Is there a relationship between schizophrenia and genius? When reading the biography of renowned personalities recognized as "geniuses" such as great mathematicians, painters or poets, I was surprised by the fact that a  great portion of this population that possess these impressive abilities tend to have Schizophrenia. (For instance the well-known mathematician John Nash).

Is there a proven correlation between schizophrenia (or other mental disorders) and  genius, and did  this disorder push the cognitive abilities of these personalities to higher limits?

One may claim that these people have developed this disorder as a consequence of being addicted to drugs or some other factors that predispose one toward mental illness. However, the cases that I am interested in are when people developed schizophrenia and then they made the greatest contributions to their fields. I have also noticed similar trends with illnesses such as epilepsy.

All three, "Nietzsche","Van Gogh" and "Guy de Maupassant" were creative despite having a mental illness. Generally speaking, however, is it possible that a mental condition modifies the way our brain functions to improve cognitive skills despite being burdened by the lows of mental illnesses and the side effects from pharmacological treatment of these disorders.
The question you have asked is not a new one. In fact, from the times of classical antiquity, Plato considered artistic creativity as a result of god-given madness. When it comes to popular figures in the arts and sciences, however, it is important to note that the illness is not restricted to them by any means.

Lord Byron and Beethoven are said to be manic-depressive, Dostoyevsky had epilepsy. In terms of madness and its impact on creativity and genius, we can merely speculate. In fact, some critics of psychiatry feel that mental illness has been given a pathological status as an 'illness' and that we tend to reverse-label these figures and diagnose them with labels that have only recently been defined. 

What I meant earlier when I said that illness is not restricted to genius was to clarify a very important point, that majority of people that suffer from mental illnesses are not creative geniuses, or even accomplished in the traditional sense. Since mental illness has a strong genetic component, it is not surprising that Albert Einstein and Bertrand Russel's son were diagnosed with schizohprenia and James Joyce's daughter was also diagnosed with it. It is important to note that sensationalized examples can distort the statistical importance of certain correlations, so it is important not to focus too extensively on them.

Your hunch is correct though, take this from the Wikipedia page on the relation between mental illness and creativity.


  A study looking at 300,000 persons with schizophrenia, bipolar
  disorder or unipolar depression, and their relatives, found
  overrepresentation in creative professions for those with bipolar
  disorder as well as for undiagnosed siblings of those with
  schizophrenia or bipolar disorder. There was no overall
  overrepresentation, but overrepresentation for artistic occupations,
  among those diagnosed with schizophrenia. There was no association for
  those with unipolar depression or their relatives.
  
  Another study involving more than one million people, conducted by
  Swedish researchers at the Karolinska Institute, reported a number of
  correlations between creative occupations and mental illnesses.
  Writers had a higher risk of anxiety and bipolar disorders,
  schizophrenia, unipolar depression, and substance abuse, and were
  almost twice as likely as the general population to kill themselves.
  Dancers and photographers were also more likely to have bipolar
  disorder.




However, let me deal with the some of the evidence that psychology and neuroscience provide us. This evidence may come from different sources but they are attempts at determining a connection between the two phenomenon, and by no means is there any rock-solid connection that has been documented by any one study. Furthermore, I must warn you that some experiments are 'reductive' in the sense that they will ascribe the entire connection between mental illness and genius to a single factor, whereas many factors (or a very complex system) are actually the reason behind the apparent correlation. Here are some theories:

Dopermergic Theory

Since medical treatments that block dopamine receptors have show to reduce symptoms of schizophrenia, there is a consensus in the psychiatric community that schizophrenia is caused by an overactive dopamine system. The particular receptor I will discuss is the D2 receptor. For instance, a lower thalamic (from the Thalamus brain region) D2 binding potential is observed in those patients who do not take antipsychotic medications. Furthermore, a significantly larger number of D2 receptors are present in the striatum of schizophrenic patients.

As I am trying to answer your question, I will try avoiding getting trapped within semantic arguments about the definition of genius. Lets assume that they are individuals with great creative abilities and divergent thinking patterns that allow them to come up with novel solutions to difficult problems, or create original pieces of art. The study quoted found a negative correlation between divergent thinking and D2 receptor binding potential not in the thalamus, but present in the striatum. Similar to patients with schizophrenia, creative people had lower D2 receptor density in their thalamus. This phenomenon may relate to creativity because it might be because when compared to normal subjects, schizophrenics 'block out' less information from the external world. These abnormal thoughts might occasionally present themselves as novel ideas.

