• About SNS

    The Scuola Normale Superiore is a public institute for higher education that in its two centuries of life has earned itself a special place, both in Italy and abroad, a place characterised by merit, talent and scientific rigour.  Two types of course are available: the undergraduate course and the PhD course.The teaching activity is distributed among three academic structures: the Faculty of Humanities, the Faculty of Mathematical and Natural Sciences, placed in Pisa, and the Institute of Humanities and Social Sciences, located in Palazzo Strozzi in Florence. 

  • Admission

    The evaluation for entrance to the first year of the undergraduate course does not include the high school leaving certificate, and the bachelor's degree is not taken into consideration in the entrance examination for the fourth year course. For each PhD course, candidates’ level of competence, talent, motivations and aptitudes to scientific research will be assessed on the basis of their qualifications and research project and an interview.

  • Academics

    The Scuola Normale Superiore offers two types of course: the undergraduate course, leading to first and second level university degrees, and the PhD course, the international equivalent of the Italian Dottorato di ricerca.The teaching and research activity is distributed among three academic structures: the Faculty of Humanities, the Faculty of Mathematical and Natural Sciences, and the Institute of Humanities and Social Sciences.The first two academic structures, housed at the Pisa site, organize courses for both the  undergraduate course and the PhD course. The Institute of Humanities and Social Sciences, situated in Palazzo Strozzi in Florence, deals only with the PhD course.

  • Research

    A highly qualifying feature of the Normale way is the strong link between teaching and research that is a characteristic of both the undergraduate and the graduate programmes of the Scuola. The research structures of the two Faculties welcome students with a relevant study interest, enabling them to collaborate in a mature way with the activities of the researchers.

  • International

    The Scuola Normale is an institute of a decidedly international nature. Examinations for admission to the undergraduate degree course and for the PhD course are open to all citizens worldwide. A certain number of places on the PhD course are reserved for students from other countries. During the pre laurea and  post lauream teaching courses, study and research programmes are made available at overseas universities and research centres with which the Scuola forms an intense network of collaboration.  The doctorate course in particular is taught in a veritable graduate school in line with the highest international standards. 

Research group in Complex Analysis and Analytical Geometry

Research group in Complex Analysis and Analytical Geometry

Research Areas and Themes:

A) Existence of holomorphic chains and Levi flat hypersurfaces with assigned boundary, in the context of complex structures and, more generally, in the context of quasi complex structures .

Levi flat hypersurfaces with assigned boundary come out in the study of the holomorphic envelope of real subvarieties in a complex space. Research about this has primarily been carried out in the case in which the ambient space is a two dimension Stein manifold using the analytic discs method of Bishop or the Gromov's compactness theorem for J-holomorphic curves.

The geometric problem can be translated in a Dirichlet problem for a quasi-linear, degenerate elliptic equation, the Levi equation for the quasi complex structure.

The problem has been studied recently in the context of certain non calibrated, quasi complex structures on R4. The existence theorem is obtained by using fairly sophisticated regularization techniques for the weak viscose solution of the Levi operator.

When the ambient space is a complex manifold of higher dimension, different from the case above, the problem is overdetermined.

As yet unpublished results have been obtained recently for the space Cn.

B) Existence of Levi-flat hypersurfaces with a partly-assigned boundary
Research on this topic has begun very recently, and partial results have been proved for the space C2. They are particularly interesting and can be considered a first step towards a “theory of domains of existence” for flat Levi hypersurfaces.

C) Geometric Structure of the weakly complete complex spaces.
The study of the geometric properties of a “weakly complete” complex X space, i.e., one which allows a weakly plurisubharmonic exhaustion function, can be quite difficult even if X is of dimension 2. It presents a number of problems of the existence on X of global analytical objects (holomorphic functions, complex subspaces, Levi-flat hypersurfaces, etc.). Preliminary results have been recently obtained with the hypothesis in which (X is of dimension 2 and) the plurisubharmonic exhaustion function is real-analytic. It is a strong hypothesis and sufficient conditions to guarantee its existence are unknown.

D) Existence of the envelope of holomorphy for open complex spaces and the Levi problem.
It is a classic from the theory of functions of several complex variables. The last significant results date back to the beginning of the ‘60s.

E) Evolution of compact subsets of C2 and CP2 by Levi form
Among the motivations for research on this theme there is on one hand the possibility of obtaining information on the various envelopes of a compact K and on the other the analysis of the singularities which can be obtained for evolution, beginning with regular sets. The problem is closely connected to A) above.
It has been shown that the evolution by Levi form of a graph ∑ in C2, over a bounded domain D in C x R, has bounded the Levi flat hypersurface with boundary b as a limit.


Scuola Normale Superiore,
P.zza dei Cavalieri, 7
56126 Pisa, italy
ph. 39 050 509252
e-mail: tomassini@sns.it
Head of the group: Giuseppe Tomassini