Department of Mathematics
Stony Brook University
Office: Math 4-121
Geometry, esp. moduli problems, degenerations, singularities, special
classes of varieties (K3s, Calabi-Yau, Hyperkahler manifolds).
Recent & Selected Publications:
- Hodge theory of degenerations, (II): Vanishing cohomology and geometric applications, (w. M. Kerr), preprint 2020.
- A conjectural bound on the second Betti number for hyper-Kaehler manifolds
(w. YJ Kim), Bull. Soc. Math. France 148 (2020), no. 3.
- Smoothing of rational singularities and Hodge structure (w. M. Kerr and M. Saito), preprint 2019.
- The LLV decomposition of hyper-Kaehler cohomology (with M. Green, YJ Kim, and C. Robles), preprint 2019.
- Automorphisms and Periods of Cubic Fourfolds (with Z. Zheng), preprint 2019.
Cohomology of the moduli space of cubic threefolds and its smooth models (with S. Casalaina-Martin, S. Grushevsky, and K. Hulek), to appear in Mem. Amer. Math. Soc.
- The Euler number of hyper-Kaehler manifolds of OG10 type (w. K. Hulek and G. Sacca), to appear in Proceedings of the ICM 2018 Satellite conference "Moduli Spaces in Algebraic Geometry and Applications", Campinas.
- Hodge theory of degenerations, (I): Consequences of the decomposition theorem , (w. M. Kerr; with an Appendix by M. Saito), preprint 2019.
- Maximally algebraic potentially irrational cubic fourfolds, preprint 2018.
- GIT versus Baily-Borel compactification for K3's which are double covers of P^1xP^1, (with K. O'Grady), preprint 2018.
- On the moduli space of pairs consisting of a cubic threefold and a hyperplane, (w. G. Pearlstein and Z. Zhang), Adv. Math. 340 (2018), 684-722.
- Completion of period mappings and Ampleness of the Hodge bundle, (w. M. Green, P. Griffiths, and C. Robles), preprint 2017.
- Remarks on degenerations of hyper-Kaehler manifolds, (with J. Kollár, G. Saccà and C. Voisin), Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2837-2882.
- GIT versus Baily-Borel compactification for quartic K3 surfaces, (with K. O'Grady), in "Geometry of Moduli" (Abel Symposia), Springer, 2018, 217-283.
- Birational geometry of the moduli space of quartic K3 surfaces, (with K. O'Grady), Compositio Math. 155 (2019), no. 9, 1655-1710.
- A hyper-Kaehler compactification of the Intermediate Jacobian fibration associated to a cubic fourfold (with G. Saccà and C. Voisin), Acta Math. 218 (2017), no. 1, 55-135.
- Complete moduli of cubic threefolds and their intermediate Jacobians (with S. Casalaina-Martin, S. Grushevsky, and K. Hulek), to appear in Proc. Lond. Math. Soc.
- Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with S. Casalaina-Martin, S. Grushevsky, and K. Hulek), J. Eur. Math. Soc 19 (2017), no. 3, 659-723.
- The KSBA compactification for the moduli space of degree two K3 pairs, J. Eur. Math. Soc. 18 (2016), no. 2, 225-279.
- Log canonical models and variation of GIT for genus four canonical curves (w. S. Casalaina-Martin and D. Jensen), J. Algebraic Geom. 23 (2014), 727-764.
- Semi-algebraic horizontal subvarieties of Calabi-Yau type (w. R. Friedman), Duke Math. J. 162 (2013), no. 12, 2077-2148.
- Simultaneous semi-stable reduction for curves with ADE singularities (w. S. Casalaina-Martin), Trans. Amer. Math. Soc. 365 (2013), no. 5, 2271-2295.
- Moduli space of cubic fourfolds via the period map, Ann. of Math. 172 (2010), no. 1, 673-711.
- Moduli space of cubic fourfolds (the GIT compactification), J. Algebraic Geom. 18 (2009), 511-545.
- The moduli space of cubic threefolds via degenerations of the intermediate Jacobian (w. S. Casalaina-Martin), J. Reine Angew. Math. 633 (2009), 29-65.
Expository lecture series given at: Luminy (Jan 2017), Guanajuato (CIMPA-CIMAT-ICTP school, Feb 2016), Angers (June 2014), KAIST (Mar 2014), Fields Institute (Aug and Nov 2013), Vancouver (Jul 2013), Barcelona (May 2013).
- Classical Period Domains (with Z. Zhang), in Recent Advances in Hodge Theory (London Mathematical Society Lecture Note Series), Cambridge Univ. Press, 2016.
- Perspectives on the construction and compactification of moduli spaces, in Compactifying Moduli Spaces (Advanced Courses in Mathematics, CRM Barcelona), Birkhauser, 2016, 1-35.
- GIT and moduli with a twist, in "Handbook of Moduli" vol. 2, Adv. Lect. Math. 25 (2013), Int. Press, 259-297.
My papers on arXiv. My Scholar profile.
My research is partially supported by NSF (CAREER DMS-1254812, DMS-1802128).
- Hodge Theory, Moduli and Representation Theory (final conference for the FRG project), Stony Brook, August 14-18, 2017.
- Positivity in Arithmetic and Geometry (Spring School), Orsay (France), May 29-June 2, 2017.
- Hyper-Kaehler Manifolds, Hodge Theory and Chow Groups (Sanya, China, Dec 18-23, 2016)
- Calabi-Yau varieties: Arithmetic, geometry and physics (Herstmonceux Castle, East Sussex, UK, June 20-25, 2016)
- Algebraic Cycles and Moduli (CRM Montreal, June 2-8, 2016)
- Program on Complex, p-adic, and logarithmic Hodge theory and their applications (SCGP, Mar-Apr, 2016)
- Program on
Moduli spaces and singularities in algebraic and Riemannian geometry (SCGP, Aug-Nov, 2015)
- Mini-school on Invariants of Singularities in zero and positive characteristics (December 4, 2015, Stony Brook)
- Collapsing Calabi-Yau Manifolds (SCGP, Aug 31-Sept 4, 2015)
- Topology of algebraic varieties (IAS, 2014-2015)
- Perspectives on complex algebraic geometry (Columbia University, May 22-25, 2015)
- New techniques in birational geometry (Stony Brook, April 6-10, 2015)
- K3, Enriques Surfaces and Related Topics (Nagoya, Nov 10-14, 2014)
- Thematic Program on Calabi-Yau varieties (Fields Institute, Fall 2013).
- Yoonjoo Kim
- Alexandra Viktorova
- Lisa Marquand
- Francois Greer (RTG postdoc)
- Adrian Brunyate (NSF postdoc)
- Giulia Sacca (postdoc) - now at Columbia
- Zheng Zhang (geometric and motivic realizations of VHS) - now at U. Colorado (Boulder)
- Patricio Gallardo (moduli of surfaces of general type, esp. quintics) - now at Washington Univ.
- Ken Ascher (undergraduate, Honors Thesis) - now at Princeton.
- Dave Jensen (postdoc) - now at U. Kentucky.
Stony Brook University
Stony Brook, NY 11794-3651
Office Phone: (631) 632-4506
Last Modified: June 2020