CNCS Grant PN-II-PCE-2011-3-0288

Title: Vector bundle techniques in the geometry of complex varieties

Contract no. 132/05.10.2011

Starting date: October 5, 2011

Initial time frame: 36 months / extended to: 60 months

Budget: 1.381.849 RON

Team members (in alphabetical order):

• 1. Marian Aprodu

  2. Vasile Brinzanescu

  3. Paltin Ionescu

  4. Stefan Papadima

  5. Victor Vuletescu

Short description:

A vector bundle is a geometric object that models the concept of continuous family of vector spaces. In algebraic geometry, relevant classes of varieties have been constructed using vector bundle techniques, for instance, either as jump loci or degeneracy loci, or by taking directly various moduli spaces. The classification program for projective varieties was revolutionized by Mori's solution to a conjecture on vector bundles. Vector bundles are basic ubiquitous objects in geometry, and hence it is important to achieve a better understanding of their influence and their relations with intrinsic properties of the base manifolds. The present projects addresses some problems in this research field and reflects the current scientific interests of the team members.

Research objectives:

• 1. Vector bundles and geometry of projective curves

  2. Vector bundles on non-Kaehler complex manifolds

  3. Homology with local coefficients

  4. Vector bundles and geometry of special varieties

Scientific reports:

2011 - romanian, english

2012 - romanian, english

2013 - romanian, english

2014 - romanian, english

2015 - romanian, english

2016 - romanian, english

Intermediate evaluation report: link

Activities in 2011-2012:

Foreign visitors (in chronological order):

• 1. Gavril Farkas (Humboldt Universitaet Berlin, Germany) - February 2012

  2. Angela Ortega (Humboldt Universitaet Berlin, Germany) - February 2012

  3. Stephane Druel (Universite Joseph Fourier Grenoble, France) - April 2012

  4. Alexandru Suciu (Northeastern University Boston, USA) - June 2012

  5. Christian Peskine (Institut Math. Jussieu Paris, France) - October 2012

  6. Edoardo Sernesi (Universita Roma Tre, Italy) - December 2012

Publications:

• 1. Marian Aprodu, Ruxandra Moraru, Matei Toma. Two-dimensional moduli spaces of vector bundles over Kodaira surfaces. Advances in Mathematics 231 (2012) 1202-1215. DOI:10.1016/j.aim.2012.06.024 (objective 2) link file

  2. Stefan Papadima, Alexander I. Suciu. Homological finiteness in the Johnson filtration of the automorphism group of a free group. Journal of Topology 5 no. 4 (2012) 909-944. DOI:10.1112/jtopol/jts023 (objective 3) link file

  3. Victor Vuletescu. Exemples de faisceaux coherents sans resolution localement libre en dimension 3. Comptes Rendus Mathématique (Académie des Sciences, Paris) 350 (2012) no. 7-8, 411-412. DOI:10.1016/j.crma.2012.03.020 (objective 2) link file

  4. Marian Aprodu, Vasile Brinzanescu, Marius Marchitan. Rank-two vector bundles on Hirzebruch surfaces. Central European  Journal of Mathematics 10 No. 4 (2012) 1321-1330. DOI:10.2478/s11533-012-0046-2 (objectives 1 and 4) link file

Dissemination:

• 1. S. Papadima. 120 years of Analysis Situs. Journee Henri Poincare, IMAR Bucharest, March 23, 2012

  2. S. Papadima. Classifying spaces and cohomology jump loci. Singularites d'equations differentielles en geometrie algebrique, CIRM Luminy, June 4-8, 2012

  3. M. Aprodu. Vector bundles and syzygies. Workshop on Syzygies in Algebraic Geometry, with an exploration of a connection with String Theory (12w5117),  Banff International Research Station, August 12-17, 2012

  4. S. Papadima. Algebre Koszul si grupuri de omotopie, I, Seminarul de Geometrie Algebrica - Topologie, IMAR Bucharest, October 11, 2012

  5. M. Aprodu. Two-dimensional moduli spaces of vector bundles over Kodaira surfaces. Forschungsseminar "Algebraische Geometrie", Humboldt University Berlin, October 17, 2012

  6. S. Papadima. Algebre Koszul si grupuri de omotopie. II, Seminarul de Geometrie Algebrica - Topologie, IMAR Bucharest, October 19, 2012

  7. V. Brinzanescu. Bundles on non-Kaehler elliptic fibrations. Math-Physics Joint Seminar, University of Pennsylvania, October 25, 2012

