The syntactic pivot is the verb argument

Verb argument

In linguistics, a verb argument is a phrase that appears in a syntactic relationship with the verb in a clause. In English, for example, the two most important arguments are the subject and the direct object....

 around which sentence

Sentence (linguistics)

In the field of linguistics, a sentence is an expression in natural language, and often defined to indicate a grammatical unit consisting of one or more words that generally bear minimal syntactic relation to the words that precede or follow it...

s "revolve", in a given language

Language

Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...

. This usually means the following:

• If the verb
  Verb
  A verb, from the Latin verbum meaning word, is a word that in syntax conveys an action , or a state of being . In the usual description of English, the basic form, with or without the particle to, is the infinitive...
  
   has more than zero arguments, then one argument is the syntactic pivot.
• If the verb agrees with at least one of its arguments, then it agrees with the syntactic pivot.
• In coordinated propositions, in languages where an argument can be left out, the omitted argument is the syntactic pivot.

The first two characteristics have to do with simple morphosyntax, and from them it is quite obvious the syntactic pivot in English

English language

English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

 (and most other European languages) is what we call the subject

Subject (grammar)

The subject is one of the two main constituents of a clause, according to a tradition that can be tracked back to Aristotle and that is associated with phrase structure grammars; the other constituent is the predicate. According to another tradition, i.e...

. An English verb cannot lack a subject (even in the imperative mood

Imperative mood

The imperative mood expresses commands or requests as a grammatical mood. These commands or requests urge the audience to act a certain way. It also may signal a prohibition, permission, or any other kind of exhortation.- Morphology :...

, the subject is implied to be "you" and is not ambiguous or unspecified), and cannot have just a direct object

Object (grammar)

An object in grammar is part of a sentence, and often part of the predicate. It denotes somebody or something involved in the subject's "performance" of the verb. Basically, it is what or whom the verb is acting upon...

 and no subject; and (at least in the present tense, and for the verb to be) it agrees partially with the subject.

The third point deserves an explanation. Consider the following sentence:

I shot the deer and killed it.

There are two coordinated propositions, and the second proposition lacks an explicit subject, but since the subject is the syntactic pivot, then the second proposition is assumed to have the same subject as the first one. One cannot do this with a direct object (in English). The result would be either ungrammatical or with a different meaning:

*I shot the deer and I killed.

The syntactic pivot is a feature of the morphosyntactic alignment

Morphosyntactic alignment

In linguistics, morphosyntactic alignment is the system used to distinguish between the arguments of transitive verbs and those of intransitive verbs...

 of the language. In nominative–accusative languages the syntactic pivot is the so-called "subject" (the argument marked with the nominative case

Nominative case

The nominative case is one of the grammatical cases of a noun or other part of speech, which generally marks the subject of a verb or the predicate noun or predicate adjective, as opposed to its object or other verb arguments...

). In ergative–absolutive languages the syntactic pivot may be the argument marked with the absolutive case

Absolutive case

The absolutive case is the unmarked grammatical case of a core argument of a verb which is used as the citation form of a noun.-In ergative languages:...

, but this is not always so, since ergative languages are often not "pure" and show a mixed behaviour (e. g. ergative morphology and accusative syntax).

Languages with a passive voice

Passive voice

Passive voice is a grammatical voice common in many of the world's languages. Passive is used in a clause whose subject expresses the theme or patient of the main verb. That is, the subject undergoes an action or has its state changed. A sentence whose theme is marked as grammatical subject is...

construction may resort to it in order to allow the default syntactic pivot to shift its semantic role (from agent to patient) in a coordinated proposition:

He worked hard and was awarded a prize.