Brain Hemispheric Ambidexterity

Like mentioned below, divergent thinking can be associated with creativity. Studies have shown that bilateral activation of the prefrontal cortex is observed in people with divergent thinking patterns, and similar results were found for schizophrenic patients - however, they had greater activation in their right prefrontal cortex, showing that they can access both hemispheres. Simultaneous access to both hemispheres allows for greater ability to create associations. This greater ability to associate ideas may result in many of the multidisciplinary or revolutionary ideas advocated by exceptional figures.

Overrepresentation of Siblings in Creative Professions

Siblings of schizophrenic patients were found in a study to be overrepresented in creative processions. It has been hypothesized that these siblings have higher levels of schizotypal personality traits when compared to normal subjects. It is important to note the difference between Schizotypy and Schizophrenia, the second is a clinical case, the first is a personality category. It is highly probable, however, that both have a similar genetic basis. Schizotypal traits include weak mental boundaries between the self and external world, strange experiences, impulsive nonconformity and magical beliefs. As mentioned in this excerpt from this blog piece from Scientific American, this has implications for creativity:


  This has important implications for creativity. Mark Batey and Adrian
  Furnham found that the unusual experiences and impulsive nonconformity
  dimensions of schizotypy, but not the cognitive disorganization
  dimension, were significantly related to self-ratings of creativity, a
  creative personality (measured by a checklist of adjectives such as
  “confident,” “individualistic,” “insightful,” “wide interests,”
  “original,” “reflective,” “resourceful,” “unconventional,” and
  “sexy”), and everyday creative achievement among thirty-four
  activities (“written a short story,” “produced your own website,”
  “composed a piece of music,” and so forth).


However, it explores both sides of the camps and makes some important observations that question the validity of the hypothesis:


  In 2014, [Albert] Rothernberg [Professor of Psychiatry at Harvard] published a book, “Flight of Wonder: an
  investigation of scientific creativity”, in which he interviewed 45
  science Nobel laureates about their creative strategies. He found no
  evidence of mental illness in any of them. He suspects that studies
  which find links between creativity and mental illness might be
  picking up on something rather different.
  
  “The problem is that the criteria for being creative is never anything
  very creative. Belonging to an artistic society, or working in art or
  literature, does not prove a person is creative. But the fact is that
  many people who have mental illness do try to work in jobs that have
  to do with art and literature, not because they are good at it, but
  because they’re attracted to it. And that can skew the data,” he said.
  “Nearly all mental hospitals use art therapy, and so when patients
  come out, many are attracted to artistic positions and artistic
  pursuits.”
  
  


Important Note: Since the question you asked was very broad. I tried to focus on the particular mental condition you stated, schizophrenia. I have not discussed other conditions such as manic-depression, unipolar depression and so forth, and there is extensive literature on the connection between those conditions and genius. For example, this book: 

Kay Redfield Jamison (1996). Touched with Fire: Manic-Depressive Illness and the Artistic Temperament. 



Sources:

“Thinking Outside a Less Intact Box: Thalamic Dopamine D2 Receptor Densities Are Negatively Related To Psychometric Creativity in Healthy Individuals.”  Örjan de Manzano, Simon Cervenka, Anke Karabanov, Lars Farde, Fredrik Ullén. 
PLoS One. 2010; 5(5).

The relationship between creativity, schizotypy and intelligence.  
Batey, M. Furnham, A. (2009).  
Individual Differences Research, 7: 272-284.

Creativity and mental disorders: Family study of 300 000 people with severe mental disorders 
Kyaga, S.; Lichtenstein, P.; Boman, M.; Hultman, C.; Långström, N.; Landén, M. (2011)
The British Journal of Psychiatry 2011, 199(5): 373–379.

Cognitive training for divergent thinking in schizophrenia: A pilot study
Takahiro Nemotoa, Ryoko Yamazawaa, Hiroyuki Kobayashia, Nobuharu Fujitaa, Bun Chinoa, Chiyo Fujiid, Haruo Kashimaa, Yuri Rassovskye, Michael F. Greenc and Masafumi Mizunof (2009). 
Progress in Neuro-Psychopharmacology and Biological Psychiatry 33 (8): 1533–1536.

Creativity in Offspring of Schizophrenic and Control Parents: An Adoption Study 
Dennis K. Kinney, Ruth Richards, Patricia A. Lowing, Deborah LeBlanc, Morris E. Zimbalist, Patricia Harlan. (2001)  
Creativity Research Journal, Vol. 13, Issue 1.