraic Geometry 4. Some Applications of the Mirror Theorem for Toric Stacks (with Tom Coates, Hiroshi Iritani and Hsian-Hua Tseng), arXiv:math.AG/1401.2611, 23 pp., submitted Journal articles 5. Erratum to \Conifold transitions and Mori theory (with Ivan Smith), Math. Res. Lett. 23, No. 3, 2016, 733-734 6. Quantum Periods for 3-Dimensional Fano Manifolds (with Tom Coates, Sergey Galkin and Alexander Kasprzyk), Geom. Topol. 20, no. 1, 2016, 103{256 7. Mirror symmetry and the classification of orbifold del Pezzo surfaces (with Mohammad Akhtar, Tom Coates, Liana Heuberger, Alexander Kasprzyk, Alessandro Oneto and Andrea Petracci), Proc. Amer. Math. Soc. 144, No. 2, 2016, 513{527 ∗Last modified: July 2016 1 8. A Mirror Theorem for Toric Stacks (with Tom Coates, Hiroshi Iritani and Hsian-Hua Tseng), Comp. Math. 151, 2015, 1878{1912 9. G2-manifolds and associative submanifolds via semi-Fano 3-folds (with Mark Haskins, Johannes Nordströmand Tommaso Pacini), Duke Math. J. 164, No. 10, 2015, 1971{2092 10. Asymptotically cylindrical Calabi{Yau 3-folds from weak Fano 3-folds (with Mark Haskins, Johannes Nordströmand Tommaso Pacini), Geom.
[Show full text]
Arxiv:0809.2579V2 [Math.AG]
EXERCISES IN THE BIRATIONAL GEOMETRY OF ALGEBRAIC VARIETIES JANOS´ KOLLAR´ The book [KM98] gave an introduction to the birational geometry of algebraic varieties, as the subject stood in 1998. The developments of the last decade made the more advanced parts of Chapters 6 and 7 less important and the detailed treatment of surface singularities in Chapter 4 less necessary. However, the main parts, Chapters 1–3 and 5, still form the foundations of the subject. These notes provide additional exercises to [KM98]. The main deﬁnitions and theorems are recalled but not proved here. The emphasis is on the many examples that illustrate the methods, their shortcomings and some applications. 1. Birational classification of algebraic surfaces For more detail, see [BPVdV84]. The theory of algebraic surfaces rests on the following three theorems. Theorem 1. Any birational morphism between smooth projective surfaces is a composite of blow-downs to points. Any birational map between smooth projective surfaces is a composite of blow-ups and blow-downs. Theorem 2. There are 3 species of “pure-bred” surfaces: (Rational): For these surfaces the internal birational geometry is very compli- cated, but, up to birational equivalence, we have only P2. These frequently appear in the classical literature and in “true” applications. (Calabi-Yau): These are completely classiﬁed (Abelian, K3, Enriques, hyper- elliptic) and their geometry is rich. They are of great interest to other mathematicians. (General type): They have a canonical model with Du Val singularities and ample canonical class. Although singular, this is the “best” model to work arXiv:0809.2579v2 [math.AG] 21 Oct 2008 with.
[Show full text]
Higher Dimensional Complex Geometry
Astérisque HERBERT CLEMENS JÁNOS KOLLÁR SHIGEFUMI MORI Higher dimensional complex geometry, A Summer Seminar at the University of Utah, Slat Lake City, 1987 Astérisque, tome 166 (1988) <http://www.numdam.org/item?id=AST_1988__166__1_0> © Société mathématique de France, 1988, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce ﬁchier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 166 ASTÉRISQUE 1988 HIGHER DIMENSIONAL COMPLEX GEOMETRY Herbert CLEMENS, János KOLLÁR, Shigefumi MORI A Summer Seminar at the University of Utah, Salt Lake City, 1987 SOCIÉTÉ MATHÉMATIQUE DE FRANCE Publié avec le concours du CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE A.M.S. Subjects Classification : 14-02, 14-C-30, 14-C-99, 14-E-05, 14-E-30, 14-E-35, 14-J-30, 32-J-25, 53-B-35, 53-C-35. Introduction These notes originated a t a seminar that was held during Jul y and August o f 198 7 at Salt Lake City. Th e original aim of the seminar was to get an overview o f the following three topics: 1. Recent advances i n the classification program o f three (an d higher) dimensional algebraic varieties. 2. Existence o f rational curves and other specia l subvarieties. 3. Existence an d nature o f special metrics o n varieties.