  8. V. Brinzanescu. Vector bundles on non-Kaehler elliptic principal bundles. Algebraic Geometry Seminar, Ohio State University, November 20, 2012

  9. S. Papadima. Classifying spaces and cohomology jump loci, Mini-workshop on Geometry and Topology, University of Sydney, December 10, 2012

  10. V. Brinzanescu. Vector bundles on non-Kaehler elliptic principal bundles. Algebraic Geometry Seminar, University of Illinois Chicago, December 11, 2012

Activities in 2013:

Foreign visitors (in chronological order):

• 1. Alexandru Suciu (Northeastern University Boston, USA) - June 2013

  2. Lucian Badescu (Universita degli Studi di Genova, Italy) - June 2013

Publications:

• 1. Alexandru Dimca, Stefan Papadima. Arithmetic group symmetry and finiteness properties of Torelli groups. Annals of Mathematics 177 no. 2 (2013) 395-423.  DOI:10.4007/annals.2013.177.2.1 (objectives 1 and 3) link file

  2. Alexandru Dimca, Stefan Papadima. Nonabelian cohomology jump loci from an analytic viepoint. Communications in Contemporary Mathematics 16 (2014) 1350025, 47 pp. DOI:10.1142/S0219199713500259 (objectives 3 and 4) link file

  3. Marian Aprodu, Matei Toma. Boundedness of some rationally-connected threefolds in P6. Communications in Algebra 42:9 (2014) 3876-3882. DOI:10.1080/00927872.2013.796963 (objective 4) link file

  4. Marian Aprodu. Lazarsfeld-Mukai bundles and applications. Commutative Algebra, Ed.: Irena Peeva, ISBN: 978-1-4614-5291-1 (Print) 978-1-4614-5292-8 (Online), Springer Verlag (2013) 1-23. DOI:10.1007/978-1-4614-5292-8_1 (objectives 1 and 4) link file

  5. Victor Vuletescu. A restriction theorem for torsion-free sheaves on some elliptic manifolds. Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie. Nouv. Ser. tome 54 (104) no 2 (2013) 237-242 (objectives 1 and 2) link file

Dissemination:

• 1. M. Aprodu. Fibrate Ulrich. Seminarul de Geometrie Algebrica, IMAR Bucharest, February 7, 2013

  2. M. Aprodu. Equations and syzygies of projective curves. Curves and their Jacobians: State of the Art Bellavista Relax Hotel, Levico Terme (Trento), February 13-16, 2013

  3. S. Papadima. To work or not to work in characteristic 0? this is the question. IMAR Bucuresti Monthly Lectures, March 20, 2013

  4. S. Papadima. Despre matrice si alte curiozitati. Faculty of Mathematics, University of Bucharest,  April 25, 2013

  5. M. Aprodu. Fibrate vectoriale in geometria varietatilor proiective. Workshop for Young Researchers in Mathematics, Constanta, May 9-10, 2013

  6. V. Brinzanescu. On the relationship between D'Angelo q-type and Catlin q-type. International Conference "Experimental and Theoretical Methods in Algebra, Geometry and Topology", Eforie Nord, June 21-24 , 2013

  7. P. Ionescu. Some special Fano manifolds. International Conference "Experimental and Theoretical Methods in Algebra, Geometry and Topology", Eforie Nord, June 21-24 , 2013

  8. M. Aprodu. Chow forms of K3 surfaces and Ulrich bundles. International Conference "Experimental and Theoretical Methods in Algebra, Geometry and Topology", Eforie Nord, June 21-24 , 2013

  9. S. Papadima. Geometry of homology jump loci and topology. Invited Address at the Joint American Mathematical Society-Romanian Mathematical Society International Meeting, Alba Iulia, June 27-30, 2013

  10. V. Brinzanescu. Vector bundles on non-Kaehler elliptic principal bundles. Algebraic Geometry Section, Joint American Mathematical Society-Romanian Mathematical Society International Meeting, Alba Iulia, June 27-30, 2013

  11. M. Aprodu. Chow forms of K3 surfaces and Ulrich bundles. Forschungsseminar "Algebraische Geometrie", Humboldt University Berlin, July 17, 2013

  12. V. Brinzanescu. Moduli spaces of vector bundles on elliptic fibrations. IBS, Center of Geometry and Physics, Pohang, Korea, August 19, 2013

  13. V. Brinzanescu. Vector bundles on non-Kaehler elliptic fibrations. Anniversary Conference, Faculty of Sciences - 150 Years, University of Bucharest, August 31, 2013

