Punctuation – Using Slash (/) as an Abbreviation

cur.

Just add a period to hint that it is an abbreviation.

See http://www.thefreedictionary.com/cur:

1. currency.
2. current.

You're asking a question about style, as such you're more likely to find a satisfactory answer outside of the dictionary.

The Associated Press Style Manual doesn't address century specifically, but says you should "Never use an abbreviation that will not be easily understood."

Wikipedia's Manual of Style has a variety of related statements:

• "Centuries and millennia not in quotes or titles should be either spelled out (eighth century) or in Arabic numeral(s) (8th century). The same style should be used throughout any article."
• "To indicate around, approximately, or about, the unitalicised abbreviation c. is preferred over circa, ca, ca., approximately, or approx., and should be spaced (c. 1291). Do not use a question mark for this function (1291?), as this may imply to the reader an uncertainty on the part of Wikipedia editors rather than on the part of reliable historians."
• "Standard symbols for units are undotted; e.g. m for the metre (not m.), kg for the kilogram (not kg.), in for the inch (not in., the quotation mark ", or the double prime ″), and ft for foot (not ft., the apostrophe ', or the prime ′).
  • Non-standard abbreviations should be dotted."

The Association of American Colleges and Universities Style Guide discusses this issue directly: "Centuries are spelled out (CMS 9.33): the twenty-first century"

I'm sure you can find others, but to my way of thinking, your best bet is to either not abbreviate century, or use something unambiguous (such as "cent.") if you have to use an abbreviation.