[Show full text]
Professor, Mathematics Department, Ohio State University, Colum- Bus, OH 43210 Visiting Research Professor, University of Utah, Salt Lake City, UT 84112
VITA AND BIBLIOGRAPHY C. HERBERT CLEMENS Position: Professor, Mathematics Department, Ohio State University, Colum- bus, OH 43210 Visiting Research Professor, University of Utah, Salt Lake City, UT 84112 Birthdate: August 15, 1939 Birthplace: Dayton, Ohio Academic Degrees: A.B. 1961 Holy Cross College M.A. 1964 University of California, Berkeley Ph.D. 1966 University of California, Berkeley (Advisor: Phillip A. Griﬃths) Research Area: Complex Geometry Professional Experience: Research and University Teaching: 1966 Peace Corps training for Peru and Chile 1966-68 Profesor Regular de Matematicas, Universidad T´ecnicadel Estado, San- tiago de Chile 1968-70 Member, School of Mathematics, Institute for Advanced Study, Prince- ton 1969 (summer) Visiting Assistant Professor, University of California, Berkeley 1970 Senior Fulbright Lecturer (Chile) 1970-72 Assistant Professor, Columbia University 1972-73 Profesor UTE, Santiago, Chile, Project Coordinator, Ford Foundation grant to UTE mathematics program 1973 (winter) Visiting Assistant Professor, Harvard University 1973-75 Associate Professor, Columbia University (tenure 1975) 1975-76 Associate Professor, University of Utah 1976-2002 Professor, University of Utah (tenure 1976) 1997-98 Visiting Professor, Scuola Normale Superiore, Pisa, Italy 2001-02 Director's Visitor, Institute for Advanced Study, Princeton 2002-19 Professor, Ohio State University 2013-17 Visiting Research Professor, University of Utah 2019-present Professor Emeritus, Ohio State University Primary and Secondary Teaching: l979-84
[Show full text]
Flips for 3-Folds and 4-Folds
Flips for 3-folds and 4-folds Florin Ambro Res. Inst. Math. Sci., Kyoto University Kyoto 606-8502, Japan Alessio Corti Department of Mathematics, Imperial College London Huxley Building, 180 Queen’s Gate London SW7 2AZ, UK Osamu Fujino Grad. School of Math., Nagoya University Chikusa-ku, Nagoya 464-8602, Japan Christopher D. Hacon Department of Mathematics, University of Utah 155 South 1400 East Salt Lake City UT 84112-0090, USA János Kollár Department of Mathematics, Princeton University Fine Hall, Washington Road Princeton NJ 08544-1000, USA James McKernan Department of Mathematics, UC Santa Barbara Santa Barbara CA 93106, USA Hiromichi Takagi Grad. School of Math. Sci., The University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153–8914, Japan Contents Chapter 1. Introduction Alessio Corti 1 1.1. Minimal models of surfaces 1 1.2. Higher dimensions and flips 1 1.3. The work of Shokurov 2 1.4. Minimal models of 3-folds and 4-folds 2 1.5. The aim of this book 3 1.6. Pl flips 3 1.7. b-divisors 4 1.8. Restriction and mobile b-divisors. 5 1.9. Pbd-algebras 5 1.10. Restricted systems and 3-fold pl flips 6 1.11. Shokurov’s finite generation conjecture 6 1.12. What is log terminal? 7 1.13. Special termination and reduction to pl flips. 8 1.14. The work of Hacon and McKernan: adjoint algebras 8 1.15. Mobile b-divisors on weak klt del Pezzo surfaces 9 1.16. The CCS Conjecture 9 1.17. Kodaira’s canonical bundle formula and subadjunction 10 1.18.
[Show full text]