  14. P. Ionescu. Some special Fano manifolds. Anniversary Conference, Faculty of Sciences - 150 Years, University of Bucharest, August 31, 2013

  15. M. Aprodu. Ulrich bundles on K3 surfaces. Anniversary Conference, Faculty of Sciences - 150 Years, University of Bucharest,  September 1st, 2013

  16. M. Aprodu. Lazarsfeld-Mukai bundles and applications. Locally Free Geometry Seminar, IMAR Bucharest, September 13, 2013

  17. V. Brinzanescu. Bundles on elliptic fibrations and connections with Yang-Mills equations. Theoretical Physics Seminar, IFIN, Magurele, September 20, 2013

  18. M. Aprodu. Towards Mq. CIRM, Trento University, October 3, 2013

Activities in 2014:

Foreign visitors (in chronological order):

None

Publications:

• 1. Marian Aprodu. Lazarsfeld-Mukai bundles and applications. II. in: Bridging Algebra, Geometry, and Topology, Springer Proceedings in Mathematics & Statistics, Vol. 96, Ibadula, Denis, Veys, Willem (Eds.) 2014, 11-20. DOI:10.1007/978-3-319-09186-0_2 (objective 1) link file

  2. Stefan Papadima, Alexandru Suciu. Vanishing of resonance and representations of Lie algebras. Journal für die reine und angewandte Mathematik, vol. 706 (2015) 83-102. DOI:10.1515/crelle-2013-0073  (objectives 1 and 3) link file

  3. Paltin Ionescu, Francesco Russo. On dual defective manifolds. Mathematical Research Letters. vol. 21 no. 5 (2014) 1137-1154. DOI:10.4310/MRL.2014.v21.n5.a10 (objectives 1 and 4) link file

Dissemination:

• 1. M. Aprodu. Holomorphic vector bundles and totally geodesic foliations. Seminarul de Geometrie Diferentiala, IMAR Bucharest, February 3, 2014

  2. S. Papadima. Affine geometry, diophantine equations and topology. Romanian Academy, Bucharest, February 28, 2014.

  3. M. Aprodu. Lazarsfeld-Mukai bundles and applications to the geometry of projective curves. Algebraic Geometry Seminar, Universita Roma Tre, Italy, February 13, 2014.

  4. M. Aprodu. Introduction to syzygies. (4 talks) KAIST Daejeon, March 2014.

  5. M. Aprodu. Syzygies and geometry of curves. (4 talks) KAIST CMC School on Algebraic Geometry, March 18-21, 2014.

  6. S. Papadima. Stratified geometry of representation varieties. Theoretical Physics Seminar, IFIN, Magurele, March 28, 2014.

  7. M. Aprodu. Ulrich bundles on K3 surfaces. K3 surfaces and their moduli, Schiermonnikoog, Netherlands, May 4-9, 2014.

  8.  M. Aprodu. Ulrich bundles on K3 surfaces. Positivity, vanishing theorems and applications, Lille, France, May 12-16, 2014.

  9. V. Brinzanescu. On the form of the universe in string theory. On Form and Pattern, Humboldt Kolleg, Bucharest, May 29-31, 2014.

  10. S. Papadima. Milnor fibrations of arrangements and modular inequalities. Univ. di Pisa, Italy, May 29, 2014.

  11. S. Papadima. Milnor fibrations of arrangements and modular inequalities. Univ. de Nice, France, June 10, 2014.

  12. M. Aprodu. Secant spaces and syzygies of special line bundles on curves. Max Planck Inst. Math. Bonn, Germany, July 17, 2014.

Activities in 2015:

Foreign visitors (in chronological order):

• 1. Sijong Kwak (KAIST Daejeon, South Korea) - April 2015

  2. Yeongrak Kim (KAIST Daejeon, South Korea) - June 2015

  3. Luigi Lombardi (University of Bonn, Germany) - June 2015

  4. Alexandru Suciu (Northeastern University Boston, USA) - June 2015

  5. Gavril Farkas (Humboldt Universitaet Berlin, Germany) - October 2015

The workshop “Interactions between Algebra, Geometry and Topology” is partly funded by the grant. The list of main speakers includes:

• 1. George Lusztig (MIT Boston, USA) - May 2015

  2. Alexandru Dimca (Nice, France) - May 2015

  3. Florian Pop (UPenn, USA) - May 2015

  4. Guenther Trautmann (Kaiserslautern, Germany) - May 2015

Publications:

