Profile
Profile

Prof. Dr. Emanuele Macrì

Former Hausdorff Postdoc
Former Bonn Junior Fellow
Current position: Associate Professor, Northeastern University

E-mail: e.macri(at)neu.edu
Homepage: http://nuweb15.neu.edu/emacri/
Institute: Mathematical Institute
Research Area: Former Research Area E
Date of birth: 17.Jun 1980
Mathscinet-Number: 756642

Academic Career

2006

PhD, International School for Advanced Studies (SISSA), Trieste, Italy

2006 - 2007

Postdoc, Max Planck Institute for Mathematics, Bonn

2007 - 2008

Postdoc, Hausdorff Center for Mathematics, University of Bonn

2008 - 2010

Assistant Professor, University of Utah, Salt Lake City, UT, USA

2010 - 2011

Professor (W2, Bonn Junior Fellow), University of Bonn

2011 - 2014

Assistant Professor (tenure-track), Ohio State University, Columbus, OH, USA

Since 2015

Associate Professor, Northeastern University, Boston, MA, USA

Research Profile

My field of interest is algebraic geometry.
In particular, I work on derived categories of coherent sheaves on algebraic varieties.

In my past research, I studied autoequivalence groups of the derived category (in [3,2,4,5]) and stability conditions in the sense of Bridgeland (in [6,7,8]).

Recently, I am studying three dimensional projective varieties. More precisely, I am working on generalizations of the classical Bogomolov-Gieseker inequality for stable sheaves on surfaces to stable complexes on threefolds, and applications of this to birational geometry.

I am also studying deformations of K3 categories embedded in derived categories of cubic fourfolds together with Martí Lahoz (University of Bonn).

Research Projects and Activities

DFG Collaborative Research Centre SFB/TR 45 “Periods, moduli spaces, and arithmetic of algebraic varieties”
Member

NSF grant, 2010 - 2013

Contribution to Research Areas

Former Research Area E
I belong to the research area E (and partly connected with C).

During my first year at HCM I wrote two preprints and completed two papers (one appeared in print and one is going to appear in 2011).

Preprints.

[11] “Bridgeland stability conditions on threefolds II: An application to Fujita's conjecture” (with A. Bayer, A. Bertram and Y. Toda), preprint 2011.

In this paper we show that the conjectural inequality for stable two-term complexes we formulate in the preprint [12] has applications towards Fujita’s conjecture on threefolds. More precisely, the conjecture implies a Reider-type theorem for threefolds, that K_X + 6L is very ample for L ample, and that 5L is very ample when K_X is trivial.

[12] “Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities” (with A. Bayer and Y. Toda), preprint 2011. Also arXiv:1103.5010.

In this paper we construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a Bogomolov-Gieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples.
Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.

Published papers.

In [1], we study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply-connected. We determine the group of autoequivalences preserving this connected component, which turns out to be closely related to <br>Gamma_1(3).
Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with <br>Gamma_1(3)-level structure. The morphism is <br>Gamma_1(3)-equivariant, and is given by solutions of Picard-Fuchs equations. This result is motivated by the notion of <br>Pi-stability and by mirror symmetry.

Finally, in [2], we study the general fibre of a formal deformation over the formal disk of a projective variety from the view point of abelian and derived categories. The abelian category of coherent sheaves of the general fibre is constructed directly from the formal deformation and is shown to be linear over the field of Laurent series. The various candidates for the derived category of the general fibre are compared.
If the variety is a surface with trivial canonical bundle, we show that the derived category of the general fibre is again a linear triangulated category with a Serre functor given by the square of the shift functor.

Selected Publications

[2] Daniel Huybrechts, Emanuele Macr\`\i, Paolo Stellari
Formal deformations and their categorical general fibre
Comment. Math. Helv. , 86: (1): 41--71
2011
DOI: 10.4171/CMH/217
[3] Daniel Huybrechts, Emanuele Macr\`\i, Paolo Stellari
Derived equivalences of K3 surfaces and orientation
Duke Math. J. , 149: (3): 461--507
2009
DOI: 10.1215/00127094-2009-043
[4] Emanuele Macr\`\i, Paolo Stellari
Infinitesimal derived Torelli theorem for K3 surfaces
With an appendix by Sukhendu Mehrotra
Int. Math. Res. Not. IMRN (17): 3190--3220
2009
DOI: 10.1093/imrn/rnp049
[5] Emanuele Macr\`\i, Paolo Stellari
Automorphisms and autoequivalences of generic analytic K3 surfaces
J. Geom. Phys.
, 58: (1): 133--164
2008
DOI: 10.1016/j.geomphys.2007.10.002
[6] Emanuele Macr\`\i
Stability conditions on curves
Math. Res. Lett. , 14: (4): 657--672
2007
DOI: 10.4310/MRL.2007.v14.n4.a10
[7] Daniel Huybrechts, Emanuele Macr\`\i, Paolo Stellari
Stability conditions for generic K3 categories
Compos. Math. , 144: (1): 134--162
2008
DOI: 10.1112/S0010437X07003065
[8] Emanuele Macr\`\i, Sukhendu Mehrotra, Paolo Stellari
Inducing stability conditions
J. Algebraic Geom. , 18: (4): 605--649
2009
DOI: 10.1090/S1056-3911-09-00524-4
[9] Claudio Bartocci, Emanuele Macr\`\i
Classification of Poisson surfaces
Commun. Contemp. Math. , 7: (1): 89--95
2005
DOI: 10.1142/S0219199705001647
[10] Emanuele Macr\`\i, Marc Nieper-Wiß kirchen, Paolo Stellari
The module structure of Hochschild homology in some examples
C. R. Math. Acad. Sci. Paris , 346: (15-16): 863--866
2008
DOI: 10.1016/j.crma.2008.05.017

Publication List

Selected Invited Lectures

2005

“Heterotic Strings, Derived Categories and Stacks”, Oberwolfach Research Institute for Mathematics (MFO), Oberwolfach

2006

“KIAS school on derived categories of coherent sheaves”, Korea Institute for Advanced Study (KIAS), Seoul, South Korea

2006

“Giornate di Geometria Algebrica e Argomenti Correlati VIII”, International School for Advanced Studies (SISSA), Trieste, Italy

2006

“Categorical constructions of primitive forms, II”, Research Institute for Mathematical Sciences (RIMS), Kyoto, Japan

2007

“Stability conditions, derived categories, etc”, Max Planck Institute for Mathematics, Bonn

2007

“Categorical aspects of algebraic geometry in mirror symmetry”, Research Institute for Mathematical Sciences (RIMS), Kyoto, Japan

2007

“Seminario di natale”, University of Milano, Italy

2008

“First CTS conference on vector bundles”, TIFR, Mumbai, India

2008

“Bundles, gerbes, and derived categories in string theory”, University of Salamanca, Spain

2009

“Derived categories and the Langlands programme”, FU Berlin

2009

“BPS state countings, stability structures, and derived algebraic geometry”, University of Hamburg

2009

“WAGS”, University of California, Los Angeles, CA, USA

2010

“School on birational geometry and moduli spaces”, University of Utah, Salt Lake City, UT, USA

2010

“Categorical methods in geometry and gauge theory”, Chern Institute, Tianjin, China

2011

“Derived Categories”, University of Tokyo, Japan

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