• 1. Marian Aprodu, Gavril Farkas, Angela Ortega. Minimal resolutions, Chow forms of K3 surfaces and Ulrich bundles. Journal für die reine und angewandte Mathematik, DOI:10.1515/crelle-2014-0124 in press. (objectives 1 and 4) link file

  2. Vasile Brinzanescu, Victor Vuletescu. Irreducible vector bundles on some elliptic non-Kahler threefolds. Journal of Geometry and Physics 91 (2015) 141-145, DOI:10.1016/j.geomphys.2014.11.012. (objective 2) link file

  3. Paltin Ionescu. Recognizing Pn in classical and modern setting. Communications in Algebra, 44:12 (2016), 5088-5094 (objectives 1 and 4) link file

  4. Marian Aprodu. Polynomial equations and vector bundles. On Form and Pattern, Ed. Acad. Române (2015), 77-82. (0bjectives 1 and 4) link file

Dissemination:

• 1. V. Brinzanescu.  Fibrați vectoriali pe varietăți complexe. (4 talks), March 2015, Faculty of Mathematics and Computer Science, University of Bucharest.

  2. S. Papadima. On the geometry and topology of partial configuration spaces of Riemann surfaces. Interactions between Algebra, Geometry and Topology, 16-17 mai 2015, IMAR Bucharest.

  3. P. Ionescu. Some special Fano manifolds. Algebraic Geometry in Ferrara (AGeFe), June 15-18, 2015, Ferrara, Italy.

  4. M. Aprodu. Syzygies schemes and the geometry of curves. (4 talks) August 12-August 14, 2015, KAIST Daejeon, South Korea.

  5. M. Aprodu. Cayley-Chow forms of K3 surfaces and Ulrich bundles. Syzygies in Algebra and Geometry, August 26-August 30, 2015, Haeundae, Busan, South Korea.

  6. M. Aprodu. Cayley-Chow forms of K3 surfaces and Ulrich bundles. September 23, 2015, Algebraic Geometry Seminar, Universita di Genova, Italy.

  7. V. Vuletescu. Holomorphic vector bundles on complex non-Kahler manifolds. October 6, 2015, Theoretical Physics Seminar, IFIN, Magurele.

  8. M. Aprodu. Cayley-Chow forms of K3 surfaces and Ulrich bundles. Moduli Spaces in Geometry, October 26-October 30, 2015, CIRM Luminy, France.

Activities in 2016:

Foreign visitors (in chronological order):

• 1. Laura Costa Farras (Universitat de Barcelona, Spain) - April 2016

  2. Rosa Maria Miro Roig (Universitat de Barcelona, Spain) - April 2016

  3. Alexandru Dimca (Univ. Nice, France) - April 2016

  4. Yeongrak Kim (KAIST Daejeon, South Korea) - May- July 2016

  5. Alexandru Suciu (Northeastern University Boston, USA) - June 2016

Publications:

• 1. Marian Aprodu, Edoardo Sernesi. Excess dimension for secant loci in symmetric products of curves. Collectanea Mathematica, DOI 10.1007/s13348-016-0166-2 in press. (objectives 1 and 4) link file

  2. Marian Aprodu, Laura Costa, Rosa M. Miro-Roig. Rank two vector bundles on non-minimal ruled surfaces. Transactions of the American Mathematical Society, to appear. (objective 4) link

  3. B. Berceanu, D. A. Macinic, S. Papadima, C. R. Popescu. On the geometry and topology of partial configuration spaces of Riemann surfaces. Algebraic and Geometric Topology, to appear (objective 3) link file

  4. Y. Kim. Ulrich bundles on blowups. Comptes Rendus Mathématique (Académie des Sciences, Paris), to appear (objectives 1 and 4) link file

  5. S. Papadima, L. Paunescu. Rank two jump loci for solvmanifolds and Lie algebras. submitted (objective 3) link

Dissemination:

• 1. M. Aprodu. Green’s conjecture for curves on K3 surfaces. February 5, 2016, Universitat de Barcelona, Spain.

  2. V. Vuletescu. Holomorphic vector bundles on some non-Kaehler manifolds. April 18, 2016, Universita degli Studi di Firenze, Italy.

  3. V. Vuletescu. Holomorphic vector bundles on elliptic principal bundles. Workshop on Complex Projective Geometry, July 8-10, 2016, Bucharest.

  4. V. Brinzanescu. Vector bundles on non-Kaehler principal elliptic bundles. Workshop on Complex Projective Geometry, July 8-10, 2016, Bucharest.

  5. M. Aprodu. Equations and syzygies. September 1, 2016, Max Planck Inst. Math. Bonn, Germany.